Waves And Wave Interactions In Plasmas 981126533X, 9789811265334

Table of contents :
Contents
Preface
Acknowledgments
Chapter 1. Introduction to Plasmas
1.1 Introduction
1.2 Saha Equation and Plasma Temperature
1.3 Basic Concepts of Plasma
1.3.1 Basic dimensionless parameters
1.3.2 Debye length and Debye shielding
1.3.3 Quasineutrality
1.3.4 Response time
1.3.5 Plasma frequency
1.3.6 Collisions and coupling limit
1.4 Criteria for Plasma
1.5 High-Temperature Plasmas
1.6 Mathematical Description
1.7 Magnetized Plasmas
1.8 Single Particle Motion in Uniform Electric and Magnetic Field
1.9 Fluid Approach
1.10 Maxwell’s Equations
1.11 Electromagnetic Wave Equation in Free Space
1.12 Plasma Kinetic Theory
1.12.1 Distribution function
1.12.2 Macroscopic variables
1.12.3 Maxwellian distribution function
1.12.4 Non-Maxwellian distribution in plasmas
1.12.5 Nonthermal distribution
1.12.6 Superthermal distribution
1.12.7 q-nonextensive distribution
1.13 Closure Form of Moment Equation
1.13.1 Equation of continuity
1.13.2 Equation of motion
1.13.3 Equation of energy
1.14 Dusty Plasma
1.15 Quantum Plasma
1.16 Quantum Plasma Models
References
Chapter 2. Introduction to Waves in Plasma
2.1 Introduction
2.2 Mathematical Description of Waves
2.3 Dispersion Relation
2.4 Linear Waves in Plasmas
2.5 Plasma Oscillation
2.6 Electromagnetic Waves
2.7 Upper Hybrid Frequency
2.8 Electrostatic Ion Cyclotron Waves
2.9 Lower Hybrid Frequency
2.10 Electromagnetic Waves with B0 = 0
2.11 Electromagnetic Waves Perpendicular to B0
2.11.1 Ordinary wave
2.11.2 Extraordinary wave
2.12 Electromagnetic Waves Parallel to B0
2.13 Hydromagnetic Waves
2.13.1 Alfven wave
2.13.2 Magnetosonic wave
2.14 Some Acoustic Type of Waves in Plasmas
2.14.1 Electron plasma waves
2.14.2 Ion acoustic waves
2.14.3 Dust acoustic waves
2.14.4 Dust ion acoustic waves
2.15 Nonlinear Wave
2.16 Solitary Waves and Solitons
2.16.1 History of solitary waves and solitons
2.17 Properties of Solitons
References
Chapter 3. Solution of Nonlinear Wave Equations
3.1 Nonlinear Waves
3.2 Direct Method
3.2.1 Korteweg–de Vries (KdV) equation
3.2.2 Cnoidal waves
3.2.3 Modified KdV (MKdV) equation
3.2.4 Schamel-type KdV (S-KdV) equation
3.2.5 Burgers’ equation
3.2.6 KP equation
3.2.7 Modified KP equation
3.3 Hyperbolic Tangent Method
3.3.1 KdV equation
3.3.2 Modified KdV equation
3.3.3 Burgers’ equation
3.3.4 KdV Burgers’ equation
3.3.5 KP equation
3.4 Tanh–Coth Method
3.4.1 KdV equation
3.4.2 Burgers’ equation
3.5 Solution of KP Burger Equation
3.6 Conservation Laws and Integrals of the Motions
3.6.1 Conserved quantity of KdV equation
3.7 Approximate Analytical Solutions
3.7.1 Damped KdV equation
3.7.2 Force KdV equation
3.7.3 Damped-force KdV equation
3.8 Multisoliton and Hirota’s Direct Method
3.8.1 Hirota’s method
3.8.2 Multisoliton solution of the KdV equation
3.8.3 Multisoliton solution of the KP equation
References
Chapter 4. RPT and Some Evolution Equations
4.1 Perturbation Technique
4.2 Reductive Perturbation Technique
4.3 Korteweg–de Vries (KdV) Equation
4.4 Modified KdV (MKdV) Equation
4.5 Gardner’s Equation
4.6 Gardner and Modified Gardner’s (MG) Equation
4.7 Damped Forced KdV (DFKdV) Equation
4.8 Damped Forced MKdV (DFMKdV) Equation
4.9 Forced Schamel KdV (SKdV) Equation
4.10 Burgers’ Equation
4.11 Modified Burgers’ Equation
4.12 KdV Burgers’ (KdVB) Equation
4.13 Damped KdVB Equation
4.14 Kadomtsev–Petviashvili (KP) Equation
4.15 Modified KP (MKP) Equation
4.16 Further MKP (FMKP) Equation
4.17 KP Burgers’ (KPB) Equation
4.18 Damped KP (DKP) Equation
4.19 Zakharov–Kuznetsov (ZK) Equation
4.20 ZK Burgers’ (ZKB) Equation
4.21 Damped ZK (DZK) Equation
References
Chapter 5. Dressed Soliton and Envelope Soliton
5.1 Dressed Soliton
5.2 Dressed Soliton in a Classical Plasma
5.3 Dressed Soliton in a Dusty Plasma
5.4 Dressed Soliton in Quantum Plasma
5.5 Dressed Soliton of ZK Equation
5.6 Envelope Soliton
5.7 Nonlinear Schrodinger Equation (NLSE)
References
Chapter 6. Evolution Equations in Nonplanar Geometry
6.1 Introduction
6.2 Basic Equations of Motion in Nonplanar Geometry
6.3 Nonplanar KdV Equation in Classical Plasma
6.4 Nonplanar KdV Equation in Quantum Plasma
6.5 Nonplanar Gardner’s or Modified Gardner’s Equation
6.6 Nonplanar KP and KP Burgers’ Equation
6.7 Nonplanar ZK Equation
6.8 Nonplanar ZKB Equation
References
Chapter 7. Collision of Solitons
7.1 Introduction
7.2 Head-on Collision
7.2.1 Head-on collision of solitary waves in planar geometry
7.2.2 Head-on collision of solitons in a Magnetized Quantum Plasma
7.2.3 Head-on collision of magneto-acoustic solitons in spin-1/2 fermionic quantum plasma
7.2.4 Interaction of DIASWs in nonplanar geometry
7.3 Oblique Collision
7.3.1 Oblique collision of DIASWs in quantum plasmas
7.4 Overtaking Collision
7.4.1 Overtaking interaction of two solitons and three solitons of EAWs in quantum plasma
7.5 Soliton Interaction and Soliton Turbulence
7.6 Statistical Characteristics of the Wavefield
7.7 Plasma Parameters on Soliton Turbulence
References
Chapter 8. Sagdeev’s Pseudopotential Approach
8.1 Nonperturbative Approach
8.2 Sagdeev’s Pseudopotential Approach
8.2.1 Physical interpretation of Sagdeev’s potential
8.2.2 Determination of the range of Mach number
8.2.3 Shape of the solitary waves
8.2.4 Physical interpretation of double layers
8.2.5 Small amplitude approximation
8.3 Effect of Finite Ion Temperature
8.4 Large-amplitude DASWs
8.5 Large-amplitude Double Layers
8.6 Effect of Ion Kinematic Viscosity
8.7 DIASWs in Magnetized Plasma
8.8 Solitary Kinetic Alfven Waves
8.9 Collapse of EA Solitary Waves
8.10 Collapse of DASWs in Presence of Trapped Ions
References
Chapter 9. Conclusion and Future Scopes
Index

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Waves and Wave Interaction in Plasmas

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Waves and Wave Interaction in Plasmas Prasanta Chatterjee Visva Bharati University, India Kaushik Roy Beluti M K M High School, India Uday Narayan Ghosh K K M College (A Constituent Unit of Munger University), India

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British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

WAVES AND WAVE INTERACTION IN PLASMAS Copyright © 2023 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.

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ISBN 978-981-126-533-4 (hardcover) ISBN 978-981-126-534-1 (ebook for institutions) ISBN 978-981-126-535-8 (ebook for individuals) For any available supplementary material, please visit https://www.worldscientific.com/worldscibooks/10.1142/13114#t=suppl Typeset by Stallion Press Email: [email protected] Printed in Singapore

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Preface Waves and wave interactions in plasmas is written in a lucid and systematic way to serve as a special book for advanced post graduate students and researchers in Applied Mathematics, Plasma Physics, Nonlinear differential equations, Nonlinear Optics, and other Engineering branches where nonlinear wave phenomena is studied. The first chapter deals with basic plasma with elementary definitions of magnetized and unmagnetized plasmas, plasma modeling, dusty plasma, and quantum plasma. In Chapter 2, we deal with linear and nonlinear waves, solitons and shocks, and other wave phenomena. In Chapter 3, solution of some nonlinear wave equations is discussed by using some standard technique. Chapter 4 starts with elementary knowledge of perturbation and nonperturbation methods. Several evolution equations are obtained in different plasma situations and properties of solitons in those environments are discussed. In Chapter 5, Higher-order correction to those equations and the improvement of the solution is are discussed. In Chapter 6, evolution equations in nonplanar geometry are obtained and the wave solution for such equation is also obtained. In Chapter 7, different type collisions of solitons in a plasma environment is discussed. The phenomena of soliton turbulence are also discussed as a consequence of multi soliton interactions. In Chapter 8, the properties of large amplitude solitary waves are obtained by using a non perturbative approach called Sagdeev’s Pseudopotential Approach. Speed and shape of solitons are also discussed in this chapter. Possible future developments of research in this area are explained in brief in Chapter 9. Prasanta Chatterjee, Kaushik Roy Uday Narayan Ghosh India April 2022 v

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Acknowledgments

We are greatly delighted to publish a book on Waves and Wave Interaction in Plasma. During writing this book, we have got help from many of our seniors and friends. We are highly obliged by the guidance we have received from Prof. Rajkumar Roychoudhury, Emeritus Professor of ISI, Kolkata. Dr. Santo Banerjee and Dr. Amar Prasad Misra have supported and motivated us in our endeavor. We are also grateful to Dr. Malay Gorui, Dr. Asit Saha, Dr. Pankaj Kumar Mandal, Dr. Ganesh Mandal, Dr. Rustam Ali, and Laxmikanta Mandi. Last but not the least, we are thankful to Dr. Sriparna Chatterjee and Diya Chatterjee for helping us in improving the language of the book.

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Contents

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Preface

v

Acknowledgments Chapter 1. 1.1 1.2 2 1.3 4

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Introduction to Plasmas

1

Introduction ................................................................. 1 Saha Equation and Plasma Temperature . . . . . . . . . . Basic Concepts of Plasma . . . . . . . . . . . . . . . . . .

1.3.1 Basic dimensionless parameters . . . . . 4 1.3.2 Debye length and Debye shielding . . . 4 1.3.3 Quasineutrality . . . . . . . . . . . . . 7 1.3.4 Response time 7 1.3.5 Plasma frequency . . . . . . . . . . . . 8 1.3.6 Collisions and coupling limit . . . . . . 9 1.4 Criteria for Plasma 10 1.5 High-Temperature Plasmas . . . . . . . . . . . 1.6 Mathematical Description . . . . . . . . . . . . 1.7 Magnetized Plasmas 1.8 Single Particle Motion in Uniform Electric and Magnetic Field 1.9 Fluid Approach 1.10 Maxwell’s Equations . . .ix. . . . . . . . . . . . 1.11 Electromagnetic Wave Equation in Free Space . 1.12 Plasma Kinetic Theory . . . . . . . . . . . . . . 1.12.1 Distribution function . . . . . . . . . .

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1.12.3 Maxwellian distribution function . . . . 1.12.4 Non-Maxwellian distribution in plasmas 1.12.5 Nonthermal distribution . . . . . . . . . 1.12.6 Superthermal distribution . . . . . . . . 1.12.7 q-nonextensive distribution . . . . . . . 1.13 Closure Form of Moment Equation . . . . . . . 1.13.1 Equation of continuity . . . . . . . . . . 1.13.2 Equation of motion . . . . . . . . . . . 1.13.3 Equation of energy . . . . . . . . . . . . 1.14 Dusty Plasma . 32 1.15 Quantum Plasma 1.16 Quantum Plasma Models . . . . . . . . . . . . References Chapter 2. 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11

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36 39 41

Introduction to Waves in Plasma

Introduction ....................................................................... Mathematical Description of Waves . . . . . . . . . . . . . Dispersion Relation 46 Linear Waves in Plasmas . . . . . . . . . . . . . . . . . . . Plasma Oscillation 51 Electromagnetic Waves . . . . . . . . . . . . . . . . . . . . Upper Hybrid Frequency . . . . . . . . . . . . . . . . . . . Electrostatic Ion Cyclotron Waves . . . . . . . . . . . . . 55 Lower Hybrid Frequency . . . . . . . . . . . . . . . . . . .  0 = 0 . . . . . . . . . . . . 58 Electromagnetic Waves with B Electromagnetic Waves Perpendicular to B0 . . . . . . . . 2.11.1 Ordinary wave 60 2.11.2 Extraordinary wave . . . . . . . . . . . . . . . . .  0 . . . . . . . . . . . 63 Electromagnetic Waves Parallel to B Hydromagnetic Waves . . . . . . . . . . . . . . . . . . . . 2.13.1 Alfven wave 66 2.13.2 Magnetosonic wave . . . . . . . . . . . . . . . . . Some Acoustic Type of Waves in Plasmas . . . . . . . . . 2.14.1 Electron plasma waves . . . . . . . . . . . . . . . 2.14.2 Ion acoustic waves . . . . . . . . . . . . . . . . . . 2.14.3 Dust acoustic waves . . . . . . . . . . . . . . . . . 2.14.4 Dust ion acoustic waves . . . . . . . . . . . . . . . Nonlinear Wave

43 43 44 50 52 53 56 60 61 66 69 71 72 73 76 77 78

Contents

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2.16 Solitary Waves and Solitons . . . . . . . . . . . . . . . . . 2.16.1 History of solitary waves and solitons . . . . . . . 2.17 Properties of Solitons . . . . . . . . . . . . . . . . . . . . . References

82 82 84 85

Chapter 3.

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3.1 3.2

Solution of Nonlinear Wave Equations

87

Nonlinear Waves .............................................................. 87 Direct Method ................................................................. 87 3.2.1 Korteweg–de Vries (KdV) equation . . . . . . . . 88 3.2.2 Cnoidal waves 90 3.2.3 Modified KdV (MKdV) equation . . . . . . . . . . 93 3.2.4 Schamel-type KdV (S-KdV) equation . . . . . . . 94 3.2.5 Burgers’ equation . . . . . . . . . . . . . . . . . . 95 3.2.6 KP equation 96 3.2.7 Modified KP equation . . . . . . . . . . . . . . . . 97 3.3 Hyperbolic Tangent Method . . . . . . . . . . . . . . . . . 98 3.3.1 KdV equation 100 3.3.2 Modified KdV equation . . . . . . . . . . . . . . . 101 3.3.3 Burgers’ equation . . . . . . . . . . . . . . . . . . 102 3.3.4 KdV Burgers’ equation . . . . . . . . . . . . . . . 103 3.3.5 KP equation 105 3.4 Tanh–Coth Method 105 3.4.1 KdV equation 106 3.4.2 Burgers’ equation . . . . . . . . . . . . . . . . . . 109 3.5 Solution of KP Burger Equation . . . . . . . . . . . . . . 109 3.6 Conservation Laws and Integrals of the Motions . . . . . . 112 3.6.1 Conserved quantity of KdV equation . . . . . . . 116 3.7 Approximate Analytical Solutions . . . . . . . . . . . . . . 117 3.7.1 Damped KdV equation . . . . . . . . . . . . . . . 117 3.7.2 Force KdV equation . . . . . . . . . . . . . . . . . 118 3.7.3 Damped-force KdV equation . . . . . . . . . . . . 119 3.8 Multisoliton and Hirota’s Direct Method . . . . . . . . . . 121 3.8.1 Hirota’s method . . . . . . . . . . . . . . . . . . . 121 3.8.2 Multisoliton solution of the KdV equation . . . . 123 3.8.3 Multisoliton solution of the KP equation . . . . . 127 References 129

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Chapter 4.

RPT and Some Evolution Equations

4.1 Perturbation Technique . . . . . . . . . . . . . . 4.2 Reductive Perturbation Technique . . . . . . . . . 4.3 Korteweg–de Vries (KdV) Equation . . . . . . . . 4.4 Modified KdV (MKdV) Equation . . . . . . . . . 4.5 Gardner’s Equation 148 4.6 Gardner and Modified Gardner’s (MG) Equation 4.7 Damped Forced KdV (DFKdV) Equation . . . . 4.8 Damped Forced MKdV (DFMKdV) Equation . . 4.9 Forced Schamel KdV (SKdV) Equation . . . . . . 4.10 Burgers’ Equation 166 4.11 Modified Burgers’ Equation . . . . . . . . . . . . 4.12 KdV Burgers’ (KdVB) Equation . . . . . . . . . . 4.13 Damped KdVB Equation . . . . . . . . . . . . . 4.14 Kadomtsev–Petviashvili (KP) Equation . . . . . . 4.15 Modified KP (MKP) Equation . . . . . . . . . . . 4.16 Further MKP (FMKP) Equation . . . . . . . . . 4.17 KP Burgers’ (KPB) Equation . . . . . . . . . . . 4.18 Damped KP (DKP) Equation . . . . . . . . . . . 4.19 Zakharov–Kuznetsov (ZK) Equation . . . . . . . 4.20 ZK Burgers’ (ZKB) Equation . . . . . . . . . . . 4.21 Damped ZK (DZK) Equation . . . . . . . . . . . References Chapter 5.

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Dressed Soliton and Envelope Soliton

201

5.1 Dressed Soliton ................................................................. 5.2 Dressed Soliton in a Classical Plasma . . . . . . . . . . . . 5.3 Dressed Soliton in a Dusty Plasma . . . . . . . . . . . . . 5.4 Dressed Soliton in Quantum Plasma . . . . . . . . . . . . 5.5 Dressed Soliton of ZK Equation . . . . . . . . . . . . . . . 5.6 Envelope Soliton 224 5.7 Nonlinear Schrodinger Equation (NLSE) . . . . . . . . . . References 238

201 201 208 213 218

Chapter 6. 6.1 6.2 6.3

Evolution Equations in Nonplanar Geometry

225

239

Introduction 239 Basic Equations of Motion in Nonplanar Geometry . . . . 240 Nonplanar KdV Equation in Classical Plasma . . . . . . . 244

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Contents

6.4 6.5

Nonplanar KdV Equation in Quantum Plasma . . . . . . Nonplanar Gardner’s or Modified Gardner’s Equation 6.6 Nonplanar KP and KP Burgers’ Equation . . . . . . . . . 6.7 Nonplanar ZK Equation . . . . . . . . . . . . . . . . . . . 6.8 Nonplanar ZKB Equation . . . . . . . . . . . . . . . . . . References .................................................................................. Chapter 7.

Collision of Solitons

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Introduction ....................................................................... Head-on Collision .............................................................. 7.2.1 Head-on collision of solitary waves in planar geometry ................................................... 7.2.2 Head-on collision of solitons in a Magnetized Quantum Plasma ................................................ 7.2.3 Head-on collision of magneto-acoustic solitons in spin-1/2 fermionic quantum plasma . . . . . . . 7.2.4 Interaction of DIASWs in nonplanar geometry .............................................................. 7.3 Oblique Collision .............................................................. 7.3.1 Oblique collision of DIASWs in quantum plasmas ................................................................. 7.4 Overtaking Collision . . . . . . . . . . . . . . . . . . . . . 7.4.1 Overtaking interaction of two solitons and three solitons of EAWs in quantum plasma . . . . . . . 7.5 Soliton Interaction and Soliton Turbulence . . . . . . . . . 7.6 Statistical Characteristics of the Wavefield . . . . . . . . . 7.7 Plasma Parameters on Soliton Turbulence . . . . . . . . . References ..................................................................................

8.1 8.2

Sagdeev’s Pseudopotential Approach

Nonperturbative Approach . . . . . . . . . . . . . . . Sagdeev’s Pseudopotential Approach . . . . . . . . . 8.2.1 Physical interpretation of Sagdeev’s potential 8.2.2 Determination of the range of Mach number 8.2.3 Shape of the solitary waves . . . . . . . . . . 8.2.4 Physical interpretation of double layers . . . 8.2.5 Small amplitude approximation . . . . . . .

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7.1 7.2

Chapter 8.

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8.3 Effect of Finite Ion Temperature . . . . . . . . . 8.4 Large-amplitude DASWs . . . . . . . . . . . . . . 8.5 Large-amplitude Double Layers . . . . . . . . . . 8.6 Effect of Ion Kinematic Viscosity . . . . . . . . . 8.7 DIASWs in Magnetized Plasma . . . . . . . . . . 8.8 Solitary Kinetic Alfven Waves . . . . . . . . . . . 8.9 Collapse of EA Solitary Waves . . . . . . . . . . 8.10 Collapse of DASWs in Presence of Trapped Ions References Chapter 9.

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Index

Conclusion and Future Scopes

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Chapter 1

Introduction to Plasmas

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1.1 Introduction The word “Plasma” has originated from the Greek word, which indicates anything formed or shaped. Sir William Crookes first identified it as a radiant matter in 1879. In 1897, Sir J. J. Thomson studied the nature of electron discharge in a cathode-ray tube and pointed the character of matter. In 1929, the word “Plasma” was first coined by two American Physicists Tonks and Langmuir to explain the inner zone of a luminous gas that is ionized and produced by electricity discharged in a tube. Besides the three states of matter, i.e., solid, liquid, and gas, there is also a fourth state called plasma. The transition of the state of matter from solid to liquid and then to gas takes place with the temperature rise. With a further increase in temperature, some or all of the atoms of neutral gas are ionized. The gas turns out to be completely or partially ionized. Eventually, a plasma state is formed. The electromagnetic forces predominate the behavior of plasma. It has a tendency to become electrically neutral. So roughly speaking, a volume of plasma is a group of an equal number of particles charged oppositely at least on a certain level. Research shows that plasma constitutes 99% of the visible universe. We belong to the rest 1% of the matter on earth which is something different from plasma. Plasmas are found in and around the earth, in lightning channels of the ionosphere, in the aurora, and the earth’s magnetosphere. Plasmas are found in the solar wind, in the magnetosphere, and in comets. Around Jupiter and Saturn, we have plasmas in the form of gigantic plasma toroids. The sun and the other stars are nothing but enormous plasma balls. Not only the stars but also the nebula within the galaxies is also composed of plasma and so on.

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2

Waves and Wave Interactions in Plasmas

Plasmas consist of positively and negatively charged particles, so at a larger length scale, the number of positively charged particles and the number of negatively charged particles in plasmas are the same. But locally, they may be different. This is what we call quasineutrality. So plasma can be characterized as a gas that is quasineutral and which contains charged and neutral particles. Moreover, plasma can also display unified behavior. Here, unified behavior indicates the movements that depend upon not only topical circumstances but also the whole plasma. Usually, a single particle (neutral) at rest will move if it takes a blow. But the situation is different in the plasma world. Plasma comprises charged particles. When these charges start moving, they generate a local concentration of positive or negative charges, which give rise to electric fields. Motions of charges also generate electric currents and hence magnetic fields. Electromagnetic fields affect the movement of other charged particles even far away. The constituent parts of plasma wield a force among them even when the distance is large. One of the most exciting features of plasma is that the forces that arise from common local collisions can be ignored if the long-range electromagnetic forces are very large. Such plasmas are called “collision-less plasmas”. 1.2 Saha Equation and Plasma Temperature The thermal ionization rule was discovered by the Indian physicist M. N. Saha (1920). He showed that the quantity of ionization can be expected in a gas that is in thermal equilibrium by the following equation. This can be represented as T 3/2 ni  2.4 × 1021 exp(−Vi /KB T ) ni n0

(1.1)

where ni , n0 , Vi , and T denote the ion density, the neutral density, the ionization energy of the gas, and the temperature (in degrees Kelvin), respectively. KB is the Boltzmann constant and ni /n0 is a balance between the ionization rate (time-dependent) and the recombination rate (densitydependent). Equation (1.1) is the famous Saha equation. Let it be explained in brief. Suppose the number of particles with energy level Ei under temperature T is given by the Boltzmann distribution as ni = gi e−Ei /KB T

Introduction to Plasmas

3

where gi is the ion degeneracy state. The ratio of the particle number of similar species having energy level Ei and ground state E0 is explained as

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ni gi = e−Ei0 /KB T , n0 g0

Ei0 = Ei − E0 .

The ionization potential Vi [eV ] is the energy difference of two states and the p2 kinetic energy of the electron as Ei0 = eVi + 2m , where p is the momentum e and me is the electron mass. Then, the ratio of the densities of the charged and neutral particles in a weakly ionized gas can be obtained by the Saha equation  ∞ 2 gi 2 ni = e−(eVi +p /2me )/KB T d p 3 n0 g0 h ne −∞  3/2 T 3/2 −eVi /KB T 2gi 1 2πme KB T = e−eV /KB T  2.4 × 1021 e . 2 g 0 ne h ne Consider 2gi /g0 ∼ 1 and ni and ne are the ion and electron densities, respectively. If an ion is charged when it comes in contact with an electron, then ni ∝ n1e . This is because it is in an equilibrium state where the combination rate is proportional to the density of the electron. Now, the ionization ratio ri = nn0i  nne0 and the degree of ionization ri i α = n0n+n . As ri  1, so = 1+r i i α2  ri2 =

2.4 × 1021 T 3/2 exp[−eVi /KB T ]. n0

The degree of ionization and temperature are proportional. But the degree of ionization is inversely proportional to the neutral density and ionization potential. At thermal equilibrium, the distribution of particle velocity of a gas is given by 3/2    m mv 2 n exp − f (v) = . 2πKB T 2KB T The following relation will give the average kinetic energy:  3 1 ∞ 1 mv 2 f (v)dv = KB T. E= n −∞ 2 2 In plasma physics, temperatures are given in units of energy. Now, 1 eV = 1.6×10−19 1.6 × 10−19 J = KB T , so T = 1.38×10 −23 = 11600 K.

Waves and Wave Interactions in Plasmas

4

1.3 Basic Concepts of Plasma 1.3.1 Basic dimensionless parameters The criteria for analysis without dimension are time t and length l. Let us look at the following three important parameters related to our criteria: p1 = n = particle number density ∼ l−3 p2 = vth = particle thermal velocity ∼ lt−1

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p3 =

e2 = interaction strength ∼ l3 t−2 . m0

Note that the final quantity contains only l and t and measures the strength of the coulomb interaction between the free particles charged in the plasma. We are now combining the three physical quantities to obtain a dimensionless parameter that characterizes the creature β γ 3(γ−α)+β (−2γ−β) P = pα t . 1 p2 p3 ∼ l

Now, −2γ − β = 0 and γ − 3α = 0. So, β = −2γ and γ = 3α, where α is an arbitrary constant. If we consider α = 1, we get the dimensionless quantity as P0 ∼  where Λ = n

3 vth ne2 1/2 ( m )

p1 p33 1 ≡ 2 6 p2 Λ

and m = mi or m = me . Choosing a different

0

value of α would result in a power of P0 which would nevertheless remain dimensionless. Λ is used to measure plasma parameters. This is dimensionless and consists of unmagnetized plasma systems. For strongly coupled plasma, the value of Λ  1. On the other hand, for weakly coupled plasma, Λ  1. In the former case, the potential energy of the interacting particles is more important than their kinetic motions. Also, for the latter case, the particle thermal motions are more significant. 1.3.2 Debye length and Debye shielding The Debye length is a basic unit of length measurement in plasma physics. This unit of length appeared in the electrolyte theory developed by P. Debye and was, therefore, given the name Debye length. Debye and Huckel (1923) have shown, using the equations from Boltzmann and Poisson, that under

Introduction to Plasmas

5

equilibrium conditions in plasma, the average potential in the vicinity of a charged plane varies according to the law φ = φ0 exp(−x/λD ).

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Here, x is the distance from the plane and λD is represented by  1/2 KT λD = . 4πnq 2 Here, n is the number density of the particle per cubic centimeter, each carrying an identical charge q. The particles have a thermal movement characterized by a temperature T . If we consider a charged particle emerge in a plasma volume, it attracts oppositely charged particles, and a charge cloud around the test charge builds up. If we assume that the plasma is cold, then there is no thermal movement, and the shielding is perfect, and the number of negative charges is equal to the number of positive charges in the cloud. However, when the temperature is finite, the charged particles at the edge of the cloud (where the electric field is weak) have enough thermal energy to escape from the static potential wall of the electron. Then, the edge of the cloud appears at the radius where the potential energy is equal to the thermal energy. This radius is the Debye length and the spherical cloud is called the Debye sphere. To find the expression of Debye length, and to understand Debye shielding, consider a plasma of a uniform density (n0 ) of both electrons and ions. Initially, there is no net charge density and no electric field. If we consider a positive test charge +q that is introduced into the plasma, which is at the origin of a spherical polar coordinate system, then the positive charge attracts negatively charged electrons and repels positively charged ions. In the steady state, the number densities of the electrons ne (r) and the ions ni (r) differ slightly in the vicinity of the test charge, but they are the same at a large distance, i.e, ne (∞) = ni (∞) = n0 . When the plasma reaches a thermodynamic equilibrium, the charged particles are distributed according to the Boltzmann law. The number density of the electrons and ions can thus be written as follows: ne (r) = n0 exp(eφ(r)/KB Te )

(1.2)

ni (r) = n0 exp(−eφ(r)/KB Ti )

(1.3)

where e, Ti , Te , and KB are the electron charge, ion temperature, electron temperature, and the Boltzmann constant, respectively. The total charge

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Waves and Wave Interactions in Plasmas

density ρ(r) with the test charge can be given as ρ(r) = e[ni (r) − ne (r)] + qδ(r)

(1.4)

where δ(r) is the Dirac delta function. Now from Poisson’s equation with the help of Equations (1.2)–(1.3), we get      ρ(r) en0 eφ eφ q 2 − δ(r) (1.5) ∇ φ=− = exp − exp − 0 0 KB Te KB Ti 0

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where 0 is the absolute permittivity for free space. We also assume the disturbed electrostatic potential is weak so that the electrostatic dynamic energy is much less than the average thermal energy, i.e., eφ  KB Te  KB Ti .

(1.6)

Using (1.6) in (1.5) and considering terms up to O(φ), we get   n0 e 2 1 1 q 2 ∇ φ= + φ − δ(r). 0 K B T e Ti 0 Assuming φ is spherically symmetric around the test charge, we get   1 d 1 d2 2 2 dφ ∇ φ= 2 (rφ). r = r dr dr r dr2 So, for r = 0, d2 1 (rφ) − 2 (rφ) = 0 2 dr λD

(1.7)

where 1 n0 e 2 = 2 λD 0 K B



 1 1 + . Te Ti

Solution of Equation (1.7) that remains finite at r → ∞, i.e.,   A r φ = exp − r λD

(1.8)

where A is a constant. Near the test charge, the electrostatic potential should be the same as it is generated by an isolated test charge q in free

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1 q space. Then, the initial condition will be φ → 4π as r → 0. Using this 0 r q condition in (1.8), we get A = 4π0 . Thus,   q r φ= exp − . (1.9) 4π0 r λD

This φ(r) is commonly known as Debye potential. It shows that if r  λD , then φ → 4πq 0 r . So, the effective potential is almost the same as the bare Coulomb potential of the test charge. While at points r  λD , the potential decreases exponentially (not as 1/r) due to the shielding effect of surrounding charges. Obviously, in free space, n0 → 0, λD → ∞, and φ fall off as 1/r. Roughly, the test charge is shielded by the plasma particles, located in a sphere with the radius λD . The test charge only interacts with these particles and has a negligible influence on particles at intervals r > λD . λD gives a rough measure of the distance above which the electron density can differ significantly from the ion density. The length λD as defined by Equation (1.8) is known as Debye length, and the sphere of radius λD is called Debye sphere. 1.3.3 Quasineutrality Quasineutrality is a fundamental concept of plasma. It means that it creates an equilibrium between both positive and negative charges in the microscopic volume element. In the plasma production process, it is evident that the number of positive and negative charges must be the same. Since the velocities of electrons and ions are different, the former is more likely to leave the regions in which they occurred. So, the number of charges of one or the other sign is initially the same. It is because the electrons are faster than the ions, and they move much more quickly towards the walls of the tube consisting of discharge and leftover ions. With the loss of electrons, there will be an excessive charge of opposite sign, which will tend to equalize the current of electrons and ions and reduce the concentration of particles of opposite sign. This tendency is called the quasineutrality of plasma. 1.3.4 Response time An example of the collective behavior of plasma is the timescale according to which the electrons establish a shielded equilibrium. The ions need much longer time than electrons to reach their equilibrium position. If we have

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Waves and Wave Interactions in Plasmas

|eφ|  KB Te , where KB is the Boltzmann constant, the electron energy does not change significantly from its thermal value. Therefore, the elec tron velocity remains close to a thermal velocity Ve = KB Te /me . To establish the new balance, the electron must reach its unique position at a typical distance λD . This time can be estimated as τ = λDe /Ve and known as response time. The reciprocal of this response time is called plasma frequency.

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1.3.5 Plasma frequency In the Debye screening analysis, the plasma is assumed to be at equilibrium, i.e., the plasma charges do not move. Screening is, therefore, an example of static collective behavior. We will now give an example of dynamic collective behavior. Suppose the plasma comprises electrons moving freely in an immobile neutralization background where the charge of the electron is q, the density is n, and the mass is m. Let the electrons move a distance d to the right and leave a background of charge density ρ = −nq and width d. Hence, the electric field produced on the edges is E = 2πρd = −2πnqd (for the right edge) and E = 2πρd = 2πnqd (for the left edge). So, the total force acted on the electrons is F = qE = −2πnq 2 d, which accelerates the right electron towards the left and vice versa. The relative acceleration of the edges of the electrons would be a = 2(qE/m) = −4πnq 2 d/m. As a = d¨, one gets d¨ = −ωp2 d where ωp2 = 4πnq 2 /m. It shows that the oscillation occurs with the frequency ωp , known as Plasma frequency. These periods of oscillation must be much smaller than the typical lifetime of the system. From the fundamental length λD and the basic velocity of the particles v = (KB T /m)1/2 , we have v ωp = = λD



4πnq 2 m

1/2 rad/s

where ωp is the characteristic oscillation rate for electrostatic disturbances in plasmas. For electron oscillations, the numerical value of ωp can be given by √ fp ≡ ωp ≈ 9000 ne Hz.

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9

1.3.6 Collisions and coupling limit The concept of weakly coupled plasma comes from the requirement of many electrons inside the electron-Debye sphere. Let us define the number of electrons inside the electron-Debye sphere by

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NDe =

4π 3 λ ne . 3 De

The borderline between weakly and strongly coupled plasma is defined ear4π0 2 3 3 1 3 lier. Also, NDe = 1 ⇒ 4π 3 λDe n3 = 1 ⇒ ne = 4π . λ3D = ( 3e2 ) Te . Due to the occurrence of different temperatures of electrons and ions, different 2/3 coupling space occurs. For NDe = NDi , Γi = 13 NDi . Due to the long range of the Coulomb force, collisions between charged particles in a plasma differ from the collisions of molecules in a neutral gas. However, collisions in weakly coupled plasmas are present due to the collective effects of the many-particle processes of Debye shielding. However, collision frequencies are estimated qualitatively as follows. An impact parameter b is set by equating kinetic energy equal to potential energy as T e2 ∼ . b 2 The collision cross-section is obtained by σ ∼ πb2 ∼

4πe4 T2

and so the collision time τ is estimated by √ 1 1 ωp ∼ nσν ∼ nmσ T ∼ 4π nλ3D τ where λD is the Debye length and ωp is the plasma frequency. For hightemperature plasmas nλ3D  1, which indicates that plasma is collisionless if ωp τ  1. Collision frequency 1/ταβ of a charged particle of species α against a particle of β is approximately given by  ωpe 1 1 1 1 me 1 1 me 1 , , , . ∼ ∼ ∼ ∼ τei τee τei τii τie nλ3D mi τee mi τei The ratio of the magnitude of the collision frequencies is given by  1 1 1 me me : : =1: : . τee τii τie mi mi

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Waves and Wave Interactions in Plasmas

When the plasma reaches the thermal equilibrium, the thermal equilibrium of the electrons arises first, followed by the ions in between, and finally, the thermal balance between the electrons and ions gets completed. Thus, a two-temperature state occurs long before the thermal equilibrium is reached. So, the plasma may have several temperatures at the same time. The electrons and ions may have different temperatures even when both are Boltzmann distributed. 1.4 Criteria for Plasma

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An ionized gas becomes plasma if it satisfies the following conditions: (i) the Debye length λD is much smaller than the dimensions L of the system, i.e., λD  L, (ii) the number of particles within the Debye sphere is more than 1, i.e., ND  1, (iii) ωτ > 1, then neutral gas becomes plasma where ω is the normal plasma oscillations frequency and τ represents the mean time between collisions with neutral atoms. 1.5 High-Temperature Plasmas The condition for the existence of plasma is that the average electron kinetic e2 , where energy should be larger than the Coulomb potential, that is, T  r r is the average particle distance and is obtained by 4 πr3 n = 1. 3 A high-temperature plasma is characterized by the smallness of the ratio of the average potential energy to the electron kinetic energy  −2/3 e2 /r 1 4π 3 Γ= = nλD  1. T 3 3 In a high-temperature plasma, the number of particles in the Debye sphere is much greater than one, i.e., nλ3D  1. It indicates that the total number of charged particles in the Debye sphere is enough to be an effective shield. 1.6 Mathematical Description To explain the state of a plasma mathematically, we have to write down all particles’ positions and velocities to describe the electromagnetic field

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11

acting in the plasma. It is, however, neither practical nor necessary to trace all the particles in a plasma. Therefore, plasma physicists typically use less detailed descriptions known as plasma models. There are two main types of models: (a) kinetic model and (b) fluid model. Sometimes, the motion of individual particles under electromagnetic force can also provide us important information. The fluid model of plasma consists of smoothed quantities such as density and average speed around each position. In the simple fluid model, a single charged species follows the basic fluid equation in an electromagnetic field, and the motion is controlled by Maxwell’s equations, Lorentz force, and other forces that may occur. A more general description is the twofluid model, in which the ions and electrons are described separately. Fluid models are more accurate if the collision is high enough to keep the plasma velocity distribution close to a Maxwell–Boltzmann distribution. As the fluid models usually describe the plasma as a single flow at a specific temperature in any spatial location, it fails to give an accurate result for velocity space structures, such as beams or double layers, or can they resolve wave– particle effects. But a kinetic model describes the particle velocity distribution function at every point in the plasma. Therefore, it does not have to assume a Maxwell–Boltzmann distribution. Generally, there are two common approaches to the kinetic description of a plasma: one is based on the representation of the smoothed distribution function on a grid in speed and position and the other technique is known as particle-in-cell (PIC) technology. It contains kinetic information by following the trajectories of a large number of individual particles. Kinetic models are usually more computationally intensive than fluid models. The Vlasov equation can be used to describe the dynamics of a system of modified particles that interact with an electromagnetic field.

1.7 Magnetized Plasmas A plasma in which the magnetic field is strong enough to influence the movement of the charged particles is called magnetized plasmas. A general quantitative criterion is that an average particle completes at least one gyration around the magnetic field before it collides (i.e., ωce /νcoll ≥ 1, where ωce is the electron gyrofrequency and νcoll is the electron collision rate). It is often the case that the electrons are magnetized while the ions are not. Magnetized plasmas are anisotropic, meaning their properties in the direction parallel to the magnetic field differ from those perpendicular

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Waves and Wave Interactions in Plasmas

to them. While the electric field in plasmas is usually small due to its high conductivity, the electric field associated with a plasma moving in a  = −v × B  (where E  is the electric field, v is magnetic field is given by E  is the magnetic field) and is not affected by the Debye the speed, and B shield. Lorentz Force: The amalgamation of the electric force and magnetic force on a point charge produces electromagnetic force, which is also known as Lorentz force. It happens because of the electromagnetic field. The equation of motion is m

dv  + v × B  )q = F = (E dt

(1.10)

 , and B  are the charge of the particle, particle mass, velocwhere q, m, v , E ity, electric field, and magnetic field, respectively. Gyrofrequency and Gyroradius: Let us consider that particle is moving in a homogeneous and stationary magnetic field which is working along the  = 0 and B  = (0, 0, Bz ). From (1.10), we get z-axis. Let E m

dv  ) = (ivy − jvx )Bz . = q(v × B dt

(1.11)

Therefore, dvx q = vy Bz , dt m dvy q = − vx Bz , dt m dvz = 0. dt

(1.12) (1.13) (1.14)

Integrating Equation (1.14), we obtain vz = constant. The velocity com. ponent along the z-axis remains constant. So, it is not affected by B Differentiating Equation (1.12), we get  2 qBz d2 vx qBz dvy = = − vx dt2 m dt m = −ωc2 vx .

(1.15)

Similarly, from Equation (1.13), we get d2 vy = −ωc2 vy dt2

(1.16)

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Introduction to Plasmas

13

z where ωc = |q|B m is known as cyclotron frequency or gyrofrequency which is always positive. Equations (1.15) and (1.16) represent the simple harmonic motion of the charged particle separately. Solving Equations (1.15) and (1.16), we have

vx = A sin(ωc t + φ0 )

(1.17)

vy = A cos(ωc t + φ0 )

(1.18)

where A and φ0 are integrating constants. Consider v = v + v⊥

(1.19)

, where v and v⊥ are the components of v parallel and perpendicular to B respectively. Since |v | and |v| are constants, so |v⊥ | is also constant. From Equations (1.18) and (1.19), we get 2 v⊥ = vx20 + vy20 = A2

⇒ A = v⊥ and vx0 = tan(φ0 ) ⇒ φ0 = tan−1 vy0



vx0 vy0



where vx0 and vy0 are initial velocity components. Integrating Equations (1.17) and (1.18) and using the initial conditions, we get v⊥ v⊥ x − x0 = − cos(ωc t + φ0 ) + cos(φ0 ) (1.20) ωc ωc v⊥ v⊥ y − y0 = sin(ωc t + φ0 ) − sin(φ0 ) (1.21) ωc ωc where (x0 , y0 , z0 ) is the initial position of the particle. Equations (1.17) and (1.18) represent the trajectory of the charged particles. Squaring and adding Equations (1.20) and (1.21), we get  2 v⊥ 2 2 (x − x1 ) + (y − y1 ) = (1.22) ωc where x1 = x0 +

v⊥ ωc

cos(φ0 ) and y1 = y0 −

v⊥ ωc

sin(φ0 ).

Case 1: If v = 0, the particle describes a circular orbit with radius rL such ⊥  that rL = vω⊥c = mv |q|B in XY plane perpendicular to B . The center (x1 , y1 ) of the circular orbit is fixed. The center and radius of the circular orbit are known as guiding center and gyroradius or cyclotron radius, respectively.

Waves and Wave Interactions in Plasmas

14

The time period of rotation is T = for nonrelativistic motion.

2π ωc

=

2πm qB

which is independent of speed

 with Case 2: If v = 0, then the particle advances along or opposite to B  . The orbit a constant speed v , and at the same time, it gyrates about B  as an axis. The pitch angle φ of the helix will be will be a helix with B given by tan φ =

v T v = . v⊥ T v⊥

(1.23)

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1.8 Single Particle Motion in Uniform Electric and Magnetic Field Let us consider that the charged particle moves in an electromagnetic field  = (Ex , 0, Ez ) and B  = (0, 0, Bz ). Now, where E dv q  ˆ  ) = q (Ex + vy Bz )ˆi − vx Bz ˆj + Ez k. = (E + v × B dt m m

(1.24)

Therefore, from (1.24), we get q dvx = (Ex + vy Bz ), dt m dvy q = − vx Bz , dt m dvz q = Ez . dt m

(1.25) (1.26) (1.27)

Differentiating Equation (1.25) with respect to t and using Equation (1.26), we get  2 q d2 vx qBz dvy = =− Bz vx = −ωc2 vx (1.28) dt2 m dt m where ωc =

|q|Bz m .

Similarly, from (1.26), we obtain   d2 vy qBz q Ex 2 =− (Ex + vy Bz ) = −ωc vy + . dt2 m m Bz

(1.29)

A harmonic oscillation occurs in the x-direction

with frequency ωc . Let us Ex consider a linear transformation v¯y = vy + Bz , which means the frame is   Ex moving with a constant velocity − Bz in the −y direction. Thus, the

Introduction to Plasmas

15

motion can be considered as the superposition of a circular orbit, and a  ×B  drift. Solving (1.28) and (1.29), we get constant motion is called E vx = A cos ωc t + B sin ωc t



vy = C cos ωc t + D sin ωc t +

−

Ex Bz



(1.30) (1.31)

where A, B, C, and D are the integration constants. If initially the particle is at rest at origin and the particle has a constant dy dx x acceleration E Bz ωc , the initial conditions are x = y = 0, dt = dt = 0, and d2 x dt2

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x = E Bz ωc at t = 0. Using the initial conditions, from (1.30), we get A = 0 and B = Thus,

vx =

Ex Bz .

dx Ex = sin(ωc t). dt Bz

(1.32)

Ex cos ωc t + C1 Bz ω c

(1.33)

On integration, we get x=−

where C1 is the integration constant. Using the same set of initial conditions, we get C1 = BEz xωc . Thus, Ex (1 − cos ωc t). Bz ω c

(1.34)

Ex (sin ωc t − ωc t). Bz ω c

(1.35)

x= Similarly, we obtain y=

Hence, (1.34) and (1.35) are the required trajectories of the charged particles. 1.9 Fluid Approach Plasma is characterized by a huge number of particles. As observed, the single-particle approach is unnecessarily complex. As it is difficult to keep track of each particle, a comprehensive statistical approach is required. The fluid model is extremely helpful in explaining the majority of the plasma phenomena because the movement of fluid elements is only considered and the uniqueness of single particles is ignored. Electric charges characterize

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Waves and Wave Interactions in Plasmas

plasma fluid. For any random fluid, the recurrent collisions between particles maintain the fluid component in continuous motion. But the presence of a magnetic field in plasma prevents frequent collisions. The fluid description method is advantageous because it is simple to use and results in seven-dimensional phase space instead of three spatial dimensions and time. However, the velocity dependence effect like Landau damping cannot be explained by this method.

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(i) Continuity equations The equation of continuity is born from the conservation of mass. The conservation of matter requires that the rate of decrease of the number of particles in a volume V must be equal to that leaves the volume V per unit time through the surface S bounding the volume V . Therefore,    ∂n ∂N = dV = − nu.dS = − ∇.(nu)dV (1.36) ∂t V ∂t V where N is the total number of particles in a volume V and u is the fluid velocity. Using divergence theory, we get ∂n + ∇.(nu) = 0. ∂t

(1.37)

This is the equation of continuity. (ii) Equation of motion Let us consider the equation of motion for a single particle with velocity v m

dv  + v × B ) = F = q(E dt

(1.38)

where F is the Lorentz force. If all the particle fluid elements will move together with average velocity u and we neglect collisions and thermal effect, the equation is nm

du  + u × B  ). = qn(E dt

(1.39)

In the above equation, the time derivative is with respect to a reference frame moving with the average velocity u. Therefore, the time derivative operator is written as d ∂  = + u · ∇ dt ∂t

(1.40)

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17

∂  correwhere ∂t is the time derivative in a fixed frame and the term u · ∇ sponds to change as the observer moves with the fluid. Hence, the convective derivative is

du ∂u  )u. = + (u · ∇ dt ∂t

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Therefore, from Equation (1.39), we get   ∂u  + u × B  ).  )u = qn(E mn + (u · ∇ ∂t

(1.41)

(1.42)

When the thermal motion of the particle is considered, an additional force of pressure gradient is added to the right-hand side of Equation (1.42). This force arises due to the random motion of particles in and out of a fluid element. This pressure gradient force per unit volume may be calculated  p, where p is the scalar kinetic pressure. So, the fluid equation of as −∇ motion is written as   ∂u  + u × B ) − ∇  p.  )u = qn(E mn + (u · ∇ (1.43) ∂t For generalization (like anisotropic distribution and shearing forces), the  p is replaced by −∇  · P , where P is the pressure effect of viscosity −∇ tensor. (iii) Pressure equations Let us consider the thermodynamic equation of state p = Cργ

(1.44)

where C is a constant and γ is the ratio of specific heats Cp /Cv . Therefore, ∇p = Cγργ−1 ∇ρ = γp ⇒

∇p ∇n =γ p n

∇ρ ρ

(as ρ = mn).

(1.45)

Now, (i) for isothermal compression, ∇p = ∇(nkT ) = kT ∇n, so γ = 1, ∇T ∇n ∇T ∇n and (ii) for adiabatic compression, ∇n n + T = γ n , so T = (γ − 1) n . Generally, γ = (2 + d)/d

(1.46)

where d is the number of degrees of freedom. It is valid only for negligible heat flow, i.e., for low thermal conductivity.

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18

1.10 Maxwell’s Equations Maxwell’s equations are considered one of the fundamental equations of electromagnetism. It is a combination of four basic equations such as Gauss’s law in electrostatics, Gauss’s law in magnetostatics, Faraday’s law of electromagnetic induction, and Ampere’s law with Maxwell’s modification. Differential Form: Maxwell’s equations can be expressed in differential form as  = ρ or ∇  = ρ  ·D  ·E ∇ 0

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 =0  ·B ∇

(Gauss’s law in electrostatic),

(Gauss’s law in magnetostatics),

  ×E  = − ∂B ∇ ∂t

(1.47) (1.48)

(Faraday’s law of electromagnetic induction), (1.49)

  ×H  = J + ∂D ∇ ∂t

(Ampere’s law with Maxwell’s modification) (1.50)

 , ρ, B  , E,  and H  are electric displacement vector (coulomb/m2 ), where D charge density (coulomb/m3 ), magnetic induction (weber/m2 ), electric field intensity (volt/m), and magnetic field intensity (amp/m-turn), respectively. Equation (1.47) is derived from Coulomb’s law. Equation (1.48) means that the magnetic monopole does not exist in our physical world. Maxwell’s equations represent the mathematical expression of certain experimental results. These equations cannot be checked directly. However, their application to any situation can be checked. Due to extensive experimental work, Maxwell’s equations are now considered one of the guiding principles such as the conservation of momentum and energy. Integral form: (i) Gauss’s law in electrostatics is  = ρ.  ·D ∇ Integrating over volume space V ,      . ρdV ∇ · DdV = V

V

Introduction to Plasmas

Using Gauss divergence theorem, we get     S = ρdV Dd S

19

(1.51)

V

 = q, the net where the surface S bounds the volume V . Since V ρdV charge is contained in V . The physical significance of Equation (1.51) is that the total flux of electric displacement vector through the surface enclosed by the volume is equal to the total charge contained within that volume. (ii) Gauss’s law in magnetostatics is

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 = 0.  ·B ∇ Integrating over volume V, 

 · Bd  V  = 0. ∇

V

Using Gauss divergence theorem, we get   S  = 0. Bd

(1.52)

S

The signification of the equation is that the total outward flux of magnetic induction through any closed surface is equal to zero, i.e., a magnetic monopole does not exist. (iii) Faraday’s law of electromagnetic induction equation is   ×E  = − ∂B . ∇ ∂t Integrating the equation over a surface S bounded by a curve C, we get    ∂B     dS. ∇ × EdS = − S S ∂t Using Stokes’ theorem, we get    ∂B  l = −  dS. (1.53) Ed S ∂t C  l and the magnetic flux is φ = The electromagnetic force is C Ed  S  . Equation (1.53) states that the e.m.f around a closed path is Bd S equal to the negative rate of change of magnetic flux linked with the path.

20

Waves and Wave Interactions in Plasmas

(iv) Ampere’s law with Maxwell’s modification equation is   ×H  = J + ∂D . ∇ ∂t Integrating the above equation, we get     ∂D  × Hd  S =  J + dS. ∇ ∂t S S

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Using stokes’ theorem, we get     ∂D     J+ dS. Hdl = ∂t C S

(1.54)



 l around This equation states that the magnetomotive force = C Hd a closed path is equal to the conduction current plus displacement current through any surface bounded by the path. 1.11 Electromagnetic Wave Equation in Free Space In free space, the volume charge density ρ = 0 and current density J = 0, hence Maxwell’s equations become  = 0,  ·D ∇

(1.55)

 = 0,  ·B ∇

(1.56)

  ×E  = − ∂B , ∇ ∂t   ×H  = ∂D ∇ ∂t

(1.57) (1.58)

 = 0 E  and B  = μ0 H  , and 0 and μ with D  0 are the absolute permittivity and permeability of the free space, respectively. From (1.57), we get  × (∇  ×E  ) = − ∂ (∇  ×B ) ∇ ∂t ) − ∇  (∇  ·E  2E  = ∂ (∇  × μ0 H ) ⇒∇ ∂t  2E  = μ0 ∂ (∇  ×H  ) (using (1.55)) ⇒∇ ∂t

Introduction to Plasmas

21

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  ∂ ∂D 2  ⇒ ∇ E = μ0 (using (1.56)) ∂t ∂t 2  2E  = μ0  0 ∂ E . ⇒∇ ∂t2

(1.59)

Similarly,  × (∇  ×H  ) = ∂ (∇  ×D ) ∇ ∂t )−∇  (∇  ·H  2H  = ∂ (∇  × 0 E ) ⇒∇ ∂t 2  2H  = 0 ∂ B (using (1.56)) ⇒∇ ∂t2 2  2H  = μ0  0 ∂ H . ⇒∇ ∂t2

(1.60)

Equations (1.59) and (1.60) are called electromagnetic wave equations in free space.

1.12 Plasma Kinetic Theory 1.12.1 Distribution function In plasma kinetic theory, distribution functions are used extensively. Suppose f (r, v , t) is the distribution function1 in a six-dimensional phase space (r, v ). Here, r and v represent the position vector and velocity vector in phase space. The particle position in phase space is defined by the coordinates (r, v ). f (r, v , t) is a function of seven independent variables: 3 for the position, 3 for velocity, and the last is time. Let r = xˆi + yˆj + zkˆ and

 = dv dr, dr = v = vxˆi + vy ˆj + vz kˆ. Volume element in phase space is dV 3 3 d r = dxdydz, and dv = d v = dvx dvy dvz . Accordingly, f (r, v , t)drdv is  (= drdv ) in phase space. the number of particle in a volume element dV It means f (r, v , t) is space particle number density in phase space at time t. 1 If f (r,  v , t) depends on  r , the distribution is inhomogeneous and f (r,  v , t) is homogeneous if it is vice versa. Again, if f (r, v, t) depends on the direction of  v , then f (r,  v , t) is anisotropic, and the distribution is isotropic if it is vice versa.

Waves and Wave Interactions in Plasmas

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1.12.2

Macroscopic variables

Macroscopic parameters like density, average velocity, and pressure are used. (a) Density: The density n(r, t) is obtained as   ∞ ∞ ∞ f (r, v , t)dvx dy dvz = n(r, t) = −∞

−∞

−∞

∞

−∞

f (r, v , t)dv .

ˆ v , t) is the normalization of f (r, v , t), then it means If ∞ f (r, fˆ(r, v, t)dv = 1. Thus, fˆ(r, v, t)d vd r is the probability of −∞ finding a particle in a volume element dV = dvdr. So, fˆ(r,v, t) is the probability per unit volume of phase space. Therefore, fˆ(r,v, t)n(r, t) = f(r,v, t). (b) Average velocity: The probability of finding a particle at r and time t with velocity v and v + dv is ∞ fˆ(r,v, t)dv. So, an average velocity fˆ(r, v , t)vdv u(r,relation t) = can be obtained from the −∞

or 1 u(r, t) = n(r, t)



∞ −∞

f (r, v , t)vdv.

(c) Average random kinetic energy: Average random kinetic energy is defined as  ∞ 1 1 Eav = mv 2 f (r, v , t)vdv. n(r, t) −∞ 2 (d) Pressure tensor: Pressure tensor is defined as  ∞  P (r, t) = m (v − u)(v − u)f (r, v , t)dv . −∞

1.12.3 Maxwellian distribution function A particularly important distribution function is the Maxwellian function  3/2

m 2 fm (v) = n exp − v 2 /vth 2πKB T where v 2 = vx2 + vy2 + vz2 , vth = (2KB T /m)1/2 , and KB is the Boltzmann constant.

Introduction to Plasmas

23

To show Maxwellian distribution as a distribution function, we need the values of three necessary integrals, as follows: √ ∞ 2 π (i) −∞ e−αv dv = √α √ ∞ 2 −αv2 π (ii) I = −∞ v e dv = 2α3/2 ∞ −αv2 (iii) I = ∞ ve dv = 0. Using the results of (i), (ii), and (iii), we get the result of fm (v) =   32 2 2 m 2 n 2πKB T e−v /vth where v 2 = vx2 + vy2 + vz2 and vth = 2KB T /m, then ∞ f¯ (v)dv = 1. −∞ m

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1.12.4 Non-Maxwellian distribution in plasmas It has been observed that in space and astrophysical plasmas, the charged particles do not follow the Maxwellian distribution. The distribution functions have a non-Maxwellian high-energy tail. In a uniform magnetic field, the particle trajectories are altered. The power-law tails are produced. It is known from Boltzmann’s theory that the velocity distributions are Maxwellian due to random collision. But the rapid fall off of the collision cross-section with particle speed shows that the high-energy particles in the tail can deviate strongly from Maxwellian. Then, the distribution of tail particles can be metastable and becomes non-Maxwellian. Maxwell distribution describes the long-range interactions in collisionless unmagnetized plasma, where the nonequilibrium stationary state exists. But from space plasma experiments, it is seen that ion and electron populations are far away from their thermodynamic equilibrium, so a non-Maxwellian distribution is adopted. 1.12.5 Nonthermal distribution Space plasma research suggests that in many situations, electron and ion populations are not in thermodynamic equilibrium. Various research clearly shows the presence of energetic electrons in astrophysical plasma environments like Earth’s bow shock and foreshock, the upper Martian ionosphere, and the vicinity of the moon. The distribution functions are considered nonthermal [1]. An energetic electron distribution is also observed in different regions of the magnetosphere. Accordingly, the nonthermal electron–ion distribution is proved to be very widespread and characteristic of space plasmas, in which coherent nonlinear waves and structures [2, 3]

Waves and Wave Interactions in Plasmas

24

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are expected to play a crucial role. Cairnes et al. [4] used the nonthermal distribution of electrons to examine the ion-acoustic single structures observed by the FREJA satellite. It has been observed that solitons with both positive and negative density disturbances could exist. The nonthermal distribution [4] for electrons is given by     ne0 αv 4 v2  1+ 4 exp − 2 (1.61) fe (v) = ve 2ve (3α + 1) 2πve2 where ne0 is the electron density at equilibrium, ve is the thermal speed of the electron, and α is a parameter that determines the population of energetic nonthermal electrons. Here, assume that the speed of the structure is low compared to the thermal speed of the electrons. Therefore, one can neglect the effect of the flow velocity on the electrons since the electron distribution in the steady state is a function of the electron energy. The distribution of electrons inthe presence  of nonzero potential can be determined by replacing

v2 ve2

by

v2 ve2

− 2Φ , where dimensionless potential

Φ = meφ 2 . φ, e, and me are the electrostatic potential of the wave or e ve structure, electron charge, and electron mass, respectively. Thus, integration over the resulting distribution function results in the following expression for the electron density: ne = ne0 (1 − βΦ + βΦ2 ) exp(Φ) where β =

4α 1+3α .

(1.62)

It is clear that β → 4/3 for very large α.

1.12.6 Superthermal distribution Plasmas can hold a considerable amount of high-energy particles both in space and in laboratories. Known as superthermal particles, these highenergy particles may spring up from external forces on the natural space environment plasmas or wave–particle interactions. A long tail is found in the region with high energy in the plasmas with a surplus of superthermal electrons. To model such plasmas, which is a generalized Lorentzian [5] distribution, appropriate particle distribution is used. It is popularly known as Kappa Distribution. The speed of resonant energy transmission between particles and plasma waves can notably vary if a significantly bigger number of superthermal particles is present. This fact differentiates kappa distribution from Maxwellian distribution. For the macroscopic ergodic equilibrium condition, Maxwell distribution is taken to be suitable. But in the

Introduction to Plasmas

25

case of long-range interactions in unmagnetized collisionless plasma where the nonequilibrium stationary condition is present, Maxwell distribution may fail to explain. Thus, in this situation, the kappa distribution can be a better option, which has been introduced by the distribution function as  −k−1 eφ vx + , (1.63) fe (x, vx ) = Ce 1 − 2 2 kme θthe 2kθthe where the normalization is given for any value of the spectral index k > 1/2 by

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ne0 Γ(k + 1) . Ce =

1/2 2 Γ(k + 1/2) 2πkθthe Here, k forms mainly the superthermal tail of the distribution, the quantity Γ signifies for the gamma function, and  1/2 k − 1/2 Te θthe = . k me In the limit k → ∞, distribution (1.63), we get Maxwell–Boltzmann velocity distribution. Hence,   ne0 vx2 − 2eφ/me fe (x, vx ) = exp 1 − (1.64) 2 2 )1/2 2vthe (2πKvthe where vthe = (Te /me )1/2 . Integrating fe (vx ) over all velocity space, we get  −k−1/2 eφ 1 ne (φ) = ne0 1 − . (1.65) k − 1/2 Te The kappa distribution function fe (v) against v for various values of κ is given in Figure 1.1. It is visible from this figure that the height and shape of the distribution differ considerably with a variation of κ. It also proves that the height decreases along with the increase in the parameter κ. It leans to the Maxwell (solid line) distribution. 1.12.7 q-nonextensive distribution Tsallis statistics or nonextensive statistics is a new advance towards statistics, and this is rooted in Boltzmann–Gibbs–Shannon (BGS) entropic measure. This new statistical approach studies the cases where Maxwell distribution fails. Renyi [6] first gave recognition to this, and later Tsallis [7] propounded this in cases where entropic index q describes the degree of

26

Waves and Wave Interactions in Plasmas

fe v k 1

40

k 1.5 k 5

30 20

k 1000

10

4

2

2

4

v

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Fig. 1.1: Plot of kappa distribution function fe (v) vs. v for several values of k = 1.0, 1.5, 5.0, and 1000.0.

nonextensivity of the above approach. The stricture q that supports the generalized entropy of Tsallis is connected to the fundamental dynamics of the system. It determines the quantity of its nonextensivity. The quality of nonextensivity marks the systems in both statistical mechanics and thermodynamics. Even the entropy of the whole is dissimilar to the total of the entropies of the individual parts. If q < 1 (superextensivity), the generalized entropy of the whole is bigger than the total of the entropies of the parts. If q > 1 (subextensivity), the generalized entropy of the method is lesser than the sum of the entropies of the parts. Thus, accordingly, qentropy may offer a suitable structure for the study of numerous astrophysical scenarios [8]. For example, stellar polytropes, solar neutrino problem, and velocity distribution of galaxy clusters. The q distribution function, which is a one-dimensional equilibrium in nature, establishes the electron nonextensivity. This is shown by fe (v) = Cq {1 − (q − 1)[(me vx2 )/(2Te ) − (eφ)/Te ]}1/(q−1) where normalization is

  1 Γ 1−q me (1 − q)  Cq = ne0 1 1 2πTe Γ 1−q − 2 

 Γ 1 +1  q−1 2 1+q me (q − 1)

 Cq = ne0 1 2 2πTe Γ q−1

(1.66)

⎫ ⎪ f or − 1 < q < 1 ⎪ ⎬ f or q > 1.

⎪ ⎪ ⎭

Introduction to Plasmas

27

For q < −1, the q distribution cannot be normalized, and if q → 1, the distribution diminishes to Maxwell–Boltzmann velocity distribution. For −1 < q < 1, high-energy states are more likely than in the extensive Maxwell case. High-energy states are less likely for q > 1 than in the extensive Maxwell case, and q distribution function shows a thermal shutdown of the extreme value vmax for the speed of the particles as  2Te 2eφ vmax = − . me (q − 1) me We get 

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ne (φ) =

∞

 ne (φ) =

∞

fe (vx )dvx

vmax

−vmax

fe (vx )dvx

⎫ f or − 1 < q < 1 ⎪ ⎬ ⎪ ⎭

f or q > 1.

So,   (q+1) eφ 2(q−1) ne (φ) = ne0 1 + (q − 1) . KTe

(1.67)

Figure 1.2 shows the q distribution function fe (v) against v for different values of q = −0.5, 1.0, 1.5. From this figure, it is seen that the amplitude and width of the distributions change remarkably as q increases. It means that the nonextensive character of the plasma has become essential. It is fe v 2.5 2.0 1.5 1.0 0.5 4

2

2

4

v

Fig. 1.2: Plot of f (v) vs. v for q → 1 (dashed line), q = −0.5 (dotted line), and 1.5 (solid line).

28

Waves and Wave Interactions in Plasmas

also clear that the amplitude increases along with the rise of the parameter q and tends to the Maxwell distribution (dashed line) when q → 1. Boltzmann equation: Ludwig Boltzmann (1844–1906) used an equation in his works and it has come to be known as the Boltzmann equation. It explained the statistical character of a thermodynamic system that is not in an equilibrium state. Let f (r, v , t) be a distribution function and satisfy the Boltzmann equation in the six-dimensional phase space (r, v )    ∂f  f + F · ∂f = ∂f + v · ∇ ∂t m ∂v ∂t coll

(1.68) 



where F is the external force acting on the particle. The term coll   ∂f explains the rate of change of f due to collisions. If ∂t = 0, then the

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∂f ∂t

coll

particle does not clash. Collisionless Boltzmann equation, where self-clash long-range interactions are neglected, is known as Vlasov equation. Vlasov equation: The Vlasov equation explained the time progression of the distribution function f (r, v , t) in the phase space (r, v ). This equation can be found from the collisionless Boltzmann equation. Moreover, the self-consistent electromagnetic fields can be produced by the existence and movement of all the charged plasma particles. ∂f  f + q (E  + v × B  ) · ∂f = 0 + v · ∇ ∂t m ∂v

(1.69)

 and B  are the charge of the particle, the mass of the particle, where q, m, E, and the self-consistent electric field and magnetic field, respectively. 1.13 Closure Form of Moment Equation The continuity equation arises from the zeroth velocity moment of the Boltzmann equation, which links the number density with the flow velocity. Two independent equations are needed to verify these two variables. The equation of motion connecting number density, flow velocity, and pressure tensor results from the first velocity moment of the Boltzmann equation, and these need to be reflected on. Thus, three variables are connected by two transport equations. Commonly, an equation for nth-order velocity moment of f has (n + 1)th-order velocity moment of f . By considering various moments of the Boltzmann equation, the groups of equations derived

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Introduction to Plasmas

29

fail to structure a closed set because they constantly embrace extra variables than the number of equations. But a closed group of equations can be obtained by producing some suppositions for the maximum moment of the distribution function emerging in the system. If the distribution function f (r, v , t) is known, plasma macroscopic variables, such as number density n(r, t), average velocity u(r, t), and temperature, can be obtained by taking appropriate velocity moments of the distribution function. Taking the zeroth velocity moment of f (r, v , t), we get number density as  (1.70) n(r, t) = f d3 v. v

The first velocity moment of f (r, v , t) gives the average velocity as  1 u(r, t) = v f d3 v. (1.71) n v The secocnd velocity moment of f (r, v, t) gives the pressure tensor as   P (r, t) = m (v − u)(v − u)f d3 v. (1.72) v

The transport equations satisfied by these variables can be obtained by taking various moments of the Boltzmann equation. 1.13.1 Equation of continuity To derive the equation of continuity, we consider the Boltzmann equation as   F ∂f ∂f ∂f  · + v · ∇f + = (1.73) ∂t m ∂v ∂t coll  + v × B  is the Lorentz force. where F = E Integrating Equation (1.73) over velocity space, we get       ∂f q  ∂f  f dv +  ) · ∂f dv = dv + v · ∇ (E + v × B dv . ∂v ∂t coll v v m v ∂t v (1.74) Now,

 v

∂f ∂ dv = ∂t ∂t

 f dv = v

∂n , ∂t

(1.75)

Waves and Wave Interactions in Plasmas

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30

since v is an independent variable.    f dv = ∇  · v f dv = ∇  · (nu) v · ∇ v

(using Equation (1.71)).

(1.76)

v

Also, 

 · ∂f dv = E ∂v v

 v

∂  )dv = · (f E ∂v



 s = 0. f E.d

(1.77)

S∞

Here, S∞ is the v-space at v → ∞. Since f → 0 faster than v −2 as v → ∞, for any distribution with finite energy,    ∂f ∂ ∂    )dv (v × B ) · dv = · (fv × B )dv − f × (v × B ∂v v v v ∂ v v ∂       ⊥ ∂ = 0. Since v × B . (1.78) ∂v The collision cannot change the net amount of the number of particles. Therefore,    ∂f dv = 0. (1.79) ∂t coll v Using results (1.75)–(1.79) in Equation (1.74), we obtain ∂n  + ∇.(nu) = 0. ∂t

(1.80)

This is the required equation of continuity. 1.13.2 Equation of motion To obtain the equation of motion, we multiply Equation (1.73) by mv and integrating the resulting equation over the velocity space, we get    ∂f   + v × B  ) · ∂f dv mv dv + mv (v · ∇)f dv + q(E ∂t ∂v v v v    ∂f = mv d3 v. (1.81) ∂t v coll Now,   ∂ ∂f ∂ v f dv = m (nu) [using Equation (1.71)]. (1.82) mv dv = m ∂t ∂t ∂t v v

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Introduction to Plasmas

Since v is an independent variables. So,      · nvv ). mv (v · ∇)f dv = m∇ · vv f dv = m(∇ v

31

(1.83)

v

 , where u is the average velocity and U  is the thermal Let us write v = u + U velocity. U  ) + 2m∇  ).  · n(vv ) = m∇  · (nuu) + m∇  · (nU  · (nuU m∇

(1.84)

  is obviously zero and the quantity mnU U   is the Here, the average U     kinetic pressure tensor P . Also, ∇· (nuu) = mu∇· (nu)+ n(u · ∇)u. Therefore,   )f dv = mu∇  )u + ∇  · (nu) + mn(u · ∇  · P . (1.85) mv (v · ∇ v

Again,

 ∂f ∂    + v × B  )]dv v (E + v × B ) · dv = · [fv (E ∂v v v ∂ v    + v × B  )dv − f (E  + v × B  ) · ∂ v dv − fv · (E ∂v v v   + v × B  )f dv = − (E



v

 + v × B  ). = −n(E

(1.86)

∂  +v ×B  )]dv =  +v×B  )]dS  = 0 as the quantity Here, v ∂ v (E [fv (E v · [f S∞ −2  + v × B  )] tends to zero faster than v as v → ∞. Also, fv · (E + [fv (E v    v × B )dv = 0 because both  E and B are independent of the velocity  forces ∂f component. Now, v mv ∂t dv is the change of momentum due to coll

collisions and is denoted by Pc . Using results (1.82)–(1.86) in Equation (1.81), we get   ∂u  + u × B ) − ∇  · u)u = qn(E  · P + Pc . + (∇ (1.87) mn ∂t Equation (1.87) is the equation of motion. When collision is neglected, then Equation (1.86) can be written as   ∂u  + u × B ) − ∇   · P . mn + (∇ · u)u = qn(E (1.88) ∂t

32

Waves and Wave Interactions in Plasmas

The physical significance of Equation (1.87) is that the time rate of change of mean momentum in each fluid element is due to externally applied forces, plus the pressure forces of the fluid itself, and the forces are also associated with collisional interactions. 1.13.3 Equation of energy To derive the energy equation, we multiply Equation (1.73) by mv 2 /2 and integrate it. The resulting energy equation is obtained. If we neglect the effects of velocity, the effects of the collision, and thermal conductivity, then we find an adiabatic equation of state as

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P = Cργ

(1.89)

where C is a constant and γ = Cp /Cv . So, ∇P = γ∇ P n . For isothermal process, γ = 1 (∇p = ∇(nT ) = T ∇n), and for the adiabatic process, γ = (2 + D)/D, where D is the degree of the freedom. 1.14 Dusty Plasma The interaction of plasmas and charged dust particles have opened a new and fascinating research area called dusty plasma. Dusty plasmas are a special form of plasmas that originated from completely or partially ionized gases, which consist of electrons, ions, and extremely massive charged dust particles in the micrometer size. It is a normal electron–ion plasma with an additional charge component of macroparticles. The different characteristic lengths in dusty plasmas are (i) dust grain size length “a”, (ii) Debye length “λD ”, and (iii) the average interparticle distance “d” (where nd d3 ∼ 1); nd is the dust number density. A plasma with dust particles or grains can either be called “dust in a plasma” or “dusty plasma”. If a  λD < d, the plasma is called “dust in a plasma”, and the situation a  d < λD corresponds to “a dusty plasma”. Electron–ion plasma and dusty plasma are dissimilar to each other as the dust particle’s charge to mass ratio is relatively small for the latter compared to the former. Moreover, the frequencies of the dust particles are lesser compared to electrons and ions. There are differences between usually electron–ion (e-i) plasma and dusty plasma. (i) Neutrality: Like e-i plasma, a dusty plasma is also globally neutral. So, at equilibrium, the charge neutrality condition becomes qi ni0 = ene0 − qd nd0 , where ns0 are unperturbed number density of the plasma

Introduction to Plasmas

33

species s (s = e, i, d for electrons, ions, and dust, respectively) and qi = Zi e. qd = Zd e (−Zd e) is the dust particle charge when the grains are positively (negatively) charged, where Zi = 1, and Zd is the number of charges residing on the dust grain surface. (ii) Debye shielding: Considering the ions and  electrons obey

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s and ni = ni0 Boltzmann distribution, i.e., ne = ne0 exp Keφ B Te   s exp − Keφ , and also considering Poisson’s equation as ∇2 φs = B Ti

4π(ene − ni − qd nd ) and calculating the same as before, one can early  obtain the dusty plasmaDebye length λD as λD = (λ λ )/ λ2De + λ2Di , where λDe = (KB Te )/(4πne0 e2 ) and λDi = De Di (KB Ti )/(4πni0 e2 ) are electron and ion Debye length. (iii) Plasma frequency: In plasma, the collective motion is characterized by plasma frequency ωp . In dusty plasma also the same situation arises, if we treat the same as that in e-i plasma. In plasmas, electrons  oscillate around ions with the electron plasma frequency ions also oscillate around dusts ωpe = 4πne0 e2 /me , and here the  with the ion plasma frequency ωpi = 4πni0 e2 /mi , and the dust particles also oscillate around  their equilibrium position with the dust plasma frequency ωpd = 4πnd0 e2 /md . In dusty plasma, the formation of dusty plasma crystal is another critical phenomenon and it is characterized by Coulombs coupling parameter Γc . It is defined as the ratio of dust potential energy to dust thermal energy. If the dust q2 potential energy Σc = ad exp(− λaD ) and dust thermal energy is KB Td , Z 2 e2

then Γc = aKdB Td exp(− λaD ). A dusty plasma is called weakly coupled if Γc  1 and strongly coupled if Γc  1. Charged dust grains demonstrate a group behavior and are significant from the standpoint of wave dynamics. In dusty plasma, the ion numbers and electron numbers are asymmetrical. Charge process: The crux of the dusty plasma research is to understand the charging of dust grains in plasma. It depends on the environment of the dust grains. The charging process causes the interaction of dust grains with gaseous plasma, with charged particles, or with photons. When dust particles are immersed in an electron–ion plasma, electrons reach the dust grain surface more rapidly than the ions as the electron thermal speed is higher than the ion thermal speed. As a result, the surface potential becomes negative. Then, eventually, the electrons are repelled, and the ions

Waves and Wave Interactions in Plasmas

34

are attracted. On the other hand, if the surface potential becomes positive, it attracts electrons and repelled ions. Assuming the maxwell velocity dis 3/2   mj vj2 mj exp − 2KB Tj , one tribution for plasma species fj (vj ) = nj 2πKB Tj can obtain the charging current Ij for attractive (qj φd < 0) and repulsive (qj φd > 0) potentials as Ij =

4πrd2 nj qj



KB Tj 2πmj

1/2   qj φd 1− for qj φd < 0 KB Tj

and

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Ij =

4πrd2 nj qj



KB Tj 2πmj

1/2



qj φd exp − KB Tj

 for qj φd > 0.

Accordingly, one can obtain the electron current Ie and ion current Ii as Ie =

−4πrd2 ne e



KB Te 2πme

1/2



eφd exp KB Te



and Ii =

4πrd2 ni e



KB Ti 2πmi

1/2 

 eφd 1− . KB Ti

The time evolution of the charge on a dust grain in plasma can be written as dqd = Ie + Ii dt dZd Ie + Ii or =− . dt e Fluid equation: Like electron–ion plasma, the fluid equations for dusty plasma can also be written in the equation of continuity, equation of motion, and Poisson’s equation. The equation of continuity is ∂nd + ∇ · (nd ud ) = Sd ∂t where the source or sink term Sd may arise in dust–ion collision and vanish in equilibrium. In general, dust number density is not affected by dust

Introduction to Plasmas

35

losing or picking up some charge, so generally, Sd is considered zero. The equation of momentum is   ∂ 1 qd + u d · ∇ nd + ∇ · Pd = (E + ud × B) − ∇φ + nd , ∂t md n d md and Poisson’s equation is ∇2 φ = 4π



mα n α ,

α

and charge fluctuation is

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dI = Ie + Ii . dt The reason behind this is that the dust particles are charged and get their electric charge by (i) electron emission, (ii) electron and ion capture, and (iii) other charging procedures. The significant basic dust grain charging procedures are (i) interaction of dust grains with gaseous plasma particles, (ii) interaction of dust grains with energetic particles (electrons and ions), and (iii) interaction of dust grains with photons. Quasineutrality condition  of the plasma p (qp np0 + Q0 nd0 ) = 0. Here, Q0 = eZd is the dust charge at equilibrium position and  = 1 or −1 according to positive and negative dust charge, respectively. As the electric potential of the adjacent plasma environments resolves the electric charge of dust grains, the electric charge of the dust particle is irregular. Thus, if a wave controls this potential, the electric charge of dust particles is influenced. As there is a variation of time in the dust charge, it is suitable to illustrate the plasma as fd = fd (r, p, t, q). Here, q is the electric charge of the grains. As in the dusty plasma, the shielding of the electric charge of dust particles by the other plasma particles is not exponential; the Debye shielding in dusty plasma is somewhat dissimilar from that in electron–ion plasma. The dust grain charge q is established by dq dt = I, where I is the sum of charging currents that are attained by the grain surface. This total charge results from two parts: an external current I1 and another part Ip with p = e, i. Ip are the currents  by the electrons and ions from the plasma. Thus, I = I1 + p=e,i Ip . The external current can be obtained from various processes like photo emission by the incident UV radiation, secondary electron emission by the impact of the energetic particles, thermionic emission, and radioactive rays. Since the electrons move faster than the ions, the dust grains mainly attain their negative charge by the dust grains while considering Ip with p = e, i. The

36

Waves and Wave Interactions in Plasmas

total current that flows on the surface of the dust grains in an equilibrium state is zero.

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1.15 Quantum Plasma Quantum mechanics was born and claimed to describe physics in the first quarter of the 20th century in a completely new way. The interpretation of quantum theory seems confusing but has a wide range of applications in astrophysics, microelectronics, and nanotechnology. Many bodily systems must be described by a so-called density matrix, which has no counterpart in classical theory. Many scientists like Max Planck, A. Einstein, N. Bohr, W. Heisenberg, E. Schrodinger, and P. Dirac have made enormous efforts to develop the subject. The development was done in two stages. On December 14, 1900, Max Plank made a presentation at a meeting of the German Physical Society. He stated that radiation is emitted or observed by matter in discrete or quanta packets of energy hν, where h is the frequency of the radiation and ν is Planck’s constant. The first step started with Max Planck’s hypothesis, which is a combination of classical and nonclassical theory. The second stage of quantum mechanics began in 1925, where Werner Heisenberg introduced a special form of quantum mechanics, also called matrix mechanics. Next year, Erwin Schrodinger introduced another mathematical form of quantum mechanics and is called wave mechanics. This mechanics combines the theory of classical waves and the relation of wave particles by Louis de Broglie. Mathematically, the ideas of wave mechanics and matrix mechanics are different. But they are identified according to the concept of physics. A classical charged particle system can be called plasma if it is almost neutral and if collective effects play a vital role in the dynamics. Traditional plasma physics has intensely focused on regimes characterized by high temperatures and low densities for which the effects of quantum mechanics have practically no impact. However, with the advancement of recent technology in semiconductors, it has become possible to imagine practical applications of plasma physics. It is here that the quantum nature of particles has a vital role to play. The quantum causes cannot be disregarded both at standard metal densities and room temperature. To examine the active characteristics of quantum plasma, electron gas is considered quantum fluid. But while reviewing particular astrophysical objects under severe density and temperature like the neutron stars and the white dwarf stars where the density is almost 10 times magnitudes bigger than ordinary solids, the quantum aspect must be

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Introduction to Plasmas

37

considered along with plasma. In the 1960s, Pines first examined quantum plasmas consisting of electrons, ions, positrons, and holes. The concept of the quantum plasma was developed with the help of the well-known mathematical model, namely (i) Schrodinger–Poisson model, (ii) Winger–Poisson model, and (iii) quantum hydrodynamics model. If the macroscopic properties are considerably influenced by the quantum nature of plasma particles, it can be taken as quantum plasma. The quantum plasma differs significantly from classical plasmas in the following ways: (a) For classical plasmas, the system temperature is much higher than the Fermi temperature because it is fully quantum or fully degenerate. (b) When we consider the quantum effects on plasmas, the density is extremely high, and the temperature is comparatively low. In contrast to traditional plasmas, these have a high temperature and a low density. Plasmas exist on the sun’s surface, in the earth’s magnetosphere, and the interplanetary and interstellar media. Both fusion and space plasmas are characterized by high-temperature and low-density regimes, for which quantum effects are entirely negligible. However, physical systems in which both plasmas and quantum effects coexist occur in nature, the most obvious example being the electron gas in a common metal. Quantum plasma also occurs in some astrophysical objects such as white dwarf stars and neutron stars, where the density is greater than that of regular solids. The interaction and the quantum effects are also crucial in dense plasmas, which are relevant for intensive plasma experiments with a fixed laser density. For microscale and nanoscale objects such as quantum diodes, quantum-free electron lasers, quantum dots and nanowires, nanophotonics, and ultra-small electronic devices, for microplasmas and semiconductors, quantum correction to plasma is essential. The thermal de Broglie wavelength (λB ) of the particles comprising plasma can be used to measure the quantum effects and is defined as λB =

 . mvT

Here,  is the Planck  constant divided by 2π, m is the mass of the charged BT species, and vT = Km . The thermal de Broglie wavelength is so small that the charged particles can be compared to the shape of a point for classical plasmas. However, when de Broglie wavelength is equal to or bigger than the average interparticle distance d = n−1/3 , then both wave functions and quantum interference overlap with each other. The quantum effects can be considered when nλ3B ≥ 1. Similarly, the system is notably influenced by the quantum features of the charged species when the system

Waves and Wave Interactions in Plasmas

38

temperature is lower than the Fermi temperature TF of the species, where TF is defined as TF =

EF KB

where EF is the Fermi energy of the charged species and is explained as p2F 2 EF = 2m (3π 2 )2/3 n2/3 . pF is called the Fermi momentum and is = 2m defined by pF = (3π 2 n)1/3 . When the system temperature T → TF of the species, then the proper distribution functions of the species transform from the Maxwell–Boltzmann distribution to the Fermi–Dirac distribution. Now,

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χ=

2/3 3 2/3 1 TF . = 3π 2 nλB T 2

Thus, plasma can be called a classical plasma when χ  1, i.e., T  TF . On the other hand, plasma can be defined as a quantum plasma if χ  1, i.e., if T  TF . In a quantum system, the thermal speed (vT ) can be changed by the Fermi velocity (vF ) because vT is pointless in extremely low temperatures. The Fermi velocity is explained as  1/2  1/3 2EF  2 vF = = . 3π n m m Now, the quantum analog of the Debye length is indicated by the symbol λF . So, vF λF = . ωp Thus, λF is the Fermi screening length for quantum plasmas. gQ is the quantum coupling parameter and can be defined as the ratio of interaction Table 1.1: Comparison between classical plasmas and quantum plasmas. Classical Plasma (i) Debye Length

λD =

vT ωp

 = 

(ii) Velocity

vT =

(iii) Temperature (iv) Coupling Parameter

 gc =

0 KB T ne2

KB T m

T 1 nλ3 D

1/2

1/2

2/3

Quantum Plasma λB = vF =

 mvT

 (3π 2 n)1/3 m

TF  gQ =

1 nλ3 F

2/3

Introduction to Plasmas

39

energy Eie to the Fermi energy EF . Eie 2e2 m ∼ gQ = = 2 EF  0 n1/3 (3π 2 )2/3



1 nλ3F

2/3

 ∼

ωp EF

2

≡ H 2.

Thus, it is obvious that gQ and gC are equivalent when λF → λD . gQ has no classical counterpart, and it represents the ratio between the plasmon energy ωp with the Fermi energy EF .

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1.16 Quantum Plasma Models The important problem of macroscopic observability of quantum phenomena in plasmas is connected with the question of when and which quantum phenomena are important in quantum plasmas. The answer to this question, of course, depends upon the models and approximations used to describe the quantum plasma. The complete description of quantum plasma as a system of many particles is a hopeless task because it is impossible to solve the Schrodinger equation for the N particle wave function of the system. This problem can be simplified by assuming that plasma is nearly ideal, i.e., two and higher-order correlations between the particles are neglected. In this view, plasma is considered as a collection of quantum particles that interact only with the collective field. Let us consider the Schrodinger equation for the N -particle wave function ψ as a function of q1 , q2 , . . . , qn , where qj (rj , sj ) is the coordinate (space, spin) of the j particle, and it gives the dynamics of the N -body problem. Let the jth particle have energy Ej . We also neglect the correlation particles, and so the wave function can be written as a product of a single particle wave function. Let the particle be identical, and so each wave function satisfies the Schrodinger equation. Let us consider a one-dimensional quantum plasma, where the electrons are described by a statistical mixture of M pure states, each with a wave function ψj , j = 1, . . . , M obeying the Schrodinger–Poisson system ∂ψj 2 2 + ∇ ψj + eφψj = 0 ∂t 2me ⎞ ⎛ N  ∇2 φ = 4πe ⎝ |ψj |2 − Zi ni ⎠

i

j =1

(1.90)

(1.91)

Waves and Wave Interactions in Plasmas

40

where φ(x, t) is the electrostatic potential, m is the electron mass, and −e is the electron charge. Electrons are globally neutralized by a fixed ion background with density n0 . We assume periodic boundary conditions, with spatial period L. Finally, in the context of this chapter, it is convenient to adopt the normalization  (1.92) dx|ψj |2 = N/M

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where N is the number of particles in a length L so that n0 = N/L and global charge neutrality is assured. Let us introduced the amplitude Aj = Aj (x, t) and the phase Sj = Sj (x, t) associated with the pure state psij = ψj (x, t) according to ψj = Aj exp(iSj /).

(1.93)

Both Aj and Sj are defined as real quantities. The density nj and the velocity uj of the j th stream of the plasma are given by nj = A2j ,

uj =

1 ∂Sj . m ∂x

(1.94)

Introducing (1.93) and (1.94) into (1.90) and (1.91) and separating the real and imaginary parts of the equation, we find ∂ ∂nj + (nj uj ) = 0 ∂t ∂x

(1.95)

  √ ∂uj ∂uj e ∂φ 2 ∂ ∂ 2 ( nj )/∂x2 + uj = + √ nj ∂t ∂x m ∂x 2m2 ∂x ∂2φ e = ∂x2 0 where

2 ∂ 2m2 ∂x



 M

(1.96)

 nj − n0

(1.97)

j =1

√ ∂ 2 ( nj )/∂x2 √ nj

 is called the Bohm potential. The continu-

ity equation (1.95) and the quantum Euler equation (1.96) are the fluid dynamic representation of the Schrodinger equation. In this context, (1.93) can be termed the Madelung decomposition of the wave function. In the resulting set of equations, quantum effects are contained in the pressure like, -dependent term in (1.96). If  = 0, we simply obtain the result of the classical multistream model. Therefore, we shall refer to (1.95)–(1.97) as the quantum multistream or quantum Dawson model.

Introduction to Plasmas

41

The classical analog of this system is the “cold plasma” model because there is no pressure term in the momentum equation. However, in the quantum plasma, the Bohm potential term plays a role similar to the pressure. It does not correspond to the gradient of a function of the density only. Although it arises directly from the Schrodinger equation and is responsible for typical quantum-like behavior involving tunneling and wave packet spreading, it also contributes to extra dispersion of the small wavelengths and is relevant when we compare the propagation of nonlinear waves in the classical and quantum cases.

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References [1] M. V. Goldman, M. M. Oppenheim, and D. L. Newman, Nonlinear Proc. Geophys. 6, 221 (1999). [2] R. C. Davidson, Methods in Nonlinear Plasma Theory. New York: Academic (1972). [3] M. Tribeche, R. Amour, and P. K. Shukla, Phys. Rev. E 85, 037401 (2012). [4] R. A. Cairns, A. A. Mamum, R. Bingham, R. Bostrom, R. O. Dendy, C. M. C. Nairn, and P. K. Shukla, Geophys. Res. Lett. 20, 2702 (1995). [5] V. M. Vasyliunas, J. Geophys. Res. 73, 2839 (1968). [6] A. Renyi, Acta. Math. Hung 6, 285 (1955). [7] C. J. Tsallis, Stat. Phys. 52, 479 (1988). [8] J. A. S. Lima, R. Silva, Jr. Santos, and J. Santos, Phys. Plasmas 19, 104502 (2010).

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Chapter 2

Introduction to Waves in Plasma

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2.1 Introduction A wave can be defined as a disturbance that traverses a medium by transferring the energy of a particle or a point to another devoid of any permanent dislocation of the medium. Certain waves do not need the medium to move. Waves can be found in the environment in various forms and can be treated as carriers of energy and information from one place to another. Linear waves propagate with an unchanging form at a constant speed. With absorption, the waves will decrease in size as it moves. But in the presence of a dispersive medium, various frequencies travel at a different speed, in two or three dimensions. As the waves spread, their amplitude also decreases. Studying nonlinear waves and oscillations in various fields is important since wave phenomena always maintain a close relationship between theories and experiments. In nature, waves exist in different forms: waves on the ocean surface, acoustic waves from a beautiful piece of music that reaches our ears, ancient electromagnetic waves from a distant star, etc. If a wave propagates through a continuous medium, it is called a mechanical wave. Here, the medium could be air, water, a spring, the earth, or even people. Sound, water waves, a pulse that travels in the spring, and earthquakes are all mechanical waves. A wave of people in a soccer stadium like “Mexican Wave” is also a mechanical wave. The particles in the medium can travel either perpendicularly towards the wave or parallel to the path of the wave. If the medium oscillates at right angles to the direction of the waves, it is called the transverse wave. Again, if the medium shifts backward and forward in the same direction, the wave is called the longitudinal wave. The sound wave is an example of a longitudinal wave. A combination of the two is known as a surface wave. The movement of

43

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44

Waves and Wave Interactions in Plasmas

the particles is generally circular or elliptical on the surface of the medium. Seismic waves can be defined as waves that either move through the interior of the earth or near the surface of the earth. Seismic waves are the result of an earthquake, explosion, or volcano that emits low-frequency acoustic energy. Although there are many different seismic waves in nature, there are four types of basic waves: (i) primary wave (P-type), (ii) secondary wave (S-type), (iii) love wave, and (iv) Rayleigh wave. The electromagnetic waves can propagate without a medium. These waves occur due to the vibration of the electromagnetic field. Light is an electromagnetic wave, and, therefore, light can travel without a medium. Matter wave and gravitational wave are also important in today’s science. Matter wave is generally used to denote particles like electrons that have wavy properties. These waves are a part of quantum mechanics. A gravitational wave is a ripple in the fabric of space–time itself. Mathematically, the energy of a wave is proportional to the square of its amplitude, i.e., E = CA2 , where E and A are the energy and amplitude of the wave, respectively. The constant C is dependent on the medium. 2.2 Mathematical Description of Waves Suppose a progressive wave is moving along the x-axis. Its motion and all other similar kinds of wave motions can be explained quantitatively through the famous wave equation ∂2u ∂2u = c2 2 2 ∂t ∂x

(2.1)

where u is the amplitude function and is a measurement of the variation of displacement along the y-axis at a particular distance x along the x axis, c is the wave speed or phase speed with which the wave is traveling, and t is the time. This is a linear second-order partial differential equation. This equation can be rewritten as    ∂ ∂ ∂ ∂ −c +c u = 0. ∂t ∂x ∂t ∂x Introducing the new variable v = equations

∂u ∂t

+ c ∂u ∂x , we therefore obtain the two

∂u ∂u ∂v ∂v +c = v, = 0. −c ∂t ∂x ∂t ∂x

Introduction to Waves in Plasma

45

 ∂v

 ∂v − c ∂x is the directional derivative of the function v in the direction  ∂v (−c, 1). Also, ∂v − c express that v is constant on the lines x + ct = ∂t ∂x constant. In other words, v(x, t) = f (x + ct). Then, the first equation now becomes ∂t

∂u ∂u +c = f (x + ct). ∂t ∂x After some calculation, we can easily obtain the solution as u(x, t) = F (x + ct) + G(x − ct).

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This solution is the general solution of the sum of a wave moving to the left, F (x + ct), and a wave moving to the right, G(x − ct), with velocity c. It has many solutions, one of which in terms of the sine function is u = A sin 2π(x/λ − ωt) where λ, ω, and A are the wavelength, the frequency, and the amplitude of the wave, respectively. So, a linear wave in mathematical theory is usually with sine (or cosine) wave of the form u(x, t) = a sin(kx ± ωt).

(2.2)

Here, x is the space and t is the time, while a, k, and ω are positive parameters. a, k, and ω are the amplitude of the waves, wavenumber, and angular frequency, respectively. Moreover, u is periodic in both space and 2π time with periods λ = 2π k and T = ω . λ is called the wavelength and T the periods. The wave has crests (local maxima) when kx ± ωt is an even multiple of π and troughs (local minima) when it is an odd multiple of π. We see that as t changes, the fixed shape X → sin(kX) changes at constant speed c along the x-axis, and X = x ± ct is obtained as a phase. The number c = ω/k is called the speed of the wave phase or phase speed. If the sign is positive, the waves move to the left and vice versa if the sign is negative. A wave of permanent shape moving at constant speed is called a traveling wave or sometimes a steady wave. For linear waves, if two waves u1 and u2 cross each other, then the resultant amplitude is the sum of the amplitudes of each separate wave at the point of crossing. The resultant amplitude function U is equal to the linear combinations of two separate amplitude functions u1 and u2 . U = a1 u 1 + a2 u 2 where a1 and a2 are arbitrary constants, called mixing coefficients.

46

Waves and Wave Interactions in Plasmas

If we consider two equivalent waves one traveling along the +x axis and the other traveling exactly in the opposite direction, i.e., along the −x axis, then u1 = A sin 2π(x/λ − ωt) u2 = A sin 2π(x/λ + ωt) and also u1 = u2 and a1 = a2 = 1. We have the linear combination of two waves whose resultant amplitude function will be

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U1 = A sin 2π(x/λ − ωt) + A sin 2π(x/λ + ωt). The new resultant wave, which does not move either forward or backward, is known as a standing wave or stationary wave. As per the example, the linearized Korteweg–de Vries equation ∂u ∂ 3 u + 3 =0 ∂t ∂x appears as a model for water waves with a small amplitude and long wavelength. The equations are sometimes referred to as Airy’s equation although this name is also used for the closely related ordinary differential equation  y − xy = 0. Substituting u(x, t) = ei(kx−ω(k)t) into the equation, we find ω(k) = −k 3 and hence c(k) = −k 2 . Here, c is negative means that the wave travels to the left. 2.3 Dispersion Relation The relationship between the energy of a system and its momentum is known as a dispersion relation. Equation ω = ck is an example of a dispersion relation as energy in waves is proportional to frequency ω and the wavenumber k is proportional to momentum. This relationship is important while studying how quickly a particular kind of Fourier components in the initial profile of wave travel and how quick energy dissipates for a certain system. To differentiate between these concepts, two types of velocities are used: phase velocity and group velocity. Phase velocity: Let us assume a one-dimensional wave u(x, t) = f (kx − ωt), where k and ω are constants. While we observe a peak of the wave, we find that at a certain point in time t0 and in space x0 , the wave has a height H = u(x0 , t0 ) = f (C0 ) where C0 = kx0 − ωt0 . Following this, it is seen that after time Δt the point H has traveled a small distance Δx so

Introduction to Waves in Plasma

47

that u(x0 + Δx, t0 + Δt) = H. For Δt and Δx small, this can only be the case if kx0 − ωt0 = C0 = k(x0 + Δx) − ω(t0 + Δt), e.g., the point that gets mapped to H is the same after the wave has traveled a small distance. It will happen if Δx ω = . Δt k This is known as the phase velocity of the wave. So for a wave of the form u(x, t) = f (kx − ωt) = e(kx−ωt) where k and ω are constants, the phase velocity Vp is defined as the constant ω Vp = . (2.3) k Although the phase velocity can also be defined more generally, where Vp determines the speed at which anyone frequency component travels, in three dimensions, the exponent (k.r − ωt) is called the phase of the disturbance. The temporal derivative of the phase is given by the frequency ω and the spatial derivative of phase is given by the wave vector k that specifies the direction of propagation. A surface of a constant phase is called a wave surface. The velocity of the constant phase is called phase velocity which can be determined from ω d (k.r − ωt) = 0 ⇒ vp = . dt k It can be noted that phase velocity may exceed the speed of light.

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kΔx = ωΔt ⇒

Group velocity: The propagation of energy in a system is given by the velocity of wave packets, known as the group velocity. The modulation information does not travel at the phase velocity but at the group velocity, which is always less than the velocity of light c. Let us consider two waves of nearly equal frequencies whose equations are given as follows: E1 = E0 cos[(k + Δk)x − (ω + Δω)t] E2 = E0 cos[(k − Δk)x − (ω − Δω)t]. E1 and E2 differ in frequency by 2Δω. Since each wave must have the phase velocity ω/k appropriate to the medium in which they propagate, one must allow for a difference 2Δk in propagation constant. E1 + E2 = 2E0 cos[(Δk)x − (Δω)t] cos(kx − ωt). This represents a sinusoidally modulated wave. The envelope of the wave, given by cos[(Δk)x − (Δω)t], carries information; it travels at the

Waves and Wave Interactions in Plasmas

48

velocity

Δω Δk .

Taking the limit Δω → 0, we obtain the group velocity as Vg =

dω . dk

(2.4)

In the case of right-traveling waves f (x − ct), we determined the linear dispersion relation ω = ck. Applying the definitions for the group and phase velocities, we see that in this instance

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Vp =

ω ck dω = =c= = Vg . k k dk

It does not contradict the theory of relativity because an indefinite long wave train cannot carry information. Information can be carried by a modulated wave, on which variation in frequency or amplitude is superimposed. We can regard each beat of information in this modulated signal as a wave packet that moves with a velocity called group velocity, Vg =

dω . dk

We get the group velocity by the slope of the dispersion relation where Vg is less than the velocity of light, and it determines the velocity with which the energy of the wave is transmitted. Classification: Propagation of any disturbance is classified by the dispersion function D(ω, k) = 0 relating the frequency ω with the wave vector k. This relation helps us to determine phase velocity and group velocity. If ∂ω ∂k → 0, then there is a resonance at this frequency, the wave will not propagate, and the wave energy is used for stationary oscillation. A linear dispersion relation dominates a nondispersive wave. For linear equations, the differentiation of ω produces the coefficient of k, which is attained likewise by division. A linear relation thus entails the equality of Vp and Vg . On the other hand, let us imagine that the relation holds Vp = Vg . Now, if we take ω as a function only of k, we can segregate variables and solve the differential equation as follows: dω ω dω dk = ⇒ = . dk k ω k On integration, we get log ω = log k + c.

Introduction to Waves in Plasma

49

The linear dispersion relation is ω = Ck where C = ec . Dispersive waves consist of unequal phase and group velocities, while nondispersive waves contain equal phase and group velocities. Let L(φ) = 0 be the partial differential equation and the dispersion relation is given by D(ω, k, Ai ) = 0 which can be expressed as

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ω = ω(k; Ai ).

(2.5)

Each root of this equation is called mode and each mode generates different waves. For each mode, the phase velocity Vp = ωk is the characteristic velocity for the mode. If an observer moves with that velocity, he can see the mode as a static disturbance. In a dispersive medium, the group velocity Vg has more importance due to this velocity of transfer of energy being through the medium. Let φ(x) ∝ ei(kx−ω(k)t) ,

(2.6)

and the temporal evaluation of φ depends on the nature of ω(k). (i) If ω(k) is real, then φ(x) represents a harmonic wave. (ii) If ω(k) is purely imaginary = iω2 (k), then φ(x, t) = eikx .eω2 (k)t . It represents a nonpropagating standing wave. (iii) Also, if ω(k) = ω1 (k) + iω2 (k), then φ(x) ∝ ei(kx−ω1 (k)t) .eω2 (k)t . In that case, two cases arise, namely (a) if ω2 (k) > 0, the solution is unstable and (b) if ω2 (k) < 0, the solution is stable. The wave is called diffusive and nondiffusive according to the value of ω = ω(k) and is complex and real, respectively. A wave will be dispersive and 2 2 nondispersive depending on the value of ∂∂kω2 . If ∂∂kω2 = 0, then the wave is 2 dispersive. On the other hand, if ∂∂kω2 = 0, then the wave is nondispersive. In the case of dispersive wave, vg = vp and vg depends on k. So, various wavelength waves travel with various group velocities, and after a certain time, the disturbance spreads over a certain length. Here, we can assume that the wave undergoes dispersion. The nondispersive wave, vg , coincides with vp , and there is no dispersion. The energy in a dispersive wave moves with the group velocity. A wave initially is propagating with phase velocity ω and after a sufficiently large time it will propagate with group velocity ∂k  and satisfy the equation kt + ωx (k) = 0 ⇒ ∂k ∂t + ω (k) ∂x = 0, i.e., along dx  with the characteristic dt = ω (k), where k is constant. So, an observer

50

Waves and Wave Interactions in Plasmas

moving with group velocity of wavenumber k.

dx dt

= ω  (k) = Vg (k) will always be with a wave

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2.4 Linear Waves in Plasmas The fluid model is used to elucidate 80% and more plasma phenomena. Here, we neglect the individual identity of a particle, and we only consider the fluid element motions. Electric charges characterize the fluid in plasma. In any random fluids, the constant collisions between particles aid the fluids’ elements to progress together. However, the same model in plasma functions as magnetic fields is present and not due to collisions. In 1928, Langmuir first explained the electron oscillations in plasma. Two types of oscillations are possible: (i) oscillations of electrons which are very fast so the ions cannot follow them and (ii) oscillations of ions which are so slow that the electrons may satisfy the Boltzmann law. In plasma, electron oscillations arise, as the plasma wants to remain neutral. When electrons are shifted in a background of ions that are uniform in nature, it gives rise to electric fields. It, in turn, pulls back and reinstates plasma in a neutral state by putting back the electrons to their original position. The electrons exceed and oscillate around the stable position with a characteristic frequency, defined as the plasma frequency due to their inertia. The charged particles move haphazardly. The charged particles in a plasma move haphazardly, intermingle with each other with the help of their own electromagnetic forces and even reciprocate to perturbations that are externally applied. This consistent movement of plasma particles gives birth to different kinds of collective wave phenomena. Both longitudinal and transverse waves are supported by electron–ion plasma. Langmuir waves and ion acoustic waves evolve due to density and potential fluctuations. In an unmagnetized plasma, traverse waves are electromagnetic and do not go with density fluctuations. The existence of an external magnetic field in plasma affords the possibility of such a variety of longitudinal and transverse waves. When neutral dust grains are placed in electron–ion plasma, they are charged due to various processes. The wave propagation can be dominated or even changed in the presence of dust grains that are charged. The wave phenomena occur due to the inhomogeneity related to an arbitrary distribution of the charged particles and the digression from the traditional quasineutrality state in electron–ion plasma. It is because of charged dust grains, and the importance of the dust particle dynamics is present. Primarily, the fluid equations in various states are linearized to obtain linear

Introduction to Waves in Plasma

51

waves. After that, the perturbation quantity is proportional to ei(kx−ωt) , and the relation between ω and k is derived. It is known as a dispersion relation. We get different waves from the dispersion relation. The steps that we follow to obtain the dispersion relation are (i) finding linearized fluid equations using the restriction for each wave to exist, (ii) using Maxwell’s equations as and where needed, and (iii) considering the dependent variable of the partial differential equations for ei(kx−ωt) . Replace partial equation to algebraic equation by replac∂ ∂ ing ∂x → ik and ∂t → iω.

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2.5 Plasma Oscillation If the electrons are dislocated from the equilibrium position, an electric field is created, and in time, it wants to withdraw the electron at the point of equilibrium for neutrality. But for the inertia of the electrons, this will exceed and oscillate around the equilibrium position, and oscillations will occur. This is called plasma oscillation. Let us neglect the magnetic field and the thermal movement for the expression of the plasma oscillation and assume that the electron movement takes place only in the x direction, the ions are fixed uniformly in space, and the plasmas are homogeneous and infinitely extended. The fluid equations are ⎫   ∂ve ∂ve ⎪ + ve = −ene E me n e ⎪ ∂t ∂x ⎬ ∂ne ∂(ne ve ) (2.7) + =0 ⎪ ∂t ∂x ⎪  ∂E ⎭ 0 = e(ni − ne ) ∂x  are electron number density, ion number denwhere ne , ni , ve , and E sity, electron velocity, and electric field, respectively. In equilibrium, let  =E  0 and there is no oscillation. To linearize ni = ne = n0 , ve = v0 , and E Equation (2.7), we separated the dependent variable into two parts, equilibrium part and perturbation part, and write ne = n0 + n1 , ve = v0 + v1 ,  =E 0 + E  1 , where the subscript is  0 for equilibrium part and  1 for and E perturbation part. The equilibrium quantities convey the state of plasma ∂n0 ∂v0 ∂E0 0 without oscillation. Hence, ∂n ∂x = 0 = v0 = E0 . So, ∂t = ∂t = ∂t = 0.

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From (2.7), we get me

 ∂v1  1 , ∂n1 + n0 ∂v1 = 0, 0 ∂E1 = −en1 . = −eE ∂t ∂t ∂x ∂x

(2.8)

Let us assume that the oscillating quantity behaves sinusoidally. Hence, v1

n1 = n1 ei(kx−ωt) , E1 = E1 ei(kx−ωt) x,

and so ∂∂t v1 = v1 ei(kx−ωt) x, = ∂ v1 ∂ ∂ −iωv1 , ∂x = −ikv1 . So, ∂t may be replaced by −iω and ∂x is replaced by ik. From (2.8), we get ime ωv1 = eE1 , n1 ω = n0 kv1 , 0 ikE1 = −en1 .

(2.9)

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From (2.9), we get   e 2 n0 2 ω − v1 = 0.  0 me Since v1 = 0 for all space and time, so  e 2 n0 ω= = ωp (say).  0 me

(2.10)

(2.11)

ωp = (e2 n0 )/(0 me ) rad/sec is called Plasma frequency. The plasma fre√ quency depends on n0 , i.e., ωp ∝ n0 . It is seen that ω does not depend upon k, then ∂ω ∂k = 0. So, there is no group velocity and it has only oscillation. Thus, the disturbance does not propagate and there is no wave. 2.6 Electromagnetic Waves The electromagnetic fields in plasmas can have two parts: (i) static or undisturbed part and (ii) oscillating part. Generally, the static part (or equilibrium part) is written with suffix 0 and the oscillating part (or per =B 0 + B  1 and E  =E 0 + E 1 turbed part) with suffix 1 (for example, B  0, B  1, E  0 , and E  1 are undisturbed magnetic field, oscillating magwhere B netic field, undisturbed electric field, and oscillating electric field, respectively). All the oscillatory quantity varies sinusoidally, i.e., ∝ ei(kx−ωt) , so ∂ ∂ ∂x → ik and ∂t → −iω. The presence of an oscillating magnetic field determines the classification of plasma waves. In the existence of oscillating magnetic fields, plasma waves are referred to electromagnetic waves, and when there are no oscillating magnetic fields, electrostatic waves are generated. Plasma waves are also classified as follows:

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Parallel wave: If the direction of wave vector k is parallel to the undis 0 , then the wave is called a parallel wave. turbed magnetic field B Perpendicular wave: If the direction of wave vector k is the perpendicular undisturbed magnetic field B0 , i.e., if B0 is along the direction of z, then k will be either in the direction of y or x, then the wave is called a perpendicular wave. Longitudinal wave: If the direction of wave vector k is parallel to the oscillatory electric field E1 with no oscillatory magnetic field, then a wave that exists is called a longitudinal wave. Transverse wave: If the direction of k is perpendicular to E1 , then the wave is a transverse wave. For the transverse wave, B1 is finite, and the wave is electromagnetic. 2.7 Upper Hybrid Frequency In a magnetized plasma, different modes of oscillations and waves are possible. Let us consider that the electrons oscillate perpendicular to B0 , the ions are static in the uniform background of positive charge, and the thermal motions of the electrons are neglected (i.e., KB Te = 0). The equilibrium  0, E  0 , and v0 , where E  0 = v0 = 0. Again, if we constates are n0 , B   sider longitudinal waves only, i.e., k  E1 , then the oscillation occurring in the plane is called upper hybrid oscillation. On the other hand, if the ion  0 , i.e., k · B  0 = 0, the acoustic wave considers, then k is perpendicular to B oscillation is called the lower hybrid frequency. The basic fluid equations are ∂ne  + ∇ · [ne ve ] = 0 ∂t   ∂  + ve × B ]  me n e + ve · ∇ ve = −ene [E ∂t  ·E  = e(ni − ne ) 0 ∇

(2.12) (2.13) (2.14)

 e, and ne are the electron mass, electron velocity, electric where me , ve , E, field, electron charge, and electron number density, respectively. If there is no thermal motion of electrons, KB Te = 0. We assume at equilibrium the density of plasma is a constant, and static electric field, static magnetic field, and oscillation magnetic field velocity at equilibrium are all zero.  =B 0 + B  1 , v = ve1 , and E  =E  1. Accordingly, n = n0 + ne1 , B

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54

So, the linearized equations are ∂ne1  · ve1 = 0 + n0 ∇ ∂t ∂ve1  1 + ve1 × B  0] me = −e[E ∂t  ·E  1 = −ene1 . 0 ∇

(2.15) (2.16) (2.17)

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 1 . Without loss of generLet us consider only longitudinal wave, i.e., k  E   1 in the direction of the x-axis and B  0 is in the ality, we take k and E  = (Ex , 0, 0), ve1 = (vx , vy , vz ), and z-direction, i.e., k = (kx , 0, 0), E  = (0, 0, B0 ). We also assume that the oscillating quantities behave sinuB ∂ ∂ soidally and hence replacing ∂x by ik and ∂t by −iω, we obtain from Equations (2.15)–(2.17) −iωne1 + in0 kvx = 0 ⇒ n1 =

k n0 vx ω 

(2.18)

  1 + ve1 × B 0 −iωme ve1 = −e E

(2.19)

0 ikEx = −ene1 .

(2.20)

Equation (2.19) can be written into the component forms −iωme vx = −eEx − evy B0

(2.21)

−iωme vy = evx B0

(2.22)

−iωme vz = 0.

(2.23)

From Equations (2.22) and (2.23), we get eEx

vx = imω 2  ω 1 − ωc2 where ωc = eB0 /m is the cyclotron frequency. Again, from Equations (2.18), (2.20), and (2.24), we get   ωp2 ωc2 1 − 2 Ex = 2 Ex ω ω where ωp =

e2 n0 0 m

is the plasma frequency. Thus,  ω = ωc2 + ωp2 .

(2.24)

(2.25)

(2.26)

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This ω is called higher hybrid frequency. The frequency of this oscillation is higher than the plasma frequency ωp . The reason behind this here is that two restoring forces are acting on the electrons, namely the electrostatic force and the Lorentz force. If ωc = 0, then ω = ωp . Thus, the higher hybrid frequency is equal to the plasma frequency if the cyclotron frequency is zero. 2.8 Electrostatic Ion Cyclotron Waves

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Electrostatic ion cyclotron waves are longitudinal waves of the ions and electrons in magnetized plasma which propagate perpendicular to the magnetic field. Suppose the electrostatic ion waves are propagating perpendicular to  . The fluid equations are the magnetic field B ∂ vi  + vi × B ] = e[E ∂t ∂ni ∂ + (ni vi ) = 0 ∂t ∂x

M

(2.27) (2.28)

where M is the mass of ions. To linearize Equations (2.27) and (2.28), as usual, we separate the dependent variable into an equilibrium part and a  φ, B  =B 0 + B  1,  = −∇ perturbation part and write vi = v0 + v1 = v1 , E φ = φ0 + φ1 = φ1 , and ni = n0 + n1 . Assume v0 = 0 and φ0 = 0. Then, the linearized equations are ∂v1  φ1 + ev1 × B 0 = −e∇ ∂t ∂n1 ∂v1 + n0 = 0. ∂t ∂x

M

(2.29) (2.30)

Now assuming v1 = (vx , vy , vz ) and B0 = (0, 0, Bz ), Equation (2.29) can be written as M v˙ x = −e

∂φ1 + evy B0 ∂x

M v˙ y = evx B0 .

(2.31) (2.32)

Assuming all perturbed terms are proportional to e−i(kx−ωt) and replacing ∂/∂x by ik and ∂/∂t by −iω, from (2.31) and (2.32), we have −M iωvx = −eikφ1 + evy B0

(2.33)

−iωM vy = −evx B0 .

(2.34)

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Eliminating vy from Equations (2.33) and (2.34), we get ekφ1 vx = ωM

 −1 Ω2c 1− 2 ω

(2.35)

where Ωc = eB0 /M . Also from Equation (2.30), we get n1 =

n0 kv0 . ω

(2.36)

 0 and therefore, We have already assumed that electrons can move along B it is assumed that the electrons obey the Boltzmann distribution. eφ1

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ne = n0 e KB Te

  eφ1 or. n0 + ne1 = n0 1 + + O(φt) KB Te ⇒ ne1 =

n0 eφ1 . KB Te

(2.37)

Assuming the neutrality n1 = ne1 , we have from (2.36)–(2.37) ω 2 = Ω2c + k 2 vs2

(2.38)

(2.38) is called the dispersion relation for electrostatic cyclotron waves, where vs = KB Te /M is the ion acoustic speed. If vs = 0, then ω = Ωc , then cyclotron waves become cyclotron frequency. 2.9 Lower Hybrid Frequency A longitudinal oscillation of ions and electrons in a magnetized plasma is known as lower hybrid oscillation. Let us consider the propagation angle θ  0 to be π . So, electrons obey the fluid equations of motion. between k and B 2 We also consider the electron temperature Te = 0 and consequently pressure gradient becomes zero, i.e., ∇pe = 0. Like previous case we shall get (2.35) vix

 −1 ek Ω2c = φ1 1 − 2 . Mω ω

(2.39)

Introduction to Waves in Plasma

As electrons are also considered mobile, similarly we have,  −1 ekφ1 ωc2 vex = − 1− 2 mω ω

57

(2.40)

where Ωc and ωc are the ion cyclotron frequency and the electron cyclotron frequency, respectively. The equation of continuity for ions is ∂ ∂ni + (ni vi ) = 0. ∂t ∂x

(2.41)

To linearize (2.41), we use ni = n0 + ni and vi = 0 + vi1 and get the linearized equation as

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∂vi1 ∂ni1 + n0 = 0. ∂t ∂x

(2.42)

As all perturbed components ni1 and vi1 are proportional to ei(kx−ωt) , Equation (2.42) becomes (−iω)ni1 + n0 (ik)vi1 = 0 k or. ni1 = n0 vi1 . ω

(2.43)

Similarly, the equation of continuity of electrons gives k ne1 = n0 ve1 . ω

(2.44)

At equilibrium, ni = ne or. ni1 = ne1 , or. vi1 = ve1 or. vix = vex . Therefore,  −1  −1 ekφ1 ekφ1 Ω2 ω2 = − 1 − 2c 1 − c2 Mω ω mω ω     2 Ωc ωc2 ⇒ M 1− 2 +m 1− 2 =0 ω ω ⇒ ω 2 = Ωc ω c 1

⇒ ω = (Ωc ωc ) 2 = ωl (say)

(2.45)

where ωl is called Lower hybrid frequency. Lower hybrid frequency is the geometric mean of ion cyclotron frequency and electron cyclotron frequency, √ i.e., ωl = Ωc ωc .

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Waves and Wave Interactions in Plasmas

0 = 0 2.10 Electromagnetic Waves with B  1 = 0. We also assume electromagnetic waves with B 0 = Let us consider B     0, i.e., B = B0 + B1 = B1 . The relevant Maxwell’s equations are  ·E 1 = ρ ∇ 0

(2.46)

  ×E  1 = − ∂B1 ∇ ∂t   ×B  1 = ∂E1 + J c2 ∇ ∂t 0

(2.47) (2.48)

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where c2 = (μ0 0 )−1 and μ0 and 0 are the permittivity in free space. In free space, J = 0, and the density ρ must be equal to zero.  ·E 1 = 0 ∇

(2.49)

  ×E  1 = − ∂B1 ∇ ∂t   ×B  1 = ∂E1 . c2 ∇ ∂t

(2.50) (2.51)

From (2.51) and using (2.50), we get  × c2 ∇

1 1 ∂B ∂2E = ∂t ∂t2  2  2 ∂ 2 ∂  1 = 0. E ⇒ −c ∂x2 ∂t2

(2.52) 

When J = 0 is considered in plasma, it includes a term J01 to account for current. This is because of first-order charged particle motion. Let J = 0 and J = J0 + J1 (J0 is the equilibrium part and J1 is the perturbed part). Then, Maxwell’s equations of motion are   ×E  1 = − ∂B1 ∇ ∂t    ×B  1 = ∂E1 + J1 c2 ∇ ∂t 0  × ⇒ c2 ∇

1 1 ∂B ∂2E 1 ∂J1 = + 2 ∂t ∂t 0 ∂t

2   × (∇  ×E  1 ) = ∂ E1 + 1 ∂J1 . ⇒ −c2 ∇ ∂t2 0 ∂t

(2.53)

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 by Assuming the oscillating quantity behaves sinusoidally and replacing ∇ ik and ∂/∂t by −iω, we get  1 ) − c2 k 2 E  1 = −ω 2 E  1 − 1 J1 iω. c2 k(k · E 0

(2.54)

 1 = 0, hence Since the wave is transverse, k.E  1 = − 1 iωJ1 (ω 2 − c2 k 2 )E 0  2  2  ∂ 2 ∂  1 = − iωJ1 . E − c ∂x2 0 ∂t2

(2.55) (2.56)

If we consider equivalently high-frequency waves, like light waves or microwaves, the ions are considered to be fixed. Now, from electron motion, we have J = en(vi − ve ). For linearization, we write J = J0 + J1 = J1 , n = n0 + n1 , vi = vi0 + vi1 , and ve = ve0 + ve1 (J0 = 0 = vi0 = ve0 ). Thus, we get J1 = e(n0 + n1 )(vi1 − ve1 ). Assuming vi1 = 0 and deleting the nonlinear term, we get J1 = −en0 ve1 . So, ve1 = −

J1 . en0

(2.57)

The electron motion equation is m

∂ve  + v × B  ]. = −e[E ∂t

(2.58)

 =E 0 + E  1, B  =B 0 + B  1, For linearization at one place ve = ve0 + ve1 , E   and v = v0 + v1 (B0 = v0 = E0 = ve0 = 0), and neglecting the nonlinear ∂ term and replacing ∂t by −iω, we obtain  1. imωve1 = eE

(2.59)

Now, from (2.55),  1 = − iω J1 = ωp2 E 1 (ω 2 − c2 k 2 )E 0

(2.60)

where ωp = (e2 n0 )/(0 m). It is known as the plasma frequency. Therefore, ω 2 = ωp2 + c2 k 2 .

(2.61)

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Equation (2.61) is the dispersion relation for electromagnetic waves. The 2 phase velocity = vφ2 = ω = c2 + ωp2 /k 2 > c2 and the group velocity is vg .  k2 Now, 2ω

dω dk

= 2c2 k ⇒ vg =

c2 . vφ

2.11 Electromagnetic Waves Perpendicular to B0 Let us now consider the electromagnetic waves. When the unperturbed  0 is present, then the waves are called electromagnetic. At magnetic field B  0 , i.e., k ⊥ B  0 . If first, we consider the direction of k is perpendicular to B  1 , then two cases arise: either E 1  B 0 we take transverse waves with k ⊥ E   or E1 ⊥ B0 . In the first case, the waves are called ordinary waves, and in the second case, it is called extraordinary waves. Now, we evaluate the dispersion relation of ordinary waves and extraordinary waves. 2.11.1 Ordinary wave  1 is parallel to B  0 , we choose B  0 = B0 z , E  1 = E1 z , and k = kx. When E

From (2.54), we get   2  1 = iω (−n0 eve1 ) k 2 − ω E c2  0 c2  1 = 0. Since E  1 = E1 z, where J1 = −n0 eve1 and k · E

we only need the component vez . The equation of motion is m

∂ve  + ve × B ] = −e[E ∂t e ⇒ vez = E1 . imω

 =B 0 + B  1 , and E  =E 0 + E  1 and Taking perturbation vez = ve0 + ve1 , B using the same way, we have ω 2 = ωp2 + c2 k 2 . The above relation is the dispersion relation for ordinary waves. In the absence of ck, ω = ωp , and if ωp = 0, then ω = ±ck.

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2.11.2 Extraordinary wave Extraordinary wave is an electromagnetic wave which is partially longitu0 dinal and partially transverse. This wave propagates perpendicularly to B 1 ⊥ B  0 . So, we consider E 1 ⊥ B  0 . Let E  1 = Ex x  0 = B0 z, with E

+ Ey y , B

   and k = kx.

However, it turns out that waves with E1 ⊥ B0 tend to be elliptically polarized instead of being plane polarized. While propagating into a plasma, the wave develops a component Ex along k. Thus, becoming partly longitudinal and partly transverse. The equation of motion of electrons is m

∂ve  + ve × B  ]. = −e[E ∂t

(2.62)

 =B  0 , where ve1 = To get the linearized form, we write ve = ve0 +ve1 and B  0 = (0, 0, B0 ). So, we have a set of linearized equations of (vx , vy , vz ) and B motion in x and y directions, and solving it we get vx = −

ie ie (Ex + vy B0 ), vy = − (Ey − vx B0 ). mω mω

Thus,  e  ωc  vx = −iEx − Ey 1 − mω ω  e  ωc  vy = −iEy + Ex 1 − mω ω

ωc2 ω2 ωc2 ω2

−1 (2.63) −1 .

(2.64)

The wave equation is given by (2.54) 2  1 ) + k 2 E  1 = iω J1 + ω E  1. −k(k · E 2 2 0 c c

(2.65)

 1 = kEx , and using this relation in Equation For longitudinal wave, k · E (2.65), we get  1 + kc2 kEx = iωn0 eve1 . (ω 2 − c2 k 2 )E 0 The above equation can be written into the component forms, and then using (2.63) and (2.64), we get ω 2 Ex =

iωn0 e vx 0

  −1 iωn0 e e ωc ωc2 ⇒ ω Ex = − iEx + Ey 1 − 2 0 mω ω ω 2

(2.66)

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(ω 2 − c2 k 2 )Ey =

iωn0 e vy 0

⇒ (ω 2 − c2 k 2 )Ey = −

  −1 iωn0 e e ωc ω2 . iEy − Ex 1 − c2 0 mω ω ω (2.67)

In (2.66), using ωp2 = n0 e2 /0 m we get 

   ωp2 ωc ωc2 2 ω 1 − 2 − ωp Ex + i Ey = AEx + iBEy = 0 ω ω 2

ω2

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where A = (ω 2 (1 − ωc2 ) − ωp2 ) and B = ωp2 ωc /ω. Similarly, from (2.67), we get     ωp2 ωc ωc2 2 2 2 2 ω −c k 1 − 2 − ωp Ey − i Ex = CEx + iDEy = 0 ω ω where C = ωp2 ωc /ω and D = [(ω 2 − c2 k 2 )(1 − form

ωc2 ω2 )

− ωp2 ], which is of the

AEx + iBEy = 0 CEx + iDEy = 0. For nontrivial solution,  det

AB CD

 =0

⇒ AD = BC. Now using ωh2 = ωc2 + ωp2 , where ωh is the upper hybrid frequency, then the condition AD = BC can be written as 

2ω ωp c

2

ω

ω 2 − ωh2 − (ω2 −ω2 ) c2 k 2 h = . ω2 ω 2 − ωc2

(2.68)

With the help of a few algebraic calculations, this can be made simple. In Equation (2.68), substituting ωh2 in place of ωc2 + ωp2 and multiplying

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2ω ωp c

2

ω

ω 2 − ωc2 − ωp2 − (ω2 −ω2 ) ωp2 ω 2 − ωp2 c2 k 2 h = = 1 − . ω2 ω 2 − ωc2 ωc2 ω 2 − ωh2 Thus, ωp2 ω 2 − ωp2 c2 k 2 c2 = = 1 − . ω2 vφ2 ωc2 ω 2 − ωh2

(2.69)

This is the dispersion relation. 0 2.12 Electromagnetic Waves Parallel to B  0 , let k lie along the To consider the electromagnetic waves parallel to B  z-axis and let E1 have both transverse components Ex and Ey , i.e., k = k

z  1 = Ex 

and E x + Ey

y . Now, we consider the equation of motion m

dv  + v × B  ]. = −e[E dt

(2.70)

We shall linearize the equation such that the dependent variables can be written into two parts: the equilibrium part and the perturbation part, i.e.,  =E  0 +E  1 and B  =B  0 , where v0 = E  0 = 0. From Equation v = v0 +v1 , E (2.70), we get m

dv1  1 + v1 × B  0 ). = −e(E dt

 0 = vy B0 i − vx B0

So, v1 × B j. Now, equating along the component wise, we get mv˙ x = −e[Ex + vy B0 ]

(2.71)

mv˙ y = −e[Ey − vx B0 ]

(2.72)

mv˙ z = 0.

(2.73)

From (2.71), we have −iωmvx = −e(Ex + vy B0 ) or

vx =

e (Ex + vy B0 ). imω

(2.74)

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Similarly, from (2.72), we obtain vy =

e [Ey − vx B0 ]. imω

(2.75)

From (2.74)–(2.75), we have  e ωc  vx = −iEx − Ey 1− mω ω  e ωc  vy = −iEy + Ex 1− mω ω

ωc2 ω2 ωc2 ω2

−1 (2.76) −1

where ωc = B0 e/m. From Maxwell’s equations, we have     ×B  0 = μ 0 ∂E1 + J1 ∇ ∂t

(2.77)

(2.78)

and   ×E  1 = − ∂B0 . ∇ ∂t

(2.79)

Differentiating (2.78) with respect to t, we get   0 1 ∂B ∂ 2E ∂J1  ∇× = μ 0 + . ∂t ∂t2 ∂t Substituting (2.79), we get   1 ∂2E ∂J1 2     + −∇(∇ · E1 ) + ∇ E1 = μ 0 . ∂t2 ∂t  1 and J1 oscillate sinusoidally and replace ∇  → ik and ∂/∂t → −iω, and E thus we have  1 ) + (ik)2 E  1 = μ0 [0 (−iω)2 E1 + (−iω)J1 ]. −ik(ik · E  1 · k = 0. Therefore, Since, E1 and

k are perpendicular, so E  1 = iωen0 v1 . (ω 2 − c2 k 2 )E 0 After this equating along the x and y axis, we get (ω 2 − c2 k 2 )Ex =

iωen0 vx 0

(2.80)

(ω 2 − c2 k 2 )Ey =

iωen0 vy . 0

(2.81)

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From (2.80) and (2.76), we get    −1  iωen0 e ωc ω2 ω 2 − c2 k 2 Ex = 1 − c2 − iEx − Ey 0 mω ω ω   ωp2 ω  Ex − i c Ey = (2.82) ωc2 ω 1 − ω2 where ωp2 = e2 n0 /(0 m). Similarly, from (2.81) and (2.77), we have

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   2  ωp2 ωc 2 2  Ey + i Ex . ω − c k Ey = ω2 ω 1 − ωc2

(2.83)

Let α = ωp2 /(1 − ωc2 /ω 2 ). Thus, from (2.82) and (2.83), we obtain ωc Ey = 0 ω ωc (ω 2 − c2 k 2 − α)Ey − iα Ex = 0. ω

(ω 2 − c2 k 2 − α)Ex + iα

(2.84) (2.85)

For a nontrivial solution of Ex and Ey , we have ωp2 . ω 2 − c2 k 2 =  1 ∓ ωωc

(2.86)

The sign ± means that there are two possible solutions to Equations (2.84)  0. and (2.85). They can propagate along B n 2 = (ck/ω)2 = 1 −

(ωp2 /ω 2 ) . (1 − ωc /ω)

The above relation is dispersion relation for the R-wave. n 2 = (ck/ω)2 = 1 −

(ωp2 /ω 2 ) . (1 + ωc /ω)

The above relation is the dispersion relation for the L-wave. The R and L waves turn out to be circularly polarized. The designations R and L mean right-hand circular polarization and left-hand circular polarization, respectively. The electric field vector for the R-wave rotates clockwise in  0 and vice versa for the L-waves. time as viewed along the direction of B

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Waves and Wave Interactions in Plasmas

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2.13 Hydromagnetic Waves Hydromagnetic waves are electromagnetic ion waves when undisturbed  0 and oscillating magnetic field B  1 are both nonzero, then magnetic field B  0 , then it is called Alfven wave and when this wave exists. When k  B k ⊥ B  0 , then it is called Magnetosonic wave. 2.13.1 Alfven wave An Alfven wave, named after Hannes Alfven, is a low-frequency magnetohydrodynamic wave (compare to the ion cyclotron frequency) where ions oscillate due to the tension of the magnetic field. The ion mass density and the magnetic field line tension provide the inertia and the restoring force, respectively. The motion of the ions and the oscillating magnetic field act in the same direction and transverse to the direction of propagation. To find the dispersion relation of the Alfven waves in plane geometry,  0, E  1 , and J1 perpendicular to B  0, B  1, let us assume k acts along with B   and v1 perpendicular to both B0 and E1 .  1 = 0 and B  0 = 0 in a vacuum J = 0 and In electromagnetic waves, B 0 μ0 = c−2 , so the relevant Maxwell’s equations are  ×E  1 = −B ˙ 1 ∇  ×B 1 = ∇  ×E  1. c2 ∇

(2.87) (2.88)

Taking the curl of Equation (2.87) and substituting into the time derivative of Equation (2.88), we have  × (∇  ×B  1) = ∇ ¨1 .  ×E ˙ 1 = −B c2 ∇

(2.89)

 is replaced Assuming plane waves varying as ei(kx−ωt) and accordingly ∇ by ik and ∂/∂t is replaced by −iω. From (2.89), we get  1 = −c2 [k(k · B  1 ) − k2 B  1 ]. ω2B  ·B  = 0 gives ω 2 = k 2 c2 and c is the phase velocity ω/k of light Also, ∇ waves.  0 = 0, Equation (2.87) is unchanged. But we In plasma, if we consider B add a term J1 /0 to the right-hand side of Equation (2.88) to account for current due to first-order charged particle motions. Hence, (2.88) becomes  ˙ 1 .  ×B  1 = J1 + E c2 ∇ 0

(2.90)

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Taking time derivative of this, we get  ¨1 .  ×B ˙ 1 = 1 ∂J1 + E c2 ∇ 0 ∂t Taking curl of Equation (2.87), we have  (∇  ·E  1) − ∇  2E  1 = −∇  ×B ˙ 1 . ∇ ˙ 1 and assuming an exp[i(k · r − ωt)] dependence,  × B Eliminating ∇ we have 2  1 ) + k 2 E  1 = iω J1 + ω E  1. −k(k · E 2 2 0 c c

(2.91)

by assumption, only the x-component of this  1 = E1 x Since k = k

z and E equation is nontrivial. The current J1 is contributed from both ions and electrons. So, J1 = n0 (vex − vix ). Since we are considering low frequencies  1 = 0, the x-component of Equation (2.91) becomes and k · E k 2 E1 = 0 (ω 2 − c2 k 2 )E1 = −iωn0 e(vix − vex ).

(2.92)

The ion equation of motion is M

∂vi1 = −e∇φ1 + evi1 × B0 . ∂t

Assuming that the wave is propagating in the x-direction and separating into components, we get −iωM vix = −eikφ1 + eviy B0 , − iωM viy = −evix B0 . Thus, after some calculations, we get vix =

ek Mω

 −1 Ω2 φ1 . 1 − 2c ω

(2.93)

Here, we neglect the thermal motion for this wave, i.e., the solution of the ion equation of the motion is considered with Ti = 0. For completeness,

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68

we also include the component viy and  −1 Ω2 E1 1 − 2c ω  −1 e Ωc Ω2 = E1 . 1 − 2c Mω ω ω

vix = viy

ie Mω

(2.94) (2.95)

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Similarly, the components of velocity of electrons are found by considering M → m, e → −e and Ωc → −ωc and taking the limit ωc2  ω 2 , we get vex = −

ie E1 ω 2 ie ω 2 = E1 → 0 mω ω 2 − ωc2 mω ωc2

vey = −

e −ωc E1 ω 2 e ωc ω 2 E1 = − E1 = − . 2 2 2 2 mω ω ω − ωc m ω ωc B0

Hence, the Larmor gyrations of the electrons are neglected, and the elec ×B  drift in the y-direction. Now, from (2.92), we get trons have an E  −1 ie Ω2c 0 (ω − c k )E1 = −iωn0e E1 1− 2 Mω ω  −1 Ω2c 2 2 2 2 ω − c k = Ωp 1 − 2 ω 2

2 2

(2.96)

where Ωp is the ion plasma frequency. Also, assuming ω 2 Ω2c , we get ω 2 − c2 k 2 = Ω2p or.

Ω2 ω2 M n0 ρ 2 p = −ω = −ω 2 = −ω 2 2 2 2 2 2 ω − Ωc Ωc  0 B0 0 B 0

ω2 c2 c2 = = 2 ρ 2 k 1 +  B 2 1 + ρμB02c 0 0

(2.97)

0

where μ0 0 = c12 and ρ is the mass density. The denominator is the relative dielectric constant for low frequency perpendicular motions and can be written as R = 1 +

μ0 ρc2 . 2 B 0

The phase velocity for an electromagnetic wave in a dielectric medium is ω c =√ R k

Introduction to Waves in Plasma

(as μR = 1). As for most laboratory plasmas,   1, then R = hence,

69 ρμ0 c2 B02

and

0 B ω = vφ = √ . k μ0 ρ  0 at a constant velocity vA called These hydromagnetic waves travel along B the Alfven velocity. This is written as  B vA = √ . μ0 ρ

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2.13.2 Magnetosonic wave In magnetized plasma, magnetosonic waves are longitudinal waves that  0 . It is driven propagate perpendicular to the undisturbed magnetic field B both by pressure and magnetic tension and are observed recently in solar corona. Let us consider low-frequency electromagnetic waves propagating  0 = B0 z and released in the course of oscillation. Here, we consider across B  p term in the equation of motion. the ∇ For the ions, the equation of motion is M n0

∂vi1  1 + vi1 × B  0 ) − γi K B T i ∇  n1 . = en0 (E ∂t

(2.98)

 1 = (Ex , 0, 0), vi1 = (vix , viy , viz ), B  0 = (0, 0, B0 ), and ∂/∂t is Now, E replaced by −iω, then along the component wise,  0) −M n0 iωvix = en0 (Ex + viy B ⇒ vix =

ie  0 ). (Ex + viy B Mω

(2.99)

Similarly, we have viy =

ie  0 ) + k γi K B T i n 1 . (−vix B Mω ω M n0

(2.100)

The equation of continuity yields ∂ni ∂ + (ni vi1 ) = 0 ∂t ∂x n1 k ⇒ = viy . n0 ω

(2.101)

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ie 0 + So, viy = − Mω vix B

k2 γi KB Ti viy . ω2 M

Assuming A =

viy (1 − A) = −

k2 γi KB Ti , ω2 M

iΩc vix ω

we get (2.102)

 0 /M . Using (2.102) in (2.99), we get where Ωc = eB vix =

ie Ex . Mω

(2.103)

This is the only component of vi1 . The only nontrivial component of the wave equation is

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0 (ω 2 − c2 k 2 )Ex = −iωn0 e(vix − vex ).

(2.104)

To obtain vex , we have to take the limit of small electron mass so that 2 ω 2 ωc2 and ω 2 k 2 vth . So,   ie ω 2 k 2 γe K B T e ik 2 γe KB Te vex = 1 − Ex . (2.105) Ex → − 2 2 2 mω ωc ω m e ωB 0 Now, (2.103)–(2.105) together give 0 (ω 2 − c2 k 2 )Ex     ie 1−A ik 2 M γe KB Te = −iωn0 e Ex + Ex . (2.106) Ω2 2 mω eM ωB 1 − A − ω2c 0 We shall again assume ω 2 Ω2c so that (1 − A) can be neglected relatively to Ω2c /ω 2 . With the help of the definition of Ωp and Alfven velocity vA , we have Ω2p 2 K 2 c 2 γe K B T e ω (1 − A) + 2 Ω2c vA M     Ω2p γe KTe 2 2 2 2 2 γi K B T i ⇒ ω −c k 1+ + 2 ω −k = 0. 2 M vA ωc M (2.107)

(ω 2 − c2 k 2 ) =

2 Since Ω2p /ωc2 = c2 /vA , Equation (2.107)    c2 2 2 2 ω 1+ 2 = c k 1+ vA    c2 ω 2 1 + 2 = c2 k 2 1 + vA

becomes γe K B T e + γi K B T i 2 M vA  vs2 . 2 vA



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Table 2.1: Summary of elementary electrostatic wave. Initial Condition

Oscillating Species

Dispersion Relation

 0 = 0 or k  B 0 B

Electrons

k ⊥ B 0

2 ω 2 = ωp2 + 32 k 2 vth

Electrons

ω2

 0 or k  B 0 B

Ions

ω 2 = k 2 vs2

=

ωp2

+

ωc2

=

2 ωh

Name Plasma oscillations Upper hybrid oscillations Ion acoustic waves

+γi KB Ti = k 2 γe Kb TeM

k ⊥ B  0 (nearly)

Ions

ω 2 = Ω2 + k 2 vs2

k ⊥ B 0

Ions

ω 2 = Ω2 ωc = ωl2

Electrostatics ion cyclotron waves Lower hybrid oscillations

Table 2.2: Summary of elementary electromagnetic wave. Initial Condition

Oscillating Species

0 = 0 B

Electrons

Dispersion Relation ω 2 = ωp2 + k 2 c2 2 2

k  B 0

Electrons

c k ω2

=1−

k  B 0

Electrons

c2 k 2 ω2

=1−

k ⊥ B  0, E 1  B 0

Electrons

c2 k 2 ω2

=1−

k ⊥ B 0  0, E 1 ⊥ B

Electrons

0 = 0 B k  B 0

c2 k 2 ω2

Ions Ions

k ⊥ B 0

Ions

=1−

2 ωp ω2 ω 1− ωc 2 ωp ω2 ω 1+ ωc 2 ωp ω2 2 2 ω 2 −ωp ωp ω 2 ω 2 −ω 2 h

ω 2 = k 2 v2 ω2 k2

v 2 +V 2

= c2 cs2 +v2A

Name Light waves R wave L wave O wave X wave Waves does not exist Alfven wave Magnetosonic wave

A

Here, vs denotes the acoustic speed. Finally, we get the dispersion relation as 2 ω2 v 2 + vA = c2 s2 . 2 2 k c + vA

2.14 Some Acoustic Type of Waves in Plasmas In a plasma, the particles move randomly and interact among themselves under electromagnetic forces. They also respond to the externally applied perturbations. As a result, we have a great variety of waves in plasma.

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Waves and Wave Interactions in Plasmas

The existence of different types of waves in plasma is important in plasma research. This is because, on the one hand, they are easy to observe, and on the other hand, their theoretical background is also well established. A brief discussion on different types of waves in plasmas is given in the following: 2.14.1 Electron plasma waves In cold plasmas, there is no thermal movement of the particles. If thermal movements are taken into account, the plasmas can be treated as classical plasma consisting of electrons and ions. The thermal movement of the electron can cause the plasma oscillation to propagate. Electrons that flow into an adjacent plasma layer with their thermal velocity carry information about what is happening in the plasma region. The plasma oscillation is then called the plasma wave. Now, the electron motion is described by the following equations: ∂ne  + ∇ · (ne ve ) = 0 ∂t   ∂  −∇  p me n e + ve · ∇ ve = −ne E ∂t  ·E  ) = e(ni − ne ). 0 (∇

(2.108) (2.109) (2.110)

The equation of state is p = Cργm where C is a constant, ρm is the mass density, and γ = Cp /Cv . As  p = γKB Te ∇  ne p = ne K B T e , ∇ and considering one dimension motion, we get γ = 3. Let us assume all the gradients in the above equations become derivatives with respect to x only. Here, considering small amplitude waves, we separate the dependent variable into two parts: the equilibrium part and the perturbation part. (1) (1)  =E 0 + E  1 , where n0 , v0 , and Let ne = n0 + ne , ve = v0 + ve , and E (1) (1)   E0 are the equilibrium part and ne , ve , and E1 are the perturbation  n0 = v0 = E0 = 0 and ∂n0 = ∂v0 = ∂E0 = 0. part. At equilibrium state, ∇ ∂t ∂t ∂t The pressure term will be (1)

.  pe = 3KB Te ∂ne x ∇ ∂x

(2.111)

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Hence, Equations (2.108)–(2.110) become me n 0

(1)

∂ve ∂t

 1 − 3KB Te ∇  n(1) = −en0 E e

(2.112)

(1)

∂ne  · v (1) = 0 + n0 ∇ e ∂t (1)  1 = e(n0 − n0 − n(1)  ·E 0 ∇ e ) = −ene .

(2.113) (2.114)

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Let all the perturbation quantities be proportional to exp[i(kx−ωt)]. Then, replacing (∂/∂x) → ik and (∂/∂t) → −iω and using the same procedure discussed above, we obtain

2



3 2 2 k ω 2 = ωp2 + vth 2

2 where ωp2 = e0nm0 . It is known as plasma frequency and vth = thermal velocity. Therefore,

vg =

2 3 vth 2 vp

(2.115) 2KTe m

is the

(2.116)

where vp (= ω/k) is the velocity and vg is the group velocity. Here, √ phase  2 −ω 2 ω p 3 2 1 v vg = 32 vth . It is clear that at large k, ω 2  ωp2 , then we ω vp = 2 th  get vg ≈ 32 vth . But at small k, vg < vth . For large λd , density gradient is small and the thermal motions carry very little information into adjacent layers. As ω < ωp , there is no real solution for vg . This means that the wave cannot propagate. This situation is called a cut-off in the dispersion relation. Physically, the wave is reflected at the point at which ω = ωp . 2.14.2 Ion acoustic waves In ordinary air, the sound wave propagates from one layer to the next by collisions of the air molecules. In the absence of collisions, ordinary sound waves would not occur. But in plasma, one can observe such waves without collisions. Ions can transmit vibrations to each other because of their charge and acoustic type waves can occur in plasma. Since the motion of massive ions is involved, these are low-frequency oscillations. To obtain the dispersion relation for the ion acoustic waves, we consider two fluid models. For electrostatic oscillations, the two fluid models are as follows:

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74

For electron, ∂ne  + ∇ · (ne ve ) = 0 ∂t   ∂ve   − γe K B T e ∇  ne . me n e + (ve · ∇)ve = −ene E ∂t

(2.117) (2.118)

For ions,

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∂ni  + ∇ · (ni vi ) = 0 ∂t   ∂vi   − γi K B T i ∇  ni . mi n i + (vi · ∇)vi = eni E ∂t

(2.119) (2.120)

E  = 4πe(ni − ne ) and E  = The electric field couples these two motions: ∇  −∇φ. The electrons can move much faster than the heavier ions. Here, we are considering low-frequency waves and so the electron inertia is neglected. Using me = 0 in (2.118), we get  φ − γe K B T e ∇  ne = 0. ene ∇

(2.121)

For slow ion waves, the electrons move so fast that they have enough time to equalize their temperature everywhere. So, the electrons are considered isothermal and we can take γe = 1. Therefore, Equation (2.121) can be written in one dimension as e On integration, we have

KB Te dne dφ = . dx ne dx



φ=

KB Te e

 log ne + C1 .

In equilibrium state ne = n0 and φ = 0, we get C1 = −KB Te log n0 . Therefore,     KB Te ne φ= . (2.122) log e n0 Hence,

 ne = n0 exp

 eφ . KB Te

(2.123)

This shows that the electrons are Boltzmann distributed. We assume the perturbation to be small relative to the thermal energy, i.e., φ KB Te .

Introduction to Waves in Plasma

Therefore, from (2.124), we have

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 eφ ne ≈ n0 1 + . KB Te

75



(2.124)

We assume small-amplitude waves and so the perturbations in plasma parameters are small as compared to their equilibrium value. Let nj = n0 + nj1 , vj = vj0 + vj1 , and φ = φ0 + φ1 . Again, let j = e, and i, which stands for electrons, and ions, respectively. The subscript 0 refers to equilibrium values, and the subscript 1 refers to the perturbed part. For a uniform neutral plasma at rest in the equilibrium state, vj0 = φ0 = ∂ vj0 ∂φ0 0 0; ∂n ∂t = ∂t = ∂t = 0. So, from (2.124), the perturbation in electron density is ne1 = ne − n0 = eφn0 eφn0 KB Te . Therefore, the perturbation of ion density ni1 = KB Te . Substituting the perturbation expression in equations for ions (2.119) and (2.120), we have ∂ni1  .vi1 = 0 + n0 ∇ (2.125) ∂t ∂vi1  φ1 − γi KB Ti ∇  ni1 , mi n 0 = −en0 ∇ (2.126) ∂t where we have assumed the changes nj1 , vj1 , and φ1 . It is due to perturbations of small quantities of the first order. We also neglected the higherorder smaller terms. Now, considering the one-dimensional case and assuming that all the perturbation quantities vary as exp[i(kx − ωt)], we get from Equations (2.125) and (2.126) −iωni1 + n0 ikvi1 = 0

(2.127)

−iωmi n0 vi1 = −iken0 φ1 − ikγi KB Ti ni1 .

(2.128)

Eliminating vi1 from these two equations and then using the relation ne1 = ni1 , we get for nonzero solutions ω2 = k2 · ⇒

ω = k



K B T e + γi K B T i mi K B T e + γi K B T i mi

(2.129) 1/2 = vs (say).

(2.130)

This is the dispersion relation for ion acoustic waves and vs is the sound speed in a plasma. The dispersion curve for ion waves has a fundamentally different character from that for electron waves. Plasma oscillations

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Waves and Wave Interactions in Plasmas

are constant frequency waves, with a correction due to thermal motions. Ion waves are constant velocity waves and exist only when there are thermal motions. The reasons for this difference can be seen from the following description of the physical mechanisms involved. Ion electron plasma oscillations, the other species (ions) remain essentially fixed. In ion acoustic waves, the other species (electrons) is far from fixed; in fact, electrons are pulled along with the ions and tend to shield out electric field arising from the bunching of ions.

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2.14.3 Dust acoustic waves The propagation of IAWs in a spatial inhomogeneity is created by a distribution of immobile dust particles. The low-frequency behavior of a dusty plasma is very similar to that of a plasma consisting of negative ions. In fact, for the case in which the wavelength and the inter-particle distance are much larger than the grain size, the dust grains can be treated as negatively charged point masses (like negative ions). However, here the charge to mass ratio of a dust particle can take on any value. Thus, with minor corrections, many results from the theory of negative ion plasmas can be adapted to dusty plasma. The long-wavelength low-frequency collective oscillations can exist in dusty plasma. We shall consider modes so that the dust particle dynamics is crucial rather than the modes affected by the dust. In particular, we study the collective motion of the negatively charged dust in thermodynamics equilibrium. We find that new types of sound waves, namely dust acoustic waves, can appear. These waves are usually of low frequency. But in some cases, the latter can be compared with that of the ion acoustic wave. It is a very low-frequency acoustic mode. Here, the dust grains participate directly in the wave dynamics. The DA waves have been theoretically predicted by Rao et al. [1], in a multicomponent collisionless dusty plasma consisting of electrons, ions, and negatively charged dust grains. The phase velocity of the DA waves is smaller than the electron and ion thermal speeds. Hence, the inertialess electrons and ions establish equilibrium in the DA wave potential φ. Dust inertia is significant here. In the DA waves, the pressures of the electrons and ions provide the restoring force, while the inertia comes from the dust mass. Thus, the DA waves are very low-frequency waves. The relation is given by ω 2 = 3k 2 VT2d +

2 k 2 CD 2 1 + k λ2D

(2.131)

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where VT d is the dust thermal speed and CD = ωpd λD is the DA speed. Since ω  kVT d , the DA wave frequency is ω=

kCD . (1 + k 2 λ2D )1/2

(2.132)

In the long-wavelength limit (namely k 2 λ2D 1), it reduces to 

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ω = kZd0

nd0 ni0

1/2 

kB Ti md

1/2 

Ti 1+ Te

 −1/2 Zd0 nd0 1− , ni0

(2.133)

which shows that the restoring force in the DA waves comes from the pressures of the inertialess electrons and ions, while the dust mass provides the inertia to support the waves. The frequency of the DA waves is much smaller than the dust plasma frequency. The DA waves have been observed in several laboratory experiments (e.g., Barkan et al. [2], Pieper and Goree [3]). The observed DA wave frequencies are of the order of 10–20 Hz. Video images of the DA wavefronts are possible, and they can be viewed with the naked eye. 2.14.4 Dust ion acoustic waves The DIA waves were predicted by Shukla and Silin [4]. For DIA waves, the phase velocity is much smaller (larger) than the electron thermal speed (ion and dust thermal speeds). In this case, the electron number density perturbation is related to the DIA waves, while the ion number density perturbation is determined from the ion continuity equation. Assuming ω  kVT i , kVT d , the dispersion relation is written as 1−

2 2 2 + ωpd ωpi kDe = 0. − k2 ω2

(2.134)

As the dust grains have a large mass, the ion plasma frequency ωpi is much larger than the dust plasma frequency ωpd . Thus, we have ω2 =

k 2 CS2 1 + k 2 λ2De

(2.135)

where CS = ωpi λDe . In the long-wavelength limit, i.e., when k 2 λ2De 1, then ω = kωpi λDe .

(2.136)

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Waves and Wave Interactions in Plasmas

It shows that the phase velocity of the DIA waves in a dusty plasma is larger than cs = (kB Te /mi )1/2 . The DIA waves have also been observed in laboratory experiments (Barkan et al. [5], Nakamura et al. [6]). The frequencies of the DIA waves for laboratory plasma parameters are in the order of kHz. At low frequencies, a new ultra low-frequency mode arises from ion oscillations in static dust distribution. 2.15 Nonlinear Wave Let us consider a linear hyperbolic equation in one dimension as

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∂u ∂u +c =0 ∂x ∂x

(2.137)

where c is a constant. The physical meaning of Equation (2.137) is that the magnitude of u at every point is carried with constant velocity c so that the disturbance of u moves with the velocity c without distortion. The general solution of (2.137) is u = f (x − ct)

(2.138)

where f is an arbitrary function and can be determined by an initial or a boundary condition. If the velocity c is constant, then the wave is called progressive. Here, ξ = x − ct is called the phase. The phase ξ is invariant along the characteristic curve. Hence, u is constant along the line ξ = constant. The above expression can be written in the form of u = f (x − ct) = r(ξ)

(2.139)

where r(ξ) is the Riemann invariant. Now, we consider the second-order linear wave equation as 1 ∂2u ∂2u − 2 2 = 0. 2 ∂x c ∂t

(2.140)

The general solution of (2.140) is u = f (x − ct) + g(x + ct).

(2.141)

Equation (2.141) represents the superposition of two arbitrary progressive waves moving to the left and the right, respectively.

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If the initial disturbance is localized within a certain domain so that u(x, 0) = 0 and ∂u(x,0) = 0 for |x| > d, then we can write ∂t f (x) = 0 = g(x),

f or |x| > d.

Consequently, f (x − t) = 0,

f or |x − t| > d

g(x + t) = 0,

f or |x + t| > d.

(2.142)

 (2.143)

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It means that after the time t, only the rightward or the leftward progressive wave is observed, whenever it is at a place sufficiently far from the origin. If we consider a nonlinear hyperbolic equation ∂2u 1 ∂2u + = 0, 2 ∂x c(u) ∂t2

(2.144)

and a solution of Equation (2.144) as u = f (x − ct)

(2.145)

where c is the local phase velocity and is written as ∂u ∂t ∂u ∂x

= −c.

(2.146)

If c is constant, Equation (2.145) reduces to Equation (2.138). In the nonlinear Equation (2.144), the phase velocity of the wave is not constant and so the wave distorts as it propagates. A wave with a single phase as given by Equation (2.145) is called a simple wave. The progressive wave is a special case of a simple wave. Along with the equiphase line u = constant, i.e., along with each characteristic, the velocity is given by the following equation: dx = c. dt

(2.147)

u is constant, i.e., u = r(ξ), so that Equation (2.144) can be integrated to give a family of straight lines but each with a different slope, x − c(ξ, 0)t = ξ.

(2.148)

Here, ξ is the parameter specifying the characteristic issuing out of the point x = ξ at t = 0. Since, u is constant along each characteristic, and if “c” decreases with ξ, then the characteristics intersect for t > 0. At the

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Waves and Wave Interactions in Plasmas

intersection, we have two different values of u so that this can be interpreted physically as the breaking of a wave. The process leading up to this corresponds to steepening of the wave. Hence, in general, a solution does not exist for all time. The inviscid Burgers’ equation is a conservation equation, more generally a first-order quasilinear equation. The solution to the equation and along with initial condition is written as

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∂u ∂u +u = 0, u(x, 0) = f (x). ∂t ∂x

(2.149)

Consider the function u at each point of the (x, t) plane and let ut + uux be the total derivative of u along with the curve which has slop dx dt = u at every point of it. For any curve ψ in (x, t) plane, assume x and u to be function of t, and ∂u dx ∂u total derivative of u is du dt = ∂t + dt ∂x . Now, on the curve ψ in (x, t) plane, the characteristic equations are du = 0, dt dx = u. dt

(2.150) (2.151)

So, u is constant (c) along with the characteristic and x = ct + c1 where c1 = constant. Let x = ξ, at t = 0, then x = ct + ξ is a straight line where ξ is a point on the x − axis(t = 0) of the x − t plane and ξ is the implicit function of x and t. Now, from the initial condition, assume u(ξ, 0) = f (ξ). So, writing u = f (ξ) on that characteristic, the characteristic is x = f (ξ)t + ξ.

(2.152)

Thus, the solution is given by u(x, t) = f (ξ) = f (x−ut) where ξ = x−ut. It is an implicit relation that determines the solution of the inviscid Burgers’ equation, provided the characteristic does not intersect. Now, ∂ξ ∂u = f  (ξ) ∂t ∂t ∂u ∂ξ = f  (ξ) . ∂x ∂x

(2.153) (2.154)

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81

Differentiating Equation (2.152) with respect to t, we get ∂ξ ∂ξ + f (ξ) + f  (ξ)t ∂t ∂t ∂ξ ∂ξ 1= + f  (ξ)t . ∂x ∂x

0=

From Equations (2.153)–(2.156), we get So,

∂u ∂t



(ξ)f (ξ) = − f1+f  (ξ)t and

(2.155) (2.156) ∂u ∂x

=

f  (ξ) 1+f  (ξ)t .

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∂u ∂u +u =0 ∂t ∂x where 1 + f  (ξ)t = 0. The initial condition u = f (ξ) is satisfied as ξ = x when t = 0. On any characteristic for which f  (ξ) < 0, 1 + tf  (ξ) → 0, a classical solution to the partial differential equation does not exist and it leads to the formation of a shock wave. The breaking time before a shock wave can be formed and is given by 1 + f  (ξ)tb = 0, i.e., tb = f−1  (ξ) . For nonlinear waves, as the wave speed depends on c and u, a gradual nonlinear distortion of the wave is produced. It propagates in the medium. It indicates that some parts of the wave travel faster than others. When c (u) > 0, c(u) is an increasing function of u, and so the higher values of u propagate faster than the lower value of u. Similarly, when c (u) < 0, higher values of u travel slower than the lower ones. So, the wave profile progressively distorts itself, leading to a vertical slope, and hence, it breaks. As for the compressive part, the wave speed is a decreasing function of x, the wave profile distorts and it ultimately breaks. The solution of the nonlinear initial-value problem exists if 1 + tf  (ξ) = 0. This condition is always satisfied even for a small-time t. Also, ux and ut tend to infinity as 1 + tf  (ξ) → 0. So, the solution develops a discontinuity (singularity) when 1 + tf  (ξ) = 0. Thus, on any characteristic for which f  (ξ) < 0, a discontinuity occurs at a time t = − f 1(ξ) . Hence, f  (ξ) = c (f )f  (ξ) < 0. Let t = τ be the time when the solution first develops a singularity for some value of ξ. Then, τ =−

1 > 0. min−∞ 0

1  dφ = √ φ 6v − φ2 . dξ 6 Again, integrating the above equation, we get the solitary wave solution as   ξ φ(ξ) = Asech (3.20) W √ where A = 6v is the amplitude and W = √1v is the the width of the √

1 wave. They are related by A = W6 , i.e., A ∝ W . So, we conclude that the amplitude is inversely proportional to the width of the wave.

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3.2.4 Schamel-type KdV (S-KdV) equation To investigate the propagation of acoustic type waves in the presence of trapped charged particles, H. Schamel [8] considered a non-Maxwellian distribution for trapped particles that are known as Schamel distribution. Accordingly, a KdV like equation was derived from the model, and such equation is called Schamel-type KdV (S-KdV) equation. The S-KdV equation contains a square root nonlinearity and can be derived from a plasma model, where trapped electrons/ions are considered. We obtain exact traveling wave solutions of the S-KdV equation by employing the direct method. The S-KdV equation is

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 ∂φ ∂3φ ∂φ +a φ + b 3 = 0. ∂t ∂x ∂x

(3.21)

Using transformation and omitting the bar, Equation (3.21) reduces to ∂φ  ∂φ ∂ 3 φ + φ + = 0. ∂t ∂x ∂x3

(3.22)

To obtain the solution of Equation (3.22) by using the transformation ξ = x − vt, where v is a constant speed, we get −v

dφ  dφ d3 φ + φ + 3 = 0. dξ dξ dξ

Integrating the above equation and imposing the condition φ → 0, at ξ → ±∞, we get   2 3/2 d2 φ φ = vφ − . dξ 2 3

d2 φ dξ 2

→0

Multiplying the above equation by 2 dφ dξ and then integrating and using the boundary condition φ → 0,

→ 0 at ξ → ±∞, we get   dφ 1 = √ φ 15v − 8 φ. dξ 15 dφ dξ

Again, integrating the above equation, we get the solitary wave solution as   ξ φ = Asech4 (3.23) W 2

√4 is the the width of the wave where A = 225v 64 is the amplitude and W = v 900 1 and they are related by A = W 4 , i.e., A ∝ W 4 .

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Solution of Nonlinear Wave Equations

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3.2.5 Burgers’ equation Burgers’ equation or Bateman–Burgers equation occurs in various areas at applied mathematics and physics. This equation models the shock waves in fluid mechanics, nonlinear acoustic, gas dynamic, and traffic flow. In 1915, Harry Bateman [9] introduced a nonlinear partial differential equation where dissipation was considered with nonlinearity. After 1948, this nonlinear equation was analyzed by Johannes Martinus Burgers [10]. This equation is known as Burgers’ equation. Burgers’ equation is the simplest nonlinear differential equation for diffusive waves in fluid dynamics. This equation can also be taken as a modified form of the Navier–Stokes equation because of the nonlinear convection term and the presence of the viscosity term. Burgers’ equation also appears in many physical problems like acoustic, dispersive water, shock waves, and turbulence. Few nonlinear partial differential equations can be solved exactly like this. Now, in this section, Burgers’ equation will be solved by the direct method. Burgers’ equation is given by ∂φ ∂ 2 φ ∂φ − +φ = 0. ∂t ∂x ∂x2

(3.24)

Let us consider the wave moves with velocity v. Now, considering the traveling wave transformation ξ = x − vt, where x − vt is the phase, Equation (3.24) is converted to −v

dφ d2 φ dφ − 2 = 0. +φ dξ dξ dξ

Integrating and using the conditions φ → 0, v2

dφ dξ

(3.25)

→ 0 at ξ → ±∞, we get

dφ 1 = − dξ. 2 − (φ − v) 2

On integration and using the boundary condition φ → 0 at ξ → ∞, we get the solitary wave solution as    ξ φ(ξ) = φ0 1 + tanh (3.26) δ where φ0 = v is the amplitude and δ = −2/v is the width of the shock wave.

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3.2.6 KP equation The Kadomtsev–Petviashvili (KP) equation is a nonlinear partial differential equation with a dimension (2+1) that describes the wave phenomena in different nonlinear mediums. This equation can also be termed as the (2+1) dimensional KdV equation. Two Soviet physicists Boris Kadomtsev and Vladimir Petviashvili [11] introduced this equation in 1970. Like the KdV equation, this equation is also modeled as acoustic waves in plasma under the influences of dissipation at long transverse perturbation. So, this is a natural extension of the KdV equation to two special dimensions. It is also applicable in nonlinear optics and other relative fields. To obtain the solution of the KP equation by the direct method, let us consider a KP equation as   ∂φ ∂3φ ∂2φ ∂ ∂φ + Aφ + B 3 + C 2 = 0. (3.27) ∂ξ ∂τ ∂ξ ∂ξ ∂η To obtain the soliton solution of the equation by using the transformation χ = lξ + mη − uτ in Equation (3.27) then integrating the transformed d2 φ equation with the help of boundary condition φ → 0, dφ dχ → 0, dχ2 → 0 at χ → ±∞, we get (Cm2 − lu)

dφ dφ d3 φ + Al2 φ + Bl4 3 = 0. dχ dχ dχ

Again, integrating the above equation and imposing the boundary condition dφ d2 φ φ → 0, dχ → 0, dχ 2 → 0 at χ → ±∞, we get d2 φ lu − Cm2 A 2 = φ− φ . 2 dχ Bl4 2Bl2 dφ Multiplying both sides by 2 dχ and then integrating with the help of the

boundary condition φ → 0,

dφ dχ

→ 0 at χ → ±∞, we get

dφ √ = dχ φ A1 − B1 φ 2

A where A1 = lu−Cm , B1 = 3Bl 2. Bl4 Integrating the above equation and finally, we get the soliton solution as √    A1 A1 χ χ 2 2 φ= sech = φ0 sech (3.28) B1 2 W

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Solution of Nonlinear Wave Equations

where φ0 = 3(lu−Cm l2 A of the wave.

2

)

is the amplitude and W = 2



l4 lu−Cm2

is the width

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3.2.7 Modified KP equation Grimshaw and Melville [12] derived a modified KP equation to describe long surface or initial waves in the existence of rotation. This equation is the appropriate extension of the KdV equation that provides a correct asymptotic explanation of the waves traveling to the right. In general, it cannot be assumed that the solutions are locally confined. Hence, we are going to obtain the wave solution from the modified KP equation. Let us consider the modified KP equation as   ∂3φ ∂2φ ∂ ∂φ 2 ∂φ + Aφ + B 3 + C 2 = 0. (3.29) ∂ξ ∂τ ∂ξ ∂ξ ∂η To obtain the solution of Equation (3.29) by using the transformation χ = lξ + mη − uτ in Equation (3.27) and then integrating the transformed dφ d2 φ equation with the help of the boundary condition φ → 0, dχ → 0, dχ 2 → 0 at χ → ∞, we get (Cm2 − lu)

dφ dφ d3 φ + Al2 φ2 + Bl4 3 = 0. dχ dχ dχ

Again, integrating the above equation and imposing the boundary condition dφ d2 φ φ → 0, dχ → 0, dχ 2 → 0 as χ → ∞, we get d2 φ (lu − Cm2 ) A 3 = φ− φ . 2 dχ Bl4 3Bl2 dφ Multiplying both sides by 2 dχ and then integrating the equation and using the boundary condition, we get

dφ  = dχ φ A − Bφ2 2

) A where A1 = (lu−Cm and B1 = 6Bl 2. Bl4 Integrating the above equation and finally, we get the solution as     √ χ A1 φ= sech Aχ = φ0 sech (3.30) B1 Wc

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Waves and Wave Interactions in Plasmas

  2) l4 B where φ0 = 6(lu−Cm is the amplitude and W = c 2 l A lu−Cm2 is the width of the wave.

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3.3 Hyperbolic Tangent Method Solving the majority of the partial differential equation is not easy. To handle this, Malfliet [13] introduced a popular technique to find a traveling wave solution of the nonlinear partial differential equations. This technique is called the hyperbolic tangent or tanh method. To avoid algebraic complications, this method uses tanh as a new variable because all derivatives of tanh are represented also by the tanh function. So, this method is a bit simpler, and we can use this method to solve nonlinear evolution equations. The starting point is a nonlinear evolution equation, which describes the dynamical evolution of the waveform u(x, t). The following are the steps: Step 1 A traveling wave solution needs a single coordinate ξ = c(x − vt) and u(x, t) = u(ξ), where u(ξ) represents the wave solution, which travels with speed v. Without loss of generality, we define c > 0 and accordingly as a consequence, the derivatives are changed into d ∂ d ∂ →c . → −cv , dv ∂x dv ∂t

(3.31)

Step 2 Using (3.31), then the partial differential equation becomes an ordinary differential equation. Step 3 In the next step, the ordinary differential equation is integrated as all terms contain derivatives. One should continue this procedure unless one of the terms contains no derivatives. The integration constants are associated with the problem and are taken to be zero. Step 4 This step is the most important step. In this step, we introduce Y = tanh(ξ) as a new independent variable. The corresponding derivatives are

Solution of Nonlinear Wave Equations

99

changed to d ∂Y d d d = = sech2 (ξ) = (1 − Y 2 ) . dξ ∂ξ dY dY dY

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Therefore, d d = (1 − Y 2 ) (3.32) dξ dY   2 d d2 2 2 d = (1 − Y ) − 2Y + (1 − Y ) (3.33) dξ 2 dY dY 2    2 d d3 d 2 2 d 2 2 = −2Y (1 − Y ) − 2Y + (1 − Y ) + (1 − Y ) − 2Y 3 2 dξ dY dY dY  2 2 d d − 2Y + (1 − Y 2 ) . (3.34) 2 dY dY 2 Higher-order derivatives can also be found accordingly. Step 5 The solutions we are looking for will come in power series of Y . Although no general procedure exists at this final stage, the series expansion of the dependent variable is most preferred. S(Y ) =

M

am Y m .

(3.35)

m=0

Detection of the parameter M is the crux of the idea of finding the solution. M will be found by balancing the linear term(s) of the highest order with the nonlinear term(s). To carry out the balance method, the highest exponents for the function u and its derivatives are as follows: ⎫ u→M ⎪ ⎪ un → nM ⎬  (3.36) u →M +1 ⎪  u →M +2 ⎪ ⎭ u(r) → M + r. The linear term of the highest order is contained in the highest derivative of the equation. This can be easily obtained by using the set of relations (3.32), (3.33), and (3.34) in the equation. The Y 2 component of Equation (3.32) leads to the order (2 + M − 1) = M + 1 and similarly for Y 3 and

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Waves and Wave Interactions in Plasmas

Y 4 . On the other hand, the nonlinear terms yield a multiple of the result. In this way, the value of M can be determined. Normally, M will be a positive integer so that a closed analytical solution can be obtained. In principle, a negative or an infinite value for M is also allowed. But it can be used in those cases where a finite value of M does not lead to a solution, and such cases are not treated here. Finally, the series expansion (3.35) is substituted into the relevant equation, and recursion relations appear. Then, the coefficients am (m = 0, 1, 2, . . . , M ) are evaluated. 3.3.1 KdV equation

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Now, we are going to solve the KdV equation by tanh method. ∂u ∂u ∂ 3 u + 6u + = 0. ∂t ∂x ∂x3

(3.37)

Using the transformation ξ = c(x−vt) and then integrating the transformed equation, we get −cvu(ξ) + 3cu2 (ξ) + c3

du(ξ) = 0. dξ 2

(3.38)

Now, we introduce a new variable Y = tanh(ξ), and we get   dS(Y ) d2 S(Y ) −vS(Y ) + 3s2 (Y ) + c2 (1 − Y 2 ) −2Y + (1 − Y 2 ) = 0. dY dY 2 (3.39) 2

dS d S M−2 The highest power of S is Y M , dY is Y M−1 , and dY . The power 2 is Y 2 2 3 dS 4d S of S (Y ) is 2M , Y dY is 3 + M − 1, and Y dY 2 is 4 + M − 2. Hence, balancing the highest order of linear and nonlinear term, we get

2M = 3 + M − 1 = 4 + M − 2. So, M = 2. We are now able to proceed as before. However, from the fact that M = 2, we observe from the structure of Equation (3.38) that S(Y ) should be proportional to (1−Y 2 ). So, we can introduce S(Y ) = ν(1−Y 2 ). Dividing Equation (3.38) by ν(1 − Y 2 ), we obtain   2 2 d(1 − Y 2 ) 2 2 2 d (1 − Y ) −v + 3ν(1 − Y ) + c − 2Y + (1 − Y ) = 0. dY dY 2 Only terms proportional to Y 2 and Y 0 are left.

Solution of Nonlinear Wave Equations

101

Thus, Y 0 coefficient: −v + 3ν − 2c2 = 0 Y 2 coefficient: −3ν + 4c2 + 2c2 = 0. We have three unknowns (v, ν, c) and two equations and we may choose c as a free parameter. The other variables are then found to be ν = 2c2 , v = 4c2 .

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Finally, we obtain the solitary wave solution as u(x, t) = 2c2 (1 − tanh2 [c(x − 4c2 t)]) = 2c2 sech2 [c(x − 4c2 t)]. (3.40) 3.3.2 Modified KdV equation Now, we are going to solve the MKdV equation by the tanh method. Let us consider the MKdV equation as ∂u ∂u ∂ 3 u + 6u2 + = 0. ∂t ∂x ∂x3

(3.41)

A traveling wave solution needs a single coordinate ξ = c(x − vt), where c > 0 and v is the velocity of the wave. So, u(x, t) = u(ξ), where u(ξ) represents the wave solution, which travels with speed v. Using the transformation and then integrating the transformed equation, we get −cvu(ξ) + 2cu2 (ξ) + c3

du(ξ) = 0. dξ 2

(3.42)

Introducing the new variable Y = tanh(ξ), then Equation (3.42) becomes   dS(Y ) d2 S(Y ) −vS(Y ) + 2s2 (Y ) + c2 (1 − Y 2 ) − 2Y + (1 − Y 2 ) = 0. dY dY 2 (3.43) Following the previous procedure, we get M = 1. It is seen that for M = 1 the series expansion (3.35) does not yield a solution unless it is a trivial one (u = constant). Here, we get two options as follows: (i) let us consider a series expansion with M = ∞ and we get recursion relation between c, v, and ai (i ≥ 1). However, it is unnecessary to discuss the unknowns in this

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Waves and Wave Interactions in Plasmas

chapter. (ii) We can also use S(Y ) = ν(1 − Y 2 )1/2 in Equation (3.43) and get −vν + 2ν 2 (1 − Y 2 ) + c2 (1 − Y 2 )1/2   2 2 1/2 dS(1 − Y 2 )1/2 2 d (1 − Y ) × − 2Y + (1 − Y ) = 0. dY dY 2 Eventually, we arrive at the following relations: Y 0 coefficient: −v + 2ν 2 − 2c2 = 0

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Y 2 coefficient: −2ν 2 + 2c2 = 0. We have three unknowns (v, ν, c) and two equations and we may choose c as a free parameter. The other variables are then found to be ν = c and v = c2 . Finally, we obtain the solitary wave solution as u(x, t) = c(1 − tanh2 [c(x − c2 t)])1/2 = csech2 [c(x − c2 t)].

(3.44)

3.3.3 Burgers’ equation Burgers’ equation is given by ∂u ∂u ∂ 2 u − +u = 0. ∂t ∂x ∂x2

(3.45)

A traveling wave solution needs to single coordinate ξ = c(x−vt), where c > 0 and v is the velocity of the wave. So, u(x, t) = u(ξ), where u(ξ) represents the wave solution, which travels with speed v. Using the transformation and then integrating the transformed equation, we get du(ξ) 1 = 0. −vu(ξ) + u2 (ξ) − c 2 dξ

(3.46)

Introducing the new variable Y = tanh(ξ) in Equation (3.46), we get 1 dS(Y ) −vS(Y ) + S 2 (Y ) − c(1 − Y 2 ) = 0. 2 dY

(3.47)

Solution of Nonlinear Wave Equations

103

Following the same procedure, we get M = 1. Therefore, the solution has the form S(Y ) = a0 + a1 Y.

(3.48)

Substituting (3.48) in (3.47), we get 1 −v(a0 + a1 Y ) + (a0 + a1 Y )2 − c(1 − Y 2 )a1 Y = 0. 2

(3.49)

Each coefficient of power of Y has to vanish. So, we arrive at the following:

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Y 2 coefficient: 1 2 a + a1 c = 0 ⇒ a1 = −2c. 2 1

(3.50)

−va1 + a0 a1 = 0 ⇒ a0 = v.

(3.51)

1 −va0 + a20 − a1 c = 0, ⇒ v 2 = 4c2 . 2

(3.52)

Y 1 coefficient:

Y 0 coefficient:

These three Equations ((3.50)–(3.52)) have four unknowns (a0 , a1 , c, v). We choose v as a free parameter. So, a0 = v, a1 = −v, and c = v2 . Finally, we obtain the shock wave solution as     v u(x, t) = u(ξ) = v 1 − tanh x − vt . (3.53) 2 3.3.4 KdV Burgers’ equation We consider the KdVB equation ∂φ ∂2φ ∂φ ∂3φ − b 2 + cφ +d 3 =0 ∂t ∂x ∂x ∂x

(3.54)

where a, b, c, and d are real constants. Assume Equation (3.54) has traveling wave solution in the form of φ(x, t) = u(ξ), where ξ = x − vt. Substituting ξ = x − vt into Equation (3.54) and integrating the transformed

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Waves and Wave Interactions in Plasmas

equation, we get d

d2 u du u2 −b + c − vu = c1 2 dξ dξ 2

(3.55)

where c1 is an arbitrary constant, whose value depends on the initial conditions. We now introduce the new independent variable Y = tanh ξ and W (Y ) = u(ξ), due to which Equation (3.55) transforms to d(1 − Y 2 )2

d2 W dW c − {2dY (1 − Y 2 ) + b(1 − Y 2 )} + W 2 − vW = c1 . 2 dY 2 dY (3.56)

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Following the same procedure, we get M = 2. Thus, W (Y ) = a0 + a1 Y + a2 Y 2 .

(3.57)

Using (3.57), we get from Equation (3.56), the following system of equations: 1 2a2 d − ba1 + ca20 − va0 − c1 = 0 2 2a1 d + 2ba2 − ca0 a1 + va1 = 0 1 ba1 − 8a2 d + ca21 + ca0 a2 − va2 = 0 2 2a1 d + 2ba2 + ca1 a2 = 0 1 6a2 d + ca22 = 0. 2

(3.58) (3.59) (3.60) (3.61) (3.62)

From these equations, the unknowns are determined as a0 =

1 (v + 12d), c

a1 = −

12b , 5c

a2 −

12d c

together with a relation between b and d as b2 = 100d2 . Equation (3.58) determines v in terms of a0 , a1 , a2 , and c1 . Finally, we obtain the solution of KdV–Burgers’ equation as φ(x, t) = a0 + a1 tanh(x − vt) + a2 tanh2 (x − vt)

(3.63)

where a0 , a1 , and a2 are given above together with the relation b2 = 100d2 .

Solution of Nonlinear Wave Equations

105

3.3.5 KP equation Let us consider the KP equation ∂ ∂ξ



∂φ ∂3φ ∂φ + Aφ +B 3 ∂τ ∂ξ ∂ξ

 +C

∂2φ = 0. ∂η 2

(3.64)

To obtain the traveling wave solution, we need the transformation ζ = ξ + η − uτ , and using the boundary conditions, i.e., φ → 0, ∂φ ∂ζ → 0, ∂2 φ ∂ζ 2

→ 0,

∂3φ ∂ζ 3

→ 0, we have

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−(u − C)

∂φ ∂φ ∂3φ + Aφ + B 3 = 0. ∂ζ ∂ζ ∂ζ

(3.65)

Following the same procedure, finally, we get the solution of the KP equation as φ(ζ) =

8B (u − C) 12B + − tanh2 (ξ + η − uτ ). A A A

(3.66)

By the same procedure, one can easily obtain the solution of the modified KP and ZK equations. 3.4 Tanh–Coth Method Let us consider a nonlinear differential equation P (u, ut , ux , uxx , uxxx , . . .) = 0.

(3.67)

To find the traveling wave solution of Equation (3.67), the wave variable ξ = x − ct and then the partial differential Equation (3.67) converts to an ordinary differential equation Q(u, u , u , u , . . .) = 0.

(3.68)

Equation (3.68) is integrated as long as all terms contain derivatives and integration constants are zero. The standard tanh method is developed by Malfliet [13] where the tanh is used as a new variable. A new independent variable Y = tanh(μξ), ξ = x−ct, where μ is the wavenumber, is introduced

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that leads to the following change of derivatives: d d = μ(1 − Y 2 ) dξ dY d2 d d2 = 2μ2 Y (1 − Y 2 ) + μ2 (1 − Y 2 ) 2 dξ dY dY 2 d3 d d2 = 2μ3 (1 − Y 2 )(3Y 2 − 1) − 6μ3 Y (1 − Y 2 )2 3 dξ dY 2 dY

⎪ ⎪ ⎪ d4 d 4 2 2 4 2 2 ⎪ = −8μ Y (1 − Y )(3Y − 1) + 4μ (1 − Y ) dξ 4 dY ⎪ 2 3 4 d d d ⎪ × (9Y 2 − 2) 2 − 12μ4 Y (1 − Y 2 )3 + μ4 (1 − Y 2 )4 .⎭ 3 4 dY dY dY (3.69) + μ3 (1 − Y 2 )3

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⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

d3 dY 3

In the tanh method, the positive integer value of M is considered, whereas in the tanh–coth method, we consider both positive and negative integer values of M . Hence, the finite expansion will be written as u(μ, ξ) = S(Y ) =

M

m=0

aK Y K +

M

bK Y −K

(3.70)

m=0

where M is a positive integer, in most cases, that will be determined. For noninteger M , a transformation formula is used to overcome this difficulty and it reduces to the standard tanh method bk = 0, 1 ≤ k ≤ M . Substituting (3.67) into the reduced ordinary differential equation results in an algebraic equation to the power of Y . Following the same procedure, we determine the value of M and then collect all coefficients of power of Y in the resulting equation where these coefficients are zero. It will give a system of algebraic equations involving the parameters ak , bk , μ, and c. Having determined these parameters, we obtain solitons in terms of sech2 or kinks in terms of tanh. However, this method may give periodic solutions. 3.4.1 KdV equation The KdV equation is ut + auux + uxxx = 0.

(3.71)

Solution of Nonlinear Wave Equations

107

Substitute the wave variable ξ = x − ct in Equation (3.71), where c is the wave speed. Now, integrating the transformed equation, we get a −cu + u2 + u = 0. 2

(3.72)

We first balance the terms u2 with u . This means that the highest power of u2 is 2M and for u is M + 2. This is obtained by using the scheme for the balancing process presented in the previous section. Using the balancing process leads to 2M = M + 2 ⇒ M = 3. The tanh–coth method allows us to use the substitution u(x, t) = S(Y ) =

2

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j=0

aj Y j +

2

bj Y −1 .

(3.73)

i=1

Substituting (3.73) into (3.72), collecting the coefficient of each power of Y r , 0 ≤ r ≤ 8, setting each coefficient to zero, and solving the resulting system of algebraic equations, we find the following sets of solutions: (i) 3c 3c 1√ , a1 = a2 = b1 = 0, b2 = − , μ = c, c > 0 2 a 2

(3.74)

c 3c 1√ a0 = − , a1 = a2 = b1 = 0, b2 = , μ = −c, c < 0 a a 2

(3.75)

a0 = (ii)

(iii) a0 =

3c 3c 1√ , a1 = a2 = b1 = 0, b2 = , μ = c, c > 0 a a 2

(3.76)

(iv) c 3c 1√ a0 = − , a1 = a2 = b1 = 0, b2 = , μ = −c, c < 0. 2 a 2

(3.77)

Consequently, we obtain the following soliton solutions and shocks:   ⎫ 3c 1√ ⎪ u1 (x, t) = sech2 c(x − ct) , c > 0 ⎬ a 2 (3.78)    ⎪ c 2 1√ ⎭ u2 (x, t) = − 1 − 3 tanh −c(x − ct) , C < 0. a 2

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Moreover, the singular solitons and shocks are   ⎫ 3c 2 1√ ⎪ u3 (x, t) = − cosech c(x − ct) , c > 0 ⎬ a 2    ⎪ c 1√ u4 (x, t) = − −c(x − ct) , c < 0. ⎭ 1 − 3 coth2 a 2 We obtain the following plane periodic solutions    ⎫ 3c 1√ ⎪ u5 (x, t) = c cosech2 −c(x − ct) , c < 0 a 2 ⎪    ⎪ 1√ c u6 (x, t) = − 1 − 3 coth2 c(x − ct) , c > 0 ⎬ a 2    √ ⎪ 3c 2 1 u7 (x, t) = c sech −c(x − ct) , c < 0 ⎪ a 2    ⎪ c 1√ u8 (x, t) = − 1 + 3 tanh2 c(x − ct) , c > 0. ⎭ a 2

(3.79)

(3.80)

Similarly, we can obtain the solution of other KdV type equation. The modified KdV equation is ut + au2 ux + uxxx = 0.

(3.81)

We first substitute the wave variable ξ = x − ct, where c is the wave speed, into the MKdV equation and the following procedure as above, one can easily obtain the solution of MKdV equation. The following solution and kink solutions are ⎫ √  3c ⎪ sech c(x − ct) , c > 0, a > 0 u1 (x, t) = ⎬ a (3.82)    ⎪ 3c 1 − c(x − ct) , c < 0, a < 0 ⎭ u2 (x, t) = tanh a 2 respectively obtained. Moreover, the following traveling wave periodic solutions (shocks and singular solitons) are derived ⎫    ⎪ 3c 1 u3 (x, t) = − c(x − ct) , c < 0, a < 0 coth ⎪ a 2 ⎬ √  6c (3.83) u4 (x, t) = sech −c(x − ct) , c < 0, a < 0 ⎪ a ⎪ √  −6c ⎭ u4 (x, t) = cosech c(x − ct) , c > 0, a < 0. a

Solution of Nonlinear Wave Equations

109

It can also be derived by using the sign of the wave speed c and the parameter a. 3.4.2 Burgers’ equation Burgers’ equation is given by ∂u ∂u ∂ 2 u − +u = 0. ∂t ∂x ∂x2

(3.84)

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Let ξ = x − ct and u(x, t) = u(ξ). Using the transformation and integrating the transformed equation with the help of initial condition u(ξ) → 0, du(ξ) dξ → 0 at ξ → ±∞, we get 1 du(ξ) −cu(ξ) + u2 (ξ) − = 0. 2 dξ

(3.85)

Next, we transform to the new variable Y = tanh(μξ) such that U (ξ) = M −M S(Y ) = ΣM + ΣM . The parameter M will be found by m=0 am Y m=1 bm Y balancing the linear terms of the highest order with nonlinear terms. So, d 2 d dξ = μ(1 − Y ) dY . Form (3.85), we have 1 dS(Y ) −cS(Y ) + S 2 (Y ) − μ(1 − Y 2 ) = 0. 2 dY

(3.86)

Following the same procedure, we get M = 1. Thus, S(Y ) = a0 + a1 Y + b1 Y −1 .

(3.87)

Following the same procedure, we finally obtain the solution of Burgers’ equation as c  c  ⎫ c ⎪ tanh (x − ct) + coth (x − ct) , c > 0, S(Y ) = c − ⎬ 2 4 4      c c c ⎪ S(Y ) = c + tanh − (x − ct) + coth − (x − ct) , c < 0. ⎭ 2 4 4 (3.88) 3.5 Solution of KP Burger Equation We know that certain nonlinear second-order ordinary differential equations can be factorized under some conditions that coincide with those of integrability obtained from a Painleve analysis. These kinds of equations frequently appear when looking for traveling wave solutions of interesting nonlinear physical equations. We also know that the factorizations are

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Waves and Wave Interactions in Plasmas

directly related to the first integrals of the equation that are a kind of Bohlin’s first integrals. Now, we are going to solve the KP equation. At first, find the Abel equation, and then we shall obtain the solution of that equation. Now, let us consider the KPB equation   ∂φ ∂ ∂φ ∂3φ ∂2φ ∂ 2φ + + Aφ + B 3 − C 2 + D 2 = 0. (3.89) ∂ζ ∂τ ∂ζ ∂ζ ∂χ ∂ζ Since the KPB equation possesses the conditionally Painleve property, it can be solved if a specific method is developed. We want to solve this equation by the factorization method. Let the exact solution of the above equation in the form of a traveling wave be

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φ(ζ, χ, τ ) = φ(ξ); ξ = hζ + lχ − ωτ

(3.90)

where the constant h, l, and ω are to be determined. Then Equation (3.89) takes the form   ∂ ∂4φ ∂3φ ∂2φ ∂ 2φ ∂φ −ωh 2 + h2 A φ + Bh4 4 − Ch3 3 + Dl2 2 = 0. (3.91) ∂ξ ∂ξ ∂ξ ∂ξ ∂ξ ∂ξ Integrating twice with respect to ξ, we get  2 φ ∂2φ ∂φ 2 2 (Dl − ωh)φ + h A = R1 ξ + R2 + Bh4 2 − Ch3 2 ∂ξ ∂ξ

(3.92)

where R1 and R2 are two integration constants. Then, we make a linear transformation as ξ = −hBθ; φ(ξ) = −(2/AB)W (θ) and substituting these we get ∂W ∂2W +C − W 2 + KW = d1 θ + d2 . ∂θ2 ∂θ 2

This is the required Abel equation, where K = Dl h−ωh B; d1 = 2 AB 2 R2 . 2h2 If we consider C = 0, then the equation will be ∂2W − W 2 + KW = d1 θ + d2 ∂θ2

(3.93) AB 3 2h R1 ,

d2 =

(3.94)

which is the Abel equation corresponding to the KP equation. To solve Equation (3.93) by the factorization method, we first consider the integration constant R1 = 0. Again, the nontrivial factorization is obtained only when d2 = 0, which is a restrictive condition. To overcome this constraint, we apply a displacement on the unknown function, W (θ) = U (θ) + δ, where δ is a solution of Equation (3.93) so that

Solution of Nonlinear Wave Equations

d2 = kδ − δ 2 . To get real values of δ, k 2 > 4δ2 . Again, 2δ = k − and therefore, k − 2δ > 0. Thus, Equation (3.93) takes the form ∂U ∂U +C − U 2 (θ) + K − 2δ)U (θ) = 0. ∂θ2 ∂θ

111

√ k 2 − 4δ2

(3.95)

Then, comparing the above equation with the factorization method, where f1 and f2 are two unknown functions, which may depend on U and θ, we get     ∂f1 ∂f1 C = − f2 + U , f1 f2 = K − 2δ − U (θ) + . (3.96) ∂U ∂θ

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Here, we find a particular solution of the above equation in which f1 and f2 do not depend explicitly on θ. Using the ansatz f1 = XU p + Y

(3.97)

where X and Y are constants. f2 = −C − (p + 1)XU p − Y.

(3.98)

Then taking the value p = 1/2, X 2 2/3, Y = −(2C/5), we get K − 2δ = K2 K2 9C 4 6C 2 2 25 ⇒ d2 = −(1/4)(k − 2δ) + 4 = 4 − 625 . Then for the particular solution dU 2 3/2 2C U =± − U (3.99) dθ 3 5 whose general solution is given by U ± (θ) =

6C 2 1 . C 25 [1 ∓ e 5 (θ−θ0 ) ]2

(3.100)

We can write the solution of the KPB equation as φ(ζ, χ, τ ) = −

2 (U (θ) + δ) AB

(3.101)

and substituting the value of U (θ) and δ, we get

  12C 2 1 1 Dl2 − ωh 6C 2 φ(ζ, χ, τ ) = − B+ − −C (hξ+lχ−ωτ ) ]2 25AB [1 + z0 e 5hB AB h2 25 (3.102) −C 5 θ0 )

where z0 = e is an arbitrary constant, which is the required particular traveling solitary wave solution.

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3.6 Conservation Laws and Integrals of the Motions A special significance of the KdV equation is the existence of an infinite number of conservation laws. By conservation law, we mean an equation of the form Tt + Xx = 0,

(3.103)

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where T , the conserved density, and X, the flux, are functions of x, t, u and higher-order x-derivatives of u. We only discuss conservation laws where T and X are polynomials in u and x-derivatives of u alone. A conserved density T is called trivial if T is an x-derivative for all u and if T = Fx , then we automatically have a conservation law (Fx )t + (−Ft )x = 0 where the t-derivatives in the flux are eliminated by the repeated use of the evolution equation. The existence of conservation laws for the KdV equation and a conjectured connection to the conservation laws for a related equation motivated the inverse scattering method of the solution developed. Conservation laws can be used to obtain integrals  ∞ of the motion. For example, if the flux X is zero as |X| → ∞, then −∞ T dx = constant. Let the flux X be zero as |X| → ∞, then from conservation law, we have Tt + Xx = 0. Integrating both sides with respect to x, we have  ∞  ∞ Tt dx + Xx dx = 0, −∞



⇒

−∞

 ⇒

−∞

∞

∞

−∞



Tt dx +

∞

−∞

dX = 0,

∂T dx + [X]∞ −∞ = 0, ∂t

 d ∞ T dx = 0, dt −∞  ∞ ⇒ T dx = c (c is a constant). ⇒

−∞

(3.104)

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Solution of Nonlinear Wave Equations

113

Furthermore, the existence of infinitely many conservation laws indicates that the KdV equation is a special equation of physical interest. Here, we explain the first three conservation laws for the KdV equation. Law 1: Let us consider the KdV equation ut − 6uux + uxxx = 0.

(3.105)

Equation (3.105) can be rewritten as (u)t + (−3u2 + uxx )x = 0. ∞ Here, T = u, X = −3u2 + uxx and accordingly  ∞ −∞ T dx=constant. So, the first conservation law for KdV equation is −∞ udx=constant. Equation (3.104) must be true for all solutions of the KdV equation . However, not all solutions of the KdV equation satisfy the asymptotic relations. For example, the conservation laws do not apply to periodic solutions of the KdV equation. Law 2: Multiplying Equation (3.105) by 2u and rewriting the equation, we get (u2 )t + (−4u3 + 2uuxx − (ux )2 )x = 0. Here, T = u2 , X = −4u3 + 2uuxx − (ux )2 and the second conservation law ∞ is −∞ u2 dx=constant. Law 3: Again, multiplying Equation (3.105) by (3u2 − uxx ) and rewriting the equation, we obtain     1 9 4 1 2 2 2 2 3 + − u + 3u uxx − 6u(ux ) + ux uxxx − (uxx ) = 0. u + (ux ) 2 2 2 t x Here, T = u3 + 12 (ux )2 , X = − 92 u4 + 3u2uxx − 6u(ux)2 + uxuxxx − 12 (uxx )2 , ∞ and the third conservation law is −∞ (u3 )t + 12 ((ux )2 )dx = constant. Note: These conserved densities can be interpreted as mass, momentum, and energy for some physical systems. Due to the existence of so many conservation laws, the KdV equation plays a distinguished role, especially those of physical interest. But several other nonlinear partial differential equations also satisfy conservation laws and people are interested about the equations which has some physical applications.

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Let us consider a more general equation ut − 6up ux + uxxx = 0,

p = 1, 2, . . .

(3.106)

If p = 1, then Equation (3.106) is a KdV equation, and if p = 2, then Equation (3.106) is an MKdV equation. Let us consider an MKdV equation as ut − 6u2 ux + uxxx = 0.

(3.107)

This equation also possesses many polynomial conservation laws of which the first three conservation forms are given in the following:

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Law 1: From Equation (3.107), we have (u)t + (−2u3 + uxx )x = 0. Here, T = u, X = −2u3 +uxx, and the first conservation law for the MKdV ∞ equation is −∞ udx = constant. Law 2: Multiplying Equation (3.107) by 2v and rewriting the equation, we obtain (u2 )t + (−3u4 + 2uuxx − (ux )2 )x = 0. Here, u2 , X = −3u4 + 2uuxx − (ux )2 , and the second conservation law  ∞T = 2 is −∞ u dx = constant. Law 3: Multiplying Equation (3.107) by (4v 3 − 2vxx ) and rewriting the equation, we get (u4 + (ux )2 )t + (−4u6 + 4u3 uxx − 2ux (6u2 ux − uxxx ) − (uxx )2 )x = 0. Here, T = u4 + (ux )2 , X = −4u6 + 4u3 uxx − 2ux (6u2 ux − uxxx) − (uxx )2 , ∞ and the third conservation law is −∞ (u4 + (ux )2 )dx = constant. However, a study of Equation (3.106) with p >= 3 leads to the discovery of only three polynomial conservation laws. This apparent distinguished role is played by both the KdV equation and the modified KdV equation. Miura transformation: If v is a solution of Qv ≡ vt − 6v 2 vx + vxxx = 0. Then u = v 2 + vx is a solution of P u ≡ ut − 6uux + uxxx = 0.

Solution of Nonlinear Wave Equations

115

Let us put u = v 2 + vx in the equation P u ≡ ut − 6uux + uxxx = 0 and we have P u = (v 2 + vx )t − 6(v 2 + vx )(v 2 + vx )x + (v 2 + vx )xxx , ⇒ P u = 2vvt + vxt − 12v 3 vx − 6v 2 vxx − 12v(vx )2 − 6vx vxx + 6vx vxx + 2vvxxx + vxxxx ,

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∂ ⇒ P u = 2v(vt − 6v 2 vx + vxxx ) + (vt − 6v 2 vx + vxxx ), ∂x   ∂ ⇒ P u = 2v + (vt − 6v 2 vx + vxxx ), ∂x   ∂ ⇒ P u = 2v + (Qv). ∂x So, if Qv = 0, then u satisfies the KdV equation P u ≡ ut −6uux +uxxx = 0, provided u = ux + v 2 . Gardner’s generalization: One being with Gardner’s generalization we know that the KdV equation is Galilean invariant and so introducing the following transformation t ≡ t, x ≡ x + 232 t , u(x, t) ≡ u (x , t ) + 1 1   42 , v(x, t) ≡ ω(x , t ) + 2 , where the specific dependence on the formal parameter  has been chosen to get the following desired results.  ∂ Now, pu = 0 ⇒ ut − 6uux + uxxx = 2v + ∂x )(vt − 6v 2 vx + vxxx ) = 0. ∂ 1 1 1 Putting v = ω + 2 , we get pu ≡ (2(ω + 2 )((ω + 2 )t − 6(ω + ) + ∂x 1 1 ∂ 1 2 2 ) (ω + ) + (ω + ) ) = 0 ⇒ (1 +  + 2 ω)(ω − 6(ω + 2ω 2 )ωx + x xxx t 2 2 2 ∂x 1 ωxxx ) = 0 [neglecting 42 ]. Also where we have dropped all primes and the Galilean transformation leaves the KdV equation invariant. Now, u = v 2 + vx = ω + ωx + 2 ω 2 ,

(3.108)

which is Gardner’s generalization of u. Here, u is a function of x and t and ω is a function of x, t, and . Now, let ω be a power series in  with coefficients which are functions of u and x-derivatives of u, then ω can be written as ω(x, t; ) = ω0 + ω1 + 2 ω2 + · · ·

(3.109)

ωx = ω0x + ω1x + 2 ω2x + · · ·

(3.110)

or

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Waves and Wave Interactions in Plasmas

and ω 2 = ω02 + 2 ω12 + 2ω0 ω1 + · · · .

(3.111)

Now, using Equations (3.109)–(3.111) in (3.108), we get u = ω0 + (ω1 + ω0x ) + 2 (ω2 + ω1x + ω02 ) + · · · . Comparing the coefficient of 0 , 1 , 2 respectively we get ω0 = u, ω1 = −ω0x = −ux , ω2 = −ω1x − ω02 = uxx − u2 , etc. Using the above relation in (3.109), we get

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ω(x, t; ) = u − ux − 2 (u2 − uxx ) + · · · .

(3.112)

So,    ∂ 1+ + 2ω [Jω] = 0 ⇒ (1 + M )[Jω] = 0 ∂x ∂ where ∂x + 2ω = M and Jω = ωt + (−3ω 2 − 22 ω 3 + ωxx )x = 0. Now,

Jω = ωt + (−3ω 2 − 22 ω 3 + ωxx )x = 0

(3.113)

is called Gardner’s equation. But in the form of (3.113), the coefficient of each power of  is a conservation law for the KdV equation and there are infinitely many of them. 3.6.1 Conserved quantity of KdV equation Let us consider the KdV equation ∂φ ∂3φ ∂φ + Aφ + B 3 = 0. ∂τ ∂ξ ∂ξ

(3.114)

Let  I=

∞

−∞

φ2 dξ.

(3.115)

Solution of Nonlinear Wave Equations

117

Now, differentiating (3.115) under integration sign with respect to τ , we get  ∞ ∂φ dI = 2φ dξ dτ ∂τ −∞   ∞  ∂3φ ∂φ − B 3 dξ [from Equation (3.114)] =2 φ − Aφ ∂ξ ∂ξ −∞   ∞  2    ∞ ∂2φ 1 ∞ d ∂φ φ3 − = − 2A − 2B φ 2 dξ . 3 −∞ ∂ξ −∞ 2 −∞ dξ ∂ξ

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=0

(3.116)

dI Since φ → 0, ∂φ ∂ξ → 0 at ξ → ±∞. Therefore, from (3.116), we get dτ = 0, i.e., I = constant. There exists an infinite number of such conserved quantities like I.

3.7 Approximate Analytical Solutions For a small perturbation, we expect that the solution will remain close to the soliton for some time. Therefore, the solution we seek will be roughly a soliton with a slowly changing shape and location plus a correction. 3.7.1 Damped KdV equation Let us consider the KdV equation with damping term cφ when the constant c is small, ∂φ ∂3φ ∂φ + Aφ + B 3 + cφ = 0. ∂τ ∂ξ ∂ξ

(3.117)

Now, dI = dτ



∞

∂φ dξ ∂τ −∞   ∞  ∂φ ∂3φ = −2 φ Aφ + B 3 + cφ dξ ∂ξ ∂ξ −∞  ∞ = −2c φ2 dξ [using Equation (3.116)] 2φ

−∞

= −2cI.

(3.118)

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Waves and Wave Interactions in Plasmas

The solution of Equation (3.118) is (3.119) I = I0 e−2c(τ −τ0) [I = I0 at τ = τ0 ].   )τ Since φ = φm (τ )sech2 ξ−M(τ is a solution of (3.117), where φm (τ ) = ω(τ )  3M(τ ) 4B M(τ ) . So, A , ω(τ ) =  ξ − M (τ )τ I= φ dξ = dξ ω(τ ) −∞ −∞     1 ξ − M (τ )τ (1 − t2 )dt putting t = tanh = 2φ2m ω(τ ) ω(τ ) 0 

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=

∞



2

∞

φ2m (τ )sech4



4 2 φ ω(τ ). 3 m

Hence, 4 2 φ (τ0 )ω(τ0 )e−2c(τ −τ0) 3 m  4B 0 where φm (τ0 ) = 3M , ω(τ ) = 0 M0 . A Hence, the solution of (3.117) is   ξ − M (τ )τ φ = φm (τ )sech2 ω(τ ) I=

where φm (τ ) =

3M(τ ) A ,

ω(τ ) =



4B M(τ ) ,

(3.120) 4

and M = M0 e− 3 c(τ −τ0) .

3.7.2 Force KdV equation Let us consider the KdV equation with an external force f0 cos(ωτ ) ∂φ ∂φ ∂3φ + Aφ + B 3 = f0 cos(ωτ ). ∂τ ∂ξ ∂ξ

(3.121)

Let the solution of (3.121) be φ = φm (τ )sech where φm (τ ) =

3M(τ ) A

and ω(τ ) =



2



ξ − M (τ )τ ω(τ )

4B M(τ ) .

 (3.122)

Solution of Nonlinear Wave Equations

119

Therefore,  I= 

∞ −∞ ∞

= −∞

=

φ2 dξ φ2m (τ )sech4



 ξ − M (τ )τ dξ ω(τ )

[f rom (3.122)]

24 √ 3 BM 2 (τ ). A2

Again,

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dI = 2f0 dτ



∞

[using (3.121) and (3.116)]

cos(ωτ )φdξ −∞



= 2f0 cos(ωτ )

∞ −∞

φm (τ )sech

2



 ξ − M (τ )τ dξ ω(τ )

= 4f0 cos(ωτ )φm (τ )ω(τ ). After some elementary calculation, we get dM (τ ) 2 = f0 A cos(ωτ ). dτ 3 Integrating the above relation with respect to τ and using the initial condition M (τ ) = M0 at τ = 0, we get M (τ ) =

2 f0 A sin(ωτ ) + M0 . 3w

(3.123)

So, the solution of (3.121) is φ = φm (τ )sech2 where φm (τ ) =

3M(τ ) A

and ω(τ ) =





ξ − M (τ )τ ω(τ )

 (3.124)

4B M(τ ) .

3.7.3 Damped-force KdV equation Let us consider the KdV equation with an external force f0 cos(ωτ ) and a damped force c ∂φ ∂φ ∂3φ + Aφ + B 3 + cφ = f0 cos(ωτ ). ∂τ ∂ξ ∂ξ

(3.125)

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Waves and Wave Interactions in Plasmas

Let the solution of (3.125) be φ = φm (τ )sech

2



ξ − M (τ )τ ω(τ )

 ) 4B and ω(τ ) = M(τ where φm (τ ) = 3M(τ A ). Therefore,    ∞ 2 4 ξ − M (τ )τ I= φm (τ )sech dξ ω(τ ) −∞ 24 √ 3 = 2 BM 2 (τ ). A

 (3.126)

[f rom (3.126)]

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Similarly, we get dI = −2c dτ



∞

−∞

φ2 dξ + 2f0



∞

cos(ωτ )φdξ. −∞

So, dI + 2cI = 4f0 cos(ωτ )φm (τ )ω(τ ). dτ After some elementary calculation, we get dM (τ ) 4 2 + cM (τ ) = f0 A cos(ωτ ). dτ 3 3 Integrating the above relation with respect to τ and using the initial condition M (τ ) = M0 at τ = 0, we get M (τ ) = M0 e

−4 3 Dτ

+

  2 −1 2f0 A 4D 1 + 9ω 2 3ω

  −4 Dτ 3 × 3ω sin(ωτ ) + 4D cos(ωτ ) − 4De .

(3.127)

So, the solution of (3.125) is φ = φm (τ )sech

2



ξ − M (τ )τ ω(τ )

 (3.128)

 ) 4B and ω(τ ) = M(τ where φm (τ ) = 3M(τ A ). Similarly, one can easily obtain the approximate analytical solution of the damped and or the forced KP/ZK equation.

Solution of Nonlinear Wave Equations

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3.8 Multisoliton and Hirota’s Direct Method In 1971, Ryogo Hirota [14] established a new method called “the Hirota direct method” to find the exact solution of the KdV equation for multiple collisions of solitons. This method can also be applicable for other nonlinear evolution equations such as the MKdV, sine-Gordon (SG), nonlinear Schrodinger (NLS), and Toda lattice (TL) equations. Here, in the first step, nonlinear partial differential equations are converted to a quadratic form of dependent variables using suitable transformations. In the second step, a special differential operator called Hirota D-operator is introduced to write the bilinear form of the equation as a polynomial of D-operator. It is called the Hirota bilinear form. There is no systematic way to construct the Hirota bilinear form for given nonlinear partial differential equations. In principle, all completely integrable nonlinear partial differential equations and difference equations can be put into the Hirota bilinear form. But the converse is not necessarily true, i.e., there exist some equations which are not integrable but have Hirota bilinear forms. Finally, the Hirota method will use the perturbation expansion, in the Hirota bilinear form, considering the coefficients of the perturbation parameter and its powers separately. After that, we reach multisoliton solutions of the said equation. The Hirota direct method has taken a significant role in the study of integrable systems. Equations having Hirota bilinear form possess multisoliton solutions. Hirota’s direct method constructs multisoliton solutions to nonlinear evolution equations that are integrable. The idea was to transform new variables to enable new variable multisoliton solutions to appear in a simple form. The method turned out to be very effective and was quickly shown to give N-soliton solutions to the KdV, MKdV, SG, and NLS equations. It is also useful in constructing their B¨acklund transformations. The advantage of Hirota’s method over the others is that it is algebraic rather than analytic. Accordingly, if one wants to find soliton solutions, Hirota’s method is the fastest in producing results. 3.8.1 Hirota’s method If a function f (x) is (m + n) times differentiable, the Pade approximation of f (x) of order (m, n) is the rational function R(x) =

p0 + p1 x + p2 x2 + · · · + pm xm G[x] = 2 n q0 + q1 x + q2 x + · · · + qn x F [x]

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Waves and Wave Interactions in Plasmas

which agrees with f (x) to the highest possible order, i.e., f (0) = R(0) f  (x) = R (x) ··· ··· f m+n (0) = Rm+n (0). From the Pade approximation, we can write u =

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u=

G , F

ut =

Gt F − GFt , F2

ux =

G(exp(kx−ωt)) F (exp(kx−ωt)) .

Then,

Gx F − GFx , F2

Gxx F 3 − GF 2 Fxx − 2F 2 Fx Gx + 2F Fx2 G F4 Gxxx 3Gxx Fx + 3Gx Fxx + GFxxx = − F F2

uxx = uxxx

+6

Gx Fx2 + GFxx Fx GFx3 − . 3 F F4

Substitute these above terms in the KdV equation ux + 6uux + Uxxx = 0 ⇒

(3.129)

Gx F − GFx G Gx F − GFx 3Gxx Fx + 3Gx Fxx + GFxxx +6 − F2 F F2 F2

+

Gxxx Gx Fx2 + GFxx Fx GFx3 +6 − = 0. F F3 F4

(3.130)

Initially, Equation (3.129) might look more complicated. After some algebraic calculation, we get Gt F − GFt + Gxxx F − 3Gxx Fx − 3Gx Fxx − GFxxx = 0 and GGx F 2 − G2 F Fx + F Gx Fx2 + F GFxx Fx − GFx3 = 0. So, a great deal of work is done. We introduce a new bilinear differentiation operator, the Hirota D-operator. The Hirota D-operator is for ntimes differentiable function of f and g is defined by Dxn f · g = (∂x1 − ∂x2 )n f (x1 )g(x2 )|x1 =x2 =x .

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Solution of Nonlinear Wave Equations

3.8.2 Multisoliton solution of the KdV equation We are going to obtain the multisoliton solution of the KdV equation by using the Hirota direct method. Let us assume a standard KdV equation as ut − 6uux + uxxx = 0.

(3.131) 2

∂ At first,    we introduce  the transformation u(x, t) = −2 ∂x  2 (log f ) =  2 f ∂ G 2 ∂x = −2 f f f−f =H form of the Pade approximation . Using this 2 f

transformation in Equation (3.131), we get

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2 f fxt − fx ft + f fxxxx − 4fx fxxx + 3fxx = 0.

(3.132)

This is the bilinear KdV form. Using the Hirota D-operator in Equation (3.132). Now,    ∂ ∂ ∂ ∂ − − Dt Dx {f · f } = {f (x, t) · f (´ x, t´)}|x´=x,t´=t ∂t ∂t´ ∂x ∂x ´ = fxt f + f fxt − ft fx − fx ft = 2(fxt f − ft fx )

(3.133)

and Dx4 {f

 · f} =

∂ ∂ − ∂x ∂x ´

4 {f (x, t) · f (´ x, t´)}|x´=x,t´=t

= fxxxx f − 4fxxxfx + 6fxx fxx − 4fx fxxx + f fxxxx 2 = 2(fxxxx f − 4fxxxfx + 3fxx ).

(3.134)

Using relations (3.133) and (3.134) in (3.132), we get the Hirota bilinear form as P (D){f · f } = (Dx Dt + Dx4 ){f · f } = 0.

(3.135)

Finally, putting f = 1 + f1 + 2 f2 + · · · into Equation (3.135), we obtain P (D){f · f } = P (D){1 · 1} + P (D){f1 · 1 + 1 · f1 } + 2 P (D){f2 · 1 + f1 · f1 + 1 · f2 } + 3 P (D){f3 · 1 + f2 · f1 + f1 · f2 + 1 · f3 } + · · · = 0. (3.136)

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One-soliton solution: To obtain the one soliton solution of the KdV equation, we consider f = 1 + f1 , where f1 = eθ1 and θ1 = k1 x + ω1 t + α1 . Also, fj = 0 for all j ≥ 2. Putting f = 1 + f1 into Equation (3.136). We need to equate the coefficients of 0 , , and 2 to make them zero. The coefficient of 0 is P (D){1 · 1} = 0 (since P (0, 0){1} = 0). The coefficient of  is P (D){1 · f1 + f1 · 1} = P (∂)eθ1 + P (−∂)eθ1 = 2P (p1 )eθ1 = 0. We have the dispersion relation P (p1 ) = 0 which implies ω1 = −k13 . The coefficient of 2 vanishes trivially since

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P (D){f1 · f1 } = P (D){eθ1 · eθ1 } = P (p1 − p1 )e2θ1 = 0. Finally, without loss of generality, we may set  = 1 so f = 1 + eθ1 and the one-soliton solution of the KdV Equation (3.131) is   k12 2 θ1 u(x, t) = − sech 2 2

(3.137)

where θ1 = k1 x − k13 t + α1 . Two-soliton solution: To obtain the two-soliton solution of the KdV equation, we consider f = 1+f1 +2 f2 , where f1 = eθ1 +eθ2 and θi = ki x+ ωi t + αi for i = 1, 2. Also, fj = 0 for all j ≥ 3. Putting f = 1 + f1 + 2 f2 into Equation (3.136). Equating the coefficients of 0 , , 2 , 3 , and 4 and making them to zero, the coefficient of 0 is P (D){1 · 1} = P (0, 0){1} = 0. The coefficient of  is P (D){1 · f1 + f1 · 1} = 2P (∂){eθ1 + eθ2 } = 2[P (∂)eθ1 + P (∂)eθ2 ] = 0, which implies P (pi ) = ki4 + ki ωi = 0, i.e., ωi = −ki3 for i = 1, 2. The coefficient of 2 becomes P (D){1 · f2 + f2 · 1} + P (D){f1 · f1 } = 2P (∂)f2 + P (D){(eθ1 + eθ2 ) · (eθ1 + eθ2 )} = 2[P (∂)f2 + P (D){eθ1 eθ2 }] = 2[P (∂)f2 + P (p1 − p2 )eθ1 +θ2 ] = 0.

Solution of Nonlinear Wave Equations

125

This makes f2 to have the form f2 = A(1, 2)eθ1 +θ2 . If we put f2 in the above equation, we obtain A(1, 2) as A(1, 2) = −

P (p1 − p2 ) (k1 − k2 )2 = . P (p1 + p2 ) (k1 + k2 )2

(3.138)

Since f3 = 0, the coefficient of 3 turns out to be P (D){f1 · f2 + f2 · f1 } = 2A(1, 2)[P (D){(eθ1 )(eθ1 +θ2 )} + P (D){(eθ2 )(eθ1 +θ2 )}]

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= 2A(1, 2)[P (p2 )e2θ1 +θ2 + P (p1 )eθ1 +2θ2 ] and this is already zero since P (pi ) = 0, i = 1, 2. The coefficient of 4 also vanishes trivially. At last, we may set  = 1, thus f = 1 + eθ1 + eθ2 + A(1, 2)eθ1 +θ2 and the two-soliton solution of the KdV Equation (3.131) is given by u(x, t) = −2

{k12 eθ1 + k22 eθ2 + A(1, 2)(k22 eθ1 + k12 eθ2 )eθ1 +θ2 + 2(k1 − k2 )2 eθ1 +θ2 } (1 + eθ1 + eθ2 + A(1, 2)eθ1 +θ2 )2 (3.139)

where θi = ki x − ki3 t + αi , i = 1, 2, and A(1, 2) =

(k1 −k2 )2 (k1 +k2 )2 .

Three-soliton solution: To obtain the three-soliton solution of the KdV equation, we consider f = 1 + f1 + 2 f2 + 3 f3 , where f1 = eθ1 + eθ2 + eθ3 and θi = ki x + ωi t + αi for i = 1, 2, 3. Also, fj = 0 for all j ≥ 4. Putting f = 1 + f1 + 2 f2 + 3 f3 into Equation (3.136) and equating the coefficients of m , m = 0, 1, 2, 3, 4, 5, 6, and making them to zero. The coefficient of 0 is identically zero. By the coefficient of 1 , we have P (D){1 · f1 + f1 · 1} = 2P (∂){eθ1 + eθ2 + eθ3 } = 0, which implies P (pi )0, i.e., ωi = −ki3 for i = 1, 2, 3. From the coefficient of 2 , we get −P (∂)f2 = [(k1 − k2 )(ω1 − ω2 ) + (k1 − k2 )4 ]eθ1 +θ2 + [(k1 − k3 )(ω1 − ω3 ) + (k1 − k3 )4 ]eθ1 +θ3 + [(k2 − k3 )(ω2 − ω3 ) + (k2 − k3 )4 ]eθ2 +θ3 .

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This makes f2 to have the form f2 = A(1, 2)eθ1 +θ2 + A(1, 3)eθ1 +θ3 + A(2, 3)eθ2 +θ3 . If we put f2 in the above equation, we obtain A(i, j) as A(i, j) = −

P (pi − pj ) (ki − kj )2 = . P (pi + pj ) (ki + kj )2

(3.140)

Here, i, j = 1, 2, 3, i < j. The coefficient of 3 becomes −P (∂)f3 = eθ1 +θ2 +θ3 {A(1, 2)P (p3 − p2 − p1 ) + A(1, 3)P (p2 − p1 − p3 ) + A(2, 3)P (p1 − p2 − p3 )}.

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Hence, f3 should be in the form of f3 = Beθ1 +θ2 +θ3 . So, the above equation gives B = −[A(1, 2)P (p3 − p2 − p1 ) + A(1, 3)P (p2 − p1 − p3 ) + A(2, 3)P (p1 − p2 − p3 )]/[P (p1 + p2 + p3 )]. If we make all the simplifications by using ωi = −ki3 for i = 1, 2, 3, we see that the above expression is equivalent to B = A(1, 2)A(1, 3)A(2, 3). Since f4 = 0, from the coefficient of 4 , we have P (D){f1 · f3 + f3 · f1 + f2 · f2 } = 0. After some calculations, the above equation becomes e2θ1 +θ2 +θ3 [BP (p2 + p3 ) + A(1, 2)A(1, 3)P (p2 − p3 )] + eθ1 +2θ2 +θ3 [BP (p1 + p3 ) + A(1, 2)A(2, 3)P (p1 − p3 )] + eθ1 +θ2 +2θ3 [BP (p1 + p2 ) + A(1, 3)A(2, 3)P (p1 − p2 )] = 0. This is satisfied by B = A(1, 2)A(1, 3)A(2, 3). Finally, the coefficient of 5 and 6 also vanishes automatically. We set  = 1, therefore, f = 1 + eθ1 + eθ2 + eθ3 + A(1, 2)eθ1 +θ2 + A(1, 3)eθ1 +θ3 + A(2, 3)eθ2 +θ3 + Beθ1 +θ2 +θ3 . So, the three-soliton solution of the KdV Equation (3.131) is u(x, t) = −2

L(x, t) M (x, t)

(3.141)

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where L(x, t) = eθ1 +θ2 [2(k1 − k2 )2 + 2(k1 − k2 )2 A(1, 3)A(2, 3)e2θ3 + A(1, 2)k12 eθ2 + A(1, 2)k22 eθ1 ] + eθ1 +θ3 [2(k1 − k3 )2 + 2(k1 − k3 )2 A(1, 2)A(2, 3)e2θ2 + A(1, 3)k12 eθ3 + A(1, 3)k32 eθ1 ] + eθ2 +θ3 [2(k2 − k3 )2 + 2(k2 − k3 )2 A(1, 2)A(1, 3)e2θ1 + A(2, 3)k22 eθ3 + A(2, 3)k32 eθ2 ] + k12 eθ1 + k22 eθ2 + k32 eθ3 + Beθ1 +θ2 +θ3 [A(1, 2)k32 eθ1 +θ2 + A(1, 3)k22 eθ1 +θ3 + A(2, 3)k12 eθ2 +θ3 ]

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+ eθ1 +θ2 +θ3 [A(1, 2)(k12 + k22 + k32 + 2k1 k2 − 2k1 k3 − 2k2 k3 ) + A(1, 3)(k12 + k22 + k32 + 2k1 k3 − 2k1 k2 − 2k2 k3 ) + A(2, 3)(k12 + k22 + k32 + 2k2 k3 − 2k1 k2 − 2k1 k3 ) + B(k12 + k22 + k32 + 2k1 k2 + 2k1 k3 + 2k2 k3 )

(3.142)

and M (x, t) = [1 + eθ1 + eθ2 + eθ3 + A(1, 2)eθ1 +θ2 + A(1, 3)eθ1 +θ3 + A(2, 3)eθ2 +θ3 + Beθ1 +θ2 +θ3 ]2 for θi = ki x − ki3 t + αi , A(i, j) = B = A(1, 2)A(1, 3)A(2, 3).

(ki −kj )2 (ki +kj )2

(3.143)

for i, j = 1, 2, 3, i < j, and

3.8.3 Multisoliton solution of the KP equation To obtain the multisoliton solution for the KP equation by using the Hirota direct method, let us assume the KP equation as (ut − 6uux + uxxx )x + 3uyy = 0.

(3.144) 2

∂ At first, the transformation u(x, t) = −2 ∂x = 2 (log f )    we introduce  2 f f −f f ∂ 2 ∂x f = −2 f 2 . Using this transformation in Equation (3.144), we get 2 f fxt − fx ft + f fxxxx − 4fx fxxx + 3fxx + 3fyy − 3fy2 = 0.

(3.145)

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Waves and Wave Interactions in Plasmas

This is the bilinear KP form. After that, we use the Hirota D-operator in Equation (3.145) and we get the Hirota bilinear form as P (D){f · f } = (Dx Dt + Dx4 + 3Dy2 ){f · f } = 0.

(3.146)

Finally, putting f = 1 + f1 + 2 f2 + · · · into Equation (3.146), we obtain P (D){f · f } = P (D){1 · 1} + P (D){f1 · 1 + 1 · f1 } + 2 P (D){f2 · 1 + f1 · f1 + 1 · f2 } + 3 P (D){f3 · 1 + f2 · f1 + f1 · f2 + 1 · f3 } + · · · = 0.

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(3.147) One-soliton solution: To obtain one-soliton solution of the KP equation, we consider f = 1 + f1 where f1 = eθ1 and θ1 = k1 x + l1 y + ω1t + α1 . Also, fj = 0 for all j ≥ 2. We put f = 1 + f1 into Equation (3.147) and then equate the coefficients of 0 , , and 2 and make them zero. The coefficient of 0 is P (D){1 · 1} = 0 (since P (0, 0){1} = 0). By the coefficient of , P (D){1 · f1 + f1 · 1} = P (∂)eθ1 + P (−∂)eθ1 = 2P (p1 )eθ1 = 0. k4 +3l2

We have the dispersion relation P (p1 ) = 0 which implies ω1 = − 1 k1 1 . Finally, without loss of generality, we may set  = 1 so f = 1 + eθ1 and one-soliton solution of the KP equation (3.144) is   k12 2 θ1 u(x, y, t) = − sech 2 2 where θ1 = k1 x −



k14 +3l21 k1



(3.148)

t + l1 y + α1 .

Two-soliton solution: To obtain two-soliton solution of the KP equation, we consider f = 1+f1 +2 f2 where f1 = eθ1 +eθ2 and θi = ki x+ωi t+li y+αi (for i = 1, 2). Also, fj = 0 for all j ≥ 3. We put f = 1 + f1 + 2 f2 into Equation (3.147) and then equate the coefficient of 0 , , 2 , 3 , and 4 and make them zero. We shall only examine the nontrivial ones which are the coefficients of 1 and 2 . From the coefficient of 1 , we have P (D){1 · f1 + f1 · 1} = 2P (∂){eθ1 + eθ2 } = 0

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which implies P (pi ) = ki4 + ki ωi + 3li2 = 0, i.e., ωi = − The coefficient of 2 becomes

ki4 +3l2i ki

for i = 1, 2.

P (D){1 · f2 + f2 · 1} + P (D){f1 · f1 } = 2P (∂)f2 + P (D){(eθ1 · eθ2 ) + (eθ2 · eθ1 )} = 2[P (∂)f2 + P (p1 − p2 )eθ1 +θ2 ] = 0. This makes f2 to have the form f2 = A(1, 2)eθ1 +θ2 . If we put f2 in the above equation, we obtain A(1, 2) as

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A(1, 2) = − =

P (p1 − p2 ) P (p1 + p2 )

k1 ω2 + k2 ω1 + 4k13 k2 − 6k12 k22 + 4k1 k23 + 6l1 l2 . (3.149) k1 ω2 + k2 ω1 + 4k13 k2 + 6k12 k22 + 4k1 k23 + 6l1 l2

At last, we may set  = 1, thus f = 1 + eθ1 + eθ2 + A(1, 2)eθ1 +θ2 and the two-soliton solution of the KP equation is given by u = −2 × [{k12 eθ1 + k22 eθ2 + [(k1 − k2 )2 + A(1, 2)((k1 + k2 )2 + k22 eθ1 + k12 eθ2 )]eθ1 +θ2 }]/[(1 + eθ1 + eθ2 + A(1, 2)eθ1 +θ2 )2 ] (3.150)  4 2 k +3l where θi = ki x − i ki i t + li y + αi , i = 1, 2, and A(1, 2) is as given in (3.149). References [1] J. S. Russell, Report of the 14th Meeting of the British Association for the Advancement of Science. London: John Murray (1844), pp. 311–390. [2] J. Boussinesq, Memoires presentes par divers savants. l’Acad. des Sci. Inst. Nat. France XXIII, 1–680 (1877). [3] N. J. Zabusky and M. D. Kruskal, Phys. Rev. Lett. 15, 240–243 (1965). [4] R. M. Miura, C. S. Gardner, and M. D. Kruskal, J. Math. Phys. 9, 1204–1209 (1968). [5] M. Tabor, Ch. 7 in Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley (1989), pp. 278–321. [6] P. Lax, Comm. Pure Appl. Math. 21, 467–490 (1968). [7] D. J. Korteweg and G. de Vries, Philos. Mag. Ser. 5(39), 422–443 (1985). [8] H. Schamel, J. Plasma Phys. 9, 377 (1973). [9] H. Bateman, Mon. Weather Rev. 43(4), 163–170 (1915).

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[10] [11] [12] [13] [14]

Waves and Wave Interactions in Plasmas

J. M. Burgers, In Advances in Applied Mechanics, Vol. 1, pp. 171–199 (1948). B. B. Kadomtsev and V. I. Petviashvili, Sov. Phys. Dokl. 15, 539–541 (1970) R. Grimshaw and W. K. Melville, Stud. Appl. Math. 80, 3 (1989). W. Malfliet, Math. Meth. Appl. Sci. 28, 2031 (2005). R. Hirota, Phys. Rev. Lett. 27, 1192 (1971).

Chapter 4

RPT and Some Evolution Equations

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4.1 Perturbation Technique The perturbation technique is a special technique of applied mathematics to find the approximate solution of a complex problem by starting from the exact solution of a relatively simple problem. The methods rely on a small dimensionless parameter, called perturbation parameter, denoted by  that indicates the strength of the complexity of the problem. In this method, we break the main problem into two parts: the solvable part and the perturbation part. After finding the exact solution of the solvable part, the approximate solution of the main problem is obtained as a power series of the small perturbation parameter . The solution ultimately comes as a series of  that satisfy the problem up to a certain order of  [1]. In the perturbation method, we solve the simplified form of the original problem and then add corrections to improve the solution. Initially, we consider the solution as an infinite series in terms of perturbation parameter . The infinite perturbation series can be written as y(x) = y0 (x) + y1 (x) + 2 y2 (x) + · · · , where  is the perturbation parameter and y0 is the exact solution of the solvable part. The terms y1 (x), y2 (x), . . . are higher-order terms improving the solution. If a slight change in perturbation parameter induces a small change in solution, then the problem is called regular perturbation problem. On the contrary, if small changes in perturbation parameter induce a huge change in the solution, the problem is called singular perturbation problem. In general, the exact solutions to the problems of fluid mechanics, solid mechanics, classical mechanics, and physics cannot be obtained because of the nonlinearity, inhomogeneities, and general boundary conditions that arise in these problems. Only approximate solutions for such

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Waves and Wave Interactions in Plasmas

132

problems exist, and hence people use the perturbation technique. The beauty of the technique is that it is an analytical technique. This technique involves keeping certain elements, neglecting some, and approximating the rest. To do this, the order of magnitude of the different elements should be taken care of by comparing those with each other and with the basic elements of the system. This process is called non-dimensionalization, where we make the variable dimensionless. The variables are made dimensionless before attempting to make any approximations. Let us consider the motion of a particle of mass m is in simple harmonic motion having the constant k and a viscous damper having the coefficient ν. So, the equation of motion is

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m

du d2 u +ν + ku = 0 dt2 dt

(4.1)

where u is the displacement of the particle and t is the time. If the particle was released from rest from the position u0 , the initial conditions are u(0) = u0 ;

du (0) = 0. dt

(4.2)

Here, u is the dependent variable and t is the independent variable. Now, the variables need to be dimensionless by using proper characteristic distance and a proper characteristic time of the system. The displacement u can be made dimensionless by using the initial displacement u0 and the time t can be made dimensionless by using the inverse of the system’s nat ural frequency ω0 = k/m. Thus, by putting u∗ = uu0 and t∗ = ω0 t where an asterisk denotes dimensionless quantities, we have (from 4.1) mω02 u0

d2 u∗ du∗ + νω u + ku0 u∗ = 0. 0 0 dt∗2 dt∗

(4.3)

Dividing throughout by the factor mω02 u0 , we get d2 u∗ du∗ + ν ∗ ∗ + u∗ = 0 ∗2 dt dt where ν ∗ =

ν mω0 .

(4.4)

The initial condition becomes u∗ (0) = 1;

du∗ (0) = 0. dt∗

(4.5)

Thus, the solution to the present problem depends only on the single parameter ν ∗ , which represents the ratio of the damping force to the inertial force.

RPT and Some Evolution Equations

133

On the other hand, if we add a nonlinear function of u as a additional force term, the equation of motion becomes m

d2 u du +ν + ku + k1 u2 = 0 dt2 dt

(4.6)

where k and k1 are constants. Introducing the same dimensionless quantities as in the preceding example, we get the equation of motion as

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∗ d2 u∗ ∗ du + ν + u∗ + u∗2 = 0 dt∗2 dt∗

(4.7)

ν where ν ∗ = mω and  = k1ku0 are two dimensionless parameters. ν ∗ rep0 resents the ratio of the damping force to the inertial force or the linear restoring force and  represents the ratio of the nonlinear and linear restoring force. Equation (4.7) is weakly nonlinear if u∗ is small, i.e., k1 u0 /k is small. Even if k1 is small compared with k, the nonlinearity will not be small if u0 is large compared with k/k1 . Thus,  is the parameter that characterizes the nonlinearity. This is how the perturbation parameter  takes a lead role in this technique. Now, we shall explain the perturbation technique. Let us consider a function u(x, ) that satisfies the differential equation L(u, x, ) = 0 and the boundary condition B(u, ) = 0, where x is the independent variable and  is a parameter. In general, these problems cannot be solved exactly. However, if we consider an (= 0 ) (say) for which the above problem can be solved exactly, and let u = u0 (x) be the solution of the problem for  = 0 , then, in general, we find the solution (for small ) in the series power of  as

u(x, ) = u0 (x) + u1 (x) + 2 u2 (x) + · · · where ui are independent of  and u0 (x) is the solution of the problem for  = 0 . We first substitute this expansion into L(u, x, ) = 0 and B(u, ) = 0, and after that we collect the coefficients of each power of . This will help us in obtaining the result. Hence, in general, we can explain perturbation as follows: the full solution A can be expressed as a series in the small parameter (say ), like the following A = A0 + A1 + 2 A2 + · · ·

(4.8)

where A0 is the exact solution of the solvable part and A1 , A2 , . . . are the higher-order terms obtained by solving the perturbation problems in an iterative way. Naturally, as  is considered small, so the higher-order terms of this series become successively smaller. Ultimately, the solution is obtained

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Waves and Wave Interactions in Plasmas

as a truncated series of the above series by keeping up certain order of A. The first approximate solution is A ≈ A0 + A1 .

(4.9)

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So, the perturbation theory is considered in terms of the order to which the perturbation is carried out: first-order perturbation theory or second-order perturbation theory. If the perturbed states are degenerated, the theory is called singular perturbation theory. In the singular case, we must take some extra care and the theory is slightly more complicated (which is not our concern). As an example, let us find the solution of the following differential equation by using perturbation technique: 1 d2 x =− , dt2 (1 + x)2

f or 0 < t,

(4.10)

with initial condition x(0) = 0,

dx(0) = 1. dt

Considering the assumption for x as x ≈ x0 (t) + α x1 (t) + · · ·

(4.11)

and substituting (4.11) into (4.10), we get 1 = 1 − 2x + 32 x2 + · · · (1 + x)2 ≈ 1 − 2(x0 + α x1 + · · · + 32 (x0 + · · · )2 + · · · = 1 − 2x0 + · · · .

(4.12)

With this, the differential equation (4.10) becomes x0 + α x1 + · · · = −1 + 2x0 + · · · and the initial conditions will be x0 (0) + α x1 (0) + · · · = 0 x0 (0) + α x1 (0) + · · · = 0.

(4.13)

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135

Now, we break the above equations into problems depending on the power of  and get O(1)  x0 = −1 (4.14) x0 (0) = 0, x0 (0) = 1.

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The solution of this problem is x0 = t(1 − 12 t). With this the next highest term is left in (4.13) is 2x0 . The term available to balance with this is α x1 , and from this we conclude α = 1. This gives us the following problem O():  x1 = 2x0 (4.15) x1 (0) = 0, x1 (0) = 0. 1 3 The solution of this problem is x1 = 12 t (4 − t). Therefore, a two-term expansion of the solution is   1 1 x ≈ t 1 − t + t3 (4 − t) + · · · . 2 12

(4.16)

4.2 Reductive Perturbation Technique The reductive perturbation technique (RPT) is a particular type of perturbation method that is different from the usual perturbation method discussed in the previous section. This method is considered as a method that reduces a set of nonlinear partial differential equations (PDEs) to a single solvable nonlinear evolution equation (NEE). Starting from fluid equations, the evolution equation is obtained in two steps or more. In the first step, the original coordinates x and t are transformed to new stretched coordinates ξ and τ (say) depending on the nature of the problem. Then, the perturbation parameter  is introduced which has a connection with the wavenumber (and hence, with the frequency and the dispersion relation). In the second step, the same  is used as a perturbation parameter where the dependent variables are expanded as a power series of  in the neighbourhood of the equilibrium point. The perturbation of the parameters is considered at a near-equilibrium point not for finding the solution but for obtaining an evolution equation. It can be applied to more general systems that include dissipation/dispersion or both. It shows that for long waves, the set of equations can be reduced to Burgers’ equation or the KdV equation, for a system with dissipation or dispersion, respectively. If we consider the propagation of a modulated wave of small amplitude, then the evolution equation would be a nonlinear Schrodinger equation (NLSE).

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If the system is weakly dispersive, i.e., the wavenumber k is much less than 1/2 (k  1/2 ), the hyperbolic approximation is justified, and the expansion and the stretching of RPT may be considered as U = U (0) + U (1) + · · · ξ=

1/2

(x − λ0 t),

(4.17)

τ =

3/2

t.

(4.18)

This will enable us to reduce a general dispersive system to the KdV equation, where U, U (0) , and U (1) are the average velocity of the fluid, solution of the equilibrium point, and higher-order perturbation term, respectively. For example, we may consider

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U,t + AU,x + K1 [K2 (K3 U,x ),x ],x = 0.

(4.19)

Similarly, in case of dissipative system, we get ∂φ(1) ∂φ(1) ∂ 3 φ(1) + αφ(1) +ν =0 ∂τ ∂ξ ∂ξ 3

(4.20)

where φ(1) may be considered as the first-order perturbation value of the potential. In 1968, Taniuti and Wei [2] first applied RPT to study the propagation of the long wave in both dispersive and dissipative systems in the framework of cold plasma. Su and Gardner [3] also obtained similar results independently. They considered the system of equation as ∂U ∂U +A + ∂t ∂X

  p  s  ∂ ∂ Hαβ + Kαβ · U = 0, ∂t ∂x α=1

(p ≥ 2)

(4.21)

β =1

where A, Hαβ , and Kαβ are n × n matrices and all of which are functions of U . So, in this method, a general transformation is introduced where the expansion of a stable solution U0 in a small parameter is considered. It is assumed that at least one real and nondegenerate eigenvalue of A0 (is denoted by λ0 ) exists. This transformation is called Gardner–Morikawa transformation [4], ⎫ U = U0 + U1 + · · · ⎪ ξ = α (x − λ0 t) ⎪ ⎬ τ = α+1 t 1 α= . (p − 1)

⎪ ⎪ ⎭

(4.22)

RPT and Some Evolution Equations

137

Using (4.22), one can easily obtain any nonlinear evolution equation like the KdV equation. One such equation is ∂φ(1) ∂φ(1) ∂ p φ(1) + αφ(1) +μ = 0. ∂τ ∂ξ ∂ξ p

(4.23)

It is a KdV equation with a higher-order dispersion.

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4.3 Korteweg–de Vries (KdV) Equation The theory of soliton is a very interesting discovery in physics. In 1895, Diederik Jonannes Korteweg and his Ph.D. scholar Gustav de Vries [5] derived a prototype equation of soliton theory known as Korteweg–De Varies (KdV) equation. This equation is derived from the basic equation of hydrodynamics. KdV equation is a nonlinear partial differential equation of two variables: the space and the time. Let us derive the KdV equation from a classical plasma model. We consider a collisionless unmagnetized electron–ion (e-i) plasma where ions are mobile and electrons obey Maxwell distribution. Step 1: Basic fluid equations The basic equations are given as follows: Equation of continuity: ∂Ni Ui ∂Ni + =0 ∂T ∂X

(4.24)

∂Ui e ∂ψ ∂Ui + Ui =− ∂T ∂X mi ∂X

(4.25)

Equation of motion:

Poisson’s equation: 0

∂2ψ = e(Ne − Ni ) ∂X 2

(4.26) eφ

where the electrons obey Maxwell distribution, i.e., Ne = en0 e KB Te . Ni , Ne , Ui , and mi are the ion density, electron density, ion velocity, and ion mass, respectively. ψ is the electrostatic potential, KB is the Boltzmann constant, Te is the electron temperature, and e is the charge of the electrons.

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Step 2: Normalization To write Equations (4.24)–(4.26) in dimensionless form, we use the following dimensionless variables: X eψ Ni Ui , t = ωp T, φ = , ni = , ui = (4.27) λD KTe n0 Cs   2 is the Debye length, C = where λD = 0 KB Te /n0 e KB Te /mi is the s ion acoustic speed, ωpi = n0 e2 /0 mi is the ion plasma frequency, and n0 is the unperturbed density of ions and electrons. Hence, using (4.27) in (4.24)–(4.26), we obtain the set of normalized equations as

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x=

∂ni ∂(ni ui ) + =0 ∂t ∂x ∂ui ∂φ ∂ui + ui =− ∂t ∂x ∂x ∂2φ = e φ − ni . ∂x2

(4.28) (4.29) (4.30)

Step 3: Linearization To linearize (4.28)–(4.30), we write the dependent variable as the sum of equilibrium and perturbed parts as ni = 1 + n¯i , ui = u¯i , and φ = φ¯. Putting ni = 1 + n¯i , where the values of parameters at equilibrium position are given by n1 = 1, u1 = 0, and φi = 0 in Equation (4.28), we get ∂ ∂ (1 + n¯i ) + (u¯i + n¯i u¯i ) = 0. ∂t ∂x Neglecting the nonlinear term

∂(n¯i u¯i ) ∂x

(4.31)

from (4.31), we get

∂n¯i ∂u¯i + =0 ∂t¯ ∂x ¯

(4.32)

which is the linearized form of Equation (4.28). Putting ui = u¯i and φ = φ¯ in Equation (4.29), we get ∂u¯i ∂u¯i ∂φ¯ + u¯i =− . ∂t ∂x ∂x

(4.33)

Neglecting the nonlinear term of (4.33), we get ∂u¯i ∂φ¯ + = 0. ∂t ∂x ¯ This is the linearized form of Equation (4.29).

(4.34)

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Putting ni = 1 + n¯i and φ = φ¯ in Equation (4.30) and neglecting O(φ¯2 ) term, we get ∂ 2 φ¯ = φ¯ − n¯i . ∂x

(4.35)

Hence, Equations (4.32), (4.34), and (4.35) are the linearized form of Equations (4.28)–(4.30), respectively. Step 4: Dispersion relation

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To get dispersion relation for low-frequency wave, let us assume that the perturbation quantities are proportional to ei(kx−ωt) and in the form of n ¯ = n0 ei(kx−ωt) ,

u ¯ = u0 ei(kx−ωt) ,

φ¯ = φ0 ei(kx−ωt) .

So, ∂n ¯ = −in0 ωei(kx−ωt) , ∂t ∂u ¯ = iku0 ei(kx−ωt) , ∂x

∂n ¯ ∂u ¯ = ikn0 ei(kx−ωt) , = −iu0 ωei(kx−ωt) ∂x ∂t ∂φ¯ ∂ 2 φ¯ = ikφ0 ei(kx−ωt) , = (ik)2 φ0 ei(kx−ωt) . ∂x ∂x2

Putting these value in Equations (4.32), (4.34), and (4.35), we get ⎫ −iωn0 + iku0 = 0 ⎬ ⎪ −iωu0 + ikφ0 = 0 ⎪ ⎭ n0 − (k 2 + 1)φ0 = 0.

(4.36)

Since system (4.36) is a system of linear homogeneous equation, so for existence of nontrivial solutions, we have −iω ik 0 −iω 1

0 ik

=0

2

0 −(k + 1)

⇒ −i2 ω 2 (k 2 + 1) + i2 k 2 = 0 k . ⇒ω= √ 2 k +1 This is the dispersion relation.

(4.37)

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For small k (i.e., for weak dispersion), we can expand (4.37) as 1 1 ω = k(1 + k 2 )− 2 = k − k 3 + · 2 The phase velocity Vp =

ω 1 = . k (1 + k 2 )

(4.38)

So, Vp → 1 as k → 0 and Vp → 0 as k → ∞. The group velocity Vg = dω dk is given by

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Vg =

1 . (1 + k 2 )3/2

(4.39)

In this case, we have Vg < Vp for all k > 0. If we consider the long-wave limit, i.e., the wavenumber k → 0, then Vp = ωk or ω = k. It corresponds to a nondispersive acoustic wave with phase speed 1, i.e., in the original dimensional variable, the wave speed is equal to the ion acoustic speed Cs . Again, if the wavelength is much larger than the Debye length, i.e., kλD  1, then the long wave dispersion will be very weak. In that case, the electrons and ions oscillate, but as a whole, they behave as a fluid. The inertia is given by the ions and, restoring force is given by the electrons. At the next order in k, it is known as 1 ω = k − k 3 + O(k 5 ) 2

as k → 0.

The O(k 5 ) correction corresponds to weak KdV-type long wave dispersion. In the short wave limit, the waves have constant frequency ω = 1, corresponding in dimensional terms to the ion plasma frequency ωpi = λCDs . For such short waves, the ions oscillate in a fixed background of electrons. Now, the phase of the waves can be written as 1 kx − ωt = k(x − t) + k 3 t. 2 Here, k(x − t) and k 3 t have same dynamic status (dimension) in the phase. Assuming k to be small order of 1/2 ,  being a small parameter measuring the weakness of the dispersion, (x − t) is thus the traveling waveform and time t is the linear form.

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Step 5: Perturbation Accordingly, we consider new stretched coordinates ξ and τ such that

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ξ = 1/2 (x − λt),

τ = 3/2 t

(4.40)

where  is the strength of nonlinearity and λ is the Mach number (phase velocity of the wave).  may be termed as the size of the perturbation. Let the variables be perturbed from the stable state in the following way (considering ni = 1, ui = 0, φ = 0, and ne = eφ = e0 = 1 at equilibrium): ⎫ (1) (2) (3) ni = 1 + ni + 2 ni + 3 ni + · · · , ⎪ ⎬ (1) (2) (3) (4.41) ui = 0 + ui + 2 ui + 3 ui + · · · , ⎪ φ = 0 + φ(1) + 2 φ(2) + 3 φ(3) + · · · ⎭ where the numbers on the hat indicate the order of the perturbed quantities. (1) (1) (1) ni , ui , and φi are the first-order perturbed quantities. Step 6: Partial derivative in terms of stretched coordinates (4.40) shows that x and t are function of ξ and τ . So, partial derivatives with respect to x and t can be transformed into partial derivative in terms of ξ and τ as ∂ ∂ ∂ξ ∂ ∂τ = + , ∂x ∂ξ ∂x ∂τ ∂x

⇒

1 ∂ ∂ = 2 ∂x ∂ξ

1 ∂ 3 ∂ ∂ ∂ ∂ξ ∂ ∂τ ∂ = + , ⇒ = − 2 + 2 ∂t ∂ξ ∂t ∂τ ∂t ∂t ∂ξ ∂τ   ∂2 ∂ ∂2 ∂2 1 ∂ = =  2. 2 , ⇒ 2 2 ∂x ∂x ∂ξ ∂x ∂ξ

We can express (4.28)–(4.30) in terms of ξ and τ as ⎫ ∂ni ⎪ 1/2 ∂(ni ui ) − λ + =0  ⎪ ∂τ ∂ξ ∂ξ ⎬ ∂ui ∂ui ∂ui ∂φ 3/2 − 1/2 λ + 1/2 ui + 1/2 =0 ∂τ ∂ξ ∂x ∂x ⎪ ⎪ 2 ∂ φ  2 − eφ + ni = 0. ⎭ ∂ξ 3/2 ∂ni

1/2

(4.42)

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Step 7: Phase velocity of the wave Substituting Equations (4.40)–(4.41) in Equation (4.42) and collecting the lowest-order O(3/2 ) terms, we get (1)

−λ −λ

(1)

∂ni ∂ui + ∂ξ ∂ξ

= 0,

(1)

⎫ ⎪ ⎪ ⎬

∂φ(1) ∂ui − = 0, ⎪ ∂ξ ∂ξ ⎪ ⎭ (1) φ(1) − ni = 0.

(4.43)

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Integrating Equations (4.43), and all the variables tend to be zero as ξ → ∞. We get (1) ni (1)

ui

φ(1)

⎫ (1) ui ⎪ = , λ ⎪ ⎬ φ(1) ,⎪ = λ ⎪ (1) ⎭ = ni .

(4.44)

From Equation (4.44), we get the phase velocity as λ2 = ±1.

(4.45)

Step 8: Nonlinear evolution equation Substituting Equations (4.40)–(4.41) in Equation (4.42) and collecting order O(5/2 ), we get (1)

∂ni ∂τ

−λ

(2)

(1) (1)

∂ni ∂ni ui + ∂ξ ∂ξ (1)

(2)

+

(2)

∂ui ∂ξ

(1)

∂ui ∂u (1) ∂ui − λ i + ui ∂τ ∂ξ ∂ξ

= 0, =−

∂φ(1) 1 (1) − φ(2) − (φ(1) )2 + ni = 0. 2 ∂ξ 2

(4.46) ∂φ(2) , ∂ξ 2

(4.47) (4.48)

Differentiating Equation (4.48) with respect to ξ and substituting (2)

(2)

for

∂ni ∂ξ

from Equation (4.46) and for

∂ui ∂ξ

from Equation (4.47),

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143

we finally obtain ∂φ(1) ∂φ(1) 1 ∂ 3 φ(1) + φ(1) + = 0. ∂τ ∂ξ 2 ∂ξ 3

(4.49) (1)

Equation (4.49) is known as the KdV equation. φ(1) ∂φ∂ξ

is the nonlinear

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3 (1) 1∂ φ 2 ∂ξ 3

and is the dispersive term. To obtain the steady-state solution of this KdV Equation (4.49), we introduce a transformation η = ξ − u0 τ , where u0 is a constant velocity normalized by Cs and finally, we get the steady-state solution as   (1) 2 η φ = φm sech (4.50) Δ  where φm = 3u0 and Δ = u20 are the amplitude and width of the solitary waves. It is clear that the height, width, and speed of the pulse are proportional to u0 , √1u0 , and u0 , respectively. u0 specifies the energy of the solitary waves. So, the larger the energy, the greater the speed, the larger the amplitude, and the narrower the width. Also, a finite number of solitons emerge with heights φm1 , φm2 , . . . and hence, each of the speed u1 , u2 , . . . travels to the right. These solitons interact with nonlinearly and preserve their soliton identity. As t → ∞, the solitons are arranged in the order of increasing height with the tallest soliton in the extreme right. The solution of the KdV Equation (4.50) represents a single pulse as shown in Figure 4.1, having a peak at η = 0 and vanishing at η → ±∞. It propagates along the positive direction of the ξ-axis with a constant velocity u0 without any change of shape. Figure 4.1 also describes that as the wave speeds get larger and larger, the wavelength becomes smaller and smaller.

Fig. 4.1: (a) Plot of φ(1) vs η for u0 = 1 and (b) plot of φ(1) vs. η for u0 = 10.

144

Waves and Wave Interactions in Plasmas

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4.4 Modified KdV (MKdV) Equation For nonlinear waves in weakly dispersive media, there are two essential processes: the increase of steepness of the wave profile owing to nonlinear effects (as known from gas dynamics) and the dispersion of the profile. If the amplitude is not too large, the dispersion can compete with the nonlinearity. With the equilibrium of these two processes, the existing waves propagate with constant velocity without deformation of the profile. It would seem that with increasing amplitude, the nonlinear effects should prevail and lead to the formation of shock waves. If we take the nonlinear and dispersion terms in the equations only till the first nonvanishing approximation, we shall get the KdV equation. For smooth initial conditions, the Cauchy problem reduces to the soliton mode. As for the averaged solutions, they indeed behave like shock waves. There are many physical situations (if the coefficient of quadratic nonlinearity becomes zero) where the nonlinearity should be taken into account for up to cubic order. We then get the modified Korteweg–de Vries (MKdV) equation. The MKdV equation has a significant role of soliton theory. It was used to construct an infinite number of conservation laws of the KdV equation that triggered the discovery of the Lax pair of the KdV equation and the breakthrough of the inverse scattering transformation. The MKdV equation is known to describe acoustic type waves in plasma. To derive an MKdV equation in unmagnetized collisionless dusty plasma, we also consider a model consist of cold inertial ions, negatively stationary dusts, and q-nonextensive electrons. Accordingly, the onedimensional normalized fluid equations are ∂ni ∂(ni ui ) + = 0, ∂t ∂x ∂ui ∂ui ∂φ + ui =− , ∂t ∂x ∂x ∂2φ = (1 − μ)ne − ni + μ ∂x2

(4.51) (4.52) (4.53)

  q+1 where ne = 1 + (q − 1)φ 2(q−1) . nj (j = i, and e, for ion, and electron, respectively), ui , and φ are the number density, ion fluid velocity, and the electrostatic wave potential, respectively. Here, μ = Zdnn0d0 . n0 , Zd , and nd0 are the unperturbed density of ions, the charge state of dust, and the unperturbed density of dust, respectively. μ is the density of the stationary

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dust. The normalization is represented by ni →

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where Cs =

ni , n0 

ui → KB Te mi

ui , Cs

φ→

eφ , KB Te

x→

x , λD

t → ωpi t

   12  1 4πne0 e2 2 e , λD = 4πnTe0 , and ω = . Cs pi 2 e mi

is the ion acoustic speed, λD is the Debye length, ωpi is the ion plasma frequency, Te is the electron temperature, KB is the Boltzmann constant, e is the magnitude of electron charge, and mi is the mass of the ion. To obtain the KdV equation, we have used the same stretched coordinates as in Equation (4.40). The expansions of the dependent variables are also considered the same as in (4.41). Substituting the above expansions (4.41) along with stretching coordinates (4.40) into Equations (4.51)–(4.53) and equating the coefficients of the lowest order of , we obtain the phase velocity as  λ=

2 . (q + 1)(1 − μ)

(4.54)

Taking the coefficients of next higher order of , and after some calculation, we obtain the KdV equation as ∂φ(1) ∂φ(1) ∂ 3 φ(1) + Aφ(1) +B =0 ∂τ ∂ξ ∂ξ 3

(4.55)

3  (q+1)(3−q) q+1 λ3 where A = 2λ − bλ . a , B = 2 , a = 2 , and b = 8 The parameters q and μ may be chosen in such a way that A = 0. Then, for those values of q and μ, the nonlinear term vanishes, and Equation (4.55) is no longer a nonlinear equation. So, there are critical points where the quadratic nonlinearity vanishes. At the critical point, the stretching (4.40) is not valid, and a new stretching is necessary. Accordingly, we use the new stretched co-ordinate as ξ = (x − λt),

τ = 3 t.

(4.56)

Now, substituting Equation (4.56) with Equations (4.41) into Equations (4.51)–(4.53) and equating the coefficients of the lowest order of  (i.e., coefficients of 2 from Equations (4.51) and (4.52) and coefficients of  from

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Equation (4.53)), we obtain the following relations: (1)

ni

(1)

=

ui , λ

(1)

ui

=

φ(1) , λ

(1)

ni

= a(1 − μ)φ(1) .

(4.57)

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Equating the coefficients of next higher order of  (i.e., coefficients of 3 from Equations (4.51) and (4.52) and coefficients of 2 from Equation (4.53)), we obtain the following relations: ⎫ 1 (2) (2) (1) (1) ⎪ ni = (ui + ni ui ) λ ⎪ ⎬   2 1 1 (4.58) (2) (1) (2) +φ ui = u ⎪ λ 2 i   2  ⎪ (2) ⎭ . ni = (1 − μ) aφ(2) + b φ(1) Equating the coefficients of next higher order of  (i.e., coefficients of 4 from Equations (4.51) and (4.52) and coefficients of 3 from Equation (4.53)), we obtain the following relations: (1)

∂ni ∂τ

−λ

(3)

(3)

(1) (2)

(2) (1)

∂ni ∂ui ∂(ni ui ) ∂(ni ui ) + + + =0 ∂ξ ∂ξ ∂ξ ∂ξ (1)

(3)

(4.59)

(1) (2)

∂φ(3) ∂ui ∂u ∂(ui ui ) + =0 −λ i + ∂τ ∂ξ ∂ξ ∂ξ ∂ 2 φ(1) (3) = (1 − μ)(aφ(3) + 2bφ(1) φ(2) + c(φ(1) )3 ) − ni ∂ξ 2

(4.60) (4.61)

. where c = (1+q)(3−q)(5−3q) 48 From Equations (4.57), it is seen that the same phase velocity is obtained as (4.54). ∂ and substituting the terms Differentiating Equation (4.61) once with ∂ξ (1) other than φ from Equations (4.57)–(4.60), we can obtain MKdV equation as ∂φ(1) ∂φ(1) ∂ 3 φ(1) + A1 (φ(1) )2 + B1 =0 ∂τ ∂ξ ∂ξ 3 3

3

(4.62)

3λ c(1−μ) 15 where A1 = 4λ and B1 = λ2 . This is similar to the KdV 3 − 2 equation with a higher-order nonlinearity. The solitary wave solution of Equation (4.62) is   ξ − u0 τ (1) φ = φm sech (4.63) Δ

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      0 1 where φm = 6u and Δ = B are the amplitude and the width of A1 u0 the solitary waves, respectively, and u0 is the speed of the ion-acoustic solitary waves. Figure 4.2 shows the plot of φm against q for different values of μ. It is seen from the figure that the amplitude of the solitary waves decreases with the increase of q. So, an increase in nonextensive electron distribution decreases the amplitude of the solitary wave. It is observed that as μ increases, the amplitude of the soliton increases. It is obvious from Figure 4.3 that as we increase μ, the width of the soliton increases.

Fig. 4.2: Plot of φm vs. q for μ= 0.6, 0.3, and 0.2.

Fig. 4.3: Plot of Δ vs. q for μ= 0.4, 0.3, and 0.2.

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4.5 Gardner’s Equation

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To obtain Gardner’s equation, we consider a four components collisionfree unmagnetized dusty plasma model consisting of electrons, ions, and both positive and negative dust grains. Electrons obey the superthermal distribution. Here, the charge neutrality condition is ni0 + Zdp ndp = ne0 + Zdn ndn , where ni0 , ne0 , ndp , and ndn are ion, electron, positively dust, and negatively dust number density at equilibrium, respectively. Zdp and Zdn represents the charge state of positive and negative dust. The basic normalization equations are as follows: ∂ni ∂ + (ni ui ) = 0 ∂t ∂x ∂φ ∂ui ∂ui + ui =− ∂t ∂x ∂x ∂2φ =ρ ∂x2 ρ = (1 − jμ)ne − ni + jμ

(4.64) (4.65) (4.66) (4.67)

where μ = |Zdp ndp − Zdn ndn |/nio . Here, j = 1, − 1 for positive, negative net dust charge. The electron distribution is represented by  −k− 12 φ ne = 1 − . (κ − 1/2) We have used the stretched coordinates (4.56) in Equations (4.64)–(4.66). The expansion of the dependent variables is also considered the same as (4.41), with an additional expression as ρ = 0+

∞ 

p ρ(p) .

(4.68)

p=1

Here, ρ is the net surface charge density. Now, expressing Equations (4.64)– (4.67) in terms of ξ and τ , substituting Equations (4.41) and (4.68) into the resulting equations, and collecting the lowest-order terms, we get (1)

ni

=

φ(1) , λ2

(1)

ui

=

φ(1) , λ

(1 − jμ)(2κ + 1) (1) (1) φ = ni . (2κ − 1)

(4.69)

Hence, from the condition for nontrivial first-order perturbed quantities, we get the phase velocity as  (2κ − 1) λ= . (4.70) (1 − jμ)(2κ + 1)

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For the next order in , we get another set of equation that, after using (4.69)–(4.70), can be written as ⎫ ⎪ φ(2) (φ(1) )2 φ(2) 3(φ(1) )2 (2) (2) + 2 , ui = + , ni = ⎬ 2λ4 λ 2λ3 λ (4.71) 1 3 (1 − jμ)(2κ + 1)(2κ + 3) ⎪ (1) 2 (2) ρ = − A(φ ) = 0, A = 4 − .⎭ λ 2 (2κ − 1)2 3jμ It is clear that A = 0 (since φ = 0) and A = 0 when κ = κc = 2(2−3jμ) . It is obvious that (4.71) is satisfied for κ = κc and κc < 0 when j = −1. In that case, solitary waves or double layers are not possible. So, for |κ − κc | =  corresponding to A = A0 , one can write A0 as   ∂A A0 ≈ s |κ − κc | = sAk  ∂κ κ=κc

where Ak =

(1−jμ) (2κc −1)3 [24κc

+ 20 − 24(1 − jμ)(2κc + 1)]. Here, s = 1 for

κ > κc and s = −1 for κ < κc . So, for κ = κc , one can express ρ(2) as 2 ρ(2) = − sAk2 φ , i.e, when κ = κc ·ρ(2) must be considered in the third-order Poisson’s equation. As the next higher order in , we get (1)

∂ni ∂τ

−λ

(3)

(3)

∂ (1) (2) ∂ni ∂ui (2) (1) + + (n u + ni ui ) = 0 , ∂ξ ∂ξ ∂ξ i i (1)

(4.72)

(3)

∂ (1) (2) ∂ui ∂u −λ i + (u u ) = 0, (4.73) ∂τ ∂ξ ∂ξ i i  ∂2φ 1 (2κ + 1) (3) (2κ + 1)(2κ + 3) (2) 2 + sAk φ − (1 − jμ) φ + φφ ∂ξ 2 2 (2κ − 1) (2κ − 1)2  (2κ + 1)(2κ + 3)(2κ + 5) 3 (3) + φ + ni = 0. (4.74) 6(2κ − 1)3 Now, using (4.69)–(4.71) and (4.72)–(4.74), one finally obtains a nonlinear dynamical equation as follows: ∂φ ∂φ ∂φ ∂3φ + pφ + qφ2 + q0 3 = 0 ∂τ ∂ξ ∂ξ ∂ξ

(4.75)

where p = sAk q0 , p0 =

q = p0 q0 ,

15 (1 − jμ)(2κ + 1)(2κ + 3)(2κ + 5) − , 2λ6 2(2κ − 1)3

q0 =

λ3 . 2

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Equation (4.75) is called further modified KdV (FMKdV) equation or Gandner’s equation. To obtain the stationary solution of Gardner’s equation, we have introduced a transformation ζ = ξ − u0 τ , where u0 is the constant velocity. d2 φ Using the conditions φ → 0, dφ dζ → 0, dζ 2 → 0 at ζ → ±∞, we finally get the stationary solitary wave solution of Gardner’s equation as    −1  1 1 1 ζ 2 (1) cosh (4.76) − − φ = φm2 φm2 φm1 Δ where

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u0 =

p 1 φm1,2 + φ2m1,2 , 3 6

p φm = − , q

V0 =

   u0 φm1,2 = φm 1 ∓ 1 + , V0

p2 6q

and the width Δ as 2 Δ= √ , −γφm1 φm2

γ=

q . 6q0

4.6 Gardner and Modified Gardner’s (MG) Equation For simple wave problems, like the nonlinear description of an ion-acoustic soliton in an electron–proton plasma, most of the compositional parameters are fixed or eliminated by proper normalization. It results in the ubiquitous KdV equation with quadratic nonlinearity. After the plasma model becomes more complicated, some critical choices for the compositional parameters which annul the coefficients of the nonlinear term in the KdV equation leading to an undesirable linear equation (the combination of stretching and expansion used) must then be adapted to account for nonlinear effects of higher degree. It is usually done for the stretching and results in the MKdV equation with cubic nonlinearity. An interesting question arises here that whether one can take this procedure to a higher level. What will happen if for complicated plasma models the coefficients of both the quadratic and the cubic nonlinearities can vanish simultaneously for a specific set of compositional parameters? For many models, such supercritical compositions will be impossible. However, there are situations where supercriticality is possible for the models with many restrictions. In mathematically inclined studies, it can be used as one of the higher-degree extensions of the KdV family of equations.

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Let us consider a plasma model composed of cold ions and two temperature (cold and hot) q-nonextensive electrons. Propagation of ion acoustic waves is characterized by the normalized model equations as ∂ni ∂(ni ui ) + = 0, ∂t ∂x ∂ui ∂ui ∂φ + ui =− , ∂t ∂x ∂x

(4.77) (4.78)

∂2φ = μnc + (1 − μ)nh − ni . ∂x2

(4.79)

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(1+q)

(1+q)

T

T

q)σc φ) 2(1−q) and nh = (1−(1−q)σh φ) 2(1−q) , where σc = Tefcf and σh = Tefhf and q is the degree of efficiency of nonextensivity. Here, μ = nni00 . Again, nj (j = i, h, and e for ion, hot electron, and cold electron), ui , and φ are the number density, ion velocity, and electrostatic potential, respectively. At equilibrium, we get nc0 + nh0 = ni0 . We introduce the following dimensionless variables: x=

x , λD

t = ωpi t,

φ=

eψ , KB Tef f

ni =

ni , n0

ui =

ui Cs

(4.80)

  2 ) is the Debye length, C = where λD = Th /(4πni0 e Tef f /m is the s ion acoustic speed, ωpi = mi /4πni0 e2 is the ion plasma frequency, and n0 is the unperturbed density of ions and electrons. Tc and Th denote temTc Th peratures of cold and hot electrons, respectively. Also, Tef f = μTh +(1−μ)T c stands for effective temperature and m is the ion mass. We have derived the KdV equation using RPT with the same stretched coordinates as (4.40). We also expand the dependent variable, the same as (4.41). Using (4.41) in Equations (4.77)–(4.79), we get the relation λ12 = (1+q) 2 [μσc + (1 − μ)σh ]. After further calculations like the previous section, we get the KdV equation as follows: ∂φ(1) ∂φ(1) ∂ 3 φ(1) + Aφ(1) +B =0 ∂τ ∂ξ ∂ξ 3 3

(4.81)

3 where A = 2λ − A1 λ3 , B = λ2 , and A1 = 18 (1 + q)(3 − q)[μσc2 + (1 − μ)σh2 ]. At certain sets of critical values, the nonlinear coefficient is A = 0. This indicates the occurrence of singularity. In that case, using the new stretched

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152

coordinates ξ = (x − λt) and τ = 3 t, we obtain the MKdV equation as ∂φ(1) ∂φ(1) ∂ 3 φ(1) + C(φ(1) )2 +B =0 ∂τ ∂ξ ∂ξ 3

(4.82)

3

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3C1 λ 15 1 3 3 where C = 4λ 3 − 2 , and C1 = 48 (1 + q)(3 − q)(5 − q)[μσc + (1 − μ)σh ]. Also, there exists a set of critical values at which the nonlinear coefficient is C = 0. Furthermore, to deal with such a situation with the help of stretched coordinates ξ = 3/2 (x − λt) and τ = 9/2 t, we find the FMKdV equation as follows:

∂φ(1) ∂φ(1) ∂ 3 φ(1) + D(φ(1) )3 +B =0 (4.83) ∂τ ∂ξ ∂ξ 3  3  35 1 4 where D = λ2 2λ 8 − 4D1 , and D1 = 384 (1+q)(3−q)(5−3q)(7−5q)[μσc + 4 (1 − μ)σh ]. To examine the small amplitude supernonlinear IASW feature, we further derive the modified Gardner’s (MG) equation. To evolute the MG equation, we introduce new stretch coordinate ξ = 3/2 (x − λt) and τ = 9/2 t. We then apply the stretched coordinate and Equation (4.41) in Equations (4.77)–(4.79). Similarly, we get the other nonlinear equations. We also obtain the following relations from coefficients of lowest order of : n3 =

3φ(1) φ(2) φ(3) 5(φ(1) )3 + + , 2λ6 λ4 λ2

u3 =

(φ(1) )3 φ(1) φ(2) φ(3) + + . 2λ5 λ3 λ (4.84)

Comparing all coefficients with the next higher order of , we get ∂n(4) ∂φ(1) 2 ∂φ(1) 3 ∂ 35 = 3 + 4 (φ(1) φ(3) ) + 8 (φ(1) )3 ∂ξ λ ∂τ λ ∂ξ 2λ ∂ξ + and

15 ∂ 3 ∂(φ(2) )2 1 ∂φ(4) [(φ(1) )2 φ(3) ] + 4 + 2 6 2λ ∂ξ 2λ ∂ξ 2λ ∂ξ

(4.85)

  ∂ 2 φ(1) 1+q + n(1) + 2 n(2) + 3 n(3) + 4 n(4) −  A1 φ(1) ∂ξ 2       2 1+q (2) 3 1+q (3) 4 1+q − A1 φ −  A1 φ −  A1 φ(4) 2 2 2 1 1 − 2 (1 + q)(3 − q)A2 (φ(1) )2 − 3 (1 + q)(3 − q)A2 2φ(1) φ(2) 8 8

RPT and Some Evolution Equations

1 (1 + q)(3 − q)(5 − 3q)A3 (φ(1) )3 48 1 − 4 (1 + q)(3 − q)A2 2φ(1) φ(3) 8 1 − 4 (1 + q)(3 − q)(5 − 3q)3A3 (φ(1) )2 φ(2) 48 1 − 4 (1 + q)(3 − q)(5 − 3q)(7 − 5q)A4 (φ(1) )4 348 1 − 4 (1 + q)(3 − q)A2 (φ(2) )2 = 0 8

153

− 3

(4.86)

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where Al = μσcl + (1 − μ)σhl , l = 1, 2, 3, 4. After some calculations, we get the further modified KdV or supercritical KdV equation as follows: ∂φ(1) ∂φ(1) ∂φ(1) ∂φ(1) ∂ 3 φ(1) + aφ(1) + b(φ(1) )2 + c(φ(1) )3 +B =0 ∂τ ∂ξ ∂ξ ∂ξ ∂ξ 3 (4.87)     3 12 120 where a = λ3 λ4 (1+q)(3−q) − A2 , b = 3λ2 λ6 (1+q)(3−q)(5−3q) − A3 , and   1 8 3 3 35 c = 2λ 8λ8 − 384 (1 + q)(3 − q)(5 − 3q)(7 − 5q)A4 . A2 = (1+q)(3−q) 2λ4 − 48 5 2 2 B2 and A3 = (1+q)(3−q)(5−3q) 2λ6 −  B3 . 4.7 Damped Forced KdV (DFKdV) Equation It is observed that the soliton solutions are damped in a dusty plasma due to dust–ion collision, and accordingly, a new equation is modeled through RPT. We consider an unmagnetized dusty plasma consisting of mobile ions and κ distributed electrons. Accordingly, the basic normalized fluid equations are ∂(ni ui ) ∂ni + = 0, ∂t ∂x ∂ui ∂ui ∂φ + ui =− − νid ui , ∂t ∂x ∂x ∂ 2φ = (1 − μ)ne − n + μ ∂x2 where  ne =

φ 1− κ − 3/2

−κ+1/2 .

(4.88) (4.89) (4.90)

Waves and Wave Interactions in Plasmas

154

Here, nj (j = i and e for ions and electrons), ui , and φ are the number density, ion velocity, and the electrostatic potential, respectively. The normalization can be represented as x , t → ωpi t λD  where n0 is the unperturbed ion density, Cs (= Te /mi ) is the ion  2 /m ) is the ion plasma frequency, λ (= acoustic speed, ω (= 4πn e pi e0 i D  Te /(4πne0 e2 )) is the Debye length, Te is the electron temperature, mi is the ion mass, e is the electron charge, and KB is the Boltzmann constant. Here, νid is the dust–ion collision frequency and μ = Zdnn0d0 . We use the same stretched coordinates use in Equation (4.40) and the same expansion of the dependent variables considered in (4.41). We also consider

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ni →

ni , no

ui →

ui , Cs

φ→

eφ , KB Te

x→

νid ∼ 3/2 νid0 .

(4.91)

Substituting (4.41) and (4.91) along with stretching coordinates into Equations (4.88)–(4.90) and equating the coefficients of the lowest order of , we obtain the phase velocity as  (κ − 3/2) . (4.92) λ= (κ − 1/2)(1 − μ) Taking the coefficients of the next higher order of , we get the damped KDV equation as ∂φ(1) ∂φ(1) ∂ 3 φ(1) + Aφ(1) +B + Cφ(1) = 0 (4.93) ∂τ ∂ξ ∂ξ 3   4 3 κ−1/2 κ2 −1/4 where A = 3−2b(1−μ)λ , B = v2 , C = νid0 2λ 2 , a = κ−3/2 , and b = 2(κ−3/2)2 . The behavior of nonlinear waves changes with the change of external periodic force [17, 18]. It shows that such a force can be produced by a flexible, high-speed waveform generator. Considering an external periodic force f0 cos(ωτ ), the damped KdV Equation (4.93) takes the form ∂φ(1) ∂φ(1) ∂ 3 φ(1) + Aφ(1) +B + Cφ(1) 1 = f0 cos(ωτ ). ∂τ ∂ξ ∂ξ 3 It is termed as damped and forced KdV (DFKdV) equation.

(4.94)

RPT and Some Evolution Equations

155

When C = 0 and f0 = 0, then Equation (4.94) converts to KdV equation. The solitary wave solution of the KdV equation is written as   2 ξ − Mτ φ1 = φm sech (4.95) W  B where φm = 3M and W = 2 A M , with M as the Mach number. From the conservation properties of the KdV equation, the conserved quantity is  ∞ I= φ21 dξ. (4.96)

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−∞

Following the same procedure as in Chapter 3, Section 3.7.3, we finally get the approximate analytical solution of Equation (4.94) as   ξ − M (τ )τ φ1 = φm (τ )sech2 (4.97) W (τ ) where φm (τ ) =  M (τ ) = M −

3M(τ ) A

and W (τ ) = 2



B M(τ ) .

M (τ ) is given by the equation

   8ACf0 6Af0 4 − 43 Cτ Ccos(ωτ )+ωsin(ωτ ) . e + 16C 2 + 9ω 2 16C 2 + 9ω 2 3

4.8 Damped Forced MKdV (DFMKdV) Equation In the last case, we derived the DKdV equation due to the dust–ion collision. Damped and forced KdV equation appears if an additional force term appears on the right hand side of Poisson’s equation. Similarly, one can also obtain the damped and forced MKdV equation in the plane. To derive the DFMKdV equation, let us consider an unmagnetized collisional dusty plasma, consisting of cold inertial ions, q-nonextensive electrons, and negative stationery charged dust. Accordingly, basic normalized fluid equations are ∂ni ∂(ni ui ) + = 0, ∂t ∂x ∂ui ∂ui ∂φ + ui =− − νid u, ∂t ∂x ∂x ∂2φ = (1 − μ)ne − ni + μ + S(x, t) ∂x2

(4.98) (4.99) (4.100)

Waves and Wave Interactions in Plasmas

156

 q+1 where ne = ne0 {1 + (q − 1)φ 2(q−1) . Here, μ = Zdnn0d0 , νid is the dust–ion collisional frequency, and the term S(x, t) [6, 7] is a charged density source that may be obtained from experimental conditions for a definite purpose. We have used the same normalization as in Section 4.7. To obtain phase velocity and the nonlinear evolution equation, we introduced the same stretched coordinates as used in Equation (4.40). The expansions of the dependent variables are also considered the same as (4.41) and (4.91). We also consider

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S ∼  2 S2 .

(4.101)

Substituting expansions (4.41), (4.91), and (4.101) along with stretching coordinates (4.40) into Equations (4.98)–(4.100) and equating the coefficients of the lowest order of , the phase velocity is obtained the same as Equation (4.54). Now, taking the coefficients of the next higher order of  (i.e., coefficient of 5/2 from Equations (4.98) and (4.99) and coefficient of 2 from Equation (4.100)), we obtain the DFKdV equation as ∂φ(1) ∂φ(1) ∂ 3 φ(1) ∂S2 + Aφ(1) +B + Cφ(1) = B ∂τ ∂ξ ∂ξ 3 ∂ξ

(4.102)

3  (q+1)(3−q) q+1 νid0 λ3 where A = 2λ − bλ . a , B = 2 , C = 2 , a = 2 , and b = 8 bλ It is obvious that for some particular value of a, b, and λ, if a = 32 , then A = 0. So, at the critical point, the stretching (4.40) is not valid. In this situation, we have used stretched coordinate the same as (4.56) and expanded the dependent variables the same as (4.41). We also consider νid ∼ 3 νid0 ,

S ∼  3 S2 .

(4.103)

Now, substituting Equations (4.41), (4.56), and (4.103) into the basic equations (4.98)–(4.100) and equating the coefficients of the lowest order of  (i.e., coefficients of 2 from Equations (4.98) and (4.99) and coefficients of  from Equation (4.100)), we get (1)

ni

(1)

=

ui , λ

(1)

ui

=

φ(1) , λ

(1)

ni

= a(1 − μ)φ(1) .

(4.104)

Equating the coefficients of the next higher order of  (i.e., coefficients of 3 from Equations (4.98) and (4.99) and coefficients of 2 from

RPT and Some Evolution Equations

157

Equation (4.100)), we obtain the following relations: (2)

ni

=

1 (2) (1) (1) (u + ni ui ) λ i

=

∂φ(2) 1 (1) ∂ui + ) (ui λ ∂ξ ∂ξ

(2)

∂ui ∂ξ

(2)

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ni

(1)

= (1 − μ)(aφ(2) + b(φ(1) )2 ).

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

(4.105)

Equating the coefficients of the next higher order of  (i.e., coefficients of 4 from Equations (4.98) and (4.99) and coefficients of 3 from Equation (4.100)), we obtain the following relations: ⎫ (1) (3) (3) (1) (2) (2) (1) ⎪ ∂ni ∂ni ∂ui ∂(ni ui ) ∂(ni ui ) −λ + + + =0 ⎪ ∂τ ∂ξ ∂ξ ∂ξ ∂ξ ⎬ (1) (3) (1) (2) ∂φ(3) ∂ui ∂ui ∂(ui ui ) (1) −λ + + + νid0 ui = 0 ⎪ ∂τ ∂ξ ∂ξ ∂ξ ⎪ 2 (1) ∂ φ (3) (3) (1) (2) (1) 3 ⎭ − (1 − μ)(aφ + 2bφ φ + c(φ ) ) = n − S 2 i ∂ξ 2 (4.106) . where c = (1+q)(3−q)(5−3q) 48 From Equation (4.104), one can obtain the phase velocity same as in (4.54) and from Equations (4.104)–(4.106), one can obtain the nonlinear evolution equation as ∂φ(1) ∂φ(1) ∂ 3 φ(1) ∂S2 + A1 (φ(1) )2 + B1 + C1 φ(1) = B1 ∂τ ∂ξ ∂ξ 3 ∂ξ 3

3

(4.107)

3λ c(1−μ) 15 where A1 = 4λ , B1 = λ2 , and C1 = νid0 3 − 2 2 . It has been observed that the nature of nonlinear waves changes remarkably in the presence of external periodic force. It is important that the source term or forcing term may arise due to the presence of space debris in plasmas of a different kind. Several people have considered different types of forcing terms like Gaussian forcing term [6], hyperbolic forcing term [6] in the form of sech2 (ξ, τ ) and sech4 (ξ, τ ) functions, and trigonometric forcing term [8] in the form of sin(ξ, τ ) and cos(ξ, τ ) functions. Taking a clue from these works, we presume that S2 is a linear function of ξ such as S2 = f0 ξ cos(ωτ )+P , where P is some constant, f0 is the strength of the source, and ω is the frequency of the source, respectively. If we put the expression of S2 in Equation (4.107),

Waves and Wave Interactions in Plasmas

158

we get ∂φ(1) ∂φ(1) ∂ 3 φ(1) + A1 (φ(1) )2 + B1 + C1 φ(1) = B1 f0 cos(ωτ ). ∂τ ∂ξ ∂ξ 3

(4.108)

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Equation (4.108) is termed as damped forced modified KdV (DFMKdV) equation. When C1 = 0 and f0 = 0, then Equation (4.108) converts to the MKdV Equation (4.62) with the solitary wave solution as (4.63). Just like the previous section, we can find the approximate analytical solution of Equation (4.108) as   ξ − M (τ )τ (1) φ = φm (τ )sech (4.109) W (τ ) where M (τ ) is given by equation     πf0 A1 /6 −νid0 τ ω 2C1 e sin(ωτ ) + cos(ωτ ) M (τ ) = 2 ω 2 + 4C12 ω    2  √ 2C1 + M − πf0 B1 A1 /24 2 . ω + 4C12 The amplitude and width are as follows:      1 √ πf0 A1 /6 −νid0 τ ω 6 φm (τ ) = √ sin(ωτ ) e 2 ω 2 + 4C12 A      √ 2C1 2C1 + cos(ωτ ) + M − πf0 B1 A1 /24 ω ω 2 + 4C12 √ B1 W (τ ) = W1 + W2 where

   A1 /6 −νid0 τ ω 2C1 W1 = sin(ωτ ) + cos(ωτ ) e 2 ω 2 + 4C12 ω    √ 2C1 W2 = M − πf0 B1 A1 /24 . 2 ω + 4C12 πf0

The dependence of the amplitude of the DIA soliton of DFMKdV equation (4.108), which concerns the strength (f0 ) of the external periodic force for different values of entropic index q, is shown in Figure 4.4. From Figure 4.4, it is seen that as we increase the entropic index q, with the strength (f0 ) of the external periodic force, the amplitude of the solitary wave decreases.

RPT and Some Evolution Equations

159

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Fig. 4.4: Plot of φm (τ ) vs f0 for several values of q.

Fig. 4.5: (a) Contour plot of φ(1) in the plane (ξ, τ ) of Equation (4.108).

From Figures 4.5(a) and 4.5(b), it is seen that the outermost contour has the same value of φ(1) , and it increases with the value of the outermost contour towards the center of the solution space from both sides. It is how a solitary wave solution is represented. We can observe from the contours that the value of the maximum amplitude in Figure 4.5(a) is 0.2 and that of Figure 4.5(b) is 0.045. 4.9 Forced Schamel KdV (SKdV) Equation Schamel [9] introduced a KdV type equation in plasmas, where the trapping of particles plays a vital role in the dynamics. Trapping of particles means that some plasma particles are confined to a finite region where they bounce back and forth, describing closed trajectories in phase space. The concept of trapping in plasmas occurs when there are momentum and energy

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160

Waves and Wave Interactions in Plasmas

exchanges between waves and resonant particles. The particles whose speed is near to the phase velocity of the waves are responsible for this. Electrons that have lesser kinetic energy are trapped in the nonlinear wave potential. It leads to the confinement of electrons by the wave potential to a region of the phase space where they oscillate. The amplitude, thickness, and speed of the solitary wave depend crucially upon the population of the trapped electrons. When deriving the KdV equation for IAWs, the Boltzmann distribution for the electron density is usually considered. Hence, the isothermality and  the  electron  inertia are neglected. It is obtained by assuming fe ∝ exp − 12 v 2 − φ (where the electron velocity is normalized by the electron thermal speed) and is motivated by the consideration that a thermal electron moves with a speed much higher than an ion acoustic wave. Thus, it would not be much affected by the wave as the isothermal behavior of the electrons needs to be reasonable. The resonant particles interact strongly with the wave during its evolution and, therefore, cannot be treated on the same footing as the free ones. We remove the effects of external boundaries so that the collision-less behaviour is guaranteed. Laboratory experiments indicate that flat-topped electron distributions are typical. Two different distribution functions for the electrons (for the free and the trapped ones) must be used to generally describe the asymptotic, nearly stationary state. To study the effects of nonisothermal electrons on the nonlinear ionacoustic waves, vortex-like electron distribution function [9] is employed. The functions are as follows: ⎫  2   (v − 2φ) 1 ⎪ , |v| > 2φ, fef = √ exp ⎬ 2 2π (4.110)    ⎪ 1 β(v 2 − 2φ) fet = √ exp − , |v| < 2φ, ⎭ 2 2π where the subscript ef (et) represents the free (trapped) electron contribution. Distribution functions that are written in Equation (4.110) are continuous in velocity space and satisfy the regularity requirements for an admissible Bernstein–Greene–Kruskal (BGK) solution. The velocity is normalized by the electron thermal velocity. It is a parameter that determines the number of trapped electrons, whose magnitude is defined by the ratio of free electron temperature (Tef ) and trapped electron temperature (Tet ). Here, the velocity of the ion-acoustic wave is assumed less compared to the electron thermal velocity. Now, integrating the electron distribution

RPT and Some Evolution Equations

161

function over the velocity space, the electron number density is obtained by ⎫     eβφ ⎪ ne = eφ erf c φ +  erf βφ , (β > 0) ⎬ |β| (4.111)     ⎪ 1 ne = eφ erf c φ +  W −βφ , (β < 0) ⎭ π|β| where W is the Dawson integral and erf c(φ) = 1 − erf (φ) is the complementary error function. Expending ne for the small amplitude limit and keeping the terms up to φ2 , it is seen that ne is same for both β > 0 and β < 0, and finally, we obtain

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4 1 ne = 1 + φ − bφ3/2 + φ2 3 2

(4.112)

√ where b = (1 − β)/ π. Physically, b indicates the deviation from isothermality. If b > 0, then the contribution of the resonant electrons to the electron density is possible. Now, we consider an e-i plasma model where the ions are mobile in the presence of superthermal trapped electrons. The external force is considered in the Poisson equation. Schamel [9] and Williams et al. [10] have considered electrons both superthermal and trapped. Both the superthermal and trapped electrons are defined in different energy regions. When √ √ the energy region lies between − 2φ < v < 2φ (v is the velocity and φ is the electric potential), the electrons are considered trapped, and the corresponding κ distribution for the trapped electrons will be 1 Γ(κ) k (v, φ) = √ fe,t 2π(κ − 32 )1/2 Γ(κ − 1/2) −κ   2 v /2 − φ × 1+β f or Ee ≤ 0. κ − 32

(4.113)

It is an extension of Schamel’s distribution [11] for Maxwellian trapped electrons. If κ → ∞ (4.113), it produces Schamel’s equation [9]. Schamel considered the distribution which separates free electrons from trapped electrons. He also considered a trapped parameter β, which measures the inverse temperature of the trapped electrons. For superthermal free electrons, we have assumed the distribution as  −κ v2 1 Γ(κ) κ 2 −φ 1+ . (4.114) fe,f (v, φ) = √ 1 κ − 32 2π(κ − 32 ) 2 Γ(κ − 12 )

Waves and Wave Interactions in Plasmas

162

For simplicity, we assume κ > 3/2. So, the number density of electron ne (φ) will be given by  ne (φ) =

√ − 2φ

−∞

 κ fe,f (v, φ)dv

+

√

2φ

√ − 2φ

 κ fe,t (v, φ)dv

+

+∞

√ 2φ

κ fe,f (v, φ)dv

(4.115) κ where fe,f (v, φ) is given by (4.114) and ne is given by (4.115). After integrating with respect to velocity, we get   κ−3/2 −κ ne (φ) = (2κ − 3) (2κ − 3 − 2φ) × (2κ − 3) 2κ − 3 − 2φ

    4 1 3 2φ 2/π × φΓ[κ]2F1 , κ, , Γ[κ − 3/2] 2 2 3 − 2κ + 2φ   2 + 2/π(2κ − 3)κ−1/2 φ × (2κ − 3 − 2βφ)−κ Γ[κ] Γ[κ − 1/2]   1 3 2βφ × 2F1 , κ, , . (4.116) 2 2 3 − 2κ + 2βφ

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−

Expanding the hyper geometric function F (a, b, c, x) as a power series a(a+1)b(b+1) x2 expansion as F (a, b, c, x) = 1 + ab c x + c(c+1) 2! + · · · and keeping 2 term up to φ , the following function ne (φ) is obtained: ne (φ) ∼ 1 + pφ + qφ3/2 + rφ2 √

8

2/π(β−1)κΓ(κ)

(4.117) 2

4κ −1 where p = 2κ−1 2κ−3 , q = 3(2κ−3)3/2 Γ(κ−1/2) , and r = 2(2κ−3)2 . Sen et al. [6] have studied nonlinear wave excitation by orbiting charged space debris objects. They considered the source term S(x − vd t) on the Poisson equation arising from the charged debris, moving at speed vd . There is no work in that area where the externally applied force is solely dependent on τ . Accordingly, the one-dimensional normalized fluid equations along with the source term are given by

∂(ni ui ) ∂ni + =0 ∂t ∂x ∂ui ∂ui ∂φ + ui =− ∂t ∂x ∂x

(4.118) (4.119)

RPT and Some Evolution Equations

163

∂2φ = ne − ni + S(x, t) ∂x2  (1 − ni ) + pφ + qφ3/2 + rφ2 + S(x, t).

(4.120)

Here, ni , ne , ui , and φ represent the ion density, electron density, ion velocity, and electrostatic potential, respectively. The term S(x, t) is a charge density source derived from experimental conditions for a definite objective. Here, the same normalization is used as in Section 4.7. To obtain the phase velocity and the nonlinear evolution equation, the stretch coordinates are taken as

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ξ = 1/4 (x − λt),

τ = 3/4 t

(4.121)

where  is an infinitely small parameter. The dependent variables ni , ui , φ, and S(x, t) can be expanded as ⎫ + ·⎪ (1) (2) ui = 0 + ui + 3/2 ui + · ⎬ φ = 0 + φ(1) + 3/2 φ(2) + · ⎪ ⎭ S(x, t) ∼ 3/2 S2 (x, t) + ·. (1)

ni = 1 + ni

(2)

+ 3/2 ni

(4.122)

Substituting Equations (4.121)–(4.122) in the model Equations (4.118)– (4.120) and comparing the coefficients of different order of , one can obtain (1)

ni

⎫ ⎪ ⎪ ⎪ ⎬

= pφ(1)

∂ 2 φ(1) (2) + ni = pφ(2) + q(φ(1) )3/2 + S2 (ξ, τ ) ∂ξ 2 (1)

(1)

(1)

−λ

∂ni ∂ui + ∂ξ ∂ξ

−λ

∂ni ∂ui ∂ni + + ∂ξ ∂ξ ∂τ

(2)

(2)

= 0, λ

∂φ(1) ∂ui − =0 ∂ξ ∂ξ

(1)

= 0, −λ

(2)

(1)

∂ui ∂ui + ∂ξ ∂τ

+

∂φ(2) ∂ξ

⎪ ⎪ ⎪ = 0. ⎭

(4.123)

To obtain phase velocity, we eliminate the first-order perturbed quantities, and for the nontrivial values of the perturbed quantities, we have discussed the phase velocity of the wave as λ2 =

1 . p

(4.124)

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Waves and Wave Interactions in Plasmas

Now, considering the term of next order of  and simply as above we get the nonlinear evolution equation as  ∂φ(1) ∂φ(1) ∂ 3 φ(1) ∂S2 + A φ(1) +B =B ∂τ ∂ξ ∂ξ 3 ∂ξ where A = − 4p3q 3/2 and B =

(4.125)

1 . 2p3/2

We suppose S2 = fB0 ξcos(ωτ ), where f0 and ω have respectively denoted the strength and frequency. Using this, we obtain from Equation (4.125)

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 ∂φ(1) ∂φ(1) ∂ 3 φ(1) + A φ(1) +B = f0 cos(ωτ ). ∂τ ∂ξ ∂ξ 3

(4.126)

It is known as forced KdV-like Schamel (SKdV) equation. When f0 = 0, Equation (4.126) represents KdV-like Schamel equation, and the solitary wave solution is of the form φ(1) = φm sech4



ξ − Uτ W

 (4.127)

  2 16B where φm = 15U and W = 8A U are the amplitude and width of the ionacoustic solitary wave, respectively, and U is the speed of the ion-acoustic solitary wave. When f0 = 0, we consider the amplitude, width, and velocity of the solitary wave depending on τ , and the approximate solution of (4.126) is of the form   ξ − U (τ )τ φ(1) = φm (τ )sech4 (4.128) W (τ ) where amplitude φm (τ ) =



15U(τ ) 8A

2

and width W (τ ) =

U (τ ) have to be determined from the equation  U (τ ) =



16B U(τ ) .

Where

64A2 f0 sin(ωt) + k2 . 135 ω

So, the solution of (4.126) is of the form (1)

φ

= φm (τ )sech

4



ξ − U (τ )τ W (τ )

 (4.129)

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165

where 5f0 sin(ωτ ) 225k 2 + 3ω 64A2 ⎛ ⎞1/2 16B ⎠ . W (τ ) = ⎝  64A2 f0 sin(ωτ ) 2 +k 135ω φm (τ ) =

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We can also examine the effects of the strength (f0 ) of the external force by solitary wave solution of Equations (4.126), as shown in Figure 4.6. It is clear that as we increase f0 , the amplitude of the ion acoustic solitary wave increases. Figure 4.7 reflects the variation of the amplitude of the ion acoustic solitary wave solution of Equation (4.126) for different values of ω. It is clear

Fig. 4.6: Variation of the solitary wave solution of Equation (4.126) for f0 = 0, 0.01, 0.02 with k = 2.5, u0 = 0.2, β = 0.5, ω = 1, τ = 1.

Fig. 4.7: Variation of the amplitude of the solitary wave solution of Equation (4.126) for ω = 0.5, 1, 1.5 with other parameters the same as in Figure 4.6.

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that the amplitude of the solitary wave increases as the force increases. At the same time, when the frequency of the external periodic forces increases, the rate of change of amplitude of the solitary wave decreases.

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4.10 Burgers’ Equation In fluid mechanics, Burgers’ equation is one of the most significant partial differential equations. The equation is named after Johannes Martinus Burgers (1895–1981). It is obtained due to the combination of a nonlinear wave motion with diffusion. The presence of the viscous term helps suppress the wave breaking and smooth out shock discontinuities. To derive Burgers’ equation, we consider an unmagnetized, collisionless dusty plasma having q distributed electrons, Boltzmann distributed ions, and both positive and negative charged dust grains. Generally, negatively charged dust grains are considered in dusty plasmas as electrons move fast towards the dust and make it negatively charged. But, positively charge clouds of dust are also present in dusty plasmas. The principle of positively charged dust grains in space arises due to (i) photoemission in the presence of flux of ultraviolet photons, (ii) thermionic emission induced by radiative heating, and (iii) secondary emission of electrons from the surface of dust grains. Therefore, dust grains are both negatively and positively charged. They exist in different regions of space like comet tails, upper and lower mesosphere, Jupiter’s magnetosphere, and planetary rings. The basic fluid equations are given as follows: ∂(N1 U1 ) ∂N1 + =0 ∂T ∂X ∂U1 ∂U1 Z1 e ∂φ ∂ 2 U1 + U1 = + η11 ∂T ∂X m1 ∂X ∂X 2 ∂N2 ∂(N2 U2 ) + =0 ∂T ∂X ∂U2 ∂U2 Z2 e ∂φ ∂ 2 U2 + U2 =− + η22 ∂T ∂X m2 ∂X ∂X 2 Ne − Ni + Z1 N1 − Z2 N2 = 0

(4.130) (4.131) (4.132) (4.133) (4.134)

(1−q)   2(q+1) (− eφ ) where Ne = Ne0 1 + (q − 1) Keφ and Ni = Ni0 e KB Ti . Here, Nj B Te (j = 1, 2, e, and i stand for negative dust particle, positive dust particle, electrons, and ions) is the number density, Uj is the fluid speed, Z1 (Z2 ) is the number of electrons (protons) residing on negative (positive) dust

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167

particles, mj is the mass, e is the electronic charge, φ is the wave potential, η1 (η2 ) is the viscosity coefficient of negative (positive) dust fluid, Tj is the temperature, and KB is the Boltzmann constant. In Equation (4.134), we have assumed the quasineutrality condition at equilibrium. Here, normalization is taken as follows: n1 → N1 /n10 , n2 → N2 /n20 , u1 → U1 /c1 , u2 → U2 /c1 , φ → (eφ)/(KB Ti ), t → T wpd , x → X/λD , η1 → η11 wpd λ2D , η2 → η22 wpd λ2D

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 where nj0 is the equilibrium  value of nj , c1 = Z1 KB Ti /m1 , wpd =  4πZ12 e2 n10 /m1 , and λD = (Z1 KB Ti )/(4πZ12 e2 n10 ), and accordingly the basic normalized equations are expressed as ∂(n1 u1 ) ∂n1 + =0 ∂t ∂x

(4.135)

∂u1 ∂u1 ∂φ ∂ 2 u1 + u1 = + η1 ∂t ∂x ∂x ∂x2 ∂n2 ∂(n2 u2 ) + =0 ∂t ∂x

(4.136) (4.137)

∂u2 ∂u2 ∂φ ∂ 2 u2 + u2 = −αβ + η2 ∂t ∂x ∂x ∂x2

(4.138) q+1

n1 − μi e−φ − (1 + μe − μi )n2 = −μe [1 + (q − 1)σφ] 2(q−1)

(4.139)

where α = Z2 /Z1 , β = m1 /m2 , μe = Ne0 /(Z1 n10 ), μi = Ni0 /(Z1 n10 ), and σ = Ti /Te . For RPT, we have used the same stretched coordinates used as in Equation (4.40). The expansions of the dependent variables are considered as ⎫ ∞ ∞   k (k) k (k) ⎪ n1 = 1 +  n1 , n2 = 1 +  n2 ⎪ k=1 k=1 ⎪ ∞ ∞ ⎬   k (k) k (k) u1 = 0 +  u1 , u2 = 0 +  u2 (4.140) ⎪ k=1 k=1 ⎪ ∞  ⎪ φ = 0+ k φ(k) . ⎭ k=1

To obtain the phase velocity and nonlinear evolution equation, we substi3 tute (4.140) and (4.40) into (4.135)–(4.139) and taking the coefficient of  2

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from (4.135)–(4.138) and  from (4.139), we get (1) n1 (1)

n2 = (1)

n1

⎫ ⎪ ⎪ ⎬

(1)

φ(1) u (1) = 1 , u1 = − λ λ (1)

φ(1) u2 (1) , u2 = (αβ) λ λ

⎪ ⎪ (q + 1) (1) (1) (1) ⎭ = (1 + μe − μi )n2 − μe σ φ − μi φ . 2

(4.141)

Now, using the set of equations in (4.141), we get the phase velocity as

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λ2 =

[1 + αβ(1 + μe − μi )] . [σμe (q + 1)/2 + μi ]

(4.142)

Substituting (4.140) into (4.135)–(4.139) and equating the coefficients of 5 O( 2 ) from (4.135)–(4.138) and O(2 ) from (4.139), we obtain (1)

∂n1 ∂τ

−λ

(1)

(2)

(2)

(2)

(1)

∂u1 ∂u (1) ∂u1 − λ 1 + u1 ∂τ ∂ξ ∂ξ (1)

∂n2 ∂τ (1)

∂u2 ∂τ

−λ

−λ

(2)

(1)

=

∂φ(2) ∂ 2 u1 + η10 ∂ξ ∂ξ 2

(2)

(2)

(1)

(4.144)

(1) (1)

(2)

= −αβ

n1 = (1 + μe − μi )n2 − μe σ − μe σ 2

(4.143)

∂n2 ∂u2 ∂(n2 u2 ) + + =0 ∂ξ ∂ξ ∂ξ

∂u2 (1) ∂u2 + u2 ∂ξ ∂ξ

(2)

(1) (1)

∂n1 ∂u1 ∂(n1 u1 ) + + =0 ∂ξ ∂ξ ∂ξ

(4.145) (1)

∂φ(2) ∂ 2 u2 + η20 ∂ξ ∂ξ 2

(4.146)

(q + 1) (2) φ 2

(q + 1)(3 − q) (1) 2 μi (φ ) − μi φ(2) + (φ(1) )2 . (4.147) 8 2 (2)

(2)

(2)

(2)

Using Equations (4.141)–(4.142) and eliminating n1 , n2 , u1 , u2 , and φ(2) from the above set of equations, we finally obtain ∂φ(1) ∂φ(1) ∂ 2 φ(1) −C + Aφ(1) = 0. ∂τ ∂ξ ∂ξ 2

(4.148)

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169

(4.148) is known as Burgers’ equation where the nonlinear coefficient A and the dissipation coefficient C are given by A=

12α2 β 2 (1 + μe − μi ) − 12 + 4μi λ4 − μe σ 2 λ4 (q + 1)(3 − q) 8λ[1 + αβ(1 + μe − μi )]

C=

η10 + η20 αβ(1 + μe − μi ) . 2{1 + αβ(1 + μe − μi )}

and

To obtain the stationary solution of this Burgers’ equation (4.148), we have introduced a transformation ζ = ξ − u0 τ . Now, applying the conditions (1) φ(1) → 0, dφdζ → 0 at ζ → ∞, the stationary solution is given by

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   ζ φ(1) = φ(1) 1 − tanh m Δ

(4.149) (1)

where u0 is a constant velocity normalized by Cs and the amplitude φm (normalized by KBe Ti ) and width Δ (normalized by λD ) of the shock waves are defined as φ(1) m =

u0 , A

Δ=

2C , u0

(4.150)

respectively. It is observed, from Equations (4.149)–(4.150), that the amplitude of the shock waves increases as u0 increases, and the width of the shock waves decreases as u0 increases. From Equation (4.149), it is clear that the shock potential profile is both positive and negative when A > 0 or A < 0. To find the parametric regimes for which positive and negative shock wave (potential) profiles exist, we have numerically analyzed A and have obtained the A = 0 curve in the β − q plane. Figure 4.8 shows that we can have positive shock wave (potential) profiles for the parameters, whose values lie above the curve A = 0, and a negative shock (potential) profile can be achieved for the parameters whose values lie below the curve A = 0. Obviously, for A = 0, the shock waves do not exist. Figure 4.9(a) reflects the effects of the nonextensive parameter q on the propagation of dust acoustic shock structure. It is clear from Figure 4.9(a) that a small change in nonextensive parameters q exhibits different shock structures, and positive dust acoustic shock wave potential decreases as q increases. Also, from Figure 4.9(b), it is seen that the shock structures are different for different values of β. It is also seen that the amplitude of the positive dust acoustic shock wave potential decreases as β increases.

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Fig. 4.8: Showing A = 0 (β vs. q ) curve for α = 0.101, σ = 0.497, μi = 0.5, μe = 0.3.

Fig. 4.9: (a) Variation of DA shock wave potential φ(1) with spatial coordinates ζ for different values of q and (b) for different values of β.

4.11 Modified Burgers’ Equation Like the MKdV equation, the modified Burgers’ equation is necessary if, for a particular set of parameters, the coefficient of the nonlinear term of the Burgers’ equation vanishes. Here, we consider the propagation of an electrostatic perturbation in an unmagnetized, collisionless, dense plasma containing degenerate electron fluids (both nonrelativistic and ultra-relativistic) and inertial viscous ion fluids [12]. For understanding the different electrostatic nonlinear phenomena in astrophysical environments (where particle velocities are near the speed of light), relativistic effects should be taken into account. Many astrophysical compact objects such as white dwarfs and neutron stars have degenerate electron number densities. These are so high (in white dwarfs and neutron

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171

stars) that on the order of 1030 cm−3 and 1036 cm−3 , or even more than their cores, it consists of strongly coupled nondegenerate ion lattices. These are immersed in degenerate electron fluids that follow the Fermi–Dirac distribution function. Chandrasekhar [13] developed a general expression for relativistic electron pressures in his classical papers. The electron fluid pressure can be satisfied by the following equation: γ Pe = Ke nα e = K e ne .

The nonrelativistic limit can be written as

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α=γ=

5 , 3

Ke =

 1/3 2 3 π πh  Λc hc 5 3 m

where Λc = πh/mc = 1.2 × 10−10 cm and h is the plank constant divided by 2π, whereas in considering the ultra-relativistic case, it is 5 α= , 3

4 γ= , 3

 1/3 3 π 3 Ke = hc  hc. 4 9 4

The basic normalized fluid equations are ∂ni ∂(ni ui ) + =0 ∂t ∂x ∂φ ∂ 2 ui ∂ui ∂ui + ui =− +η 2 ∂t ∂x ∂x ∂x ∂φ ∂nγe ne =K ∂t ∂x ∂2φ = −ρ ∂x2 ρ = ni − ne .

(4.151) (4.152) (4.153) (4.154) (4.155)

nj (j = e, and i, stands for electrons, and ions, respectively) is the number density, uj is the fluid speed, mj is the mass, φ is the wave potential, η is the viscosity coefficient, x is the space variable, and t is the time variable. At equilibrium, we have ni0 = ne0 where the normalization is done as follows: nj → nj /nj0 , t → twpi ,

φ → (eφ)/(me c2 ), η η→ . wpi λ2m mi ni0

ui → ui /ci ,

x → x/λm ,

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172

 Here, nj0 is the equilibrium me c2 /mi , wpi =   value of nj , ci = 4πe2 ni0 /mi , and λm = (me c2 )/(4πe2 ne0 ). c is the speed of light in K nγ−1

e e0 vacuum, e is the magnitude of the charge of an electron, and K = m 2 . ec For perturbation analysis, let us consider new stretched coordinates ξ and τ such that

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ξ = 2 (x − λt),

τ = 4 t

(4.156)

where  is a parameter measuring the weakness of the nonlinearity and λ is the phase speed of the waves. We can expand the variables nj , ui , and ψ in a power series of  as ⎫ ∞ ∞   (k) (k) ⎪ nj = 1 +  k nj , u i = 0 + k u 1 ⎬ k=1 k=1 (4.157) ∞ ∞ ⎪   k (k) k (k) φ = 0+  φ , ρ=0+  ρ .⎭ k=1

k=1

To obtain the phase velocity and nonlinear evolution equation, we substitute (4.156)–(4.157) into (4.151)–(4.155), and equating the lowest power of , we get (1)

ni

=

φ(1) , λ2

(1)

φ(1) , λ

ui

=

λ=

 γK.

n(1) e =

φ(1) . γK

(4.158)

We get from Equation (4.158) (4.159)

Equation (4.159) is the phase velocity. We substitute Equations (4.156)– (4.157) in Equations (4.151)–(4.155), equate the coefficient of 4 from Equations (4.151)–(4.154), and take the coefficient of 2 from Equation (4.155). We obtain a set of equations as follows: ⎫ φ(2) φ(2) (φ(1) )2 3(φ(1) )2 ⎪ (2) (2) , ui = + ni = 2 + ⎬ λ 2λ4 λ 2λ3   ⎪ φ(2) (γ − 2)(φ(1) )2 (2) 1 (γ − 2) 3 (2) , ρ = + 4 (φ(1) )2 = 0. ⎭ ne = 2 − 4 4 λ 2λ 2 λ λ (4.160) For the next higher order in , we equate the coefficient of 5 from Equations (4.151)–(4.152) and 3 from Equation (4.155), thus we can derive the

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173

following sets of equations: (1)

∂ni ∂τ

(1)

−λ

(3)

(3)

∂ (1) (2) ∂ni ∂ui (2) (1) + + (n u + ni ui ) = 0 ∂ξ ∂ξ ∂ξ i i (3)

(1)

∂φ(3) ∂ (1) (2) ∂ui ∂u ∂ 2 ui −λ i + −η =0 (ui ui ) + ∂τ ∂ξ ∂ξ ∂ξ ∂ξ 2    ∂ (γ − 2)(φ(1) φ(2) ) (γ − 2)(2γ − 3)(φ(1) )3 + − =0 φ(3) − λ2 n(3) e ∂ξ λ4 6λ6 (3)

ni

(3) − n(3) . e = −ρ

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We finally obtain the modified Burgers’ equation as ∂φ(1) ∂φ(1) ∂ 2 φ(1) + A(φ(1) )2 =C ∂τ ∂ξ ∂ξ 2

(4.161)

where A = 4λ1 3 [15 − (γ − 2)(2γ − 3)] and C = η2 . The stationary shock solution of Equation (4.161) is written as      ζ (4.162) φ(1) = φm 1 − tanh Δ where the spacial coordinate is ζ = ξ − u0 τ , the amplitude is φm =  and the width is Δ = uC0 .

3u0 2A ,

4.12 KdV Burgers’ (KdVB) Equation The propagation of IAWs in a collisionless plasma is described by the KdV equation. The KdV equation exhibits solitary wave solutions as well as cnoidal wave solutions without dissipation. On the other hand, ion acoustic shock waves arise in dissipative plasmas. We consider a dissipative mechanism to explain the existence of shock waves. One possible dissipative mechanism peculiar to plasmas is due to the kinematic viscosity. It is interesting and worthwhile to examine the effects of kinematic viscosity on the propagation of ion acoustic waves, where the dissipation plays a crucial role to form shock waves (than solitary waves) due to kinematic viscosity. KdVB equation is such an equation that can contain the dissipative and dispersion of the media simultaneously. The standard RPT is used to derive the KdVB equation for IAWs. To investigate the effects of kinematic viscosity on IAWs, we use the modified ion momentum equation, equation

Waves and Wave Interactions in Plasmas

174

of continuity, and Poisson’s equation. The basic fluid normalized equations are written as ∂(ni ui ) ∂ni + =0 ∂t ∂x

(4.163)

∂ui ∂ui ∂φ ∂ 2 ui + ui =− + ηi 2 ∂t ∂x ∂x ∂x

(4.164)

∂2φ = exp φ − ni ∂x2

(4.165)

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μ ω

where ηi = iC 2pi , and μi is the ion kinematic viscosity. s To obtain the KdVB equation, we have introduced the same stretched coordinates as in Equation (4.40). The expansions of the dependent variables are also considered the same as (4.41). Substituting expansions (4.41), along with the stretch coordinates (4.40) into Equations (4.163–4.165) and equating the coefficient of the lowest order of , we obtain the phase velocity as λ2 = ±1.

(4.166)

To obtain the nonlinear evolution equation, we substitute Equation (4.41) in Equations (4.163)–(4.165) and collect the next higher order of , (1)

∂ni ∂τ

(1)

−λ

(2)

(1) (1)

∂ni ∂ni ui + ∂ξ ∂ξ (2)

(2)

+

∂ui ∂ξ

= 0,

(1)

(4.167) (1)

∂φ(2) ∂ui ∂u ∂ 2 ui (1) ∂ui − λ i + ui + = η , i0 ∂τ ∂ξ ∂ξ ∂ξ 2 ∂ξ 2 ∂φ(1) 1 (1) − φ(2) − (φ(1) )2 + ni = 0. 2 ∂ξ 2

(4.168) (4.169)

Differentiating Equation (4.169) with respect to ξ and substituting (2)

from Equation (4.167) and for

∂ui ∂ξ

(2)

∂ni ∂ξ

from Equation (4.168), we finally obtain

ηi0 ∂ 2 φ(1) ∂φ(1) ∂φ(1) 1 λ ∂ 3 φ(1) − + φ(1) + = 0. ∂τ λ ∂ξ 2 ∂ξ 3 2 ∂ξ 2

(4.170)

Equation (4.170) is known as KdVB equation. The second, third, and fourth terms of Equation (4.170) respectively are nonlinear, dispersion, and dissipative. If the system is nondissipative, then we obtain the KdV equation (1.137). With the help of tanh method, we can also get the stationary

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Fig. 4.10: The stationary shock structures (4.170) as a function of ξ and τ .

solution of Equation (4.170) as φ(1) (ξ, τ ) = a0 + a1 tanh{α(ξ − V τ )} + a2 tanh2 {α(ξ − V τ )}

(4.171)

2

ηi0 1 i0 α where a0 = A1 (V + 12Bα2 ), a1 = − 6η5A , a2 = − 12Bα A , α = ± 20B , A = λ , λ and B = 2 . Here, V is the shock wave velocity. The shock height is directly proportional to the square of ηi0 and inversely to the product AB, whereas the shock width varies inversely with ηi0 and directly with B. Figure 4.10 represents the three-dimensional view of the stationary shock structures for different values of ηi0 .

4.13 Damped KdVB Equation To derive the damped KdVB equation, we consider an unmagnetized collisional dusty plasma model. It consists of cold inertial ions, stationary negative dust charge, and q-nonextensive electrons. Here, the damping is considered because of the dust–ion collision along with the usual kinetic viscosity. The basic normalized set of equations are ∂ni ∂(ni ui ) + =0 ∂t ∂x ∂ui ∂ui ∂φ ∂ 2 ui + ui =− + η 2 − νid ui ∂t ∂x ∂x ∂x ∂2φ = (1 − μ)ne − ni + μ ∂x2

(4.172) (4.173) (4.174)

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Waves and Wave Interactions in Plasmas

  q+1 where ne = 1 + (q − 1)φ 2(q−1) . Here, we use the same normalization according as in Section 4.7. To obtain the Damped KdVB equation, we introduce the same stretched coordinates as in Equation (4.40). The expansions of the dependent variables are also considered the same as in Equation (4.41). We also consider η = 1/2 η0 ,

νid ∼ 3/2 νid0 .

(4.175)

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To obtain the phase velocity and nonlinear evolution equation, we substitute the above expansions (4.41) and (4.175) along with stretched coordinates (4.40) into Equations (4.172)–(4.174) and equate the coefficients of the lowest order of . We thus get the phase velocity the same as in Equation (4.54). Taking the coefficients of next higher order of , we obtain the damped KdV Burgers’ equation as ∂φ(1) ∂ 3 φ(1) ∂ 2 φ(1) ∂φ(1) + Aφ(1) +B +C + Dφ(1) = 0 (4.176) 3 ∂τ ∂ξ ∂ξ ∂ξ 2 3  3 where A = 2v − v 3 (1 − μ)b , B = v2 , C = − η210 , and D = νid0 with 2 (q+1)(3−q) . When C = 0 and D = 0, then Equation (4.176) converts to b= 8 the KdV equation, and the solution of the KdV equation is   2 ξ − M0 τ φ1 = φm sech (4.177) W  B 0 where φm = 3M A and W = 2 M0 are the amplitude and the width of the solitary waves, and M0 is the Mach number. Like the previous section, similarly, we find the approximate solution of Equation (4.176) in the form of   ξ − M (τ )τ φ(1) = φm (τ )sech2 (4.178) W (τ )  ) where the amplitude φm (τ ) = 3M(τ A , width W (τ ) = 2 B/M (τ ), and P M0 velocity M (τ ) can be obtained by the relation M (τ ) = M0 Q(1−e P τ )+P eP τ , 4 4 C where P = 3 D, Q = 15 B , and M (0) = M0 as τ = 0. From Figure 4.11, it is clear that the solitary wave amplitude decreases monotonically if the viscosity coefficients η10 increase slowly. In Figure 4.12, the amplitude of the solitary wave is plotted concerning the collision frequency parameter νid0 . It is obvious here that the amplitude of the solitary

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177

Fig. 4.11: Variation of the amplitude of the solitary wave from (4.176) with respect to η10 for q = 0.6, τ = 2, M = 0.1, νid0 = 0.01, and μ = 0.5.

Fig. 4.12: Variation of the amplitude of the solitary wave from (4.176) with respect to νid0 for q = 0.6, τ = 2, M = 0.1, η10 = 0.1, and μ = 0.5.

wave decreases as the dust–ion collision frequency parameter νid0 increases gradually for positive nonzero values. Thus, the solitary wave solution does not exist for νid0 = 0. Figure 4.13 shows that the solitary wave width increases monotonically if the value of the viscosity coefficients η10 increases slowly.

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Waves and Wave Interactions in Plasmas

Fig. 4.13: Variation of width of the solitary wave from (4.176) with respect to η10 for q = 0.6, τ = 2, M0 = 0.1, νid0 = 0.1, and μ = 0.5.

4.14 Kadomtsev–Petviashvili (KP) Equation Soviet physicists Boris Kadomtsev (1928–1998) and Vladimir Petviashvili (1936–1993) [14] studied the evolution of the long IAWs of small amplitude propagating in plasma in their research work and derived an evolution equation named as KP equation. The KP equation can also be applied to model water waves of long wavelength with weakly nonlinear restoring forces and frequency dispersion. Here, we derive the KP equation for DIA waves in a three-component dusty plasma, subject to an external magnetic field. In this model, the plasma consists of negatively charged ions, massive dust grains, and q-nonextensive electrons. The dust dynamics are not taken into account, and the charges of dust grains are assumed to be constant. In the previous section of Chapter 2, we have already discussed the q-nonextensive distribution. The basic equations are written as ∂ni  + ∇ · (ni ui ) = 0, ∂t

(4.179)

 ∂ ui  )ui = − e∇φ + eB0 ui × ez , + (ui · ∇ ∂t mi mi c

(4.180)

 2 φ = 4πe[ne − ni + zd nd ] ∇ where  1 +1 eφ q−1 2 ne (φ) = ne0 1 + (q − 1) . Te

(4.181)

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Here, nj (j = e, i, and d stands for electron, ion, and dust particle respectively), ui (u, v, w), and φ are the number densities, ion velocity, and the plasma potential, respectively. mi , zd , and e are the ion mass, the dust charge number, and the elementary charge, respectively. Let us consider that the wave is propagating in the xz-plane. Accordingly, the normalized equations are ∂ni ∂(ni u) ∂(ni w) + + = ∂t ∂x ∂z   ∂u ∂ ∂ + u +w u= ∂t ∂x ∂z   ∂v ∂ ∂ + u +w v= ∂t ∂x ∂z   ∂w ∂ ∂ + u +w w= ∂t ∂x ∂z  2  ∂ ∂2 + 2 φ + βδ1 ni = ∂x2 ∂z

0, −

(4.182) ∂φ + v, ∂x

(4.183)

−u, −

(4.184)

∂φ , ∂z

(4.185) q+1

β[(1 + (q − 1)φ) 2(q−1) + δ2 ]

(4.186)

where β = rg2 /λ2e , δ1 = ni0 /ne0 , and δ2 = nd zd /ne0 . Here, rg = Cs /Ω is the  ion gyroradius and λe = Te /(4πne0 e2 ) is the electron Debye length. The normalizations are taken as Ωt →t, (Cs /Ω)∇ → ∇, ui /Cs → ui , ni /ni0 → ni , eφ/Te → φ, where Cs = Te /mi is the ion acoustic velocity and Ω = (eB0 )/(mi c) is the ion gyrofrequency. Also, nj0 (j = i, e) is the unperturbed density. The stretch coordinates are taken as χ = 2 x,

ξ = (z − λt),

τ = 3 t

(4.187)

where λ is the phase velocity of IAWs. The dependent variables are expanded as ⎫ ∞ ∞   ⎪ 2k+1 (k) 2k (k) ni = 1 +  ni + · · · , u =  u + ···⎪ k=1 k=1 ⎪ ∞ ∞ ⎬   v= 2k+1 v (k) + · · · , w = 2k+1 w(k) + · · · (4.188) ⎪ k=1 k=1 ⎪ ∞  ⎪ φ= 2k φ(1) + · · · . ⎭ k=1

Waves and Wave Interactions in Plasmas

180

To obtain the phase velocity and nonlinear evolution equation, we substitute the expansions given by Equations (4.187)–(4.188) into Equations (4.182)–(4.186) and equate the coefficients of lowest power of , thus we get the phase velocity as λ2 =

2δ1 . 1+q

(4.189)

Again, equating next higher-order terms of , we get ⎫ (2) (1) ∂u(1) ∂w(2) ∂(ni w(1) ) ∂ni ⎪ + + + = 0, ⎪ ∂ξ ∂χ ∂ξ ∂ξ ⎪ (1) ∂w(1) ∂w(2) ∂φ(2) (1) ∂w −λ +w + = 0, ⎪ ⎬ ∂τ ∂ξ ∂ξ ∂ξ ⎪ ∂u(1) ∂u(2) ∂u(1) ∂φ(2) −λ + w(1) + = 0, ⎪ ∂τ ∂ξ ∂ξ ∂χ ⎪   ⎪ q + 1 (2) (q + 1)(3 − q) (1) 2 (2) φ + −β (φ ) − δ1 ni = 0. ⎭ 2 8 (1)

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∂ni ∂τ

∂ 2 φ(1) ∂ξ 2

−λ

(4.190)

From relations (4.189)–(4.190), we obtain the KP equation as   (1) ∂ ∂φ(1) ∂ 3 φ(1) ∂ 2 φ(1) (1) ∂φ + Aφ +B + C =0 ∂ξ ∂τ ∂ξ ∂ξ 3 ∂χ2

(4.191)

2

−6 λ where A = − (3−q)λ , B = β(1+q) , and C = λ2 . 4λ The tanh method is one of the famous methods to find a solution of the KP Equation (4.191). With the help of tanh method, we use the transformation ζ = ξ+η−uτ , where u is the speed of the nonlinear structure. Now applying (1) 2 (1) 3 (1) the conditions φ(1) → 0, ∂φ∂ζ → 0, ∂ ∂ζφ 2 → 0, ∂ ∂ζφ 3 → 0 as ζ → ∞, we get the analytical solution of the KP Equation (4.191) as

φ(1) (ζ) =

   12B 1 − tanh2 ξ + η − (4B + C)τ . A

(4.192)

The speed of the co-moving frame (say u) is related to the weak dispersive and diffraction coefficients such as u = (4B + C), which has been obtained using the boundary conditions, i.e., ζ → ∞, φ(1) (ζ) → 0, and tanh2 (ζ) → 1.

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Fig. 4.14: (a) Variation of solitary wave potential φ(1) with spatial coordinates χ for different values of q > 1 and (b) for different values of q < 1.

Also, the solution using the pseudo-potential approach is written as   ξ + η − uτ φ(1) (ζ) = φm Sech2 (4.193) W  where maximum amplitude φm = 3(u−C) and width W = 4B/(u − C) of A the soliton has been defined. Here, it should be noted that both solutions have the same form with the same numerical results. We illustrate the effect of the nonextensive parameter q by plotting φ(1) versus χ as shown in Figure 4.14. It is also observed from Figure 4.14(a) that, as q(> 1) increases, the amplitude and the width of the solitary waves decrease. Also, from Figure 4.14(b), it is seen that as the amplitude and the width of the solitary waves decrease accordingly, the value of q(< 1) increases. 4.15 Modified KP (MKP) Equation The KP equation is a two-dimensional analog of the KdV equation. We reconsider the derivation of the KP equation, which is modified to include the effects of rotation. The motivation for this work is that if the solution of the modified KP (MKP) equation is assumed to be locally confined, then they satisfy a certain constraint. It appears to restrict the class of allowed initial conditions considerably. Grimshaw [15] derived a MKP equation to describe long surface or initial waves in the presence of rotation. The MKP equation is the appropriate extension of the MKdV equation. The MKP equation provides a correct asymptotic description of the waves traveling to the right. In general, it cannot be assumed that the solutions are locally confined.

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182

It is required to proceed with the MKP equation by considering the higher-order coefficients of . The same set of stretched coordinate Equation (4.187) are considered, but expansion (4.188) of the dependent variables are not valid. So, a set of new expansions of the dependent variables is taken into account ⎫ ⎪ + ··· , u = 0 +  u + ··· ni = 1 + ⎪ k=1 k=1 ⎪ ∞ ∞ ⎬   v = 0+ 2k v (k) + · · · , w = 0 + k w(k) + · · · ⎪ k=1 k=1 ⎪ ∞  ⎪ φ = 0+ k φ(k) + · · · . ⎭ Downloaded from www.worldscientific.com

∞ 

∞ 

(k)  k ni

k+1 (k)

(4.194)

k=1

Substituting the above expansions (4.194), along with the same stretched coordinates (4.187) into Equations (4.179)–(4.186), and equating the coefficients of different powers of , one can obtain (1)

ni

(2)

ni

=

w(1) , λ

=

ni w(1) + w(2) , λ

(1)

w(1) =

φ(1) 1 , λ

φ(1) =

(2)

ni

=

(1)

δ1 n i , a

δ1 = 1 + δ2

aφ(2) + b(φ(1) )2 , δ1

(4.195)

∂u(1) 1 ∂φ(1) = ∂ξ λ ∂X (4.196)

⎫ (3) ∂u(1) ∂w(3) ⎪ ∂ (1) (2) ∂ni (2) (1) −λ + + (n w + ni w ) = − , ∂ξ ∂χ ∂ξ i ∂ξ ⎪ ⎪ ⎬ (2) (1) ∂w(1) ∂w(3) ∂φ(3) (1) ∂w (2) ∂w (4.197) −λ +w +w + = 0, ⎪ ∂τ ∂ξ ∂ξ ∂ξ ∂ξ ⎪ ∂ 2 φ(1) (3) (3) (1) (2) (1) 3 ⎪ ⎭ − β{aφ + 2bφ φ + k(φ ) − δ1 ni } = 0 ∂ξ 2 (1)

∂ni ∂τ

1 1 where a = (q+1) 2 , b = 8 {(q + 1)(3 − q)}, and k = 48 {(q + 1)(3 − q)(5 − 3q)}. From Equation (4.195), one can obtain the phase velocity as

λ2 =

δ1 . a

(4.198)

RPT and Some Evolution Equations

183

(3)

By eliminating ni , w(3) , and φ(3) from Equation (4.197) with the help of Equations (4.195)–(4.196), one can obtain the MKP equation as   (1) ∂(φ(1) φ(2) ) ∂ 3 φ(1) ∂ 2 φ(1) ∂ ∂φ(1) (1) 2 ∂φ +A − D(φ ) +B + C =0 ∂ξ ∂τ ∂ξ ∂ξ ∂ξ 3 ∂χ2 (4.199)

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  2 −6 3b 2a V where A = − (3−q)V , B = β(1+q) , C = V2 , and D = 3V 4V 2a k − V 2 + V 4 . There exists a set of critical values at which A = 0 and Equation (4.199) converts to the MKP equation as   ∂φ(1) ∂ 3 φ(1) ∂ 2 φ(1) ∂ ∂φ(1) − D(φ(1) )2 +B +C = 0. (4.200) 3 ∂ξ ∂τ ∂ξ ∂ξ ∂χ2 We assume the stationary wave solution of MKP Equation (4.200) as φ(1) = φ(1) (ζ) where ζ = kξ + lχ− u0 τ . Substituting this expression into the MKP Equation (4.200), we obtain   ζ φ(1) = φm Sech2 (4.201) Δ where the amplitude of the soliton is φm = 6(u0 k − Cl2 /Dk 2 )1/2 and the  1/2 B width of soliton is Δ = k 2 (u0 −cl . 2) 4.16 Further MKP (FMKP) Equation Let us consider a magnetoplasma consisting of ions and dust particles. Here, the ion obeys the nonextensive nonthermal distribution and the exterˆM0 , where xˆ is nal magnetic field (M ) acting analog the x-axis, i.e., M = x a unit vector along the x -axis. The basic normalized equations are ∂nd  + ∇ · (nd ud ) = 0, ∂t

(4.202)

∂ ud ˆ,  )ud = ∇  φ − ud × x + (ud · ∇ ∂t

(4.203)

 2 φ = α1 (nd − ni ) ∇ (4.204)   where α1 = r2 /λ2D , r = Cs /Ω, λD = Ti /(4πe2 n0 zd0 ), Cs = Ti /m, and Ω = (eM0 )/md c. r, λD , Cs , Ω, c, e, md , and zd are the dust gyroradius,

Waves and Wave Interactions in Plasmas

184

Debye length, dust acoustic velocity, dust gyrofrequency, speed of the light, elementary charge, dust mass, and the number of the charge residing on the dust grains, respectively. The dust charge qd = −ezd. nd and ud (u, v, w) are the number density and velocity of the dust particles, respectively. The ions obey the nonextensive nonthermal distribution, i.e., 1     2    q−1 + 12 eφ eφ eφ +N 1 − (q − 1) ni = ni0 1 − M Ti Ti Ti (4.205)

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16αq(2q−1) 16αq where M = − (5q−3)(3q−1)+12α and N = (5q−3)(3q−1)+12α . Here, the normalization is done as follows:

ni →

ni , ni0

nd →

nd , n0

ud →

ud , Cs

φ→

eφ , Ti

t → tΩ,

x→

x . r

Let us consider that the wave is propagating in the xy-plane. Accordingly, the normalized equations are ∂nd ∂(nd u) ∂(nd v) + + = ∂t ∂x ∂y   ∂u ∂ ∂ + u +v u= ∂t ∂x ∂y   ∂ ∂ ∂v + u +v v= ∂t ∂x ∂y   ∂w ∂ ∂ + u +v w= ∂t ∂x ∂y 

0,

(4.206)

∂φ , ∂x

(4.207)

∂φ − w, ∂y

(4.208)

v,

(4.209)

1     q−1 + 12  ∂2 ∂ 2 + 2 φ = α1 nd − 1 − M φ + N φ 1 − (q − 1)φ . ∂x2 ∂y (4.210)

To obtain the FMKP equation, we introduce the same stretched coordinates as in Equations (4.187). The expansions of the dependent variables are

RPT and Some Evolution Equations

185

considered as follows: nd = u= v=

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w= φ= νid ∼

⎫ ⎪ 1+ + ··· , ⎪ k=1 ⎪ ∞  ⎪ k (k) 0+  u + ··· , ⎪ k=1 ⎪ ∞  k+1 (k) 0+  v + ··· ,⎬ k=1 ⎪ ∞  ⎪ 0+ 2k w(k) + · · · , ⎪ k=1 ⎪ ∞  ⎪ 0+ k φ(k) + · · · , ⎪ k=1 ⎭ 3 νid0 ∞ 

(k)  k nd

(4.211)

where νido is the dust–ion collisional frequency. Substituting Equation (4.187) along with the stretched coordinates (4.211) into the system of Equations (4.206)–(4.210) and equating the coefficients of lowest order of , we get the phase velocity as λ2 =

2 . q + 1 + 2M

(4.212) (3)

Equating the coefficient of different power of  and eliminating nd , w(3) , and φ(3) , one can obtain the nonlinear equation   ∂ ∂φ1 ∂(φ1 φ2 ) ∂ 3 φ1 ∂ 2 φ1 2 ∂φ1 −A − Dφ1 +B + C = 0 (4.213) ∂η ∂τ ∂η ∂η ∂η 3 ∂χ2 where A = (q+1)(3−q) , 8

λ 2P

[3P 2 − 2Q], B =

λ λ 2P α1 , C = 2 , P q+1 3λ 2 , D = 2P (R +

=

q+1+2M , 2 3

b =

Q = b + N + aM, a = 2P − 3P Q), and R = K + bM + aN . At certain sets of critical values, the nonlinear coefficients A = 0, and Equation (4.213) reduces to the following MKP equation:   ∂ ∂φ1 ∂φ1 ∂ 3 φ1 ∂ 2 φ1 − Dφ21 +B + C = 0. ∂η ∂τ ∂η ∂η 3 ∂χ2

(4.214)

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Waves and Wave Interactions in Plasmas

If A is at the same order of , but not zero, we derive the FMKP equation using the same stretched coordinates and the expression as the MKP equation   ∂(φ1 φ2 ) ∂ 3 φ1 ∂ ∂φ1 ∂φ1 ∂ 2 φ1 −A − Dφ21 +B +C = 0. 3 ∂η ∂τ ∂η ∂η ∂η ∂χ2

(4.215)

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4.17 KP Burgers’ (KPB) Equation In a nonlinear dissipative medium, shock-like structures arise due to the interaction of nonlinearity and dissipation. However, if a medium contains both dispersion and dissipation, then the propagation properties of a small amplitude disturbance can be adequately explained by the KP Burgers’ (KPB) equation. While the viscosity, particle reflection, interparticle collisions, and Landau damping can lead to energy dissipation, the nonadiabaticity of the dust charge fluctuation provides an alternate physical mechanism causing the dissipation. Now, we shall derive the KPB equation in a pair-ion plasma system. The pair-ion plasma is important in plasma physics because of the various astrophysical environment like the pulsar magnetosphere, active galactic nuclei, and neutron star. The energies are produced in intense, and in turn, they produce electron–positron with the help of pair producing and annihilation. The physics of the pair-ion plasma became interesting as it descended from its astrophysical heights to the terrestrial laboratory. The laboratory pair plasma generates a sufficiently dense pair-ion plasma, which + consists of equal masses and positive and negative fullerene ions (C60 and − C60 ). Generally, the pair plasma is different from the standard e-i plasma in which the different masses of species break the symmetry between the components. If the pair-ion plasma is produced under identical conditions, it must remain symmetric, for example, with regard to their thermal speed and temperature [16]. Let us consider a pair-ion plasma model, consisting of positive as well as negative ions and electrons. Here, both ions have equal mass (m+ = m− = m), and the electron obeys the kappa distribution. The basic equations are ∂nα  + ∇ · (nα uα ) = 0 ∂t

(4.216)

∂ uα  )uα = q E − 1 ∇  pα + μ∇  2 uα + (uα · ∇ ∂t m nα m

(4.217)

RPT and Some Evolution Equations

187

where nα (α = +, − stands for positive ion, negative ion), vα , and pα are the number density, fluid velocity, and the pressure, respectively. q and μ are the charges and the kinematic viscosity. The isothermal pressure for ions is written as pα = nα T α .

(4.218)

∇2 φ = 4πe(αn− (1 + α)n+ + ne ).

(4.219)

Also, the Poisson equation is

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Here, ne = (1 − φ/(k − 1/2))−(k+1/2) . Tα represents the temperature with  = −∇  φ, and φ is the electrostatic wave T− = T+ . The electric field is E potential. n0 is the unperturbed number densities. The two-dimensional normalized basic equations are as follows: ∂n+ ∂ ∂ + (n+ u+ ) + (n+ v+ ) = 0 ∂t ∂x ∂y ∂n− ∂ ∂ + (n− u− ) + (n− v− ) = 0 ∂t ∂x ∂y ∂u+ ∂u+ ∂u+ ∂φ 1 ∂n+ ∂ 2 u+ + u+ + v+ + =− + η+ ∂t ∂x ∂y ∂x n+ ∂x ∂x2

(4.220) (4.221) (4.222)

∂v+ ∂v+ ∂v+ ∂φ 1 ∂n+ ∂ 2 v+ + u+ + v+ + =− + η+ ∂t ∂x ∂y ∂y n+ ∂y ∂y 2

(4.223)

∂φ β ∂n− ∂u− ∂u− ∂u− ∂ 2 u− − + u− + v− =− + η− ∂t ∂x ∂y ∂x n− ∂x ∂x2

(4.224)

∂v− ∂v− ∂v− ∂φ β ∂n− ∂ 2 v− − + u− + v− =− + η− ∂t ∂x ∂y ∂y n− ∂y ∂y 2

(4.225)

∂φ ∂φ + = αn− − (1 = α)n+ + ne . ∂x2 ∂y 2

(4.226)

Here, uα and vα are the ion fluid velocity along the x-axis and the y-axis, respectively, and normalization is taken as nα uα eφ x , uα → , φ→ , x→ , t → ωp t. n0 vs T+ λ+    Here, λ+ = T+ /(4πn0 e2 ), ωp = 4πn0 e2 /m, and vs = T+ /m. Also, β = T− /T+ , η+ = μ+ /(λ+ vs ) and η− = μ− /(λ− vs ). The viscosity coefficients of the fluids of the same mass are different since their temperatures are assumed to be different. nα →

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Waves and Wave Interactions in Plasmas

To obtain the KPB equation, we introduce the stretch co-ordinate as ξ = 1/2 (x − λt),

χ = y, τ = 3/2 t.

(4.227)

Also, we expand the dependent variable as nα = 1 + n1α + 2 n2α + · · · , uα = 0 + u1α + 2 u2α + · · ·

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vα = 0 + 3/2 vα1 + 5/2 vα2 + · · · , φ = 0 + φ1 + 2 φ2 + · · · .

 (4.228)

Assuming the value of η to be small, i.e., η± = 1/2 η0± , where η0± is o(1). Substituting Equations (4.227)–(4.228) into the normalized equation and collecting different powers of  and equating the lowest order of , we get ⎫ 1 −1 λ n1+ = 2 φ1 , n1− = 2 φ1 , u1+ = 2 φ1 ⎪ ⎬ λ −1 λ −β λ −1 (4.229) −λ 1 1 λ −λ 1 ⎪ 1 1 1 u− = 2 φ , v+ = 2 φ , v− = 2 φ .⎭ λ −β λ −1 λ −β From these set of equations, we get the phase velocity of the wave as λ=

 (1 + β)/2.

(4.230)

Taking the next higher-order equation in , we get ∂n1+ ∂n2 ∂v 1 ∂u2+ ∂ 1 1 −λ + + (n+ u+ ) + + + =0 ∂τ ∂ξ ∂ξ ∂χ ∂ξ

(4.231)

∂n1− ∂n2 ∂v 1 ∂u2− ∂ 1 1 −λ − + (n− u− ) + − + =0 ∂τ ∂ξ ∂ξ ∂χ ∂ξ

(4.232)

∂u1+ ∂u2 ∂u1 ∂n2+ ∂n1 ∂ 2 u1+ ∂φ2 − λ + + u1+ + + − n1+ + − η0+ + =0 ∂τ ∂ξ ∂ξ ∂ξ ∂ξ ∂ξ ∂ξ 2 (4.233) ∂u1− ∂u2 ∂u1 ∂n2 ∂n1 ∂ 2 u1− ∂φ2 − λ − + u1− − − + β − − βn1− − − η0− =0 ∂τ ∂ξ ∂ξ ∂ξ ∂ξ ∂ξ 2 ∂ξ (4.234) ∂ 2 φ1 k + 1/2 2 (k + 3/2)(k + 1/2) 1 2 = αn2− − (1 + α)n2+ + φ + (φ ) . ∂ξ 2 k − 1/2 2(k − 1/2)2 (4.235)

RPT and Some Evolution Equations

189

Then, eliminating n2α , u2α , and φ2 from Equations (4.231)–(4.235) and using Equations (4.229)–(4.230), we obtain the KP Burgers’ equation as   1 ∂ ∂φ1 ∂ 3 φ1 ∂ 2 φ1 ∂ 2 φ1 1 ∂φ + Aφ +B − C + D = 0 (4.236) ∂ξ ∂τ ∂ξ ∂ξ 2 ∂χ2 ∂ξ 3 where

 (1 + α)(3λ2 − 1) α(3λ2 − β) (k + 3/2)(k + 1/2) − A= − B, (λ2 − 1)3 (λ2 − β)3 (k − 1/2)2

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B=

(λ2 − 1)2 (λ2 − β)2 , 2λ[(1 + α)(λ2 − β)2 + α(λ2 − 1)2 ]

C=

(1 + α)η0+ (λ2 − β)2 − (λ2 − 1)2 αη0− , 2[(1 + α)(λ2 − β)2 + α(λ2 − 1)2 ]

D=

λ 2

where A and C are the coefficients of nonlinearity and dissipation, whereas B and D are the coefficients of the predominant and weak dispersion, respectively. With the help of tanh method, we find the solution of Equation (4.236), where we use the transformation ζ = k(ξ + χ − u0 τ ). Here, k is a dimensionless wave number and the solution is       2 6C 2 Cξ 6C 1 +D τ φ (ξ, χ, τ ) = 1 − tanh +χ− 25AB 10B 25B       2 3C 2 Cξ 6C 2 + + D τ . (4.237) Sech +χ− 25AB 10B 25B The solution will exist only when T+ = T− . But when T+ = T− , the coefficients A, B, and C become undetermined, and the soliton structure can not be obtained. 4.18 Damped KP (DKP) Equation In a two-dimensional system, the soliton was first modeled by Kadomstev and Petviashvili through a two-dimensional partial differential equation known as KP equation for a cold plasma system. It is a multidimensional extension of the well-known KdV equation. The investigations are done in the framework of the KP equation ignoring the collisional (dissipative) effects. These possesses the characteristics of an integrable system and were

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Waves and Wave Interactions in Plasmas

an extreme simplification of the real plasmas in nature. However, with the inclusion of the dissipative effects, the plasma remains no longer a Hamiltonian. The nonlinear localized structures in such types of non-Hamiltonian systems are known as dissipative solitons [17]. In addition to the interplay between nonlinearity and dispersion necessary for the formation of a solitary wave structure in an integrable system, a balance between gain and loss should exist to have a dissipative nonlinear localized structure. New features of dispersion and nonlinearity, in dissipative longitudinal solitons in a 2D system, have been pointed out in complex plasma [18]. It is, therefore, tempting to search for the role of dissipative effects due to plasma-neutral collisions on the propagation of ion solitary waves. Now, we investigate the dissipative solitons propagating in degenerate dense plasmas, inspired by the current interest in the dissipative solitons. This may link the dissipative solitons’ paradigm to plasma dynamics. To derive the DKP equation, we consider a collisional dusty plasma having negatively charged ions, massive dust grains, and isothermal electrons in presence of an external static magnetic field acting along z-axis. The dust dynamics are not considered, and dust charge is assumed to be constant. Let us also assume that the wave is propagating in the xz plane, and accordingly, the normalized basic equations are given by ∂n ∂(nu) ∂(nw) + + ∂t ∂x ∂z ∂u ∂u ∂u +u +w ∂t ∂x ∂z ∂v ∂v ∂v +u +w ∂t ∂x ∂z ∂w ∂w ∂w +u +w ∂t ∂x ∂z

= 0, =−

(4.238) ∂φ + v, ∂x

= −u,

(4.239) (4.240)

∂φ − vid w, ∂z

(4.241)

∂2φ ∂2φ + 2 = δ1 + δ2 n e − n ∂x2 ∂z

(4.242)

=−

where ne = eσφ . nj (j = e, i, and d stands for electron, ion, and dust particle, respectively) and φ are the number density and the electrostatic potential. u, v, and w are velocity along the x, y, z axis, respectively. Here, σ = Te /Ti . The charge neutrality condition is ni0 = zd nd0 + ne0 . So, δ1 + δ2 = 1. δ1 = nnd0i0zd and δ2 = nne0 , where nj0 denotes unperturbed number i0 densities. zd denotes the dust charge number, and the dust charge is qd = −ezd , where e denotes the elementary charge.

RPT and Some Evolution Equations

191

Here, the normalization is done in the following way: ni , ni0

v w eφ x , w→ , φ→ , x → , t → ωt Ci Ci Ti λ   where Ci = Ti /mi , λD = Ti /(4πe2 n0 ), ωpi , νid , and mi are the ion acoustic speed, Debye length, ion plasma frequency, dust–ion collisional frequency, and the ion mass, respectively. To obtain phase velocity and the DKP equation, we introduce the same stretched coordinates as in Equation (4.187). The expansions of the dependent variables are considered the same as Equation (4.211). Substituting the above Equation (4.211) along with the same stretched coordinates (4.187) into Equations (4.238)–(4.242) and equating the coefficients of lowest powers of , we get

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ni →

(1)

ni

=

u→

u , Ci

1 (1) w , λ

v→

w(1) =

1 (1) φ , λ

φ(1) =

1 (1) n , δ2 σ i

λ

∂u(1) ∂φ(1) = . ∂ξ ∂χ (4.243)

From relations (4.243), one can obtain the phase velocity as λ2 =

1 . δ2 σ

(4.244)

Equating the coefficients of the next higher order of  , we have (1)

(1)

∂ni ∂τ

+

(2)

∂u(1) ∂w(2) ∂ni w(1) ∂n + −λ i + = 0, ∂χ ∂ξ ∂ξ ∂ξ

(4.245)

(2)

∂u(1) ∂u(1) ∂φ(2) ∂u − λ i + w(1) + = 0, ∂τ ∂ξ ∂ξ ∂χ ∂w(1) ∂w(2) ∂w(1) ∂φ(2) −λ + w(1) =− − νid0 w(1) , ∂τ ∂ξ ∂ξ ∂ξ   ∂ 2 φ(1) σ 2 (φ(1) )2 (2) (2) = δ − ni . σφ + 2 ∂ξ 2 2 From Equations (4.245)–(4.248), we get DKP equation as   ∂φ(1) ∂ 3 φ(1) ∂ 2 φ(1) ∂ ∂φ(1) (1) + Aφ(1) +B + Cφ +D =0 3 ∂ξ ∂τ ∂ξ ∂ξ ∂χ2 where A =

λ3 2 ,

B=

3−σλ2 2λ ,

C=

νeid0 λ3 ,

and D = λ2 .

(4.246) (4.247) (4.248)

(4.249)

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Waves and Wave Interactions in Plasmas

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4.19 Zakharov–Kuznetsov (ZK) Equation In 1974, Zakharov and Kuznetsov derived a nonlinear equation named the ZK equation as a nonlinear evolution equation to study IAWs in a strongly magnetized plasma in the z dimension. The dynamics of IASWs are the silent character in the nonlinear phenomena of modern plasma physics. Nonlinear differential equations, such as the KdV equation and the MKdV equation, describe the different properties of nonlinear plasma waves in one dimension. The Zakharov–Kuznetsov (ZK) equation, and KP equation, describe the different properties of nonlinear plasma waves in more than one dimension. Zakharov and Kuznetsov [19] derived and used to study three-dimensional acoustic ion waves in magnetized plasma. This equation regulates the nature of weakly nonlinear waves in a plasma that consists of cold ions and hot superthermal electrons in a magnetic field. To derive the ZK equation, we consider a plasma model consisting of cold ions, superthermal distributed electrons in the presence of dust par = yˆB  0 along the y-axis. ticles. Also, the external static magnetic field B Accordingly, the normalized equations are as follows: ∂ni ∂(ni u) ∂(ni v) ∂(nw) + + + ∂t ∂x ∂y ∂z   ∂u ∂ ∂ ∂ + u +v +w u ∂t ∂x ∂y ∂z   ∂ ∂ ∂ ∂v + u +v +w v ∂t ∂x ∂y ∂z   ∂w ∂ ∂ ∂ + u +v +w w ∂t ∂x ∂y ∂z

= 0,

(4.250)

=−

Ωi ∂φ − w, ∂x ωpi

(4.251)

=−

∂φ , ∂y

(4.252)

=−

Ωi ∂φ + u, ωpi ∂z

(4.253)

∂ 2 φ ∂ 2φ ∂ 2 φ + 2 + 2 = ne − ni ∂x2 ∂y ∂z

(4.254)

where ne =

 1−

φ k − 1/2

−(k+1/2) .

Here, n, ne , ui (= u, v, w), Te , mi , e, φ, Ωi , ωpi , νid, and λD are the ion number density, electron number density, ion velocity, electron temperature, ion mass, electron charge, electrostatic potential, ion cyclotron frequency, ion plasma frequency, dust–ion collision frequency, and Debye length, respectively.

RPT and Some Evolution Equations

193

The normalization is done in the following way: ni →

ni , n0

ne →

ne , ne0

ui →

ui , Cs

φ→

eφ , Te

x→

x , λD

t → ωpi t.

Here, δ1 = nnd0 and δ2 = nne0 with the condition δ1 + δ2 = 1. λD =   1/2 i0  i01/2 −1 Te Te . , ωpi = 4πnme0i e2 , and Cs = m 4πne0 e2 i To obtain the ZK equation, we introduce new stretched coordinates as

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ξ = 1/2 x,

ζ = 1/2 (x − λt),

η = 1/2 y,

τ = 3/2 t.

(4.255)

λ is the phase velocity of waves. The dependent variables are expanded in the following way: ⎫ (1) (2) ⎪ ni = 1 + ni + 2 ni + · · · ⎪ u = 0 + 3/2 u(1) + 2 u(2) + · · · ⎬ (4.256) v = 0 + v (1) + 2 v (2) + · · · ⎪ 3/2 (1) 2 (2) w = 0 +  w +  w + ···⎪ ⎭ φ = 0 + φ(1) + 2 φ(2) + · · · . To obtain the phase velocity and the ZK equation, we substitute the above expansions along with stretching coordinates (4.255) into Equations (4.250)–(4.254), and equating the coefficient of , we get ⎫ ωpi ∂φ(1) (1) v (1) φ(1) ⎪ (1) (1) , w =− , v = , ni = λ Ωi ∂ξ λ ⎪ ⎬ ωpi ∂φ(1) ωpi ∂u(1) (4.257) , w(2) = λ , u(1) = ⎪ Ωi ∂η Ωi ∂ζ ⎪ ωpi ∂w(1) (1) k − 1/2 u(2) = −λ , φ(1) = ni . ⎭ Ωi ∂ζ k + 1/2 Equating the coefficient of the next higher order of , we have

⎫ (2) ⎪ ∂u(2) ∂v (2) ∂w(2) ∂ (1) (1) ∂ni −λ + + (n v ) + + = 0⎪ ∂ζ ∂ξ ∂ζ i ∂ζ ∂η ⎬ ∂v (1) ∂v (2) ∂v (1) ∂φ(2) −λ + v (1) + = 0⎪ ∂τ ∂ζ ∂ζ ∂ζ ⎪ 2 (1) 2 (1) 2 (1) ∂ φ ∂ φ ∂ φ (2) (2) (1)2 ⎭ + + − A φ − A φ + n = 0 1 2 i ∂ξ 2 ∂ζ 2 ∂η 2

(1)

∂ni ∂τ

where A1 =

(k+1/2) (k−1/2)

and A2 =

2(k−1/2) (k+1/2) .

(4.258)

Waves and Wave Interactions in Plasmas

194

From (4.257), one can obtain the phase velocity 1 − λ2 A1 = 0.

(4.259)

Expressing all the perturbed quantities in terms of φ(1) from Equations (4.257)–(4.258), the ZK equation is obtained by ∂φ(1) ∂φ(1) ∂ 3 φ(1) ∂ + Aφ(1) +B +C 3 ∂τ ∂ζ ∂ζ ∂ζ



∂ 2 φ(1) ∂ 2 φ(1) + ∂ξ 2 ∂η 2

 =0

(4.260)

where

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A=

λ 2A1



 3A1 − 2A 2 , λ2

B=

λ , 2A1

and C =

 2  ωpi λ 1 + A1 λ2 2 . 2A1 Ωi

The ZK Equation (4.260) admits higher-dimensional solutions, such as plane solitons, 2-D cylindrical solitons, and 3-D spherical solitons. For the sake of simplicity, we here consider the plane solitary wave solutions φ = φ(χ) of the ZK equation, where χ is the coordinate χ = ζ cos α + ξ sin α − u0 τ.

(4.261)

Here, α is the angle between the direction of propagation and the magnetic field B0 and u is the soliton speed in the (ξ − η − ζ) frame. Without loss of generality, we assume that the wave propagates in the (ξ − ζ) plane because of cylindrical symmetry around the ζ-axis. Using transformation (4.261) and assuming that φ(1) together with its χ-derivatives vanish at χ → ∞, the ZK Equation (4.260) yields the following solution: (1)

φ

3u Sech2 = A cos α



1 u/ cos α 2 4 B cos α + C sin2 α

1/2  χ

(4.262)

 1/2 u/ cos α 3u 1 where φm = A cos is the amplitude and W = is the 2 2 4 B cos α+C sin α α inverse of the width of the solitary waves. The effects of k on the solitons are illustrated by plotting φ(1) versus χ as shown in Figure 4.15. It is also observed from Figure 4.15 that k is directly proportional to the amplitude and the width of the solitary waves.

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RPT and Some Evolution Equations

195

Fig. 4.15: Solitary wave solution of Equation (4.260) for different values of κ.

4.20 ZK Burgers’ (ZKB) Equation Like the KdVB equation or KPB equation, ZK Burgers’ (ZKB) equation is also derived to study the two-dimensional solitons and shocks in a single equation in the presence of the magnetic field. To derive the ZKB equation, we consider a homogeneous, collisionless magnetoplasma consisting of ions and q-nonextensive distributed electrons. The viscosity is also considered. ˆ 0 , where xˆ is The plasma is confined in an external magnetic field B0 = xB the unit vector along the x-direction and B0 is the external static magnetic field. Accordingly, the basic normalized equations are as follows: ∂ni  + ∇ · (ni ui ) = 0, ∂t ∂ui ˆ) + μ∇  )ui = −∇  φ + Ω(ui × x  2 ui , + (ui · ∇ ∂t  2 φ = ne − (1 + α)ni + α ∇

(4.263) (4.264) (4.265)

where nj (j = i and e stand for ions and electrons), u, v, and w are number density, the ion fluid speed in x, y, and z directions, respectively. Here, the

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196

normalization is done in the following way: ni , ne0

ni →

ui →

ui , Cs

φ→

eφ , Te

x→

x , λD

t → ωpi t

where ne0 is the electron unperturbed equilibrium plasma density and α =  zd nd0 /ne0 . Cs = KB Te /mi and KB are  the ion acoustic velocity and −1 Boltzmann constant, respectively. ωpi = mi /(4πe2 ne0 ) is the ion plasma  frequency, and λD = KB Te /(4πe2 ne0 ) is the Debye length. Ω is the ion cyclotron frequency, c is the velocity of light, and μ is the kinematic viscosity. To obtain the ZK Burgers’ equation, we introduce the stretched coordinate as

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ξ = 1/2 (x − λt), η = 1/2 y,

ζ = 1/2 z,

τ = 3/2 t,

μ = 1/2 μ0 (4.266)

where μ is scaled in such a way that μ0 becomes the effective viscosity. The viscosity is incorporated in nonlinear equations like KdVB, KPB, and ZKB, assumed to be the order of 1/2 suggested in Equation (4.266) in the perturbation scheme. Let us expand the dependent variables as ⎫ (1) (2) ni = 1 + ni + 2 ni + · · · , ⎪ ⎪ u = u(1) + 2 u(2) + · · · , ⎬ (4.267) v = 3/2 v (1) + 2 v (2) + · · · , ⎪ w = 3/2 w(1) + 2 w(2) + · · · , ⎪ ⎭ φ = φ(1) + 2 φ(2) + · · · . By substituting Equations (4.266) and (4.267) into Equations (4.263) − (4.265) and equating the lowest power of , one gets (1)

ni

=

1 (1) u , λ

u(1) =

1 (1) φ , λ

v (1) = −

1 ∂φ(1) . Ω ∂ζ

(4.268)

From Equations (4.268), we get the phase velocity as  λ = 2(1 + α)/(q + 1). The next order of  is w

(1)

∂w(1) 1 ∂φ(1) ∂v (1) Ω Ω = , = − w(2) , = v (2) , Ω ∂η ∂ξ λ ∂ξ λ

(4.269) ⎫ ⎪ ⎬

⎪ ∂ 2 φ(1) ∂ 2 φ(1) ∂ 2 w(1) (2) (2) (1) 2 ⎭ + + − β φ = β (φ ) − (1 + α)n 1 2 i ∂ξ 2 ∂η 2 ∂ζ 2

(4.270)

RPT and Some Evolution Equations

197

where β1 = (q +1)/2 and β2 = (q +1)(3−q)/8. Considering the next higher order of , we get ⎫ (1) (2) ⎪ ∂u(2) ∂v (2) ∂w(2) ∂ (1) (1) ∂ni ∂ni −λ + (n u ) + + + = 0, ⎪ ∂τ ∂ξ ∂ξ i ∂ξ ∂η ∂ζ ⎬ (1) ∂u(1) ∂u(2) ∂φ(2) (4.271) (1) ∂u −λ +u + ⎪ ∂τ ∂ξ ∂ξ ∂ξ ⎪  2 (1)  2 (1) 2 (1) ⎭ − μ0 ∂ ∂ξu 2 − ∂ ∂ηu 2 − ∂ ∂ζu 2 = 0.

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(2)

Now, we eliminate u(2) and ni from Equation (4.271) by using Equation (4.270). After simplification and eliminating the second order of φ, we finally obtain the ZKB equation as   ∂φ(1) ∂φ(1) ∂ 3 φ(1) ∂ 2 φ(1) ∂ ∂ 2 φ(1) + Aφ(1) +B + C + ∂τ ∂ξ ∂ξ 3 ∂ξ ∂η 2 ∂ζ 2  2 (1)  ∂ φ ∂ 2 φ(1) ∂ 2 φ(1) −D + + =0 (4.272) ∂ξ 2 ∂η 2 ∂ζ 2 where

 3 2β2 v03 − , , B= 4 v0 1+α 2(1 + α)   v3 1 1 μ0 C= 0 + 2 , and D = . 2 1+α Ω 2 v3 A= 0 2



4.21 Damped ZK (DZK) Equation Like DKdV and DKP equations, now we shall derive the damped ZK (DZK) equation. For this, let us consider a plasma model consisting of cold ions, superthermal electrons in the presence of dust particles, and the external static magnetic field B = yB ˆ 0 along the y-axis. The normalized continuity, momentum, and Poisson equations are as follows: ∂ni ∂(ni u) ∂(ni v) ∂(ni w) + + + = 0, ∂t ∂x ∂y ∂z   ∂u ∂ ∂ ∂ Ωi ∂φ − + u +v +w w, u=− ∂t ∂x ∂y ∂z ∂x ωpi   ∂ ∂ ∂ ∂v ∂φ + u +v +w − νid v, v=− ∂t ∂x ∂y ∂z ∂y

(4.273) (4.274) (4.275)

Waves and Wave Interactions in Plasmas

198

  ∂w ∂ ∂ ∂ Ωi ∂φ + u +v +w + u, w=− ωpi ∂t ∂x ∂y ∂z ∂z

(4.276)

∂2φ ∂2φ ∂2φ + 2 + 2 = δ1 + δ2 n e − n i . ∂x2 ∂y ∂z

(4.277)

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The normalized electron density is written as  −(k+1/2) φ ne = 1 − k − 1/2 where ni , ne , ui (= u, v, w), φ, and νid are the ion number density, electron number density, ion velocity, electrostatic potential, and dust–ion collision frequency, respectively. Here, we use the same normalization like in Section 4.19. To obtain the DZK equation, we introduce the same stretched coordinates as in Equation (4.255). The expressions of the dependent variables are considered the same as Equations (4.256) with νid ∼ 3/2 νid0 .

(4.278)

Substituting the above expansions along with stretching coordinates into Equations (4.273)–(4.277) and equating the coefficient of , we get φ(1) = where A1 = (1)

ni

=

k+1/2 k−1/2 .

v (1) λ ,

(1)

ni δ1 A1

(4.279)

Equating the coefficient of 3/2 , we get (1)

w(1) = − Ωpii ∂φ∂ξ , ω

v (1) =

φ(1) λ ,

Considering the coefficient of 2 , we obtain w(2) = λ

ωpi ∂u(1) ωpi ∂w(1) , u(2) = −λ , Ωi ∂ζ Ωi ∂ζ

u(1) =

ωpi ∂φ(1) Ωi ∂η .

(4.280)

⎫ ⎪ ⎬

⎪ 2 ∂ 2 φ(1) ∂ 2 φ(1) ∂ 2 φ(1) (2) + + = δ1 (A1 φ(2) + A2 φ(1) ) − ni . ⎭ 2 2 2 ∂ξ ∂ζ ∂η

(4.281)

Comparing the coefficients of 5/2 , we obtain (1)

∂ni ∂τ

⎫ (2) ⎪ ∂u(2) ∂v (2) ∂w(2) ∂ (1) (1) ∂ni −λ + + (n v ) + + = 0⎬ ∂ζ ∂ξ ∂ζ i ∂ζ ∂η ⎪ ∂v (1) ∂v (2) ∂v (1) ∂φ(2) −λ + v (1) + − νid0 v (1) = 0 ⎭ ∂τ ∂ζ ∂ζ ∂ζ

(4.282)

RPT and Some Evolution Equations

where A2 =

(k+1/2)(k+3/2) . 2(k−1/2)2

199

Using relationships (4.280), we get 1 − λ2 A1 δ2 = 0.

(4.283)

Expressing all the perturbed quantities in terms of φ(1) from Equation (4.281), the damped ZK equation is obtained by ∂φ(1) ∂φ(1) ∂ 3 φ(1) + Aφ(1) +B + Dφ(1) ∂τ ∂ξ ∂ζ 3   ∂ 2 φ(1) ∂ ∂ 2 φ(1) +C + =0 ∂ζ ∂ξ 2 ∂η 2

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where A=

λ 2A1



(4.284)

 3A1 − 2A 2 , λ2

λ , 2δ2 A1   ω2 λ 2 pi C= 1 + δ2 A1 λ 2 , 2δ2 A1 Ωi νid0 . D= 2 B=

References [1] [2] [3] [4]

[5] [6] [7] [8] [9] [10] [11] [12] [13]

A. H. Nayfeh, Introduction to Perturbation Techniques. Wiley-vch (2004). T. Taniuti and C. C. Wei, J. Phys. Soc. Jpn. 24, 941 (1968). C. H. Su and C. S. Gardner, J. Math. Phys. 10, 536 (1969). C. S. Gordner and G. K. Morikawa, Coutant Institute of Mathematical Sciences Report No. NY09082, 1960 (unpublished); Comm. Pure Appl. Math. 18, 35 (1965). D. J. Korteweg and G. deVries, Phil. Mag. 39, 422 (1985). A. Sen, S. Tiwary, S. Mishra, and P. Kaw, Adv. Space Res. 56(3), 429 (2015). S. Chowdhury, L. K. Mandi, and P. Chatterjee, Phys. Plasmas 25, 042112 (2018). V. S. Aslanov and V. V. Yudintsev, Adv. Space Res. 55, 660 (2015). H. Schamel, Plasma Phys. 14, 905 (1972). G. Williams, F. Verheest, M. A. Hellberg, M. G. M. Anowar, and I. Kourakis, Phys. Plasmas 21, 092103 (2014). H. Schamel, J. Plasma Phys., 9, 377 (1973). M. A. Hossen, M. R. Hossen, and A. A. Mamun, J. Korean Phys. Soc. 65, 1883 (2014). S. Chandrasekhar, Phi. Mag. 11, 592 (1931).

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Waves and Wave Interactions in Plasmas

B. B. Kadomtsev and V. I. Petviashvili, Soviet Phys. Dokl. 15, 539 (1970). R. Grimshaw, Stud. Appl. Math. 73, 1 (1985). W. Oohara and R. Hatakeyama, Phys. Rev. Lett. 91, 205005 (2003). V. I. Karpman and E. M. Maslov, Sov. Phys. JETP 46, 281 (1977). D. Samsonov, A. V. Ivlev, R. A. Quinn, G. Morfill, and S. Zhdanov, Phys. Rev. Lett. 88, 095004 (2002). [19] V. E. Zakharov and E. A. Kuznetsov, Sov. Phys. JETP 39, 285 (1974).

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[14] [15] [16] [17] [18]

Chapter 5

Dressed Soliton and Envelope Soliton

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5.1 Dressed Soliton KdV, KP, and ZK type equations help us to describe solitary wave propagation in plasmas. Taylor and Ikezi [1] first showed that for ion acoustic solitary waves, the experimental observation data do not match with the soliton solution of KdV, KP, ZK, or similar equations. To get an accurate result, the theory needs to be improved so that the amplitude, width, and velocity of the solitary waves will be more accurate with the physical situation. Accordingly, a modification or correction of the amplitude, width, and velocity is necessary. One such modification is the inclusion of the higher-order perturbation correction of the KdV, KP, and ZK types of equations, and accordingly, the improvement of Mach number, amplitude, and width should be done properly. The solution of these equations with higher-order correction (source term) will surely give a better result. This improved solution is named Dressed Solution. In this chapter, we are going to derive the NLEEs with source term and to obtain higher-order correction of amplitude, width, and velocity for KdV or KdV type soliton in classical plasma, dusty plasma, and quantum plasma and higher-order correction of the same in the frame of the ZK equation. 5.2 Dressed Soliton in a Classical Plasma To obtain the dressed soliton (higher-order correction to KdV equation) in a classical plasma model, we derive the KdV equation from the plasma model. Here, we consider a collisionless unmagnetized electron–ion plasma, where ions are mobile and electrons obey the Maxwell distribution.

201

202

Waves and Wave Interactions in Plasmas

Step 1: Basic normalized equation The normalized fluid equations are written as ∂ni ∂(ni ui ) + =0 ∂t ∂x ∂ui ∂ui ∂φ + ui =− ∂t ∂x ∂x ∂2φ = e φ − ni ∂x2

(5.1) (5.2) (5.3)

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where ni , ui , and φ are the ion density, ion velocity, and electrostatic potential, respectively. Here, we apply the same normalization procedure as in Section 4.7. Step 2: KdV equation evolution To obtain the KdV equation, we introduce the same stretched coordinates as used in (4.40). Expansions of the dependent variables are also considered the same as in (4.41). Substituting (4.40) and (4.41) in (5.1)–(5.3), and equating the first order of , we get (1)

ni

(1)

= ui

= φ(1)

and λ = 1.

(5.4)

Taking the coefficient of the next higher-order of , we obtain the KdV equation as ∂φ(1) ∂φ(1) 1 ∂ 3 φ(1) + φ(1) + = 0. ∂τ ∂ξ 2 ∂ξ 3

(5.5)

Step 3: Higher-order approximation As stated earlier, the KdV soliton does not always match with observed data, and to overcome this discrepancy, a higher-order correction of the KdV equation has been prescribed here. So, we include the higher-order perturbation corrections to the KdV solitons. It has resulted in a better agreement with the experimental observations. When the higher-order perturbation corrections are included in the KdV soliton, then the improved soliton is the dressed soliton. However, from perturbation theory, it is known that the higher-order correction is to be considered only if the amplitude of the higher-order correction is much less than the amplitude of the original KdV soliton. If the amplitude of the higher-order correction is

Dressed Soliton and Envelope Soliton

203

greater or equal to the amplitude of the KdV soliton, then the solution obtained is not physical and is to be ignored. The KdV equation contains the lowest-order relation between the nonlinearity and dispersion. To get a better result, we include the second-order (2) (2) perturbation term φ(2) , ni , ui in (4.41). This will provide us a more accurate relationship between amplitude, width, and velocity of the wave. Now, we shall investigate the higher-order nonlinear and dispersion effect. (2) So, we take the next order of  and obtain second-order quantities ni and (2) ui , and it is expressed as (2)

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ni

= φ(2) +

1  (1) 2 ∂ 2 φ(1) φ − , 2 ∂ξ 2

(2)

ui

= φ(2) −

1 ∂ 2 φ(1) . 2 ∂ξ 2

(5.6)

The behavior of the second-order potential φ(2) is determined from the linear inhomogeneous differential equation of φ(2) , whose source term is given as a function of φ(1) . Thus, we determine the nonsecular solution of φ(2) from     ∂ φ(1) φ(2) ∂φ(2) 1 ∂ 3 φ(2) + + = S φ(1) (5.7) 3 ∂τ ∂ξ 2 ∂ξ     3 (1) 5 (1) ∂ ∂φ(1) 2 where S φ(1) = M φ(1) ∂ ∂ξφ3 + N ∂ ∂ξφ5 + P ∂ξ ; M = 12 , N = ∂ξ − 38 , P = − 58 . Step 4: Renormalization Ichikawa and his collaborators [2–4] first prescribed the corrections to the KdV soliton in the next higher order of approximation in the RPT. However, their results included the secular terms, and hence, the theory of higher-order approximation in the RPT was not established. For the general weakly dispersive system, the higher-order approximations come from the linearized KdV equation. It includes a source term inhomogeneous in nature. The reason is that the secular terms appear due to self-resonance, which in turn comes from nonlinear oscillation. Furthermore, the secular terms, resulting from the soliton part, are eliminated by adding them to the KdV equation. The derivatives of the higher-order conserved densities are expressed in terms of the conserved quantities also. The resultant equation is known as the generalized KdV equation. It is seen from the higher-order derivatives of conserved densities of the KdV equation that the overall solutions are solutions of the linearized KdV equation along with the secular solutions for the special initial data. The renormalization

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204

technique is a technique that is used to eliminate the secular terms that appeared in the one-soliton solution developed by Kodama and Taniuti [5]. Then, the nonsecular solutions are obtained in all order, as a particular solution to the higher-order equations. As a result, the solution to the original system becomes the KdV soliton with velocity shift plus nonlinear corrections, which are bounded all time. It is called the dress of the KdV soliton (dressed soliton) after Ichikawa [2]. To get a nonsecular solution for φ(2) , the method of renormalization is used. Accordingly, Equation (5.5) is modified as

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1 ∂ 3 φ(1) ∂φ(1) (1) ∂φ(1) ∂φ(1) +φ + + δΛ = 0. 3 ∂τ ∂ξ 2 ∂ξ ∂ξ

(5.8)

The inhomogeneous Equation (5.7) for φ(2) becomes  (1)  ∂φ(2) ∂(φ(1) φ(2) ) 1 ∂ 3 φ(2) ∂φ(2) ∂φ(1)  + + + δΛ = S φ + δΛ . (5.9) ∂τ ∂ξ 2 ∂ξ 3 ∂ξ ∂ξ The parameter δΛ in Equations (5.8) and (5.9)  is introduced in such a way that the secular (resonant) term in S φ(1) is canceled by the term δΛ

(1) ∂φ ∂ξ

in Equation (5.9).

Step 5: Second-order inhomogeneous differential equation evolution Now, we introduce a new stationary frame variable η as η = ξ − (Λ + δΛ)τ

(5.10)

where (Λ + δΛ) = X − 1, and X is the Mach number. Hence, Equation (5.8) can be transformed into −Λ

dφ(1) (1) dφ(1) 1 d3 φ(1) +φ + = 0. dη dη 2 dη 3

(5.11)

Integrating Equation (5.11), the boundary conditions φ(1) and its derivatives vanish as η → ±∞. We obtain the stationary renormalized solitary wave solution of Equation (5.8) (up to 1st order in Λ) as  φ(1) = φ0 sech2 (Dη)

(5.12)

Dressed Soliton and Envelope Soliton

 =D = where φ0 = 3Λ, D Equation (5.9), we obtain −Λ

 Λ 1/2 2

205

. Using Equations (5.10) and (5.12) in

 (2) ) 1 d3 φ(2) dφ(2) d(sech2 (Dη)φ + φ0 + dη dη 2 dη 3

 Λφ0 sech2 (Dη  ) tanh(Dη  ) + A1 sech2 (Dη  ) tanh(Dη)  = −2Dδ

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 ) tanh(Dη  ) + A3 sech6 (Dη  ) tanh(Dη)  + A2 sech4 (Dη

(5.13)

where A1 = −32N D5 φ0 , A2 = 480N D5 φ0 − 8M D3 φ20 − 16D3 P φ20 , A3 = −720N φ0 D5 + 24M φ20 D3 + 24P D3 φ20 . Integrating Equation (5.13) with respect to η, the boundary conditions φ(2) and its derivatives vanish as η → ±∞, and we obtain a second-order inhomogeneous differential equation for φ(2) . 2 (2)  (2) + 1 d φ = δΛφ0 sech2 (Dη)  −Λφ(2) + φ0 sech2 (Dη)φ 2 dη 2

 ) + A12 sech4 (Dη  ) + A13 sech6 (Dη)  + A11 sech2 (Dη

(5.14)

A1 A2 where A11 = − 2D = 4N Λ2 φ0 , A12 = − 4D = −90N Λ3 + 9Λ3 (M + A3 3 3 2P ), A13 = − 6D = 90N Λ − 18(M + P )Λ . To cancel the secular term, we choose A11 + δΛφ0 = 0 which gives

δΛ = −4N Λ2.

(5.15)

Hence, using Equation (5.15), and putting φ0 = 3Λ, Equation (5.14) becomes 1 d2 φ(2)  ) − 1)φ(2) = A12 sech4 (Dη  ) + A13 sech6 (Dη)  + Λ(3 sech2 (Dη 2 dη 2 (5.16) where the secular term is removed. Equation (5.16) is a second-order inhomogeneous differential equation with a solution that can be written as φ(2) = φcomp + φp

(5.17)

where φcomp is the complementary function and φp is the particular solution.

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206

Step 6: Particular solution evolution

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A particular solution to the inhomogeneous equation only plays the role of the higher-order correction. Most authors’ [6–8] remedy to an obtained particular solution of Equation (5.16), using the method of variation in parameter, will be lengthy. Chatterjee and collaborates [9, 10] developed a simpler series solution method similar to the tanh method. To obtain the particular solution to Equation (5.16), here the series considered as power series of sech2 terms unlike in the tanh method is the series considered as power series of tanh term. This technique is based on a truncated power series solution of the second-order evolution and the solution can easily be obtained as a power series of sech2 terms. Let φp be defined as φp =

K 

 ) = a1 sech2 (Dη  ) + a2 sech4 (Dη)  + ··· ai sech2i (Dη

(5.18)

i=1

and be a particular solution of Equation (5.16). The unknown K will be determined after equating the highest power of sechξ that arises in the left- and right-hand sides of Equation (5.16) after the substitution. The  and highest power term that arises on the left-hand side is sech2K+2 (Dη) 6  on the right-hand side is sech (Dη). Hence, the value of K will be 2. The appropriate series that is required to solve Equation (5.16) is then written as  ) + a2 sech4 (Dη)  φp = a1 sech2 (Dη

(5.19)

where a1 and a2 are to be calculated by equating the coefficients of the different powers of sechξ (namely, sech2 ξ, sech4 ξ, and sech6 ξ), and after substituting Equation (5.19) in Equation (5.16), we get  3  ) − 2a2 Λsech6 (Dη)  a1 + 3a2 Λsech4 (Dη 2  ) + A13 sech6 (Dη).  = A12 sech4 (Dη

(5.20)

4  6  Equating the coefficients equa 2 of sech (Dη ) and sech 2(Dη) of the above tion, we obtain a1 = 3Λ A12 + A13 /Λ = 30N Λ − 6(2M + P )Λ2 , a2 = 13 − A2Λ = −45N Λ2 + 9(M + P )Λ2 .

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Thus, the series is truncated at k = 2. Hence, the particular solution φp can be expressed as    φp = 30N Λ2 − 6(2M + P )Λ2 sech2 (Dη)    + − 45N Λ2 + 9(M + P )Λ2 sech4 (Dη).

(5.21)

Step 7: To obtain complementary solution The complementary solution ψcomp of Equation (5.16) is given by

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ψcomp = C1 P32 (z) + C2 Q23 (z)

(5.22)

where P32 (z) and Q23 (z) are the associated Legendre functions defined as P32 (z) = 15z(1 − z 2 ) Q23 (z) =

1+z 2 15 z(1 − z 2 )ln + + 5 − 15(1 − z 2 ) 2 1 − z 1 − z2

(5.23) (5.24)

 [6–8]. The coefficients C1 and C2 can be determined where z = tanh(Dη) using the boundary conditions. The first term in the complementary function is secular which can be eliminated by renormalizing the amplitude. Also, φ2 (η) → 0 as η → ∞, so, C2 = 0. One can notice that the complementary part does not affect the stationary solution.

Step 8: Dressed soliton evolution Using Equations (5.12) and (5.18), the stationary one soliton solution up to second-order in Λ, for ion acoustic waves in the classical plasma, is written as  ) + a1 sech2 (Dη  ) + a2 sech4 (Dη).  φ = φ(1) + φ(2) = φ0 sech2 (Dη (5.25) The amplitude of the KdV soliton that includes the second-order contribution φ0 , the amplitude of the dressed soliton that includes the second-order

Waves and Wave Interactions in Plasmas

208 ∗

0 , and the width W

of the dressed soliton are as follows: contribution φ φ0 = 3(Λ + δΛ) = 3(Λ − 4N Λ2 ,

0 ∗ = 3(Λ + δΛ) + a1 + a2 = 3(Λ − 4N Λ2 + a1 + a2 , φ  1/2 2

=D  −1 = W . Λ − 4N Λ2 Equation (5.5) is the standard sech2 type KdV soliton, whose amplitude  φ0 = 3Λ and width W = 2/Λ.

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5.3 Dressed Soliton in a Dusty Plasma To derive dressed soliton in dusty plasma, we consider a four-component dusty plasma model consisting of Boltzmann distributed electrons, nonthermal ions, and negatively and positively charged dust grains. The basic normalized equations are ∂n1 ∂ + (n1 u1 ) = ∂t ∂x ∂u1 ∂u1 + u1 = ∂t ∂x ∂n2 ∂ + (n2 u2 ) = ∂t ∂x ∂u2 ∂u2 + u2 = ∂t ∂x

0

(5.26)

∂φ ∂x

(5.27)

0

(5.28)

−αβ

∂φ ∂x

(5.29)

∂2φ = n1 − (1 − μi + μe )n2 ∂x2 + μe eσφ − μi (1 + β1 φ + β1 φ2 )e−φ

(5.30)

where nj (j = 1, 2 stands for negatively charged and positively charged grains, respectively) is the number density, uj is the charged dust fluid m1 2 speed, and φ is the electric potential. Here, α = Z Z1 , β = m2 , μe = ne0 ni0 Ti 4α1 Z1 n10 , μi = Z1 n10 , σ = Te , β1 = 1+3α1 . α1 is the ion nonthermal parameter that determines the number of fast (energetic) ions, and mj are masses of dust particles. Ti , Te , kB , and e are ion temperature, electron temperatures, the Boltzmann constant, and electron charge, respectively. nj0 , ne0 , and ni0 are the equilibrium number density of dust particle, electron, and ions (with Z1 and Z2 being the number of electrons and protons residing on

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209

a negative and positive dust particle). Here, we use the same normalization as in Section 4.10. Now, we derive the KdV equation from Equations (5.26)–(5.30) by employing the RPT, where the stretched coordinates are taken the same as (4.40) and the dependent variables are expanded the same as (4.140). Using Equation (4.140) in Equations (5.26)–(5.30), and equating terms of the first-order in , we get the phase velocity λ as λ2 =

1 + αβ(1 + μe − μi ) . σμe + μi − μi β1

(5.31)

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Similarly, equating terms of the next higher-order of , we obtain the KdV equation as ∂φ(1) ∂φ(1) ∂ 3 φ(1) + Aφ(1) +B =0 (5.32) ∂τ ∂ξ ∂ξ 3 2 2

1 4 2 where A = 2λ[1+αβ(1+μ 3α β (1 + μ − μ ) − 3 − λ (μ σ − μ ) , e i e i −μ )] e i 3

λ B = 2[1+αβ(1+μ . e −μi )] Next, we determine the higher-order nonlinear and dispersion effects of the KdV equation. We start by equating the next higher-order terms in , and after some standard algebra, we obtain the differential equation for the higher-order correction φ(2) as     ∂ φ(1) φ(2) ∂φ(2) ∂ 3 φ(2) +A +B = S φ(1) (5.33) 3 ∂τ ∂ξ ∂ξ

where  (1) 2 3 (1) ∂ ∂φ∂ξ  (1)   (1) 2 ∂φ(1) ∂ 5 φ(1) (1) ∂ φ S φ =L φ + Mφ +N +P , ∂ξ ∂ξ 3 ∂ξ 5 ∂ξ L=

1 (L1 + L2 + L3 ), 4λ3 [1 + αβ(1 + μe − μi )]

L1 = (−6A2 λ2 − 14Aλ − 15), L2 = −(1 − μi + μe )(6A2 αβλ2 − 20Aα2 β 2 λ + 15α3 β 3 ),  3 μi β 1 μi 6 μe δ L3 = 2λ +3 + , 2 2 2 M =

(−12ABλ2 − 16B) − (1 − μi + μe )(6ABαβ − 8Bα2 β 2 , 2[1 + αβ(1 − μi + μe )]4λ2

Waves and Wave Interactions in Plasmas

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210

N =

−3B 2 , 2λ

P =

(−9ABλ − 4B) − (1 − μi + μe )(9ABαβλ − 4Bα2 β 2 ) . 4λ2 [1 + αβ(1 + μe − μi )]

Equation (5.33) is the linear inhomogeneous differential equation in φ(2) , whose source term is given as a function of φ(1) , and the expression of φ(1) is known from Equation (5.32). Equation (5.33) describes the effect of higher-order nonlinearity and dispersion, and to get the dressed soliton, this needs to be solved. To get a nonsecular solution for φ(2) , we use the method of renormalization, developed by Kodama and Taniuti [5]. Accordingly, equations (5.32) and (5.33) are modified. Introducing the new stationary frame η = ξ − (Λ + δΛ)τ and proceeding the same way as before, we get −Λ

dφ(1) d3 φ(1) dφ(1) + Aφ(1) +B = 0. dη dη dη 3

(5.34)

Integrating Equation (5.34) under the boundary conditions φ(1) and its derivatives vanish as η → ±∞, we obtain the stationary renormalized solitary wave solution as  φ(1) = φ0 sech2 (Dη) (5.35)  Λ  where φ0 = 3Λ A and D = D = 4B . Similarly, proceeding the same way as before, we obtain a second-order inhomogeneous differential equation for φ(2) . B

d2 φ(2)  ) − 1)φ(2) = A12 sech4 (Dη  ) + A13 sech6 (Dη)  + Λ(3 sech2 (Dη dη 2 (5.36) 3

3

3

3

3

9(M+P )Λ 9Λ (M+2P ) Λ Λ + 9LΛ where A12 = − 45N , A13 = 45N 2A2 B 2AB 2 − A2 B 2AB 2 + A3 and 2 NΛ δλ is given by δΛ = − B 2 . Equation (5.36) is a second-order inhomogeneous differential equation with a solution that can be written as

φ(2) = φcomp + φp

(5.37)

where φcomp is the complementary function and φp is the particular solution. Similarly, the series solution method is used to obtain the particular

Dressed Soliton and Envelope Soliton

solution of Equation (5.36) as  15N Λ2 3(2M + P )Λ2 9LΛ2  − φp = + sech2 (Dη) 2AB 2 A2 B A3  45N λ2 9LΛ2 9(M + P )Λ2  + − + − sech4 (Dη). 4AB 2 2A2 B 2A3

211

(5.38)

The complementary solution ψcomp of Equation (5.36) is given by ψcomp = C1 P32 (z) + C2 Q23 (z),

(5.39)

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where P32 (z) and Q23 (z) are the associated Legendre functions. These are defined as   P32 (z) = 15z 1 − z 2 (5.40) Q23 (z) =

 1+z   2 15  z 1 − z 2 ln + + 5 − 15 1 − z 2 , 2 2 1−z 1−z

(5.41)

 where z = tanh(Dη). The coefficients C1 and C2 can be determined using the boundary conditions. The first term in the complementary function is secular which can be eliminated by renormalizing the amplitude. Also, φ2 (η) → 0 as η → ∞, so, C2 = 0. One can notice that the complementary part has no effect on the stationary solution. Using Equations (5.35)–(5.37), the stationary one soliton solution up to the second-order in Λ for DAWs in the four component dusty plasma is written as  ) + a1 sech2 (Dη  ) + a2 sech4 (Dη).  φ = φ(1) + φ(2) = φ0 sech2 (Dη (5.42) The amplitude of the KdV soliton that includes the second-order contribution φ0 , the amplitude of the dressed soliton that includes the second-order ∗

0 , and the width W

of the dressed soliton are as follows: contribution φ   N Λ2 3 Λ − 3(Λ + δΛ) 2 B φ0 = = , A A  2 3 Λ − NBΛ2 ∗ 3(Λ + δΛ)

φ0 = + a1 + a2 = + a1 + a2 , A A  1/2 4B −1

 W =D = . 2 Λ − NBΛ2 Equation (5.35) is the standard sech2 type KdV soliton, whose amplitude  4B 1/2 . ψ0 = 3λ Λ A and width W =

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212

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We illustrate the effects of higher-order nonlinear terms, by plotting the KdV soliton φ(1) (dotted line), the higher-order correction φ(2) (dashed line), and the dressed soliton φ (solid line) vs η as shown in Figure 5.1(a). It is seen that the amplitude of the dressed soliton φ is larger than the amplitude of the KdV soliton φ(1) . However, from the perturbation theory, it is known that the higher-order correction is to be considered only if 0 | |φ   20 |  1, i.e., |a1 + a2 | < |φ0 |. If |a1 + a2 | ≥ |φ0 |, then the solution |φ obtained is not physical. The shapes of the dressed soliton and KdV soliton are shown in Figure 5.1(b) for the same set of parameters as above, but the value of β1 = 0.75. From the above discussions, it is clear that the shape of the dressed soliton may exist, but the solution is not physical. Figure 5.2 is drawn to show this situation. In this figure, the amplitudes of the KdV soliton φ0 (dotted line), the second-order correction φ20 (dashed line), and dressed soliton (solid line) are plotted against β1 .

Fig. 5.1: The KdV soliton (dotted line), second-order correction (dashed line), and dressed soliton (solid line) are plotted (a) with β1 = 0.11 and (b) with β1 = .75.

Fig. 5.2: The amplitude of the KdV soliton (dotted line), second-order correction (dashed line), and dressed soliton (solid line) are plotted against β1 .

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213

It is seen that if β1 ≥ 0.44, the amplitude of the higher-order correction exceeds the amplitude of the first-order KdV soliton. So, the dressed soliton solution is invalid for β1 ≥ 0.44. 5.4 Dressed Soliton in Quantum Plasma

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To derive the dressed soliton in quantum plasma, we consider an unmagnetized collision-free quantum pair-ion plasma model, having electrons, positive and negative ions, and immobile negatively charged dust grains. The basic normalized equations are  √ ∂ne H 2 ∂ ∂ 2 ne /∂x2 ∂φ − 2ne + , (5.43) 0= √ ne ∂x 2μ ∂x ∂x ∂ ∂n+ + (n+ u+ ) = 0, ∂t ∂x ∂u+ ∂u+ ∂φ + u+ = −m , ∂t ∂x ∂x ∂n− ∂ + (n− u− ) = 0, ∂t ∂x ∂u− ∂u− ∂φ + u− = , ∂t ∂x ∂x

(5.44) (5.45) (5.46) (5.47)

∂2φ = μne − βn+ + n− + δ ∂x2

(5.48)

where nj , uj , and mj (j = e, d, +, and − stands for electrons, dust grains, positive ions, and negative ions, respectively) are the number density, velocity, and mass, respectively. μ = ne0 /Z− n−0 and β = Z+ n+0 /Z− n−0 are connected through the charge neutrality condition μ = β − 1 − δ with δ = Zd0 nd0 /Z− n−0 , m = Z+ m− /Z− m+ . nj0 is the equilibrium number density.  is the scaled Planck’s constant divided by 2π, φ is the electrostatic wave potential, and pe is the electron pressure. The equilibrium charge neutrality condition is ne0 + Z− n−0 + Zd0 nd0 = Z+ n+0 , where Z± and Zd0 are the charge states for positive (negative) ions and dusts. We assume that the ions are cold, and the electrons obey the pressure law pe =

me vF2 e 3 n 3n2e0 e

 2KB TF e /me is the electron Fermi thermal speed, TF e is the  2/3 2/3 ne0 /2me , and particle Fermi temperature given by KB TF e = 2 3π 2

where vF e =

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214

KB is the Boltzmann constant. Here, the normalizations are taken as

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x → ωp− x/cs ,

t → ωp− t,

nα → nα /nα0 ,

uα → uα /cs , φ → eφ/(kB TF e ),  ω nj0 e2 /0 mj is the j-particle plasma frequency, and cs = pj = KB TF e /m− is the quantum ion-acoustic speed. In the following set of normalized equations, we introduce the nondimensional quantum parameter H = ωpe /(KB TF e ) (the ratio between the electron plasma energy and the electron Fermi energy) proportional to quantum diffraction. Integrating Equation (5.43) with the boundary conditions ne → 1, ∂ne /∂x → 1 and φ → 0 at ±∞, we have √ H 2 ∂ 2 ne /∂x2 φ = −1 + n2e − . (5.49) √ ne 2μ We consider stretched coordinate the same as Equation (4.40). The dependent variables are expanded as nα = 1 +

∞  j

j n(j) α + ··· ,

uα =

∞ 

j u(j) α + ··· ,

φ=

j

∞ 

j φ(j) + · · ·

j

(5.50) where α = e, +, −, and  is a small nonzero parameter proportional to the amplitude of the perturbation. Upon substituting expressions (5.50) into Equations (5.44)–(5.49) and collecting the terms in different powers of  from the lowest-order of , we obtain the dispersion law as  2(1 + mβ) . (5.51) λ=± μ Comparing the next higher-order of  and eliminating the second-order perturbed quantities from a set of equations, we obtain the KdV equation for DIAWs as ∂φ(1) ∂φ(1) ∂ 3 φ(1) + Aφ(1) +B =0 ∂τ ∂ξ ∂ξ 3

(5.52)

where the nonlinear coefficient A and the dispersion coefficient B are written as    √   3 μ 1 − m2 β 1 + mβ 1 + mβ  A= −   16 − H 2 . 3 , B = 8μ 128μ3 8 1 + mβ

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215

Next, we determine the higher-order nonlinear and dispersion correction of the KdV equation. Equating the next higher-order terms in  and after some standard algebra, we obtain the differential equation for the higherorder correction φ(2) as   ∂ φ(1) φ(2) ∂φ(2) ∂ 3 φ(2) +A +B ∂τ ∂ξ ∂ξ 3  2 ∂φ(1) ∂ 3 φ(1) ∂ 5 φ(1) ∂ = L φ(1) + M φ(1) +N +P 3 ∂ξ ∂ξ ∂ξ 5 ∂ξ



∂φ(1) ∂ξ

2 (5.53)

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where       3 v02 − A2 4A m2 β − 1 3 5m2 β − Av0 m2 β − 2Av0 − 3 L= + − 16v0 v04 μ 2v05 μ    3 ABμ 1 B 3H 2 v0 − M = + 4 4 − 4m2 β + Amβv0 + Av0 − μ 32 v0 2 v0    1 9ABμ 2B  2 3H 2 v0 − − P = + 4 m β−1 μ 32 4v0 v0 N =

3B 2 H 4 v0 − . 128μ2 2v0

Equation (5.53) is a linear inhomogeneous differential equation in φ(2) , whose source term is given as a function of φ(1) , and we determine the nonsecular solution for φ(2) . Using the method of renormalization, to obtain a nonsecular solution for φ(2) , Equations (5.52) and (5.53) are modified as

∂φ(2) ∂τ

∂φ(1) ∂φ(1) ∂ 3 φ(1) ∂φ(1) + Aφ(1) +B + δΛ =0 (5.54) 3 ∂τ ∂ξ ∂ξ ∂ξ    (1)  ∂ φ(1) φ(2) ∂ 3 φ(2) ∂φ(2) ∂φ(1)  +A +B + δλ = S φ + δΛ . ∂ξ ∂ξ 3 ∂ξ ∂ξ (5.55)

The parameter δΛ in Equations (5.54) and (5.55) is introduced in the same way as it was introduced in the previous problems. Introducing a new stationary frame variable η = ξ − (Λ + δΛ)τ , where (Λ + δΛ) = X − 1, and X is the Mach number. The stationary renormalized solitary wave solution of

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216

Equation (5.54) up to the first order in Λ is obtained as  φ(1) = φ0 sech2 (Dη) (5.56)  λ  =D= where φ0 = 3λ and D A 4B . Similarly, proceeding in the same way as before, we obtain a second-order inhomogeneous differential equation for φ(2) as B

d2 φ(2)  ) − 1)φ(2) = A2 sech4 Dη  + A3 sech6 Dη  + λ(3 sech2 (Dη dη 2 2

3

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9λ λ where A2 = − 45N 2AB 2 + and δλ is given by

(M+2P ) B, 2A2

2

2

(5.57)

3

3

λ 9Mλ 27P λ 9Lλ A3 = − 45N 2AB 2 + A2 B + 2A3 − A3 ,

δλ = −

N λ2 . B2

(5.58)

Equation (5.57) is a second-order inhomogeneous differential equation whose solution can be written as φ(2) = φcomp + φp

(5.59)

where φcomp is the complementary function and φp is the particular solution. It can be shown easily that the complementary function φcomp has no role in the second-order correction φ2 , while the particular integral does. A series solution method developed by Chatterjee et al. [9] is adopted to determine φp , where φp is defined as a truncated power series. The particular solution φp can be obtained in a similar way as Step 6 in Section 5.2. So, the particular solution of Equation (5.57) is  ) + a2 sech4 (Dη)  φp = a1 sech2 (Dη 2

2

2

(5.60) 2

2

2

3(2M+P )λ 9(M+P )λ λ 45N λ − 9Lλ where a1 = 15N − 9Lλ A2 B 2A3 . 2AB 2 − A3 and a2 = − 4AB 2 + 2A2 B Using Equations (5.56), (5.59), and (5.60), the stationary one soliton solution up to the second order in Λ for pair-ion quantum plasma is written as

 ) + a1 sech2 (Dη  ) + a2 sech4 (Dη).  φ = φ(1) + φ(2) = φ0 sech2 (Dη (5.61) The amplitude of the KdV soliton that includes the second-order contribution φ0 , the amplitude of the dressed soliton that includes the second∗

0 , and the width W

of the dressed soliton are then order contribution φ

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expressed as

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3(λ + δλ) φ0 = , A

0 ∗ = 3(λ + δλ) + a1 + a2 , φ A

= W



4B λ + δλ

1/2 .

Equation (5.56) gives the renormalized KdV soliton, whose amplitude and width are φ0 and D. For a higher-order correction to the KdV soliton, ∗

0 and W

, respecthe speed is λ + δλ, and amplitude and width are φ tively. The quantum correction to the pair-ion plasma system is also considered. The effects of higher-order nonlinear terms on solitons are illustrated by plotting the KdV soliton φ(1) (dotted line), the second-order correction φ(2) (dashed line), and the dressed soliton φ (solid line) vs. η for H = 1 as shown in Figure 5.3. The amplitude of the dressed soliton φ is found to be larger than the amplitude of the KdV soliton φ(1) . Here, we recall the fact from the perturbation theory that one can only consider 0 | the higher-order correction if |a1|φ+a  1, i.e., |a1 + a2 | < |φ0 |. On the 2| contrary, a nonphysical solution is obtained if |a1 + a2 | ≥ |φ0 |. The shapes of the dressed soliton (solid line), second-order correction (dashed line), and the KdV soliton (dotted line) are shown in Figure 5.4 for H = 2.8. The amplitude of the dressed soliton is same as the KdV soliton, but the width is greater. Figure 5.5 shows the same plot as Figure (5.4) with the same parameters except H = 3.55. One can easily justify that this dressed soliton is not physical, as the amplitude of the second-order correction has exceeded that of the KdV soliton.

(1) (dotted line), higher-order correction φ (2) (dashed line), and Fig. 5.3: KdV soliton φ  (solid line) are plotted against η for H = 1. dressed soliton φ

218

Waves and Wave Interactions in Plasmas

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(1) (dotted line), higher-order correction φ (2) (dashed line), and Fig. 5.4: KdV soliton φ  (solid line) are plotted against η for H = 2.8. dressed soliton φ

(1) (dotted line), higher-order correction φ (2) (dashed line), and Fig. 5.5: KdV soliton φ  (solid line) are plotted against η for H = 3.55. dressed soliton φ

5.5 Dressed Soliton of ZK Equation We consider a three species dense quantum plasma composed of inertialess electrons and ions and negatively charged mobile heavy dust particles.  0 = B0 xˆ, where The plasma is in the uniform external magnetic field B ˆ is the unit vector in the B0 is the strength of the magnetic field and x x-direction. At equilibrium, the charge neutrality condition applies, namely, ni0 = ne0 + Zd nd0 , where nj0 is the unperturbed density of the jth species and Zd is the charge number of the charged dust. In the low-frequency limit ω  Ωcd , where ω is the wave frequency, Ωcd = eB0 /md c is the dust cyclotron frequency, e is the electronic charge, md is dust particle mass, and c is the speed of the light in vacuum. The normalized basic set of equations in two dimensions (xy-plane) in such a quantum plasma system

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219

are as follows:

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∇2 φ = α(μe ne + nd − μi ni )  ∂udx ∂ ∂ ∂φ + udx + udy udx = ∂t ∂x ∂y ∂x  ∂udy ∂ ∂ ∂φ − udz + udx + udy udy = ∂t ∂x ∂y ∂y  ∂udz ∂ ∂ + udx + udy udz = udy ∂t ∂x ∂y

(5.62) (5.63) (5.64) (5.65)

∂nd ∂(nd udx ) ∂(nd udy ) + + =0 ∂t ∂x ∂y

(5.66)

φ−σ

√ n2e H2 σ + √e ∇2 ne + = 0 2 2 2 ne

(5.67)

−φ −

√ n2i H2 1 + √i ∇2 ni + = 0. 2 2 2 ni

(5.68)

∂ ∂ Here, ∇ = x ˆ ∂x + yˆ ∂y (ˆ y is the unit vector in the y-direction). Furthermore, nj , ud , and φ are the number density of the j th species, dust  fluid  veloc

ity, and electrostatic potential, respectively. Hj =

Zd mj md

Ωcd c2sd

is the

quantum diffraction for the j th species (electron and ion), where mj and Csd are the mass of the jth and dust-acoustic speed, respectively.  species 2 ωpd 4πZd2 e2 nd0 TF e 2 2 2/3 , TF e,i = 2me,i α = Ω2 , σ = TF i , ωpd = , μj = md KB (3π ne,i0 ) nj0 Zd nd0

cd

(j = e, i). The diffraction effects for the dust are ignored since md  me,i . To obtain Equations (5.67) and (5.68) from the electron and ion momentum equations along the magnetic field, we use the boundary conditions ne = 1, ni = 1, and φ = 0 at infinity. Now, we use the normalization as follows: nj →

nj , nj0

Space is normalized by

ud → ωcd Csd ,

ud , Csd

φ→

where csd =

eφ , 2KB TF i 

2Zd KB TF i . md

t → Ωcd t.

(5.69)

We assume that the me,i V 2

inertia-less electrons and ions obey the pressure law pe,i = 13 n2 F e,i n3e,i e,i0  2KB TF e,i with VF e,i = being the electron and ion Fermi velocity. me,i Now, to derive the ZK equation for the considered magnetized quantum plasma, the independent variables are stretched as ξ = 1/2 (lx x − λt),

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η = 1/2 ly y, τ = 3/2 t with lx2 + ly2 = 1, and the dependent variables are expanded as ⎫ (1) (2) udx = udx + 2 udx + · · · ⎪ (1) (2) ⎪ udy = 2 udy + 3 udy + · · · ⎬ (1) (2) (5.70) udz = 3/2 udz + 5/2 udz + · · · ⎪ (1) (2) nj = 1 + nj + 2 nj + · · · ⎪ ⎭ φ = φ(1) + 2 φ(2) + · · · where j = e, i, d, and  is a small nonzero parameter proportional to the amplitude of the perturbation. Now, substituting the expression from Equation (5.70) into Equations (5.62)–(5.68) and collecting the coefficients of  in the lowest order, we obtain the dispersion relation as  1/2 σ λ = lx . (5.71) μe + σμi In the higher order of , eliminating the second-order perturbed quantities, we obtain the ZK equation for the magnetized quantum dust acoustic waves ∂φ(1) ∂φ(1) ∂ 3 φ(1) ∂ 3 φ(1) + Aφ(1) +B + C =0 (5.72) ∂τ ∂ξ ∂ξ 3 ∂ξ∂η 2   3l4x  αμ H 2 αμ H 2  λ3 μe λ3 where A = 2l , B = 2α 1 − 4σe 2 e − i4 i , and C = 2 2 − μi − λ4 x σ  λ3 l2y μe He2 μi Hi2  . 2αl2x 1 + α 1 − 4σ2 − 4 Next, we determine the higher-order correction to the nonlinear and dispersion effect of the ZK equation. We start by equating the next higherorder terms in , and after some standard algebra, we obtain the differential equation for the higher-order correction φ(2) as   ∂φ(2) ∂ 3 φ(2) ∂ 3 φ(2) ∂  (1) (2)  +A +B + C = S (2) φ(1) φ φ 3 2 ∂τ ∂ξ ∂ξ ∂ξ∂η

(5.73)

with    2 ∂φ(1) ∂ 3 φ(1) ∂ 5 φ(1) S (2) φ(1) = L φ(1) + M φ(1) + N ∂ξ ∂ξ 3 ∂ξ 5 +P

∂φ(1) ∂ 2 φ(1) ∂ 5 φ(1) ∂ 5 φ(1) ∂φ(1) ∂ 2 φ(1) + Q + R + S ∂ξ ∂ξ 2 ∂ξ 3 ∂η 2 ∂ξ∂η 4 ∂ξ ∂η 2

+T

∂φ(1) ∂ 2 φ(1) ∂η ∂ξ∂η

Dressed Soliton and Envelope Soliton

where

   λ3 3αμe 3αμi 3lx μ e − μi σ 2 + − α A − 2αlx2 2σ 3 2 2λ λσ 2  αA2 lx2 3αAlx4 9αlx6 3αA2 lx5 − − − − , λ3 λ5 2λ6 2λ5   λ3 −3αμe lx2 He2 3αμi lx2 Hi2 lx2 γ Aα lx = + − − 2αlx2 4σ 3 4 4λ σ 2 λ    Bα μe − μi σ 2 2ABαlx2 2Blx4 3lx4 − − , − − λ5 λσ 2 λ4 λ5   αBγlx2 λ3 αμe lx4 He4 αμi lx4 Hi4 αB 2 lx2 − = + − , 2αlx2 16σ 3 16 4λσ 2 λ4    lx2 γα 3A + lx2 λ3 −9αμe lx2 He2 3αμi lx2 Hi2 = + − 2αlx2 8σ 3 2 4λσ 2  3ABαlx2 lx3 αB lx4 αB − − , − λ4 λ4 λ4 σ    αμi lx2 ly2 Hi4 γα Clx + Bly2 λ3 αμe lx2 ly2 He4 + − = 2αlx2 8σ 3 8 4λσ 2  Bαly2 2BCαlx2 2 2 − − + αλ l y , λ λ4   Cγαly2 Cαly2 αμi lx ly4 Hi4 λ3 αμe ly4 He4 αlx2 C 2 − = − − + , 2αlx2 16σ 3 λ4 4λσ 2 λ 16   l2  3αμe ly2 Hi2 αγly2 A + λx λ3 −3αμe ly2 He2 = + − 2αlx2 4σ 3 4 4λσ 2

L=

M

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N P

Q

R S

Aly2 Cα 4 5 ACαlx2 αlx4 λ − − , − λ5 l x λ4 λ  3αμi ly2 Hi2 Aαly2 γ λ3 −3αμe ly2 He2 T = + − 2αlx2 4σ 3 4 2λσ 2  αlx2 ly2 2Aαly2 2ACαlx2 − + , − λ2 λ4 λ   γ = 2σ 2 − μe He2 − μi σ 2 Hi2 . −

221

222

Waves and Wave Interactions in Plasmas

Thus, Equation (5.72) is obtained from the basic set of equations which is a ZK equation for the first-order perturbed potential φ(1) , whereas Equation (5.73) is a ZK-type equation with an inhomogeneous term (source term on the right-hand side of Equation (5.73)) for the second-order perturbed potential φ(2) . Now, we use the method of renormalization [5]. Accordingly, Equations (5.72) and (5.73) become ∂φ¯(1) ∂ 3 φ¯(1) ∂ 3 φ¯(1) ∂  ¯(1)  δθ ∂φ¯(1) + Aφ¯(1) +B +C + =0 φ 3 2 ∂τ ∂ξ ∂ξ ∂ξ∂η lx ∂ξ

(5.74)

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∂φ¯(2) ∂ 3 φ¯(2) ∂ 3 φ¯(2) ∂  ¯(1) ¯(2)  δθ ∂φ¯(2) +A +B + C + φ φ ∂τ ∂ξ ∂ξ 3 ∂ξ∂η 2 lx ∂ξ   δθ ∂φ¯(1) = S 2 φ¯(1) + . lx ∂ξ

(5.75)

Here, φ¯(1) and φ¯(2) are the renormalized potentials. The parameter δθ is expanded only on the right-hand side of the added equation in the renormalization method as δθ = εθ1 + ε2 θ2 + · · · , since θn , n = 1, 2, 3, . . . , are chosen to cancel the resonant term in S (2) (φ1 ). Let us introduce a new variable ζ as ζ = lx ξ + ly η − (θ + δθ)τ

(5.76)

where the parameter θ is related to the Mach number (M) by θ + δθ = M − 1 = ΔM . Under this transformation, Equations (5.74) and (5.75), respectively, become  d3 φ¯(1) θ dφ¯(1) (1) ¯ χ + Aφ − =0 dζ 3 lx dζ χ

(5.77)

  d3 φ¯(2) dφ¯(1) d  ¯(1) ¯(2)  θ dφ¯(2) 1 (2)  ¯(1)  φ + A − = + δθ . (5.78) φ S φ dζ 3 dζ lx dζ lx dζ

Now, the first integral of the ordinary differential Equation (5.77) with the 2 ¯(1) ¯(1) boundary conditions φ¯(1) = dφdζ = d dζφ 2 = 0 as |ζ| → ∞ gives d2 φ¯(1) χ + dζ 2



A ¯(1) θ ¯(1) φ = 0. φ − 2 lx

(5.79)

Dressed Soliton and Envelope Soliton

223

Here, χ = (B − C)lx + C. One soliton solution of Equation (5.79) is written as  (1) 2 ζ φ (ζ) = φ0 sech (5.80) w where  the width w and the amplitude φ0 of the solitary wave are given by

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w=2 have

lx χ θ

and φ0 =

3θ Alx .

Using Equation (5.80) in Equation (5.78), we

   d3 φ¯(2) d ¯(2) θ dφ¯(2) 2 ζ − χ + Aφ φ sech 0 dζ 3 dζ w lx dζ      2φ0 δθ ζ ζ 2 ζ 4 ζ = A1 − sech tanh + A2 sech tanh wlx w w w w   ζ 6 ζ + A3 sech tanh (5.81) w w 8φ2

4 2 0 0 where A1 = − 32φ w 3 {(N − Q + R)lx + (Q − 2R)lx + R}, A2 = − w 3 {(M + P −

2Lφ30 w + 12φ20 720φ0 2 4 2 w 3 {(2M +P −S −T )lx +(S +T )}− w 5 {(N −Q+R)lx +(Q−2R)lx +R}. ¯(2) Integrating Equation (5.81) with the boundary conditions φ¯(2) = dφdζ = ¯(2) d2 φ dζ 2 = 0 as |ζ| → ∞ gives a second-order inhomogeneous ordinary differ¯(2)

4 2 0 S − T )lx2 + (S + T )} + 480φ w 5 {(N − Q + R)lx + (Q − 2R)lx + R}, A3 = −

ential equation in φ χ

as

 d2 φ¯(2) ¯(2) sech2 ζ − θ φ¯(2) + Aφ φ 0 dζ 2 w lx     φ0 δθ A1 w wA2 wA3 2 ζ 4 ζ 6 ζ − − = sech sech sech . − lx 2 w 4 w 6 w (5.82)

To cancel the secular term, we consider δθ =

wA1 lx 2φ0

(5.83)

3θ Hence, using Equation (5.83) and putting φ0 = Al , Equation (5.82) gives x   d2 φ¯(2) ζ θ ¯(2) −1 χ + 3sech2 φ dζ 2 w lx   wA2 wA3 4 ζ 6 ζ − =− sech sech . (5.84) 4 w 6 w

Waves and Wave Interactions in Plasmas

224

The complete solution of the second-order inhomogeneous differential Equation (5.84) can be written as φ¯(2) = φ¯comp + φ¯p

(5.85)

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where φ¯comp and φ¯p are the complementary function and the particular solution of Equation (5.84). It can be shown easily that the complementary function φ¯comp has no role in the second-order correction φ¯(2) , while the particular integral does play a role. The particular solution φ¯p can be obtained in a similar way as Step 6 in Section 5.2. So, the particular solution of Equation (5.84) is   2 ζ 4 ζ ¯ φp = a1 sech + a2 sech (5.86) w w 3

3

A3 w (A3 − A2 ) and a2 = − w48χ . Thus, Equations (5.80), (5.85), where a1 = 24χ and (5.86) together give the stationary one soliton solution for the considered magnetized quantum plasma. This can be expressed as    ζ ζ ζ φ¯ = φ¯(1) + φ¯(2) = φ¯0 sech2 + a1 sech2 + a2 sech4 . (5.87) w w w

The amplitude of the ZK soliton φ¯0 (that includes the second-order contribution), the amplitude of the dressed soliton φ¯∗0 (that includes the secondorder contribution), and the width w ¯ of the dressed soliton are as follows (δθ is given in Equation (5.83)):  3(θ + δθ) lx χ ¯ , w ¯=2 , φ¯∗0 = φ¯0 + a1 + a2 . (5.88) φ0 = Alx θ + δθ 5.6 Envelope Soliton In a dispersive system, when the leading-order nonlinear and dispersive effects are considered, then the envelope of a small amplitude narrow wave pulse may arise and satisfies the nonlinear Schrodinger equation. This equation admits envelope soliton solution under certain conditions. So, a group of waves with an envelope propagating without changing the form is known as envelope soliton. Envelope soliton survives when higher-order effects are considered. Envelope solitons are of two types: bright envelope soliton and dark envelope soliton. The ion acoustic dark or bright envelope solitons are formed for modulation of both stable or unstable waves in the plasma region.

Dressed Soliton and Envelope Soliton

225

5.7 Nonlinear Schrodinger Equation (NLSE) Now, we will derive the NLSE from a classical plasma model. Here, we consider a collisionless unmagnetized e-i plasma, where ions are mobile and electrons obey the Maxwell distribution. The normalized set of equations are ∂n ∂(nu) + =0 ∂t ∂x ∂u ∂u ∂φ +u =− ∂t ∂x ∂x

(5.89) (5.90)

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∂2φ = eφ − n. ∂x2

(5.91)

The normalization is similar to Section 4.3. To obtain the envelope soliton in the framework of NLSE, we employ a multiple scale perturbation. Taking A = A(0) +

∞ 

n A(n)

(5.92)

n=1

where A is any dependent variable (φ, n, u) and A(n) =

∞  l=−∞

(n)

Al (Xm , Tm )eil(kx−wt) ,

Here, Tm = m t, Xm = m x. So,

∂ ∂x

= m ∂X∂ m ,

∂ ∂t

m ≥ 1. = m ∂T∂m . Now,

  ∞  ∂ ∂ A= A(0) + n A(n) ∂x ∂x n=1 =

∞  n=1

n

∞  l=−∞

 ∞  ∂ (n) m Al eil(kx−wt) ilk + ∂X m m=1

Since this is true for all n and l, so we get ∞  ∂ ∂ = ilk + m ∂x ∂X m m=1 ∞ ∞    ∂2 ∂2 2 2 m ∂ = −l k + 2ilk  + . m+m 2 ∂x ∂Xm ∂Xm ∂Xm  m=1 m,m =1

(5.93)

Waves and Wave Interactions in Plasmas

226

Similarly, we get ∞  ∂ ∂ = −ilw + m . ∂t ∂Tm m=1

Accordingly, the variables are perturbed from the stable state in the following way (considering n = 1, u = 0, φ = 0 at stable state equilibrium): n = 1+

∞  n=1

v=

∞ 



n

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n =1

nv =

∞ 

∞ 

n

nnl eil(kx−wt)

l=−∞ ∞ 





vln eil (kx−wt)

l =−∞



n

∞ 

  vln eil (kx−wt)

+

l =−∞

n =1

∞ 



n+n

∞ 





nnl vln ei(l+l )(kx−wt) .

l,l =−∞

n,n =1

Now, ∂n = ∂t

 − ilw +

∞ 

m

m=1

= −ilw − ilw

∞ 

+

∞ 

− ilw

∞ 

n

n=1

∞ 

nnl eil(kx−wt)

l=−∞

nnl eil(kx−wt)

l=−∞

p

p=m+1 m=1

∂ (nv) = ∂x



∞ 

n

n=1 ∞ 

∂ ∂Tm

∞  l=−∞

∂np−m l eil(kx−wt) ∂Tm

(5.94)

   ∞ ∞ ∞      ∂ ilk + m n vln eil (kx−wt) ∂Xm   m=1 n =1

∞ 

+



n+n

∞ 



n

n =1

+ ilk

l =−∞

  nnl vln ei(l+l )(kx−wt)



l,l =−∞

n,n =1

= ilk

∞ 

∞ 

∞ 





vln eil (kx−wt)

l =−∞ ∞ 

q=n +1 n =1

q

∞   q−n n  i(l+l ) (kx−wt) vl e nl e

l,l =−∞

Dressed Soliton and Envelope Soliton

+

∞ ∞  

∞ 

+

∞   ∂vlr−m  eil (kx−wt) ∂Xm 

r

r=m+1 m=1

l =−∞

∞ 

∞ 

s

s=m+n +1 m=1 n =1

×

227

∞ 

∞ 

l=−∞ l =−∞

∂  s−m−n n  i(l+l ) (kx−wt) vl e n e . ∂Xm l

So, ∞ ∞      ∂ (nv) = ilk n vln eil (kx−wt) ∂x  

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n =1

+ ilk

l =−∞

∞ 

∞ 

q=n +1 n =1

+

∞ ∞  

r

r=m+1 m=1 ∞ 

+

∞   q−n n  i(l+l ) (kx−wt) vl e nl e

q

l,l =−∞

∞   ∂vlr−m  eil (kx−wt) ∂X m 

l =−∞

∞ 

∞ 

s

s=m+n +1 m=1 n =1

×

∞ 

∞ 

l=−∞ l =−∞

∂  s−m−n n  i(l+l ) (kx−wt) vl e n e . ∂Xm l

(5.95)

Putting the value of (5.94) and (5.95) in (5.89), and equating the coefficients of n , we get −ilw

∞ 

nnl eil(kx−wt) +

l=−∞

−∞

∞ 

+ ilk

vln eil(kx−wt)

+ ilk

∞ ∞  

∞   n−n n  il(kx−wt) nl−l vl e

n =1 l=−∞ l =−∞

l=−∞

+

∞ ∞ (n−m)   ∂nl eil(kx−wt) ∂T m m=1 l=

∞ ∞ (n−m)   ∂vl eil(kx−wt) ∂X m m=1 l= −∞

+

∞ 

∞ 

∞ 

∞ 

m=1 n =1 l=−∞ l =−∞

∂  (n−m−n ) (n )  il(kx−wt) n  vl e = 0. ∂Xm l−l

Waves and Wave Interactions in Plasmas

228

Since this is true for all l and eil(kx−wt) = 0, so, we get −ilwnnl +

∞ ∞ ∞ (n−m)     n−n n  ∂nl + ilkvln + ilk nl−l vl ∂Tm   m=1 n =1 l =−∞

+

∞ 

(n−m) ∂vl

+

∂Xm

m=1

∞ 

∞ 

∞ 

n =1 l =−∞

∂  (n−m−n ) (n )  = 0. n  vl ∂Xm l−l m=1

(5.96)

Also, 

∂v = ∂t

∞ 

∂ − ilw +  ∂T m m=1

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= −ilw

∞ 

+

∞ 

n

n=1 ∞ 

m

  ∞



n=1



∞ 

n

vln eil(kx−wt)

l=−∞

vln eil(kx−wt)

l=−∞ ∞ 

p

p=m+1 m=1

∞  l=−∞

∂ (p−m) il(kx−wt) v e . ∂Tm l

(5.97)

Now, v

∂v = ∂x

 ∞ n=1

×

∞ 

n

 ∞

l=−∞



∞ 

n

n =1 ∞ 

=

vln eil(kx−wt)

∞ 

  vln eil (kx−wt)



l =−∞

p

p=n +1 n =1 ∞ 

+

 ∞  ∂ il k + m ∂Xm m=1

∞ 

∞ 



l=−∞ l =−∞ ∞  ∞ 

q=n +m+1 n =1 m=1

p





il kvlp−n vln ei(l+l )(kx−wt)

∞ 

∞ 

l=−∞ l =−∞





vlq−n −m

∂vln i(l+l )(kx−wt) e . ∂Xm (5.98)

Again, ∂φ = ∂x

   ∞ ∞ ∞   m ∂ n n il(kx−wt) ilk +   φl e ∂Xm n=1 m=1

= ilk

l=−∞

∞  n=1

n

∞  l=−∞

φnl eil(kx−wt) +

∞ ∞   p=m+1 m=1

p

∞  ∂φp−m l eil(kx−wt) . ∂Xm

l=−∞

(5.99)

Dressed Soliton and Envelope Soliton

229

Putting the values of (5.97), (5.98), and (5.99) in (5.90) and isolating the order n , we get −ilw

∞ 

vln eil(kx−wt)

l=−∞

∞ ∞ ∞ ∞   ∂vln−m il(kx−wt)   + e + ∂Tm  m=1 l=



∞ 

+ ilk

n =1 l=−∞ l =−∞

−∞



n−n n il(kx−wt) × il kvl−l +  vl e

φnl eil(kx−wt) +

∞  ∞ ∞  

∞ 

∂Xm





n−n −m vl−l 

n =1 m=1 l=−∞ l =−∞ ∞ ∞   ∂φn−m il(kx−wt) l

m=1 l=−∞

l=−∞

∞ 

e

∂vln il(kx−wt) e ∂Xm

= 0.

(5.100)

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Since this is true for all l and eil(kx−wt) = 0, we get −ilwvln +

∞ ∞ ∞ ∞ (n−m) (n−m)     ∂vl ∂φl n−n n + ilkφnl + + il kvl−l  vl ∂Tm ∂Xm   m=1 m=1 n =1 l =−∞

+

∞ 

∞ 



∞ 



n =1 m=1 l =−∞

n−n −m vl−l 

(n )

∂vl = 0. ∂Xm

(5.101)

Again,   ∞ ∞  ∞   ∂2φ ∂2 m+m 2 2 m ∂ = −l k + 2ilk  +  ∂x2 ∂Xm m=1 m ∂Xm ∂Xm m=1 =1  ∞  ∞ ∞ ∞     × n φnl eil(kx−wt) = −l2 k 2 n φnl eil(kx−wt) n=1

+ 2ilk

n=1

l=−∞ ∞ 

∞ 

p

p=m+1 m=1

×

∞ 

∞  l=−∞

∞  ∞ 

l=−∞

∂φp−m il(kx−wt) l ∂Xm

e





q=m+m +1 m=1 m =1

q

∞  ∂ 2 φq−m−m l eil(kx−wt) ∂Xm ∂Xm

(5.102)

l=−∞

as n=1+

∞  n=1

n

∞  l=−∞

nnl eil(kx−wt) .

(5.103)

Waves and Wave Interactions in Plasmas

230

Now, φ2 φ3 + + ··· 2 6

eφ = 1 + φ + = 1+

∞ 

n

n=1

∞ ∞ 1   n+n  2 n=1 n

φnl eil(kx−wt) +

=1

l=−∞

∞ 

×

∞ 

∞ 

 φnl φnl

e

i(l+l )

e

kx−wt

l=−∞ l =−∞ ∞ 

×

∞ 

∞ ∞ ∞ 1    n+n +n +  6 n=1 n  =1 n =1

∞ 







φnl φnl φnl ei(l+l +l



)

ekx−wt + · · ·

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l=−∞ l =−∞ l =−∞

= 1+

∞ 

n

n=1

∞ 

φnl eil(kx−wt) +



∞ 

×

∞ 

∞ 

1 6

∞ 

∞  ∞ 

∞ 

p

p=n +1 n =1

l=−∞

× ei(l+l ) ekx−wt +

∞ 1  2 

∞ 

l=−∞ l =−∞





φnl φp−n l

q

q=n +n +1 n =1 n =1

∞ 

l=−∞ l =−∞ l =−∞













−n φnl φnl ei(l+l +l ) ekx−wt + · · · φq−n l

(5.104) Now, putting the values of (5.102)–(5.104) in (5.91) and isolating the n order equation, we get −l2 k 2

∞ 

φnl eil(kx−wt) + 2ilk

l=−∞

∞ ∞   ∂φn−m l eil(kx−wt) ∂X m m=1 l= −∞

∞  ∞ ∞ ∞    ∂ 2 φn−m−m l nnl eil(kx−wt) eil(kx−wt) + ∂X ∂X  m m  m=1 m 

+

=1 l=−∞

−

∞ 

l=−∞

φnl eil(kx−wt) −

l=−∞ ∞ ∞ ∞ 1    − 6  

∞ ∞ 1   2 

∞ 



n =1 l=−∞ l =−∞

∞ 

∞ 

n =1 n =1 l=−∞ l =−∞ l =−∞





n il(kx−wt) φn−n l−l φl e







−n n n il(kx−wt) − ··· φn−n l−l −l φl φl e

(5.105)

Dressed Soliton and Envelope Soliton

231

Since this is true for all l and eil(kx−wt) = 0, we get −l2 k 2 φnl + 2ilk

 ∞ ∞  ∞   ∂φn−m ∂ 2 φn−m−m l l + + nnl − φnl ∂X ∂X ∂X  m m m m=1 m=1 m

=1

−

1 2

∞ 

∞ 

n =1 l =−∞





n φn−n l−l φl −

1 6

∞ 

∞ 

∞ 

∞ 

n =1 n =1 l =−∞ l =−∞

− · · · = 0.









−n n n φn−n l−l −l φl φl

(5.106)

1. Linear Part

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A. Harmonic modes with n = 1, l = 1: Linear dispersion law From Equations (5.96), (5.101), and (5.106) and equating the coefficients of  for n = 1, l = 1, we get ⎫ ⎪ ⎪ ⎬

k 1 v w 1 k v11 = φ11 w

n11 =

n11 = (1 + k 2 )φ11

⎪ ⎪ ⎭

.

(5.107)

From Equation (5.107), we obtain the dispersion law: w2 =

k2 1+k2 .

(5.108)

2. Nonlinear Part B. Modes with n = 2, l = 1: Group Velocity For n = 2 and l = 1, we have from Equations (5.96), (5.101), and (5.106) −iwn21 + ikv12 + v12 =

∂n11 ∂v11 + =0 ∂T1 ∂X1

k ∂φ1 i ∂φ11 k 2 φ1 − i 2 1 − w w ∂T1 w ∂X1

n21 = (1 + k 2 )φ21 − 2ik

∂φ11 . ∂X1

(5.109) (5.110) (5.111)

232

Waves and Wave Interactions in Plasmas

Putting the values of n11 , v11 , n21 , and v12 in (5.109), we get ∂φ11 ∂φ1 + vg 1 = 0 ∂T1 ∂X1

(5.112)

w3 k3 .

Using (5.112), we get from

where the group velocity is defined as vg = (5.110) v12 =

∂φ1 k 2 φ1 − iw 1 . w ∂X1

(5.113)

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C. Second Harmonic Modes with n = l = 2 For the second-order quantities with n = l = 2, we obtain from equation (5.106) −(1 + 4k 2 )φ22 + n22 −

1  1 2 = 0. φ 2 1

(5.114)

From (5.101) and (5.96), v22 =

k 3  1 2 k 2 φ2 + φ , w 2w3 1

  2  2 3 n22 = 1 + k 2 φ22 + 1 + k 2 φ11 . 2

(5.115) (5.116)

Putting n22 in (5.114), we get φ22 =

2 + 6k 2 + 3k 4  1 2 φ1 . 6k 2

(5.117)

From (5.116), we get using (5.117) n22 =

2 + 15k 2 + 12k 4  1 2 φ1 . 6w2

(5.118)

Using (5.117), we get from (5.115) v22 =

2 + 9k 2 + 6k 4  1 2 φ1 . 6kw

(5.119)

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D. Zeroth Harmonic Modes with n = 2, l = 0 and n = 3, l = 0 Considering the terms corresponding to n = 3 and l = 0, we get from (5.96)  ∂  1 2 ∂v 2 k ∂n20 = − 0 − 2 1 + k2 |φ | ∂T1 w ∂X1 1 ∂X1

(5.120)

and for n = 2 and l = 0, we get from (5.106) φ20 = n20 − |φ11 |2 .

(5.121)

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Also, for n = 3 and l = 0, we get from (5.101) ∂v02 ∂φ20 k 2 ∂  1 2 + + 2 |φ | = 0. ∂T1 ∂X1 w ∂X1 1

(5.122)

Now, φ20 = n20 − |φ11 |2 ∂ 2 ∂ 2 ∂  1 2 n = φ + |φ | ∂T1 0 ∂T1 0 ∂T1 1 ∂ 2 ∂  1 2 ∂ 2 n0 = φ0 − vg |φ | . ∂T1 ∂T1 ∂X1 1 As

∂ ∂T1

(5.123)

∂ + vg ∂X = 0, from Equation (5.120), we get 1

∂n20 ∂v 2 k ∂  1 2 = − 0 − 2 (1 + k 2 ) |φ | ∂T1 w ∂X1 1 ∂X1  ∂n20 ∂ 2 12 2 =− v + |φ | ∂T1 ∂X1 0 vg 1   ∂φ20 ∂  1 2 1 − vg2 = −(3 + k 2 − vg2 ) |φ | = 0. ∂X1 ∂X1 1 Integrating, we get φ20 = −

3 + k 2 − vg2 1 2 |φ1 | . 1 − vg2

Using relation (5.124), we get from (5.121)  2 + k2 n20 = − |φ11 |2 . 1 − vg2

(5.124)

(5.125)

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From (5.122), we get

−vg

∂v02 − ∂X1



3 + k 2 − vg2 1 − vg2

Integrating, we get



∂v02 ∂φ20 k 2 ∂  1 2 + + 2 |φ | = 0 ∂T1 ∂X1 w ∂X1 1 ∂ ∂  1 2 (|φ11 |)2 + (1 + k 2 ) |φ | = 0. ∂X1 ∂X1 1

  2 + k 2 vg2  1 2  v02 = −  |φ1 | . 1 − vg2 vg

(5.126)

E. Harmonic Modes with n = 3, l = 1: The NLS equation Now, for n = 3 and l = 1, we get from (5.106)  1 ∂φ11 ∂ 2 φ11  1 2 − φ1 φ0 + φ1−1 φ22 − (|φ11 |)2 φ11 = 0 + 2 ∂X2 ∂X1 2   1 2 1 1  1 2 1 ∂φ ∂ φ1 (20) (22) −(1 + k 2 )φ31 + n31 + 2ik 1 + − −C + C + |φ1 | φ1 = 0. 3 3 ∂X2 2 ∂X12 (5.127)

−(1 + k 2 )φ31 + n31 + 2ik

From (5.96), ∂n21 ∂n11 ∂v12 ∂v11 + + + ∂T1 ∂T2 ∂X1 ∂X2 2 1

1 2 1 2 2 1 + ik n0 v1 + n1 v0 + n−1 v2 + n2 v−1 = 0

−iwn31 + ikv13 +

 ∂φ11 ∂ 2 φ11  ∂ 2 φ11 k ∂φ11 + 1 + k2 − iw + 2 2 ∂X1 ∂X1 w ∂X2 ∂T2      (22) k (22)  1 2 1 (20) k (20)  |φ1 | φ1 = 0. + ik − C1 1 + k 2 + 1 + k 2 C2 + C1 − C2 w w (5.128)

− iwn31 + ikv13 − 2ikvg

From (5.101), −iwv13 + ikφ31 +



∂v12 ∂v 1 ∂φ11 1 1 v22 − v22 v−1 + 1 + + ik v02 v11 + 2v−1 =0 ∂T1 ∂T2 ∂X2

∂ 2 φ11 k ∂φ11 ∂φ11 + + ∂X12 w ∂T2 ∂X2   k (20) k (22)  1 2 1 |φ1 | φ1 = 0. + ik − C2 + C2 w w

−iwv13 + ikφ31 + iwvg

(5.129)

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Putting v13 in (5.128), we get  k ∂ 2 φ11 k ∂φ11 ∂φ11 ikφ31 + iwvg + + 2 2 w ∂X1 w ∂T2 ∂X2   k (20) k (22)  1 2 1 2k ∂ 2 φ11 + ik − C2 + C2 |φ1 | φ1 + vg w w w ∂X12     1 + k 2 ∂φ11 ∂ 2 φ11 k ∂φ11 k (20) k (20)  − + − i + 1 + k2 − C1 − C2 2 2 iw ∂T2 w ∂X2 w ∂X1 w    (22) k (22)  1 2 1 + 1 + k 2 C2 + C1 |φ1 | φ1 . (5.130) w

n31 = −i

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Putting n31 in (5.127), we get  i

∂φ11 ∂φ1 + vg 1 ∂T2 ∂X2

+P

 2 ∂ 2 φ11 + Q |φ11 | φ11 = 0 2 ∂X1

(5.131)

where P =−

3 w5 w3 , Q= 2 4 2k 2k



1 + C322 − C320 2



  w  20  + k C220 − C222 + C1 − C122 . 2

This equation is called cubic nonlinear Schrodinger equation. Now, applying the traveling wave transformation ξ = x − vg t, τ = t, one finds the standard form of the NLS equation as i

∂ψ ∂2ψ + P 2 + Q|ψ|2 ψ = 0 ∂τ ∂ξ

(5.132)

where ψ denotes the electric correction φ11 . The group velocity  1 d2potential  ω dispersion coefficient P = 2 dk2 is a real-valued parameter, and Q is the cubic nonlinear (local) coefficient. The coefficient P is proportional to the dv group dispersion dKg , and the coefficients Q is proportional to the nonlinear frequency shift. If the amplitude of the plasma wave is not small, nonlinearities give rise to a modulation of the wave amplitude. When the amplitude varies slowly throughout oscillation, the equation describing the evolution of the wave amplitude (in certain cases) becomes the nonlinear Schrodinger equation. In plasma, it is found that under certain conditions, conversion of an initially uniform wave train into a spatially modulated wave proves to be energetically favorable. This effect is known as modulationally instability. The plane wave solutions of Equation (5.132) are modulationally unstable

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Waves and Wave Interactions in Plasmas

if P Q > 0. Let us rewrite Equation (5.132) as

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i

∂ 2ψ Q ∂ψ − |ψ|2 ψ = −  2 ∂τ ∂ξ P

(5.133)

where τ  = P τ . The Schrodinger equation for quasiparticles, the wave function of which is given by ψ, is trapped by a self-generated nonlinear potential 2 V = −Q P |ψ| . For P Q > 0, this potential will have an attractive sign. If somehow the quasiparticle density |ψ|2 increases, the depth of this potential increases; more particles are attracted, leading to a further increase in the potential depth. In this respect, the instability may be the result of a consequence of the self-trapping of the quasiparticles. For electron plasma waves, the modulational instability can be easily understood in terms of the pondermotive force. When an electromagnetic wave of vary high intensity interacts with plasma particles in a plasma nonlinear processes, the force due to radiation pressure is coupled to the plasma particles and is called the pondermotive force. Suppose we have a plasma wave with a ripple on the envelope. The gradient in wave intensity causes a pondermotive force, which pushes plasma out of the region with maximum intensity towards the region with minimum intensity. It causes a ripple, in a density depression in the region of maximum intensity. The dispersion relation is 3 2 (5.134) ω 2 = ωpe + k 2 VT2h . 2 It indicates that waves of large k can exist only in a region of small ωpe or low density. Thus, the density of the region’s depression can trap plasma waves leading to further enhancement of local field and hence, the instability. During the nonlinear evolution of the wave, when the nonlinear effects are balanced by the dispersion effect, a stable nonlinear wave structure is formed, called the envelope soliton. When both P and Q are positive, Equation (5.132) has a solitary wave solution. Let us consider ψ = f (ξ)exp.iAτ

(5.135)

where f (ξ) is real and A is an arbitrary constant. Substituting (5.135) in (5.132), we find that f satisfies the equation −Af + P ⇒P

d2 f + Qf 3 = 0 dξ 2

d2 f = Af − Qf 3 . dξ 2

(5.136)

Dressed Soliton and Envelope Soliton

237

df Multiplying both sides by 2 dξ and integrating, we get

 P

df dξ

2

1 = Af 2 − Qf 4 + C 2

(5.137)

where C is a constant. Assuming that f and its derivatives vanish at ξ → ±∞, we find that C = 0. Therefore, Equation (5.137) can be rewritten as  df A  =± dξ. (5.138) P Q 2 f 1− f 2A

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Substituting f =



2A Q sechθ

in Equation (5.138) and integrating, we get 

θ=∓

A dξ + C1 P

where C1 is a constant. Therefore,   2A A f= sech ξ ∓ C1 . Q P

(5.139)

(5.140)

If we take the origin of the coordinate for ξ in such a way that f attains maximum at ξ = 0, then C1 = 0. Therefore,   2A A f= sech ξ . (5.141) Q P Hence, the solitary wave solution of the nonlinear Schrodinger Equation (5.132) becomes   2A A ψ= sech ξ eiAτ . (5.142) Q P Suppose the wave field is described by E = Re ψ ei(kx−ωt) then using (5.142) we can write   2A A E= sech (x − vg t) cos(kx − ωt + A2 t). Q P

(5.143)

(5.144)

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238

Setting  = 1 and



2A Q

= a, we get

     Q a2 Q E = a sech a (x − vg t) cos kx − ω − t . 2P 2

(5.145)

Thus, there is a nonlinear frequency shift of the wave, and this frequency 2 shift is equal to a 2Q , where a is the amplitude of the solitary wave. Solution (5.146) shows that the amplitude of the wave moves with the group velocity vg . At t = 0, the wave profile is written as   Q E = a sech a x) cos kx. (5.146) 2P

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The wave profile resembles a simple soliton but oscillating between positive and negative values, and it is known as envelope soliton. References [1] H. Ikezi and R. J. Taylor, Phys. Rev. Lett. 22, 923 (1969). [2] Y. H. Ichikawa, T. Mitsuhashi, and K. Konno, J. Phys. Soc. Jpn. 41, 1382 (1976). [3] T. Aoyama and Y. H. Ichikawa, J. Phys. Soc. Jpn. 42, 313 (1977). [4] Y. H. Ichikawa, T. Mitsuhashi, and K. Konno, J. Phys. Soc. Jpn. 43, 675 (1977). [5] Y. Kodama and T. Taniuti, J. Phys. Soc. Jpn. 45, 298 (1978); 45, 1765 (1978). [6] R. S. Tiwari and M. K. Misra, Phys. Plasmas 13, 062112 (2006). [7] T. S. Gill, P. Bala, and H. Kaur, Phys. Plasmas 15, 122309 (2008). [8] A. Esfandyari-Kalejahi, I. Kourakis, and P. K. Shukla, Phys. Plasmas 15, 022303 (2008). [9] P. Chatterjee, G. Mandal, K. Roy, S. V. Muniandy, S. L. Yap, and C. S. Wong, Phys. Plasmas 16, 072102 (2009). [10] K. Roy and P. Chatterjee, Indian J. Phys. 85, 1653 (2011).

Chapter 6

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Evolution Equations in Nonplanar Geometry

6.1 Introduction The applications of solitons or shocks in different areas of plasmas are well known. In the past few decades, the propagation of IASWs, DASWs, DIASWs, etc., has been studied experimentally in the framework of KdV equation, Burgers’ equation, KP equation, ZK equation, or some modified form of these equations. These have been explained clearly in Chapter 4. But, these studies are limited to unbounded planar geometry and are neither realistic in laboratory plasma nor space plasma. In reality, nonplanar geometry changes the shape of the waves. In nature, distorted waves are observed in the laboratory and space and are certainly not bounded in one or two-dimensional planar geometry. Frenz et al. [1] have observed the features in the auroral region, especially at the higher polar altitudes. Maxon and Viecelli [2, 3] initiated this problem theoretically and observed the propagation of radially ingoing acoustic type waves in cylindrical geometry and spherical geometry. It is seen that as the amplitude grows, the width decreases, and the propagation speed increases, while the soliton travels inward, which is different than the usual KdV soliton. However, the square root of the peak amplitude multiplies by the width and remains constant for a long span. They have also seen that the cylindrical soliton travels slower than the spherical soliton but faster than the soliton of the same amplitude of planar geometry.

239

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Waves and Wave Interactions in Plasmas

6.2 Basic Equations of Motion in Nonplanar Geometry The basic system of equations in a plasma model is mainly governed by the equation of continuity, the equation of motion, and the Poisson equation. For a classical electron-ion plasma, where ions are mobile in the background of electrons, may be modeled as

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∂ni  + ∇ · (ni · u) = 0, ∂t ∂u  )u = −(∇  φ) + Ω(u × xˆ) + μ∇  2 u, + (u · ∇ ∂t  2 φ = ni − ne ∇

(6.1) (6.2) (6.3)

where ni and ne are the ion and electron number density, respectively. ui , vi , and wi are the ion fluid speed in x, y, and z directions. Equations (6.1)– (6.3) refer to the equation of continuity, the equation of motion, and the Poisson equation, respectively. To study nonlinear waves in nonplanar geometry, let us first express the equation of continuity in cylindrical and spherical geometry. (a) Continuity equation in cylindrical geometry: The continuity equation arises from the fundamental physical principle of the conservation of mass. To derive this equation, we choose a convenient control volume inside the fluid and assume that all the fluid particles flowing inside the control volume must flow out. In planer geometry, the cubic control volume is taken into consideration where the sides of the cube are parallel to the coordinate axes. For cylindrical coordinate, a point P is represented by the triple (r, θ, z), where r and θ are the polar coordinates of the projection of P onto the xy-plane and z has the same meaning as in cartesian coordinate. We choose the control volume dV = rdθdrdz. As the mass conservation inside the control volume is zero, so the rate of flow in = Rate of flow out + Accumulation, i.e., Rate of flow out – Rate of flow in + Accumulation = 0. If we assume the velocity field u = uer +veθ +wez and write the velocity component r, θ, z direction as u, v, w in place of ur , uθ , uz , respectively. The mass of the fluid in the control volume is M = ρdV , and the rate of change of mass or accumulation in the control volume is ∂ρ ∂t rdrdθdz. For the net flow through the control volume, let us deal with one face at a time. First, we consider the r faces. The net inflow is m ˙ r,in = ρurdθdz,

Evolution Equations in Nonplanar Geometry

241

while the outflow in the r direction is m ˙ r,out = (ρu + ∂ρu ∂r )(r + dr)dθdz. So, the net flow in the r direction is m ˙ r,out − m ˙ r,in = ρudrdθdz + ∂ρu ∂r rdrdθdz + ∂(ρu) ∂ρu 1 2 2 ∂r rdr dθdz = r ρudV + ∂r + O(dr ). We shall now complete the net flow in the θ direction. Here, the areas of the inflow and outflow faces are the same. The net flow in the θ direction is m ˙ θ,net =

1 ∂ρV dV. r ∂r

We now turn our attention to the z-direction. The face area of a sector of angle dθ is

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Az =

1 1 1 (r + dr)2 dθ − r2 dθ = rdrdθ + dr2 dθ = rdrdθ + O(dr2 dθ). 2 2 2

The inflow at the lower z face is m ˙ ( z, in) = ρwAz = ρwrdrdθ, and the outflow at the upper z face is m ˙ r,out = (ρw + ∂ρw ∂z dz). Az = ρwrdrdθ + ∂ρw rdrdθdz. ∂z Hence, the net flow in the z direction is m ˙ z,out − m ˙ z,in = ∂ρw ∂z rdrdθdz. Putting things together, we obtain the continuity equation as 1 ∂(ρu) ∂(ρv) ∂(ρw) ∂ρ dV + ρudV + dV + dV + dV = 0. ∂t r ∂r ∂θ ∂z Conventionally, we take n as the plasma number density. So, by replacing fluid density ρ by n and dividing both the sides by dV , we get the continuity equation for plasma model in cylindrical coordinate system as ∂n 1 ∂ 1 ∂ ∂ + (rnur ) + (nuθ ) + (nuz ) = 0, ∂t r ∂r r ∂θ ∂z

(6.4)

where the component of the velocity vector is written as u = (ur , uθ , uz ). (b) Continuity equation in spherical geometry: Here, we start with a spherical control volume dV = r2 sin θdrdθdφ, where r, θ, and φ stand for the radius, polar, and azimuthal angles, respectively. The azimuthal angle is also referred to as the zenith or colatitude angle, and the differential mass is dM = ρr2 sin θdrdθdφ. Let the velocity field be u = uer + veθ + weφ . Like the previous case, mass conservation is represented by accumulation, net flow, and source terms in a control volume. The accumulation term is the rate of time of the change of mass. We, therefore, have ∂ρ 2 ∂t r sin θdrdθdφ. The net flow through the control volume can be separated into that corresponding to each direction.

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Waves and Wave Interactions in Plasmas

For radial flow: We start with the radial direction. So, we have m ˙ in = ρuAin . The inflow area Ain is a trapezoid, whose area is given by Ain = 1 2 [r sin θdφ+ r sin(θ + dθ)dφ]rdθ. The midsegment is considered as the average of the bases. Expanding sin(θ + dθ) = sin θ cos dθ + cos θ sin dθ = sin θ + cos θdθ and substituting into Ain , we have Ain = r2 sin θdθdφ + 1 2 2 2 2 r cos θdθ dφ = r sin θdθdφ, where high-order terms  have been  dropped. The outflow in the radial direction is m ˙ out = ρu + ∂ρu dr Aout , but ∂r Aout = 12 [r sin θdφ + (r + dr) sin(θ + dθ)dφ]. By only keeping the lowest order (second- and third-order terms in the resulting expression), we have Aout = r2 sin θdθdφ + 2 sin θdrdθdφ. In Aout , we kept both second-order and third-order terms. The reason is that this term will be multiplied by ‘dr’ and, therefore, the overall order will be three. In principle, one must carry all those terms until the final substitution is made, and only then one can compare the terms and keep those with the lowest order. Hence, the net flow in the radial direction is 2 given by m ˙ out − m ˙ in = 2ρur sin θdrdθdφ + ∂ρu ∂r r sin θdrdθdφ. For polar flow (θ): The inflow in the polar direction is m ˙ in = ρvAin , where Ain = rsinθdrdφ. The outflow in the θ direction is m ˙ out = (ρv + ∂ρv 1 dθ)A , where A = [r sin(θ + dθ)dφ + (r + dr) sin(θ + dθ)dφ]dr. out out ∂θ 2 Expanding and keeping both second- and third-order terms, we get Aout = r cos θdrdθdφ+r sin θdtdφ. Finally, the net flow obtained in the polar direction is m ˙ out − m ˙ in = ρvr cos θdrdθdφ + ∂ρv dθ r sin θdrdθdφ. For azimuthal flow (φ): The inflow in the azimuthal direction is given by m ˙ in = ρwAin , with Ain = rdrdθ, while the outflow is m ˙ out = (ρw + ∂ρw ∂φ dφ)Aout , and Aout = rdrdθ. At the outset, the net flow in the azimuthal direction is m ˙ out − m ˙ in =

∂ρw ∂φ rdrdθdφ. Now, by collecting all mass fluxes ∂ρu ∂ρv dV ∂ρw dV dV ∂r dV + ρv cos θ r sin θ + ∂θ r + ∂φ r sin θ = 0.

dV we have ∂ρ ∂t dV + 2ρu r + Dividing by dV and combining terms and replacing fluid density ρ by plasma density n as convention, the above equation reduces to

∂n 1 ∂ 1 ∂ 1 ∂ + 2 (r2 nur ) + (nuθ sin θ) + (nuφ ) = 0, ∂t r ∂r r sin θ ∂θ r sin θ ∂φ

(6.5)

which is the continuity equation in spherical coordinates. (ii) Equation of motion: The Navier–Stokes equation for incompressible fluid motion gives   ∂u  P + μ∇   2 u + F n + u.∇u = −∇ (6.6) ∂t

Evolution Equations in Nonplanar Geometry

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where left-hand side represents inertia per volume. The first term is for unsteady acceleration, and the second term is for convective acceleration. The first two terms of the right-hand side represents the divergence of stress. The first term is for pressure gradient, the second term is for viscosity, and the third term is for other body forces. In cylindrical coordinates, the Navier–Stokes equation along r direction is written as   ∂ur uθ ∂ur ∂ur u2 ∂ur − θ + ur + + uz n ∂t ∂r r ∂θ ∂z r     ∂ 1 ∂ ∂P 1 ∂ 2 ur ∂ 2 ur 2 ∂uθ − =− +μ (rur ) + 2 + + Fr , ∂r ∂r r ∂r r ∂θ2 ∂z 2 r2 ∂θ (6.7) along θ direction is expressed as   ∂uθ ∂uθ uθ ∂uθ ur uθ ∂uθ n + ur + + + uz ∂t ∂r r ∂θ r ∂z     2 ∂ 1 ∂ 1 ∂P 1 ∂ uθ ∂ 2 uθ 2 ∂ur =− +μ (ruθ ) + 2 + + 2 + Fθ , r ∂θ ∂r r ∂r r ∂θ2 ∂z 2 r ∂θ (6.8) and along z direction is expressed as   ∂uz ∂uz uθ ∂uz ∂uz n + ur + + uz ∂t ∂r r ∂θ ∂z      1 ∂ ∂P ∂ 2 uz ∂uz 1 ∂ 2 uz =− +μ + + Fz . r + 2 ∂z r ∂r ∂r r ∂θ2 ∂z 2

(6.9)

In spherical coordinates, the Navier–Stokes equation along r direction is written as  u2φ ∂ur ∂ur uθ ∂ur uφ ∂ur u2θ − − + ur + + n ∂t ∂r r ∂θ r sin θ ∂φ r r      1 ∂ ∂P ∂ 1 ∂ur 1 ∂ 2 ur 2 ∂ur =− +μ 2 r + 2 sin θ + 2 ∂r r ∂r ∂r r sin θ ∂θ ∂θ r sin θ ∂φ2    1 ∂ 2 ur 2 ∂uθ 2uθ 2 ∂uφ 2ur − 2 cot θ − 2 + 2 2 +μ − 2 − 2 + Fr , r r ∂θ r r sin θ ∂φ r sin θ ∂φ2 (6.10)

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along θ direction is expressed as 

 u2φ cot θ ∂uθ ∂uθ uθ ∂uθ uφ ∂uθ ur uθ − n + ur + + + ∂t ∂r r ∂θ r sin θ ∂φ r r      1 ∂ 1 ∂P ∂uθ 1 ∂ 1 ∂ =− +μ 2 (uθ sin θ) r2 + 2 r ∂θ r ∂r ∂r r ∂θ sin θ ∂θ  1 ∂ 2 uθ 2 ∂ur 2 cot θ ∂uθ − 2 + 2 + 2 (6.11) + Fθ , r sin θ ∂φ2 r ∂θ r sin θ ∂φ and along φ direction is expressed as

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 ∂uφ ∂uφ uθ ∂uφ uφ ∂uφ ur uφ uθ uφ cot θ n + ur + + + + ∂t ∂r r ∂θ r sin θ ∂φ r r      1 ∂ ∂uφ 1 ∂ 1 ∂P 1 ∂ +μ 2 (uφ sin θ) =− r2 + 2 r sin θ ∂φ r ∂r ∂r r ∂θ sin θ ∂θ  1 ∂ 2 uφ 2 ∂ur 2 cot θ ∂uθ + 2 + 2 + 2 (6.12) + Fφ . r sin θ ∂φ2 r sin θ ∂φ r sin θ ∂φ The Navier–Stokes equation with no body force (i.e., F = 0) is considered if Fr = 0 = Fθ = Fz = Fφ . (iii) Poisson’s equation: In the cylindrical coordinates, Poisson’s equation is written as   1 ∂ ∂ψ 1 ∂2ψ ∂2ψ = −4πn, r + 2 2 + r ∂r ∂r r ∂θ ∂z 2

(6.13)

while in the spherical coordinates, Poisson’s equation is written as     1 ∂ ∂ ∂2ψ 1 ∂ψ 1 2 ∂ψ = −4πn. r + sin θ + 2 2 2 2 r ∂r ∂r r sin θ ∂θ ∂θ r sin θ ∂φ2

(6.14)

6.3 Nonplanar KdV Equation in Classical Plasma Now, we are going to derive the nonplanar KdV equation from a three component unmagnetized, collision-free plasma, consisting of warm ion fluid and superthermal distributed electrons and positrons in cylindrical

Evolution Equations in Nonplanar Geometry

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or spherical geometry. The basic equations will be written as ∂ni  + ∇ · (ni u) = 0 ∂t ∂u  )u = e E − 1 ∇  pi + (u · ∇ n i mi ∂t mi

(6.15) (6.16)

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where ni , u, mi , e, and E are the ion number density, ion fluid velocity, ion mass, the electron charge, and the electric field, respectively. pi is the pressure of the adiabatically hot ions, given by the following thermodynamic equation of state  γ ni pi = pi0 . (6.17) ni0 At equilibrium, pi0 = ni0 Ti0 , where ni0 is the unperturbed number density of ions, Ti0 is the ion equilibrium temperature, and the adiabatic constant is defined as γ = (NN+2) , where N represents the number of degrees of freedom of the ions. Here, N = 1. Also, Poisson’s equation is  = 4πe(ne − ni − np )  ·E ∇

(6.18)

where nj (j = e, and p, stands for electron, and positron, respectively) and Tj are the number densities and temperature, respectively. The electric  = −∇  φ so that φ is the electrostatic wave potential. In equilibrium, field E ne0 = ni0 + np0 , nj0 is the unperturbed number density. We consider u = (ur , 0, 0) (motion takes place along the direction of r only). For simplicity, we take ur = ui . Accordingly, the normalized equations are as follows: ∂ni 1 ∂ ν + ν (r ni ui ) = 0 (6.19) ∂t r ∂r ∂ui ∂ui ∂ni ∂φ + ui + 3σi ni =− (6.20) ∂t ∂r ∂r ∂r   1 ∂ ν ∂φ (6.21) r = ne − np − ni rν ∂r ∂r



−κ+ 12 −κ+ 12 p 1 where ne = 1−p 1 − φκ and np = 1−p 1 + σp φκ . p = np0 /ne0 and σp = Te /Tp , σi = Ti0 /Te . κ is the real parameter measuring deviation from the Maxwellian equilibrium (recovered for κ infinite). ν = 0 in the case of planar geometry, and ν = 1, 2 in the case of cylindrical and spherical geometries, respectively.

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The normalization is taken as nj ui r eφ nj → , ui → , r→ , t → ωpi t, φ → . nj0 Cs λD KB Te Te Te 4πni0 e2 Here, j = i, e, p. Ci = m , λ = , and ω = are the D pi mi 4πni0 e2 i ion-acoustic speed, the electron Debye length, and the ion plasma frequency, respectively. Derivation of cylindrical/spherical KdV equation: To obtain the KdV equation in nonplanar geometry, we introduce the stretched coordinates as 3

1

τ = 2 t

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ξ =  2 (r − λt),

(6.22)

where  is a small expansion parameter and λ is a wave phase velocity normalized to Cs . The expansions of the dependent variables are also considered as ∞ ∞ ∞

(k) (k) ni = 1 + k φ(k) + · · · .  k ni + · · · , u i = k u i + · · · , φ = k=1

k=1

k=1

(6.23) Substituting Equation (6.23) in Equations (6.19)–(6.21) and collecting different powers of , we obtain the following equations to the lowest order in : (1)

ni

(1)

=

ui , λ

(1)

ui

=

λ2

λ φ(1) . − 3σi

(6.24)

From (6.24), we get the value of phase velocity as  (1 − p)κ + 3σi (κ − 12 )(1 + pσp ) λ= . (κ − 12 )(1 + pσp )

(6.25)

Taking the next higher order of , we get (1)

∂ni ∂τ

(1)

−λ

(2)

(2)

∂ (1) (1) ν (1) ∂ni ∂ui + + (n u ) + u =0 ∂ξ ∂ξ ∂ξ i i λτ i (2)

(2)

(1)

(1)

∂φ(2) ∂ui ∂u ∂n (1) ∂ui (1) ∂ni − λ i + 3σi i + ui + = −3σi ni ∂τ ∂ξ ∂ξ ∂ξ ∂ξ ∂ξ ∂ 2 φ(1) (2) + ni ∂ξ 2

⎫ ⎪ ⎪ ⎬

⎪ ⎪ (κ − 12 )(1 + pσp ) (2) (κ − 12 )(κ + 12 )(1 − pσp2 ) (1) 2 φ + = (φ ) . ⎭ (1 − p)κ 2(1 − p)κ2 (6.26)

Evolution Equations in Nonplanar Geometry (2)

247

(2)

Now, eliminating ni , ui , and φ(2) from Equation (6.26) and using relations given in Equations (6.24)–(6.25), we obtain the nonplanar modified KdV equation as ∂φ(1) ∂φ(1) ∂ 3 φ(1) ν (1) + +B =0 φ + Aφ(1) ∂τ 2τ ∂ξ ∂ξ 3 where

(6.27)

ν (1) 2τ φ

is the additional term that has arisen due to the nonplanar

 (1−p)(κ+ 12 )(1−pσp2 ) 12σi (κ− 12 )(1+pσp ) 1 3+ geometry. Here, A = 2λ − (κ− 1 )(1+pσ and 2 (1−p)κ p)

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2

2

2

κ (1−p) 1 B = 2λ . A and B are the coefficients of nonlinearity and (κ− 12 )2 (1+pσp )2 dispersion, respectively. If we put ν = 0 in Equation (6.27), we get the planar KdV equation. ν = 1, 2 correspond to the KdV equations in the ν in the modified KdV cylindrical and spherical geometries. The term 2τ equation appears due to a geometrical effect, and at longer time τ , the effect of this geometrical effect is reduced. For planar geometry (ν = 0), a stationary solution of Equation (6.27) has the following form:    3V V (1) 2 Sech ξ−Vτ (6.28) φ = A 4B

where V is the constant velocity normalized by Cs . Now, we can solve Equation (6.27) numerically. The initial condition that we have used in our numerical results is the form of the stationary solution (6.28) at τ = −10. In Figure 6.1, we have plotted the negative potential solitary structure for different values of the superthermal parameter κ and at different times ranging from τ = −3 to τ = −10 in cylindrical geometry. It is clear from the figure that both the amplitude and width of the negative potential electrostatic soliton decreases with a decrease of κ (i.e., an increase in superthermal). The pulse amplitude and width of the negative soliton become larger with an increase in the time duration. Figure 6.2(a) shows the formation of the positive potential wells for several values of κ and at different times in cylindrical geometry. As the superthermal parameter κ increases, the amplitude as well as width decreases. Moreover, the amplitude of the pulses increases when |τ | decreases. The numerical plots of solitons in spherical geometry, at different times, have been plotted for several values of κ in Figure 6.2(b). As the magnitude of |τ | increases, the solution looks like those for planar KdV solitons. This is because of the extra term (ν/2τ )φ(1) , and it becomes small for large values of |τ |. As κ decreases (the superthermal character of

Waves and Wave Interactions in Plasmas

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248

Fig. 6.1: Plot of φ(1) against ξ for different values of ν for the solution of (6.27), where χ = −50, σ = 0.052, α = 0.1, κ = 0.5, γ = 3, τ = −5, and η0 = 0.98.

(a)

(b)

Fig. 6.2: The profiles of shock wave in (a) cylindrical geometry and (b) spherical geometry, where the other parameters are same as those in Figure 6.1.

the plasma increases), the potential pulse amplitude and width increase. Furthermore, it is observed that increasing the parameter κ leads to a decrease in the velocity of solitary waves, and the solitary peaks are well separated over time. However, the positive solitary pulses have the same qualitative behavior as the negative solitary waves. 6.4 Nonplanar KdV Equation in Quantum Plasma The nonlinear dynamics of IASWs in a three-component unmagnetized, collisionless quantum plasma, consisting of inertialess electrons and positrons

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249

(the phase velocity of the wave is assumed to be much less than the Fermi velocities of electrons and positrons), inertial ions, are governed by the following normalized equations: ∂ni ν ∂(ni ui ) + (ni ui ) + = 0, ∂t r ∂r

(6.29)

∂ui ∂ui ∂φ + ui =− , ∂t ∂r ∂r

(6.30)

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∂φ ∂ne He2 ∂ + − ne ∂r ∂r 2 ∂r



 ∂2 √ ∂r 2 ne √ ne

Hp2 ∂ ∂φ ∂np − − σnp + ∂r 2 ∂r ∂r



= 0,

 ∂2 √ ∂r 2 np √ np

(6.31)

= 0,

(6.32)

∂2φ = (1 + p)ne − pnp − ni . ∂r2

(6.33)

Here, we consider that the Fermi temperature of electrons or positron is much larger than that of ions, i.e., TF j  TF i (j =e, p). The electron/positron obeys the pressure law Pj = mj n3j vF2 j /3n2j0 , vF j = (2KB TF j /mj )1/2 is the electron/positron Fermi thermal speed, and KB is the Boltzmann constant. TF j (j = i, e, and p stands for ion, electron, and positron, respectively), mj , and nj are the Fermi temperature, the mass, and the number density, respectively. Also, nj0 is the unperturbed density. At equilibrium, ne0 = ni0 + np0 . The electrons/positrons are considered to be degenerative owing to their small mass compared to the ions. Here, the normalizations are taken as nj →

nj , nj0

ui →

ui , csi

t → ωpi t,

r→r

csi , ωpi

φ→

eφ 2KB TF e

  where csi = 2KB TF e /mi and ωpi = (ni0 e2 )/(mi 0 ) are the quantum  ion-acoustic speed and ion plasma frequency, respectively. Also, Hj = 2 2 ωpi mi mj c2si

is the quantum diffraction parameter, and p = np0 /ni0 . Here, 

is the Planck constant divided by 2π, e is the charge of electron, and 0 is the permeability of the free space. Finally, ν represents the geometry factor, ν = 0 for the planar geometry, whereas ν = 1 and 2 for the cylindrical and spherical geometry, respectively.

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Waves and Wave Interactions in Plasmas

Integrating Equations (6.31) and (6.32) and using the boundary conditions ne = 1, np = 1, and φ = 0 at infinity, we obtain   ∂2 √ ne ) 1/2 2( ne = 1 + 2φ + He2 ∂r √ , (6.34) ne   ∂2 √ np ) 1/2 2φ Hp2 ∂r 2( np = 1 − + . (6.35) √ np σ σ

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To derive the KdV equation for a one-dimensional quantum ion acoustic solitary waves from Equations (6.29), (6.30), and (6.33)–(6.35), we have used the RPT, and the independent variables are stretched in the the same way as in Equation (6.22). Let us expand the dependent variables as ψ = ψ (0) +

∞

n+1 ψ (n)

(6.36)

n=1

where ψ = (ne , np , ni , ui , φ) with ψ (0) = (1, 1, 1, 0, 0). Using the expressions in (6.22) and (6.36) into Equations (6.29), (6.30), and (6.33)–(6.35), at the lowest order of , we get the dispersion relation as  σ λ= . (6.37) p + (1 + p)σ In the next higher order of , we eliminate the second-order perturbed quantities, and after doing some simple algebraic operations, we obtain the nonplanar KdV equation as ∂φ(1) ∂φ(1) ∂ 3 φ(1) ν (1) + Aφ(1) +B + φ = 0, (6.38) 3 ∂τ ∂ξ ∂ξ 2τ    3  3 pH 2 H2 with A = λ2 λ34 + 1 + p − σp2 and B = λ2 1 − (1 + p) 4e − 4σ2p . A and B are the coefficients of nonlinearity and dispersion, respectively. If we put ν = 0 in Equation (6.38), we get the planar KdV equation, ν = 1, 2 corresponding to the KdV equation in cylindrical and spherical geometries. It is important to mention here that the quantum diffraction of electrons and positrons plays a vital role in the wave dispersion but not in the nonlinear properties of the waves. 6.5 Nonplanar Gardner’s or Modified Gardner’s Equation To derive the Modified Gardner’s (MG) equation in nonplanar geometry, we consider a four-component, collision-free, unmagnetized dusty

Evolution Equations in Nonplanar Geometry

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plasma, consisting of inertial ions, q-nonextensive distributed electrons, and stationary positively as well as negatively charged dust. At equilibrium, ni0 + Zdp ndp = ne0 + Zdn ndn , where ni0 , ne0 , ndp , and ndn are ion, electron, positive dust, and negative dust number density at equilibrium, respectively. Zdp and Zdn represent the charge state of positive and negative dust, respectively. The dynamics of DIAWs in such plasma can be explained by a set of normalized equations as follows: 1 ∂ ν ∂ni + ν (r ni ui ) = 0 ∂t r ∂r ∂φ ∂ui ∂ui + ui =− ∂t ∂r ∂r   1 ∂ ∂φ rν =ρ rν ∂r ∂r ρ = (1 − jμ)ne − ni + jμ

(6.39) (6.40) (6.41) (6.42)

1+q

where ne = [1 + (q − 1)φ] 2(q−1) . Here, nj (j = i, and e, stands for ion, and electron, respectively), ui , and φ are the number density, ion fluid velocity, and electrostatic potential, respectively. The normalizations are taken as eφ r ρ , r→ , t → ωpi t, ρ → ni0 KB Te λD    where Ci = Te /mi , λD = Te /(4πni0 e2 ), and ωpi = 4πni0 e2 /mi are the ion-acoustic speed, the ion Debye radius, and the ion plasma frequency, respectively. Te , e, mi , and ρ are the electron temperature, electron charge, ion mass, and the net surface charge density, respectively. Here, μ = |Zdp ndp − Zdn ndn |/ni0 . Also, j = 1 for Zdn ndn > Zdp ndp and j = −1 for Zdn ndn < Zdp ndp . j = −1 and 1 stand for the positive and negative net dust charge, respectively. Relation (6.42) is valid for stationary or static (positive and negative) dust. This is a correct approximation for the DIA waves, whose frequency (ω) is much larger than both the positive dust plasma frequency (ωpdp ), and negative dust plasma frequency (ωpdn ), i.e., ω  ωpdp , ωpdn . To study finite amplitude DIA, DLs in a dusty plasma by using (6.39)– (6.42) by RPT, we use the new stretched coordinates as ξ = (r − λt), τ = 3 t, and expand all the dependent variables in power series of  as ni →

ni , ni0

ui →

ui , Ci

φ→

ψ = ψ (0) +

∞

k=1

k ψ (k) + · · ·

(6.43)

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252

where ψ = (ni , ui , φ, ρ) with ψ (0) = (1, 0, 0, 0). Now, expressing (6.39)– (6.42) in terms of ξ and τ , substituting (6.43) into the resulting equations, and equating the lowest order in , we get (1) ui

φ(1) , = λ

(1) ni

φ(1) = 2 , λ

 ρ

(1)

= 0,

λ=

2 . (6.44) (1 − jμ)(q + 1)

Again, equating the next order in , we get (2)

ui

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ρ

(2)

=

φ(2) φ(2) φ2 3φ2 (2) + , ni = 4 + 2 , ρ(1) = 0 3 2λ λ 2λ λ

⎫ ⎪ ⎬

1 3 (1 − jμ)(q + 1)(3 − q) ⎪ = − Aφ2 = 0, A = 4 − .⎭ λ 2 4

(6.45)

It is clear that A = 0 (since φ = 0) and A = 0 when q = qc = 3{1−(1−jμ)} 1+3(1−jμ) . It is obvious that (6.45) is satisfied when q = qc . We have numerically shown how qc varies with μ. The results are displayed in Figure 6.3, which, in fact, represents the parametric regimes that correspond to A = 0, A > 0, and A < 0. So, for |q − qc | =  corresponding to A = A0 , one can write A0 as A0  s

∂A ∂q

q=qc

|q − qc | = sAq , where Aq =

(1−jμ) {3(1 − jμ)(q 2

+ 1) +

(q − 1)}. Here, s = 1 for q > qc , and s = −1 for q < qc . So, when q = qc , one can express ρ(2) as ρ(2)  − 12 sAq φ2 , i.e., when q = qc , ρ(2) , it must be considered in the third-order Poisson’s equation.

Fig. 6.3: Showing the variation of qc [obtained from A(q = qc ) = 0] with μ for j = 1 and j = −1. The solid (dash) line represents this variation for j = 1(−1).

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253

To the next higher order in , we get ⎫ (3) (3) ⎪ ∂ (1) (2) ν (1) ∂ni ∂ui (2) (1) −λ + + (n u + ni ui ) + u = 0⎪ ∂ξ ∂ξ ∂ξ i i λτ i ⎪ (1) (3) ∂φ(3) ∂ (1) (2) ∂ui ∂ui ⎪ −λ + =0 (u u ) + ⎬ ∂τ ∂ξ ∂ξ i i ∂ξ .  ⎪ ∂2φ 1 (q + 1) (3) 2 + sAq φ − (1 − jμ) φ ⎪ ∂ξ 2 2 2 ⎪  (q + 1)(3 − q) (2) (q + 1)(3 − q)(5 − 3q) 3 (3) ⎭ + φ + ni = 0 ⎪ φφ + 4 48 (6.46) (1)

∂ni ∂τ

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Now, using (6.44)–(6.46), finally, we obtain the MG equation as ∂φ ν ∂φ ∂φ ∂3φ + φ + mφ + nφ2 + n0 3 = 0 ∂τ 2τ ∂ζ ∂ζ ∂ζ

(6.47)

(1−jμ)(q+1)(3−q)(5−3q) 15 where m = sAq n0 , n = m0 n0 , m0 = 2λ , and 6 − 16 3 λ n0 = 2 . For ν = 0, Equation (6.47) converts to a standard Gardner’s equation. We have derived the MG equation using RPT, which is valid beyond the KdV limit. We now turn to Equation (6.47) with the term ν 2τ φ, which is due to the effect of nonplanar geometry. An exact analytical solution of Equation (6.47) is not possible. To study the effects of nonplanar geometry on time-dependent DIAGSNs, we have solved Equation (6.47) numerically. We have used our numerical analysis taking the initial condition in the form of the stationary solution of Equation (6.47) without ν ψ. Our aim is now to numerically analyze the MG equation. the term 2τ However, for clear understanding, we first discuss the stationary solution of this standard Gardner’s equation briefly (i.e., Equation (6.47) with ν = 0). To obtain the stationary solution of this Gardner’s equation, we use the transformation ζ = ξ − u0 τ , where u0 is the constant speed. Now, we d2 φ apply the conditions φ → 0, dφ dξ → 0, dξ 2 → 0 at ξ → ∞. Thus, we get the stationary solution of the standard Gardner’s equation as    φm ξ φ= 1 + tanh (6.48) 2 Δ

where φm = respectively.

6su0 Aq n0

and Δ =

24 −φ2m m0

are the amplitude and the width,

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Fig. 6.4: Showing the effects of cylindrical geometry on DIA GSs for j = 1, μ = 0.509, s = 1, and q = 0.298.

Fig. 6.5: Showing the effects of spherical geometry on DIA GSs for j = 1, μ = 0.509, s = 1, and q = 0.298.

We have numerically solved Equation (6.47) and have studied the effects of cylindrical and spherical geometries on the time-dependent DIAGSNs. The results are depicted in Figures 6.4 and 6.5. Figure 6.4 explores how the effect of cylindrical geometry modifies DIA GSs for j = 1, while Figure 6.5 shows that the spherical geometry effect on the DIA for j = 1. It is displayed in Figures 6.4 and 6.5 that for a large value of |τ |, i.e., at τ = −25, the spherical and cylindrical SWs are similar to one-dimensional geometry. ν ψ is no longer dominant. But, It can be concluded that for large |τ |, 2τ ν when the value of |τ | decreases, the modified term 2τ ψ becomes dominant, and cylindrical and spherical SW structures differ from the one-dimensional structure.

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6.6 Nonplanar KP and KP Burgers’ Equation

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To derive the cylindrical/spherical KPB and the same type KP equation, we consider a plasma model having superthermal electrons, Boltzmann distributed positrons, and singly charged adiabatically hot positive ions. The phase velocity of IAWs is assumed to be much smaller than the electron and positron thermal velocities and larger than the ion thermal velocity (vthi ω/k vthe , vthp ). It should be noted that typically Te = Tp . We, therefore, ignore the electron and positron inertia and write down the equation of motion for the ions. The basic system of equations in cylindrical and spherical geometry in such a plasma model is governed by ∂ni  + ∇ · (ni vi ) = 0, ∂t ∂vi  )vi = e E − 1 ∇  pi + μ∇  2 vi + (vi · ∇ n i mi ∂t mi

(6.49) (6.50)

where ni is the ion number density, vi is the ion fluid velocity, mi is the ion mass, e is the magnitude of the electron charge, E is the electric field, and μ is the kinematic viscosity, respectively. pi is the pressure of the adiabatically hot ions and is represented by the following thermodynamic equation of state:  γ ni pi = pi0 . (6.51) ni0 Here, pi0 = ni0 Ti0 is the ion pressure at equilibrium, ni0 is the unperturbed ion density, Ti0 is the ion equilibrium temperature, and the adiabatic constant is defined as γ = (NN+2) , where N represents the number of degrees of freedom of the ions. Since the kinematic viscosity represents the diffusion in momentum, therefore, we include this contribution from ions and neglect the contribution of electrons and positrons. The electrons and positrons are assumed to obey the Kappa distribution and Boltzmann distribution, respectively, on the ion acoustic timescale and are expressed  −κ+ 12   and np = np0 exp − Teφp . The parameter κ as ne = ne0 1 − κ−φ 1 2 shapes predominantly the superthermal tail of the distribution, and the normalization has been provided for any value of the κ > 1/2. The system of equations is closed with the help of Poisson’s equation,  · E = 4πe(ni + np − ne ) ∇

(6.52)

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256

where np and ne are the number densities, while Tp and Te are temperatures of positrons and electrons, respectively. The electric field E = −∇φ so that φ represents the electrostatic wave potential. At equilibrium, ni0 + np0 = ne0 , where np0 and ne0 are the unperturbed positron and electron number densities, respectively. The normalized form of continuity, momentum, and Poisson’s equations in nonplanar cylindrical and spherical geometries are

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1 ∂(rν ni ui ) 1 ∂(ni vi ) ni vi ∂ni + ν + + (ν − 1) cotθ = 0, ∂t r ∂r r ∂θ r ∂ui ∂ui vi ∂ui vi2 ∂φ σ ∂ni − − + ui + =− γnγ−2 i γ−1 ∂t ∂r r ∂θ r ∂r (1 − α) ∂r   2(ν − 1)vi νui 2 ∂vi − + η ∇2 ui − 2 − 2 cotθ , r r ∂θ r2

(6.53)

(6.54)

∂vi ∂vi vi ∂vi vi ui 1 ∂φ σ ∂ni − + ui + + =− γnγ−2 ∂t ∂r r ∂θ r r ∂θ r(1 − α)γ−1 i ∂θ   2 ∂ui vi − 2 + η ∇2 vi + 2 , (6.55) r ∂θ r (sin2 θ)ν−1  −κ+ 12 ∂φ 1 ∂(rν ∂r ) 1 ∂ 2 φ (ν − 1) ∂φ φ + 2 2 + cotθ = 1− rν ∂r r ∂θ r2 ∂θ κ − 12 − αexp(−φ) − ni

(6.56)

where ν = 1, 2 for cylindrical and spherical geometries, respectively. The expression for ∇2 is different for cylindrical and spherical geometries and, therefore, caution needs to be exercised while opening it in the said geometries. ni →

ni , ne0

vi →

vi , Cs

φ→

eφ , Te

r→

r , λDe

t→

Cs t. λDe

(6.57)

 Cs = Te /mi , ne0 , and φ are the ion acoustic speed, electron equilibrium density, and the electrostatic wave potential, respectively. λDe =  Te /(4πne0 e2 ) is the electron Debye length. Also, defined α = np0 /ne0 , σ = Ti0 /Te , and η = μ/(λDe Cs ). To investigate the IASWs in unmagnetized e-p-i plasma, we stretch the independent variables as 1

ξ =  2 (r − λt),

1

χ = − 2 θ,

3

τ = 2 t

(6.58)

Evolution Equations in Nonplanar Geometry

257

where λ is the wave phase velocity. Let us expand the perturbed quantities as ⎫ (1) (2) (3) ni = (1 − α) + ni + 2 ni + 3 ni + · · · , ⎪ ⎪ (1) (2) (3) ⎬ ui = ui + 2 ui + 3 ui + · · · , (6.59) 3 (1) 5 (2) 7 (3) ⎪ vi =  2 vi +  2 vi +  2 vi + · · · , ⎪ ⎭ φ = φ(1) + 2 φ(2) + 3 φ(3) + · · · . 1

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The value of η is assumed to be small so that we may let η =  2 η0 , where η0 is O(1). Substituting Equations (6.58)–(6.59) in Equations (6.53)–(6.56) and collecting different powers of , we obtain the following equations to the lowest order in , (1)

ni

=

(1 − α)φ(1) , λ2 (1 − γσ λ2 )

(1)

ui

=

φ(1) . λ(1 − γσ λ2 )

From (6.60), we get  (1 − α)(κ − 12 ) + γσ{(κ + 12 ) + α(κ − 12 )} λ= . (κ + 12 ) + α(κ − 12 )

(6.60)

(6.61)

The next higher-order equations in  are written as   (1) (1) (1) (2) (2) ∂ n u i i ∂ni ∂n ∂u − λ i + (1 − α) i + ∂τ ∂ξ ∂ξ ∂ζ (1)

+

(1 − α)ν (1) (1 − α) ∂vi (1 − α) (ν − 1) (1) + ui + vi = 0, λτ λτ ∂χ λτ χ

(1)

(2)

(2)

(6.62)

(1)

∂φ(2) γσ ∂ni ∂ui ∂u (1) ∂ui −λ i + + ui + ∂τ ∂ξ (1 − α) ∂ζ ∂ξ ∂ξ (1)

(1)

γ(γ − 2)σ (1) ∂ni ∂ 2 ui − η0 = 0, ni 2 (1 − α) ∂ξ ∂ξ 2  (1)  (1) ∂φ 1 ∂vi γσ − 2 1+ 2 = 0, ∂ξ λ τ λ (1 − γσ ) ∂χ λ2      κ + 12 (2) κ + 12 κ + 32  (1) 2 (2) φ − ni −  φ  2 κ − 12 2 κ − 12 +

− αφ(2) +

α  (1) 2 ∂ 2 φ(1) φ = 0. + 2 ∂ξ 2

(6.63) (6.64)

(6.65)

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Waves and Wave Interactions in Plasmas (2)

(2)

Now, eliminating ni , ui , and φ(2) from Equations (6.62)–(6.65) and using relations given in Equations (6.60)–(6.61), we obtain KPB equation as   ∂φ(1) ∂ 3 φ(1) ∂ 2 φ(1) ∂ ∂φ(1) ν (1) + Aφ(1) +B − C + φ ∂ξ ∂τ ∂ξ ∂ξ 2 2τ ∂ξ 3  2 (1)  D ∂ φ Ξ(ν − 1) ∂φ(1) + 2 + =0 (6.66) 2 τ ∂χ χ ∂χ

 γσ(γ+1){(κ+ 12 )+α(κ− 12 )} (1−α){α(κ− 12 )2 −(κ+ 12 )(κ+ 32 )} 1 3+ + , where A = 2λ (1−α)(κ− 1 ) {(κ+ 1 )+α(κ− 1 )}2 2

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(κ− 1 )2

2

2

η0 1 2 B = (1−α) 2λ {(κ+ 12 )+α(κ− 12 )}2 , C = 2 , and D = 2λ . Here, Ξ = 0, 1 for planar and nonplanar geometries, respectively. A and C are the coefficients of nonlinearity and dissipation, respectively, whereas B and D are the coefficients of dispersion. If we put ν and Ξ equal to zero in Equation (6.66), we get the planar KPB equation with the only difference that the coefficient D 1 in that case. The factor τ 2 comes due to the nonplanar effects. would be 2λ ν = 1, 2 together with Ξ = 1 and corresponds to the KPB equations in the cylindrical and spherical geometries, respectively. When η0 = 0, i.e., the value of C = 0, then we find the KP equation in nonplanar geometry. Now putting η0 = 0 in Equation (6.66), we get the KP equation as   ∂φ(1) ∂ 3 φ(1) ∂ ∂φ(1) ν (1) φ + Aφ(1) +B + ∂ξ ∂τ ∂ξ ∂ξ 3 2τ  2 (1)  D ∂ φ Ξ(ν − 1) ∂φ(1) + 2 + = 0. (6.67) 2 τ ∂χ χ ∂χ

The planar KPB equation is   ∂φ(1) ∂ 3 φ(1) ∂ 2 φ(1) ∂ 2 φ(1) ∂ ∂φ(1) + Aφ(1) +B − C +D = 0. (6.68) 2 3 ∂ξ ∂τ ∂ξ ∂ξ ∂χ2 ∂ξ Using the transformation ζ = k(ξ + χ − U τ ), where k is the dimensionless nonlinear wavenumber and U is the velocity of the nonlinear structure in the moving frame, the solution of the KPB Equation (6.68) is written as [?]    2   6 C2 C 6C φ(1) (ξ, χ, τ ) = 1 − tanh ξ+χ− +D τ 25 AB 10B 25B      2 2 3 C C 6C + +D τ (6.69) sech2 ξ+χ− 25 AB 10B 25B where k = C/10B and U =

6C 2 25B

+ D.

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259

We have numerically solved the nonplanar KPB Equation (6.68). The initial profile that we have used in our numerical analysis is (6.69). As Equation (6.69) is just a particular solution of Equation (6.68), it is necessary to plot numerical solutions of Equation (6.68) to clearly understand the behavior of the shock wave. The initial condition that we have used in our numerical results, in the form of the stationary solution (6.68) at τ = −20. Figure 6.6 shows the cylindrical and spherical shock structures at different spectral indexes (κ). The shock height and width decrease as the value of the spectral index increases. Therefore, spectral index κ has a significant impact on the formation of shock structures. This result indicates that for small values of κ, the superthermal electrons in the tail of the distribution function increase, which plays a role to increase the height and width of the nonlinear structure. Figure 6.7 depicts the change in shock wave structure due to the variation of η0 . It is found that an increase in η0 (meaning an increase in dissipation) considerably enhances the amplitude as well as the steepness of the shock front. By increasing the value of viscosity, i.e., dissipation in the system, the strength of shocks increases. It acquires more propagation speed in the case of spherical geometry.

Fig. 6.6: Cylindrical and spherical shock structures evolved at τ = −10 for several κ.

Fig. 6.7: Cylindrical and spherical shock structures evolved at τ = −10 for several η0 .

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260

6.7 Nonplanar ZK Equation Now, we are going to investigate the characteristics of nonlinear wave structures in a homogeneous, collisionless magnetized plasma, comprising of nonextensive electrons and inertial ions in the framework of the cylindrical  0 = zB ˆ 0 , where zˆ is ZK equation. Here, the external magnetic field is B the unit vector along the z-direction and B0 is the external static magnetic field. Accordingly, the basic system of normalized equations are as follows:

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∂ni 1 ∂ 1 ∂ ∂ + (rni ui ) + (ni vi ) + (ni wi ) = 0, ∂t r ∂r r ∂θ ∂z ∂ui ∂ui vi ∂ui ∂ui v2 ∂φ − i =− + ui + + wi + Ωvi , ∂t ∂r r ∂θ ∂z r ∂r ∂vi ∂vi vi ∂vi ∂vi ui vi 1 ∂φ − + ui + + wi =− − Ωui , ∂t ∂r r ∂θ ∂z r r ∂θ ∂wi ∂wi vi ∂wi ∂wi ∂φ + ui + + wi =− , ∂t ∂r r ∂θ ∂z ∂z   2 2 1 ∂ ∂φ 1 ∂ φ ∂ φ r + 2 2 + 2 = ne − np − ni r ∂r ∂r r ∂θ ∂z

(6.70) (6.71) (6.72) (6.73) (6.74)

1+q

where ne = [1 + (q − 1)φ] 2(q−1) and nj (j = i, e, and p stands for ion, electron, and positron, respectively) is the number density. Here, ui , vi , and wi are the ion fluid speed in r, θ, and z directions, respectively. The normalizations are taken as ni →

ni , ni0

φ→

eφ , KB Te

ui →

ui , vs

r→

r , λD

vi →

vi , vs

wi →

wi , vs

t → ωi0 t

 where nj0 is the unperturbed equilibrium plasma density, vs = KB Ti /mi −1 is the ion acoustic speed, and KB is the Boltzmann constant. ωpi =   mi /(4πe2 ni0 ), λD = KB Te /(4πe2 ni0 ), and Ω = (eB0 /mi c)/ωpi are the ion plasma frequency, the Debye radius, and the ion cyclotron frequency, respectively. To derive the cylindrical ZK equation, we introduce the stretch coordinates in the following way: ξ = 1/2 (r − λt),

η = −1 θ,

ζ = 1/2 z,

τ = 3/2 t.

(6.75)

Evolution Equations in Nonplanar Geometry

261

Expanding the dependent variables as ⎫ + ··· ,⎪ (1) (2) ⎪ = ui + 2 ui + · · · , ⎬ 3/2 (1) 2 (2) =  vi +  vi + · · · , ⎪ (1) (2) = 3/2 wi + 2 wi + · · · , ⎪ ⎭ = φ(1) + 2 φ(2) + · · · . (1)

ni = 1 + ni ui vi wi φ

(2)

+  2 ni

(6.76)

Substituting Equations (6.75)–(6.76) into Equations (6.70)–(6.74) and equating the lowest power of , we get (1)

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ni

=

1 1 ∂φ(1) 1 (1) (1) (1) φ , ui = φ(1) , vi = − . 2 λ λ Ω ∂ζ

(6.77)

From (6.77), we get  λ=

2(1 − p) . (qe + 1)(qp + 1)σp

(6.78)

To the next order of , one can get (1)

wi

(1)

=

(1)

(1)

Ω (2) ∂wi ∂vi = − wi , ∂ξ λ ∂ξ

=

⎫ ⎪ ⎪ ⎬

(1)

1 ∂φ(1) 1 ∂vi ∂wi , + λΩτ ∂η λτ ∂η ∂ζ

= 0,

Ω (2) v , λ i (1)

∂ 2 φ(1) 1 ∂ 2 φ(1) ∂ 2 wi + + ∂ξ 2 λ2 τ 2 ∂η 2 ∂ζ 2

⎪ ⎪ (2) ⎭ (2) (1) 2 = β1 φ + β2 [φ ] − n

(6.79)

i

1 where β1 = λ12 and β2 = 4(1−p) [(1 + qe )(3 − qe ) − (1 + qp )(3 − qp )pσ 2 ]. The next higher order of  gives ⎫ (1) (2) ∂ (1) (1) ∂ni ∂ni ⎪ −λ + (n u ) = 0, ⎪ ∂τ ∂ξ ∂ξ i i ⎬ (2) (1) (2) (2) 1 ∂vi ∂ui ui ∂wi (6.80) + + + = 0, ⎪ ∂ξ λτ λτ ∂η ∂ζ ⎪ (1) (2) (1) ∂φ(2) ⎭ ∂ui ∂u (1) ∂ui − λ i + ui =− . ∂τ ∂ξ ∂ξ ∂ξ

Eliminating the second-order derivatives of ni , ui , and φ from the Equations (6.79)–(6.81) with the help of Equation (6.77), we finally obtain the

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ZK equation in cylindrical geometry as ∂φ(1) φ(1) ∂φ(1) ∂ 3 φ(1) + + Aφ(1) +B ∂τ 2τ ∂ξ ∂ξ 3   ∂ 2 φ(1) ∂ 1 ∂ 2 φ(1) +C + =0 ∂ξ v02 τ 2 ∂η 2 ∂ζ 2 where A =

1 2v0 [3

− 2λ4 β2 ], B =

λ3 2 ,

and C =

λ3 2

 1+

1 Ω2

(6.81)

 . Equation (6.81) (1)

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differs with the ZK equation (in planar geometry) in terms of φ2τ and v21τ 2 0   2 (1) φ the coefficient of ∂ ∂η . These mentioned terms occur in the equation 2 due to the cylindrical geometry, which does not appear in planar geometry. 6.8 Nonplanar ZKB Equation We consider a homogeneous, collisionless magnetoplasma, consisting of ions and q-nonextensive distributed electrons, by taking into account the vis 0 = xB ˆ 0, cosity. The plasma is confined to an external magnetic field B where x ˆ is the unit vector along the x-direction and B0 is the external static magnetic field. Accordingly, the basic normalized equations are ∂ni  + ∇ · (ni · u) = 0, ∂t ∂u  )u = −(∇  φ) + Ω(u × xˆ) + μ∇  2 u, + (u · ∇ ∂t  2 φ = ne − (1 + α)ni + α ∇

(6.82) (6.83) (6.84)

where nj (j = i and e for ion and electron, respectively) is the number density and u(ui , vi , wi ) is the ion fluid speed. Here, the normalizations are taken as ni →

ni , ne0

u→

u , vs

φ→

eφ , KB Te

r→

r , λD

t → ωpi t

(6.85)

where nj0 is the unperturbed equilibrium plasma density and α =  zd nd0 /ne0 . Here, vs = KB Te /mi , KB are theion acoustic velocity −1 and the Boltzmann constant, respectively. ωpi = mi /(4πe2 ne0 ), λD =  KB Te /(4πe2 ne0 ), and Ω = (eB0 /mi c)/ωpi are the ion plasma period, the Debye radius, and the ion cyclotron frequency, respectively. c is the velocity of light and μ is the kinematic viscosity.

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To derive the ZK Burgers’ equation, we use the stretched variables as ξ = 1/2 (x − λt),

η = 1/2 y,

ζ = 1/2 z,

τ = 3/2 t,

μ = 1/2 μ0 (6.86)

where μ is scaled in such a way that μ0 becomes the effective viscosity. Expanding the dependent variables as ⎫ + ··· ,⎪ (1) (2) ⎪ = ui + 2 ui + · · · , ⎬ (1) (2) = 3/2 vi + 2 vi + · · · , ⎪ (1) (2) = 3/2 wi + 2 wi + · · · , ⎪ ⎭ = φ(1) + 2 φ(2) + · · · . (1)

ni = 1 + ni ui vi Downloaded from www.worldscientific.com

wi φ

(2)

+  2 ni

(6.87)

Substituting Equations (6.86)–(6.87) into Equations (6.82)–(6.86) and equating the lowest power of , we obtain (1)

ni

1 (1) = ui , λ

(1)

ui

1 = φ(1) , λ

(1)

vi

1 ∂φ(1) =− , Ω ∂ζ

 λ=

2(1 + α) . q+1 (6.88)

To the next order of , one gets (1) wi

(1)

(1)

1 ∂φ(1) ∂vi Ω (2) ∂wi = , = − wi , Ω ∂η ∂ξ λ ∂ξ (1)

∂ 2 φ(1) ∂ 2 φ(1) ∂ 2 wi + + 2 2 ∂ξ ∂η ∂ζ 2

(2)

= β1 φ

Ω (2) = vi , λ (1) 2

+ β2 (φ

) − (1 +

⎫ ⎪ ⎬ ⎪

(6.89)

(2) α)ni ⎭

(q+1)(3−q) where β1 = q+1 . 8 2 and β2 = The next higher order of  gives

⎫ (2) (2) (2) (2) ∂ (1) (1) ∂ni ∂ui ∂vi ∂wi ⎪ −λ + + + = 0, (n u ) + ⎬ ∂ξ ∂ξ i i ∂ξ ∂η ∂ζ   (1) (2) (1) (1) (1) (1) ⎪ ∂φ(2) ∂ui ∂u ∂ 2 ui ∂ 2 ui ∂ 2 ui (1) ∂ui ⎭ − λ i + ui + = μ0 + + . ∂τ ∂ξ ∂ξ ∂ξ ∂ξ 2 ∂η 2 ∂ζ 2 (6.90) (1)

∂ni ∂τ

Waves and Wave Interactions in Plasmas

264 (2)

(2)

Eliminating ui and ni from Equation (6.90) by using Equations (6.88)– (6.89), we get  ∂φ(1) 3 1 ∂φ(2) 2 ∂φ(1) + 4 φ(1) + 2 (1 + α) 3 λ ∂τ λ ∂ξ λ ∂ξ  2 (1)    ∂ 2 φ(1) ∂ 2 φ(1) ∂ 3 φ(1) 1 ∂ ∂ φ ∂ ∂ 2 φ(1) + 2 + + + + 2 2 2 2 Ω ∂ξ ∂η ∂ζ ∂ξ ∂η ∂ζ ∂ξ 3   ∂φ(2) ∂φ(1) ∂ 2 φ(1) ∂ 2 φ(1) μ0 ∂ 2 φ(1) − 3 − β1 − 2β2 φ(1) + + = 0. ∂ξ λ ∂ξ 2 ∂η 2 ∂ζ 2 ∂ξ (6.91)

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Thus, we finally obtain

  (1) ∂φ(1) ∂ 3 φ(1) ∂ 2 φ(1) ∂ ∂ 2 φ(1) (1) ∂φ + Aφ +B +C + ∂τ ∂ξ ∂ξ 3 ∂ξ ∂η 2 ∂ζ 2  2 (1)  ∂ φ ∂ 2 φ(1) ∂ 2 φ(1) −D + + = 0, (6.92) ∂ξ 2 ∂η 2 ∂ζ 2   2β2 λ3 3 λ3 which is the ZKB equation, where A = 2 λ4 − 1+α , B = 2(1+α) ,   3 1 C = λ2 1+α + Ω12 , and D = μ20 . References [1] J. R. Franz, P. M. Kintner, and J. S. Pickett, Geophys. Res. Lett. 25, 2041 (1998). [2] S. Maxon and J. Viecelli, Phys. Rev. Lett. 32, 4 (1974a). [3] S. Maxon and J. Viecelli, Phys. Fluids 17, 1614 (1974b).

Chapter 7

Collision of Solitons

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7.1 Introduction In theoretical studies of nonlinear science, propagation and collision of solitary waves occupy an important place. The interesting features of soliton collision are as follows: when two solitons collide, they interact, exchange energies and positions with each other, then separate, and finally regain their original waveforms. Solitons ensure their asymptotic preservation of form following an interaction. We can realize this by applying two consecutive voltage pulses in plasmas: the first pulse generates a small amplitude soliton and the second pulse produces a large amplitude one. Since the large amplitude soliton travels faster, it will overtake the smaller one. The larger pulse overtakes the smaller pulse, and the amplitude of the front pulse increases when the amplitude of the larger pulse decreases. Generally, two kinds of one-dimensional interaction occur. First, when two or more different amplitude solitons propagate in the same direction, they will overtake each other after some time. This phenomenon is called an overtaking collision. Second, when two solitons propagate in opposite directions, they interact for a relatively short time, emerge, and finally separate with phase shifts. This phenomenon is known as a head-on collision. Moreover, solitons may also interact obliquely while moving in a plane at an arbitrary angle to each other. This interaction occurs at some event altering the trajectories of colliding waves, and this kind of interaction is known as an oblique collision. 7.2 Head-on Collision Following voltage pulse generation between two plasmas, two types of scenarios may be seen. As a consequence, the generation of the voltage pulse 265

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Waves and Wave Interactions in Plasmas

is made from the same end of the plasma and then the so-called overtaking collision is depicted. When a third independent discharge is put at the opposite end of the plasma, in which the waves are detected, then a soliton is excited at each end of the main plasma simultaneously. The solitons propagate towards each other and interact near the center of the plasma. It is a relatively new concept in plasma dynamics. Initially, they interact, exchange their energies and positions with one another, then separate off, and finally regain their original waveforms. Throughout the whole process of collision, the solitary waves are stable and preserve their identities. The main effect due to the collision is their phase shift. This property of solitary waves can only be preserved in a conservative system. Let us take two independent moving solitary waves: aS(ξ) and bS(η). S(x) = sech2 ( x2 ) is a progressive wave of the permanent type that satisfies the KdV equation. The variables ξ and η denote the right and left going wave frame coordinates, respectively. The constants a and b specify the height of the waves. The experimental results show no observable nonlinear behavior upon interaction. The two peaks add up linearly when they overlap and penetrate each other. No deformation of the trajectories of the solitons is observed by H. Ikezi et al. [1]. T. Maxworthy [2] tested the higher-order effects in the head-on collision of two solitons, demonstrated by numerical computation. To illustrate this theoretically, let us take two solitons φα and φβ in the plasma. They are asymptotically far apart in the initial state, then travel toward each other, and after a certain time, they interact, collide, and finally depart. Hence, we expect that the collision will be quasielastic, so it will only cause shifts of the post-collision trajectories (phase shift). To analyze the effects of a collision, we employ an extended version of the PLK method which is discussed in the following: PLK method: Poincar´e developed a method for finding the periodic solution of a system of the first-order equation. The equation is written as dxi = Xi (x1 , x2 , ......xi , .....xn ; ε) (i = 1, 2, ....n) dt where t is the time variable and ε is a small parameter representing the perturbation influence. The equation with ε = 0 corresponds to the unperturbed system, and a periodic solution with period T (0) can be easily obtained from it. The essence of Poincar´e’s method is that not only the variables xi = x0i ∞ k k + k=1 ε xi + · · · are expanded but also the  ∞ 0 k k period T = T + k=1 ε T + · · · is expanded. This method has many applications in the theory of nonlinear oscillation. Poincar´e’s method of

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perturbation is based on the concept of expansion of the exact solution in a power series of small parameter ε, where the zeroth-order solution being independent of ε, the first-order solution is proportional to ε, etc. Lighthill’s method removes these difficulties. In this principle, the dependent variable u is expanded, and the independent variables x and y are also expanded ∞ in the power series of ε. By letting u = u0 (ξ, η) + k=1 εk uk (ξ, η) + · · · , ∞ k k ∞ k k x = ξ + k=1 ε x (ξ, η) + · · · , and y = η + k=1 ε y (ξ, η) + · · · , where ξ and η take the place of the original independent variables x and y. u0 (ξ, η) is simply the zeroth-order solution of the classical perturbation method with ξ and η replacing x and y. When we neglect the higher-order terms in u, the approximate solution is simply the zeroth-order perturbation solution with the stretched coordinates. Lighthill applied this method to solve the problems involving partial differential equations when the zeroth-order solution was obtained from a reduced linear equation of equal order. In many problems, a good zeroth-order approximation can be obtained only if a boundary layer solution is used. Kuo first recognized this necessity in his solution to the problem of the laminar incompressible boundary layer on a flat plate. It constitutes a further extension of Poincar´e’s original concept. Extended PLK method: In PLK method, the dependent variables are expanded and the independent variables are also stretched. The extended PLK method in one-dimensional plasma evolves and the dependent variables are expanded as earlier. The independent variables are stretched in opposite directions (one in positive direction and another in negative direc  k+1 εk+1 nk + · · · , ui = u0 + ∞ uk · · · , tion). We consider ni = 1 + ∞ k=1 k=1 ε ∞ k+1 and ψ = k=1 ε ψk + · · · . The independent variables are expressed as ξ = ε(x − c1 t) + ε2 P0 (η, τ ) + ε3 P1 (η, ξ, τ ) + · · · , η = ε(x + c2 t) + ε2 Q0 (ξ, τ ) + ε3 Q1 (η, ξ, τ ) + · · · , τ = ε3 t where ξ and η denote the trajectories of two solitons traveling towards each other and c1 and c2 are the unknown phase velocities. Perturbation solution for two solitary waves colliding head-on: Let us consider two solitary waves initially far apart, of small but finite amplitude and moving towards each other. The time evolution of their interaction and the final state after their collision will be our main concern.

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Waves and Wave Interactions in Plasmas

We introduce the following co-ordinates transformations (wave frames) as 1

ξ0 = ε 2 k(x − CR t),

1

η0 = ε 2 l(x + CL t)

(7.1)

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where 0 < ε < 1, and ε is the dimensionless parameter representing the order of magnitude of the wave amplitude. The scaling of the horizontal 1 wavelength following Ursell’s relationship is taken as ε 2 , leaving k and l as the wavenumbers of order unity for the right and left going waves, respectively. The right and left going wave speeds are CR and CL and are related to the amplitudes of the waves. Anticipating that difficulty (might show up in our perturbation method), we introduce the following transformations of wave-framed co-ordinates with phase functions as ξ0 = ξ − εkθ(ξ, η),

η0 = η − εlφ(ξ, η)

(7.2)

where θ(ξ, η) and φ(ξ, η) are to be determined in the process of the perturbational solution. These functions, introduced to make asymptotic approximations, allow us to calculate phase changes due to collision. Using (7.1) and (7.2), we obtain the transformation between derivatives as    ⎫ 1 ∂θ ∂ ∂ ∂ ε2 ∂ ∂θ ∂ ⎪ − + CR = (CR + CL ) l + εkl ⎬ ∂t ∂x D ∂η ∂η ∂ξ ∂ξ ∂η (7.3)    1 ⎪ ∂ ∂ ε2 ∂ ∂φ ∂ ∂φ ∂ ⎭ = − (CR + CL ) k + εkl − CL − ∂t ∂x D ∂ξ ∂ξ ∂η ∂η ∂ξ   ∂θ ∂φ 2 where D = 1 − εk ∂θ 1 − εl ∂φ ∂ξ ∂η − ε kl ∂η ∂ξ , θ(ξ, η) = θ0 (η)+ εθ1 (ξ, η)+ · · · , φ(ξ, η) = φ0 (ξ) + εφ1 (ξ, η) + · · · . Phase shifts in head-on collisions: We know that the general solution 2 2 of the standard wave equation ∂∂tφ2 = c2 ∂∂xφ2 is given by φ(x, t) = f (x − ct) + g(x + ct), where φ has some property associated with the wave and f and g are arbitrary functions. f (x − ct) represents a progressive wave, moving in the positive direction of the x-axis with a constant speed c, while g(x + ct) represents a progressive wave, moving in the negative direction of the x-axis with the same speed c. The arguments x − ct and x + ct are called the phase of f and g waves, respectively. Both of these are constants for space–time. The phase shift is any change that occurs in the phase due to collision. In a head-on collision, the change of position occurs due to the interaction

Collision of Solitons

269

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of two solitons. Let the trajectories of the two solitary waves be given by

ξ = (x − ct) + 2 P (0) (η, τ ) + 3 P (1) (ξ, η, τ ) + · · · (7.4) η = (x + ct) + 2 Q(0) (ξ, τ ) + 3 Q(1) (ξ, η, τ ) + · · · where ξ and η denote the trajectories of two solitons traveling toward each other (i.e., to the right and left, respectively), both with the same phase velocity c. Then, clearly P (0) (η, τ ) and Q(0) (ξ, τ ) are the leading phase changes due to head-on collision. The corresponding phase shifts P (0) and Q(0) and the difference between the post-collision and past-collision leading phase changes are ⎫ (0) (0) P (0) = Ppost-collision − Ppast-collision ⎪ = (x − ct)|ξ=0,η=∞ − (x − ct)|ξ=0,η=−∞ ⎬ (7.5) (0) (0) ⎪ Q(0) = Qpost-collision − Qpast-collision ⎭ = (x + ct)|ξ=∞,η=0 − (x + ct)|ξ=−∞,η=0 . 7.2.1 Head-on collision of solitary waves in planar geometry Now, we shall study the variation of phase shift and the trajectories of the two solitary waves after the collision. To obtain the phase shift and the trajectories, a couple of KdV equations have to be derived by employing the extended PLK method. Derivation of KdV equations: Let us consider two-component plasma consisting of cold ions and nonextensive electrons. Let us analyze the headon collision of IASWs in this model. Accordingly, the set of normalized equations are ∂ni ∂(ni ui ) + = 0, ∂t ∂x ∂ui ∂ui ∂φ + ui =− , ∂t ∂x ∂x ∂2φ = ne − ni ∂2x 1+q

(7.6) (7.7) (7.8)

where ne = [1+(q −1)φ] 2(q−1) . The normalization and notation are accordingly the same as in Section 4.4. Now, we assume that the two solitons α and β in the plasma, which are asymptotically far apart in the initial state,

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270

travel towards each other. After some time, they interact, collide, and then depart. Let the solitons have small amplitudes ∼ε, where ε is a small parameter characterizing the strength of nonlinearity, and the interaction between two solitons is weak. Hence, we expect that the collision will be quasielastic, and so it will only cause shifts of the post-collision trajectories (phase shift). To analyze the effects of a collision, let us employ the extended PLK method. According to this method, the dependent variables are expanded as ni = 1 +

∞ 

εk+1 nk ,

ui = 0 +

k=1

∞ 

εk+1 uk ,

φ=0+

k=1

∞ 

εk+1 φk .

(7.9)

k=1

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The independent variables are written as ⎫ ξ = ε(x − λt) + ε2 P0 (η, τ ) + ε3 P1 (η, ξ, τ ) + · · · , ⎪ ⎬ η = ε(x + λt) + ε2 Q0 (ξ, τ ) + ε3 Q1 (η, ξ, τ ) + · · · , ⎪ ⎭ τ = ε3 t

(7.10)

where ξ and η denote the trajectories of two solitons traveling towards each other and λ is the unknown phase velocity of IASWs to be determined. The variables of P0 (η, τ ) and Q0 (ξ, τ ) are also to be determined. Substituting Equations (7.9) and (7.10) into Equations (7.6)–(7.8) and equating the quantities with equal power of ε, we obtain the coupled equations in different orders of ε. For the leading order, we have   ⎫ ∂n1 ∂u1 ∂u1 ∂n1 +λ + + −λ = 0, ⎪ ∂ξ ∂η ∂ξ ∂η ⎪   ⎬ ∂φ1 ∂u1 ∂φ1 ∂u1 (7.11) +λ + + −λ = 0, ⎪ ∂ξ ∂η ∂ξ ∂η   ⎪ 1+q n1 − φ1 = 0. ⎭ 2 Solving Equation (7.11), we get φ1 = φξ (ξ, τ ) + φη (η, τ ),   1+q n1 = [φξ (ξ, τ ) + φη (η, τ )], 2 u1 =

1 1 φξ (ξ, τ ) − φη (η, τ ). λ λ

(7.12) (7.13) (7.14)

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With the solvability condition [i.e., the condition required to obtain a uniquely defined n1 and u1 from Equations (7.13) and (7.14), when φ1  1/2 2 is given by (7.12)], the phase velocities λ = q+1 + u0 and λ =  1/2 2 − u0 are also obtained. Now, we shall determine the unknown q+1 function φξ and φη in the next orders. Relations (7.12)–(7.14) imply that, at the leading order, we have two waves: one φξ (ξ, τ ) that travels to the right and another φη (η, τ ) that travels to the left. The next higher order leads to    ∂ 2 u3 ∂ ∂φξ ∂φξ ∂ 3 φξ ∂ ∂φη ∂φη − λ1 =− + Aφξ +B − Aφη 3 ∂ξ∂η ∂ξ ∂τ ∂ξ ∂ξ ∂η ∂τ ∂η      2 ∂ 3 φη ∂P 0 ∂ 2 φξ ∂Q0 ∂ φη − C −B − Dφη + C − Dφξ ∂η 3 ∂ξ 2 ∂η 2 ∂η ∂ξ (7.15)     1/2  2  1 where A = 12 −(λ1 )3/2 1 − q−1 + 3 , B = 12 (λ1 )3/2 , C = 2 λ1      2  1 2 1/2 , D = 12 (λ1 )3/2 1 − q−1 + (λ1 )1/2 , and λ1 = q+1 . 2 2 (λ1 ) On integration of the above equation with respect to the variables ξ and η, one can get    ∂φξ ∂φξ ∂ 3 φξ λ1 u3 = − + Aφξ +B dη ∂τ ∂ξ ∂ξ 3    ∂ 3 φη ∂φη ∂φη −B − − Aφη dξ ∂η ∂η 3 ∂τ  2    ∂P 0 ∂ φξ − C − Dφη dξdη ∂ξ 2 ∂η  2    ∂Q0 ∂ φη + C − Dφξ dξdη. (7.16) ∂η 2 ∂ξ The integrand of the first integral on the right-hand side of Equation (7.16) will be proportional to η because the integrand is independent of η. The second term in the right-hand side of Equation (7.16) is dependent only on ξ and τ and, therefore, if the integrand is not identically equal to zero, then the integral will be proportional to η, which will give rise to secular terms. Hence, we must have the integrand to be identically zero. The same

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argument holds for the second integral. Hence, we obtain the following equations: ∂ 3 φξ ∂φξ ∂φξ + Aφξ +B = 0, ∂τ ∂ξ ∂ξ 3

(7.17)

∂φη ∂φη ∂ 3 φη − Aφη = 0. −B ∂τ ∂η ∂η 3

(7.18)

Equations (7.17) and (7.18) are the famous KdV equations. The solitary waves are traveling in opposite directions in the reference frames of ξ and η, respectively, and their expressions are  1/2   AφA 1 φξ = φA sech2 ξ − AφA τ , (7.19) 12B 3  1/2   AφB 1 2 φη = φB sech η + AφB τ (7.20) 12B 3 where φA and φB are the amplitudes of the two solitons in their initial positions. The soliton-like solutions are formed because of the balance between the nonlinearity and dispersion in nonlinear dispersive media. The soliton-like solutions are obtained from Equations (7.17) and (7.18). The coefficients A and B, appearing in the soliton solutions, are represented by Equations (7.17) and (7.18). These may create two possibilities: (i) AB > 0 and (ii) AB < 0. Case (i): When AB > 0, Equations (7.19) and (7.20) give the compressive IASW solutions (φA > 0 and φB > 0). Case (ii): When AB < 0, Equations (7.19) and (7.20) give the rarefactive IASW solutions (φA < 0 and φB < 0). The third and fourth terms in Equation (7.16) are not secular in this order though they could be secular in the next order. Hence, we have C

∂P0 = Dφη , ∂η

C

∂Q0 = Dφξ . ∂ξ

(7.21)

The leading phase changes due to the collision can be calculated by integral Equation (7.21) and are written as D P0 (η, τ ) = C



12BφB A

1/2 

 tanh

AφB 12B

1/2    1 η + AφB τ + 1 , 3 (7.22)

Collision of Solitons

Q0 (ξ, τ ) =

D C



12BφB A

1/2 

 tanh

AφB 12B

273

1/2    1 ξ − AφA τ − 1 . 3 (7.23)

Therefore, up to O(ε2 ), the trajectories of the two solitary waves for head-on interactions are 1/2 12BφB ξ = ε(x − λt) + ε C A   1/2    AφB 1 × tanh η + AφB τ + 1 + · · · , 12B 3  1/2 D 12BφB η = ε(x + λt) + ε2 C A   1/2    AφB 1 × tanh ξ − AφA τ − 1 + · · · . 12B 3

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2D



(7.24)

(7.25)

To obtain the phase shift after a head-on collision of the two solitons, let us assume that in the initial state (t = −∞) the solitons are asymptotically far from each other (first soliton is at ξ = 0, η = −∞ and the second soliton is at η = 0, ξ = +∞). After the collision (t = +∞), the first soliton is far to the right of the second soliton (soliton α is at ξ = 0, η = +∞ and soliton β is at η = 0, ξ = −∞). Using Equations (7.24) and (7.25), we obtain the corresponding phase shifts ΔP0 and ΔQ0 as follows from (7.5) and finally we get ΔP0 = −2ε

2D

C



12BφB A

1/2 ,

ΔQ0 = 2ε

2D

C



12BφA A

1/2 . (7.26)

From (7.21), it is clear that the phase shift will be positive or negative depending on the coefficient D/C in Equation (7.21). It is also seen from Equation (7.26) that the magnitudes of the phase shifts are directly related to the physical parameters ε, φA , and φB . Since the first soliton is traveling to the right and the second soliton is traveling to the left, it is seen from Equation (7.26) that due to the collision, each soliton has a negative phase shift in its traveling direction. The negative phase shift means that the solitons reduce their velocities during the interaction stage. It is clear from Figure 7.1 that for several values of τ , the time evolution of the head-on collision solitary wave solution φξ will be shifted towards the right with the

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Fig. 7.1: Plot of φξ (solid line) and φη (dashed line) against ξ and η, respectively, for several values of τ .

progress of time. But whereas the solution φη will be shifted towards the left with the progress of time.

7.2.2 Head-on collision of solitons in a Magnetized Quantum Plasma We have studied the linear and nonlinear wave propagation in classical e-p-i plasmas using a variety of plasma models. The subfield quantum plasma of plasma physics is also enriched with various investigations on e-pi plasma. Chatterjee et al. [3] have investigated the nonlinear propagation of quantum ion acoustic waves in dense quantum plasma containing electrons, positrons, and positive ions. When an e-p-i plasma is cooled under extremely low temperatures, the de Broglie wavelength of the charge carriers becomes comparable to the dimension of the system. In such a situation, ultra-cold e-p-i plasma behaves like a Fermi gas, and quantum mechanical effects are expected to play a vital role in the behavior of charged particles. All the works we have discussed earlier on head-on collision intend to find the KdV solitons with their phase shifts and the effects of different parameters on it, avoiding the detailed discussion on the critical composition. However, after the works of Verheest et al. [4, 5], there has been a surge of interest in the mKdV solitons for the critical composition, where the cubic nonlinearity, rather than the quadratic nonlinearity of the KdV equations, will appear in the evolution equations.

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Let us consider a three species dense quantum plasma composed of electrons, positrons, and singly charged positive ions. The plasma is con 0 = B0 zˆ, where B0 is sidered to be in the uniform external magnetic field B the strength of the magnetic field and zˆ is the unit vector in the z-direction. At equilibrium, the charge neutrality condition is ne0 = ni0 + np0 , where nj0 (j = e, p, and i stand for electrons, positrons, and ions, respectively) is the unperturbed number density. We assume that the Fermi temperature of electrons or positron is much larger than that of ions (i.e., TF e,p  TF i ). The electron/positron obeys the pressure law Pe,p = mn3e,p VF2e,p /3n2e,p0 , where VF e,p = (2KB TF e,p /m)1/2 is the electron/positron Fermi thermal speed, KB is the Boltzmann constant, TF e,p (TF i ) is the electron/positron (ion) Fermi temperature, m is the mass of electron and positron, and the number density of electron/positron is ne,p with the equilibrium value ne,p0 . The basic normalized equations are ∂ni  + ∇ · (ni ui ) = 0 ∂t ∂ ui  ui = −∇  φ + ui × zˆ + ui · ∇ ∂t  2 φ = ne − np − ni Ω∇

(7.27) (7.28) (7.29)

  √ 1/2 ∇2 n where ne = μe 1 + 2φ + He2 √ne e and np = μp 1 − 2σφ + √ 1/2 ∇2 n σHe2 √np p . ui , mi , and φ are the ion fluid velocity, the ion mass, and the electrostatic potential, respectively. σ = TF e /TF p is the electron √ to positron Fermi temperature ratio, and He = eB0 /2c mi mKB TF e is the quantum diffraction parameter. Here, the normalizations are taken as ∇Cs eφ , ∇→ , t → tωci ωci 2KB TF e  where ωci = eB0 /mi c and ωpi = 4πe2 ni0 /mi are the iongyrofrequency 2 2 and the ion plasma frequency. Also, Ω = ωci /ωpi and Cs = 2KB TF e /mi , where  is the Planck constant divided by 2π and c is the speed of light in vacuum. As usual, two solitary waves S1 and S2 are considered. To analyze the effect of collision, we employ the extended PLK method, and hence, the dependent variables are expanded in different powers of in nj →

nj , ni0

ui →

ui , Cs

φ→

ψ = ψ0 +

∞  n=1

n ψ (n)

(7.30)

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where ψ = (ne , ni , np , uix , uiy , uiz , φ) with ψ0 = (μe , 1, μp , 0, 0, 0, 0). The stretched independent variables are expanded as ⎫ ξ = (lx x + ly y + lz z − λt) + 2 P (ξ, η, τ ) + · · · , ⎪ ⎬ (7.31) η = (lx x + ly y + lz z + λt) + 2 Q(ξ, η, τ ) + · · · , ⎪ ⎭ 3 τ = t

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where ξ and η denote the trajectories of the two solitary waves during a head-on collision in the direction having direction cosine (lx , ly , lz ), with equal but opposite directional velocities. The variables P , Q, etc., will be determined later. Using the extended PLK method, and after some calculation, we get [the same as the previous section] the coupled KdV equation as (2)

∂φξ

+

∂τ

(2)

(2) (2) ∂φξ Aφξ

∂ξ

(2) 3

+B

(2)

∂ 3 φξ

=0

∂ξ 3

(7.32)

(2)

∂φη ∂ 3 φη −B =0 (7.33) ∂η ∂η 3    3l4z He2 λ3 λ3 2 2 2 where A = 2l and B = 2l 2 μe − σ μp + λ4 2 1 + Ω − lz − 4μ μ (μp + σ μe ) . e p z z Also, we get ∂φη ∂τ

− Aφ(2) η

l2 ∂P − z2 φ(2) = 0, ∂η 4λ η

l2 (2) ∂Q − z2 φξ = 0. ∂ξ 4λ

(7.34)

Equation (7.32) is a KdV equation. This wave is traveling in the ξ direction. Equation (7.33) is another KdV equation. This wave is propagating in the η direction, which is opposite to ξ. Their one soliton solutions are given as follows:  1/2   AφA 1 (2) 2 ξ − AφA τ (7.35) φξ = φA sec h 12B 3  1/2   AφB 1 2 φ(2) = φ sec h η + Aφ τ (7.36) B B η 12B 3 where φA and φB are the amplitudes of the two solitary waves in their initial positions. The leading phase changes P (η, τ ) and Q(ξ, τ ) are due to the head-on collision and are calculated from Equation (7.34). To obtain the phase shifts after the head-on collision of the two solitary waves, we assume that the waves S1 and S2 are asymptotically far from

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each other at the initial time τ = −∞, S1 is at ξ = 0, η = −∞ and S2 is at ξ = ∞, η = 0. After the collision, at time τ = +∞, S1 is far to the right of S2 , i.e., S1 is at ξ = 0, η = ∞ and S2 is at ξ = −∞, η = 0. We obtain the corresponding phase shifts P =

l2 − 2 z2 λ



3BφB A

 12

,

Q =

l2

2 z2 λ



3BφA A

 12

.

(7.37)

Derivation of phase shifts at critical compositions: At the critical composition when A = 0, the quadratic nonlinearity in the KdV Equations (7.32) and (7.33) will disappear, and the cubic nonlinearity will appear. (2) (2) Then, we consider φ(2) − φξ − φη = 0. We can cancel [4] the solutions of Downloaded from www.worldscientific.com

(2)

(2)

the linear operator without loss of generality. Hence, φξ = φη = 0 and according to the same procedure, we get the mKdV equations as (1)

∂φξ

+

∂τ (1)

∂φη ∂τ 

3

λ where A1 = 2l 2 z  2 σ μe ) . Also, we get

15l6z 2λ6

−

(1) (1) 2 ∂φξ A1 (φξ )

∂ξ

2 + A1 (φ(1) η ) 3(μe +σ3 μp ) 2



λ2 8l2z



μe + σ 2 μp −

l6z λ6

+B

∂ξ 3

(1)

and B =

=0

(7.38)

=0

(7.39)

(1)

∂φη ∂ 3 φη +B ∂η ∂η 3

∂P 2 − C(φ(1) η ) = 0, ∂η where C =

(1)

∂ 3 φξ

λ3 2l2z

 1 + Ω − lz2 −

∂Q (1) − C(φξ )2 = 0 ∂ξ

He2 4μe μp (μp

+

(7.40)

 .

Now the mKdV Equations (7.38) and (7.39) will keep the same form, (1) (1) whatever may be the sign of φξ or φη . The one soliton solutions of Equations (7.38) and (7.39) are (1)

φξ

φ(1) η

  A1 φ2a φa ξ − τ 6   1/2  A1 A1 φ2b = ±φb sec h φb η − τ . 6B 6 

= ±φa sec h

A1 6B

1/2

(7.41) (7.42)

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Now, proceeding as in the case of KdV solitons, we obtain the corresponding colliding phase shifts as

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P = −2 2 Cφb



6B A1

1/2 ,

Q = 2 2 Cφa



6B A1

1/2 .

(7.43)

Since, the balance between the nonlinear and the dispersion terms in Equations (7.32), (7.33), (7.38), and (7.39) creates the KdV and the mKdV soliton-like solutions, respectively. So, the coefficients A, B, and A1 , (with 3l4 the restriction μe − σ 2 μp + λ4z = 0) are responsible for such situations. Here, two possibilities are created: (i) If AB > 0 (i.e., both φA and φB are positive), the solutions of Equations (7.35) and (7.36) are the compressive IASWS solutions (CIASWs). (ii) If AB < 0 (both φA and φB are negative), the solutions of Equations (7.35) and (7.36) are rarefactive IASWs (RIASWs). But the mKdV solitons are compressive as well as rarefactive if A1 B > 0 for positive, as well as negative polarities. To illustrate the statements (i) and (ii), we draw Figure 7.2, where we plot the regions of CIASWS and RIASW solutions in the generic case in the He -Ω plane with μp = 1, σ = 1.585, and lz = 0.9. We plot the region of CIASWs and RIASWs for critical case and also the region where the mKdV soliton does not exist in the He -Ω plane (see Figure 7.3) for lz = 0.89 and μp = 0.249. Thus, there

Fig. 7.2: The regions for CIASWs and RIASWs in Ω − He plane for σ = 1.585, lz = 0.9, and μp = 1 in the generic case.

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Fig. 7.3: The region in the Ω − He plane for lz = 0.89 and μp = 0.249, where the two types of MKdV solitons exist.

is no limitation for the existence of KdV solitons on the parameters, but limitations are there for the mKdV solitons. 7.2.3 Head-on collision of magneto-acoustic solitons in spin-1/2 fermionic quantum plasma In dense plasmas, degenerate electrons follow the Fermi–Dirac pressure law, and the quantum force is connected with the Bohm–de Broglie potential, and as a product of which the waves disperse at nanoscales. Not only that, the effects of the electron spin appear themselves in terms of a magnetic dipole force and spin precession. It can be obtained by transforming the Pauli equation to fluid-like variables. Hence, the dynamics of electrons in Fermi’s degenerate plasmas will be intervened not only by the Lorentz force but also by the effects of quantum statistical pressure, the Bohm force, and the intrinsic spin of electrons. Marklund and Brodin [6] introduced the dynamics of spin-1/2 quantum plasma in the nonrelativistic magnetoplasma. They [7] showed that the spin properties of the electrons and positrons might lead to interesting collective effects in quantum magnetoplasma by the equation of Schrodinger Pauli. Marklund, Eliasson, and Shukla [8] also showed the existence of magneto solitons in fermionic quantum plasma. They found that if one neglects the magnetic diffusivity, the magnetic field satisfies an equation identical to

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the continuity equation. Hence, one can take the magnetic field linearly proportional to the density of plasma fluid by applying a simplification of the governing equations. Later, Misra and Ghosh [9] obtained the spin magnetosonic shock-like waves in quantum plasma considering the magnetic diffusivity. Linear and nonlinear compressional magnetosonic waves in magnetized degenerate spin-1/2 Fermi plasmas are investigated [10]. Relativistic corrections to the Pauli Hamiltonian in the context of a scalar kinetic theory for spin-1/2 quantum plasmas have also been established by Asenjo et al. [11]. Here, we are going to study the head-on collision of magnetosonic solitons in a fermionic quantum plasma, taking into account spin effects. The total mass density, the center-of-mass fluid flow velocity, and the  = current density are defined respectively as ρ = (me ne + mi ni ), V (me ne ve + mi ni vi )/ρ, and j = (−e ne ve + e ni vi ). mk , nk , vk , and e (k = e, and i, stands for electron, and ion, respectively) are the mass, the number density, the fluid velocity, and the electron charge, respectively. Assuming the quasineutrality condition (ne = ni ) and taking the mag = B(x, t)ˆ netic field along the z -axis so that B z and taking the velocity  V = V (x, t)ˆ x and the density as ρ(x, t), we get the following system of normalized basic equations: ∂b ∂(bv) + =0 ∂t ∂x

(7.44)

√  2 2ωpe ∂v ∂v ∂b ∂ 1 ∂2 b 2 ∂(ln b) √ − cs +v =− + ∂t ∂x ∂x ∂x ωc |ωce | ∂x b ∂x2 2 + vB



∂ [ln(cosh(ze b)) + ze b(tanh(ze b))] ∂x

(7.45)

where ρ = ρ0 b, b = B/B0 , cs = Cs /CA , and v = V /CA . Also, CA =  B02 /(μ0 ρ0 ) isthe Alfven speed, Cs = KB (Te + Ti )/mi is the sound speed, ωpk = (nk0 e2 )/( 0 mi ) is the plasma frequency, ωc = 2me c2 /h is the Compton frequency, ze = (μB B0 )/(KB Te ) is the temperature2 2 normalized Zeeman energy, and vB = (μB B0 )/(εmi CA ), where μ0 is the permeability of the vacuum, B0 is the magnetic field strength, ρ0 is the total mass density of the charged plasma particles, Ti and Te are ion and electron temperatures, KB is the Boltzmann constant, and μB is the magnitude of Bohr magneton. The normalized variables are taken as t → ωci t and x → (ωci x)/CA . In deriving Equations (7.44) and (7.45), the magnetic resistivity is neglected.

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Now, we assume that the two solitons α and β apart in the initial state and travel towards each effects of a collision, we employ an extended PLK this method, the dependent variables are expanded b = 1 + b1 + 3/2 b2 + 2 b3 + · · ·

are asymptotically far other. To analyze the method. According to as

(7.46)

v = v1 + 3/2 v2 + 2 v3 + · · · and the independent variables are given by

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⎫ ξ = 1/2 (x − λt) + 3/2 P0 (η, τ ) + 5/2 P1 (ξ, η, τ ) + · · · ⎪ ⎬ η = 1/2 (x + λt) + 3/2 Q0 (ξ, τ ) + 5/2 Q1 (ξ, η, τ ) + · · · ⎪ ⎭ τ = ε3 t.

(7.47)

Here, ξ and η denote the trajectories of two solitons traveling towards each other and λ is the unknown phase speed normalized by CA . P0 (η, τ ) and Q0 (ξ, τ ) are also needed to be determined. Using extended PLK method, and after some calculations, we get ∂b11 ∂b11 ∂ 3 b11 + A1 b11 + B1 =0 ∂τ ∂ξ ∂ξ 3

(7.48)

∂b12 ∂b12 ∂ 3 b12 − B1 − A1 b12 =0 ∂η ∂η 3 ∂τ

(7.49)

3λ2 −c2 −z 2 v 2 sec h2 z (3−2z tanh ze )

e e s e B where A1 = 2λ  2 (2 tanh z + z sec h2 z ). λ = 1 + c2s − ze vB e e e Also, we get

C1

∂P0 = D1 b12 , ∂η

C1

ω2

, B1 = − 2|ωcepe|λωc , and

∂Q0 = D1 b11 ∂ξ

λ2 +c2 +z 2 v 2 sec h2 z (3−2z tanh z )

(7.50)

e e e s e B where C1 = 2λ and D1 = . 2λ Equations (7.48) and (7.49) are the two side of traveling wave KdV equations in the reference frames of ξ and η, respectively. Their special solutions are  1/2   A1 bA 1 b11 = bA sec h2 ξ − A1 bA τ (7.51) 12B1 3  1/2   A1 bB 1 2 b12 = bB sec h η + A1 bB τ (7.52) 12B1 3

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Fig. 7.4: Graphs of the variation of phase shift Q0 against the Zeeman energy ze for phase velocities vp = 0.0098 (solid line), 0.0152 (dotted line), and 0.0198 (dashed line).

where bA and bB are the amplitudes of the two solitons in their initial positions. We obtain the corresponding phase shifts as P0 = −2

2 D1

C1



12B1 bB A1

1/2 ,

Q0 = 2

2 D1

C1



12B1 bA A1

1/2 . (7.53)

2 To draw Figure 7.4, we consider = 0.1, BA = 0.1, ωpe /(ωc |ωce |) = 1, and vB = 0.2. Figure 7.4 represents the variations of the phase shift Q0 with the Zeeman energy (ze ) for different values of vp , when cs = 0.1. The phase shift Q0 is (i) positive and increasing when 0 < ze < 0.8 and 3 < ze < 7, (ii) positive and decreasing when 0.8 < ze < 2, (iii) negative and decreasing when 2 < ze < 2.5, (iv) negative and increasing for 2.5 < ze < 3, and (v) positive and constant for ze > 7. Thus, the phase shift is positive and negative for different domains for the Zeeman energy. With a particular value of ze , the magnitude of the phase shifts increases as the speed of the wave decreases. Qualitatively, one can say that the phase shifts become flat when the Zeeman energy is large. We consider the case where the Zeeman energy ze ≥ 1 so that the spin contribution to the soliton dynamics is enhanced, and the Zeeman energy plays a crucial role in the phase shifts. The KdV-type soliton-like solutions are formed due to the balance between the nonlinearity and dispersion in nonlinear dispersive media. So, the condition for the existence of soliton-like solutions is A1 = 0 and B1 = 0. Since, the soliton α is traveling from the left and β is traveling from the right, Equation (7.53) imply that each soliton has a positive or a negative phase shift, depending upon the sign of the coefficient D. Moreover, from Equation (7.53), it is clear that both B1 bB /A1 and B1 bA /A1 must be positive, if both P0 and Q0 are real. It follows logically that bB and bA

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Fig. 7.5: The leading phase changes P (η, τ ) due to head-on collision against η and τ for the parametric values as in Figure 7.4 and for ze = 0.498.

must have the same sign. Hence, both the solitons will be either hump types or dip types. Thus, the positive or negative phase shift does not depend on the type of mode (i.e., ion-acoustic, dust-ion-acoustic, dust-acoustic, magneto-acoustic, and electrostatic waves). In Figure 7.5, we consider the three-dimensional plot of the leading phase changes due to head-on collision against the space–time variables for the parametric values as in Figure 7.4 and three different values of Zeeman energy. It is clear from these three figures that the leading phase changes due to the head-on collision that is initially positive, and after a certain value of Zeeman energy, it is negative. Finally, it remains positive after a certain value of Zeeman energy although the other plasma parameters are the same. 7.2.4 Interaction of DIASWs in nonplanar geometry In this section, we will consider the nonlinear propagation of finiteamplitude DIA waves in a four-component collisionless, unmagnetized, dusty plasma consisting of q-nonextensive distributed electrons, stationary positive and negative dust charge, and inertial ions in a nonplanar geometry. At equilibrium, we have ni0 +Zdp ndp =ne0 +Zdn ndn , where ni0 , ne0 , ndp , and ndn are, respectively, ion, electron, positive dust, and negative dust number density at equilibrium, and Zdp (Zdn ) represents the charge state of positive (negative) dust. The dynamics of DIAWs in such plasma

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284

can be described by the following set of normalized equations:

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∂ni 1 ∂ ν + ν (r ni ui ) = 0 ∂t r ∂r ∂ui ∂ui ∂φ + ui =− ∂t ∂r ∂r   1 ∂ ∂φ rν = (1 − jμ)ne − ni + jμ rν ∂r ∂r

(7.54) (7.55) (7.56)

where ν = 0 in case of 1D planar geometry and ν = 1, 2 in case of nonplanar (cylindrical or spherical) geometries, respectively. All the notations and normalizations are the same as in Section 6.5. Suppose that two solitary waves in nonplanar geometry, R and L, have been excited in the system. The solitary wave R(L) is traveling outward (inward) from (to) the initial point of the coordinate system. The initial position (at time t = 0) of the solitary wave R(L) is at r = rR (r = rL ), rL  rR . Since 0 < r < ∞, this consideration is different from cartesian solitons, where −∞ < x < ∞ in one dimension. For the head-on collision between two Cartesian solitons, one can consider one soliton is at x ≈ −∞ and the other one at x ≈ ∞ (at the initial time t = 0). After some time, they interact, following a collision, and then depart from each other. To investigate the head-on collision between the two solitary waves in a nonplanar geometry, we extend the PLK method to the nonplanar geometry. We anticipate that the collision will result in phase shifts in their postcollision trajectories. Thus, we introduce the following transformations: ⎫ ξ = ε(r − ct − rR ) + ε2 A1 (η, L) + ε3 A2 (η, ξ, L) + · · · , ⎪ ⎬ (7.57) η = ε(r + ct − rL ) + ε2 B1 (ξ, L) + ε3 B2 (η, ξ, L) + · · · , ⎪ ⎭ 3 L=ε r where ξ and η denote the trajectories of two solitons traveling towards each other and c is the unknown phase velocity of DIASWs. The variables of A1 (η, L) and B1 (ξ, L) are also to be determined. Introducing the asymptotic expansion ni = 1 +

∞ 

εk+1 nk + · · · ,

k=1

φ=

∞  k=1

εK+1 φk + · · ·

ui = u0 +

∞  k=1

εK+1 uk + · · · , (7.58)

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where u0 is the drift fluid velocity. Proceeding the same way as the previous problem, we have ∂φξ ∂ 3 φξ ν ∂φξ + AφA +B + φξ = 0, ∂L ∂ξ ∂ξ 3 2L

(7.59)

∂φη ∂φη ∂ 3 φη ν + Aφη +B + φη = 0 3 ∂L ∂η ∂η 2L

(7.60)

where A = 34 (q + 1)(1 − jμ) − 14 (3 − q) and B = Also, we get

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2

∂A1 = Dφη , ∂η

2

1 (q+1)(1−jμ) .

∂B1 = Dφξ ∂ξ

(7.61)

where D = 14 (q + 1)(1 − jμ) + 14 (3 − q). Equations (7.59) and (7.60) are the two side traveling wave KdV equations in the reference frames of ξ and η, respectively. Their special solutions are    1 2ν/3 1/2 ν/3 2ν/3 φξ = UA ΘA sec h2 ΓA ΘA ξ − AUA ΘA L , (7.62) 3    1 2ν/3 1/2 ν/3 2ν/3 2 φη = UB ΘB sec h ΓB ΘB η − AUB ΘB L (7.63) 3 where UA and UB are the amplitude of two DIASWs R, and L, respectively. Here, ΘA = LA /L, ΘB = LB /L, ΓA = AUA /12B, and ΓB = AUB /12B with LA = ε3 rR and LB = ε3 rL . It is noted that the amplitude decreases drastically as L(r) increases in cylindrical and spherical geometries. In Equations (7.59) and (7.60), A and B are the coefficients of nonlinearity and dispersion, respectively. If we set ν = 0, Equations (7.59) and (7.60) will become the planar KdV equations. ν = 1 and ν = 2 correspond to the cylindrical and spherical KdV equations, respectively. We obtain the corresponding phase shifts ΔP0 and ΔQ0 if the initial separation between the two solitons is large enough, i.e., rL  rR , and the observation time is much larger than the collision time, i.e., t  tc = (rL − rR )/2, as  1/2  ν/3  1/2  ν/3 12BUB 12BUA rL rL 2 2 ΔP0 = −ε D , ΔQ0 = ε D . A r A r (7.64) For cylindrical case, Equations (7.59) and (7.60) are the outward (R) and inward (L) traveling wave KdV equations, respectively. Now, we want to show how the interaction takes place. At the initial position, these two

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solitary waves R and L are at r = rR = 57.5 and r = rL = 137.5, respectively, at time t = 0. With the progress of time, as seen in Figures 7.6, the outward solitary waves R and inward solitary waves L come closer and closer, and ultimately they collide. After colliding with each other, they interchange their position, and, finally, they depart from each other so that the soliton R has the radius r = rL = 137.5 and soliton L has the radius r = rR = 57.5, as depicted in Figure 7.6. For fixed physical parameters and initial soliton position, the phase shifts are proportional to r−ν/3 . So, the collision-induced phase shifts in cylindrical/spherical geometry decrease with r according to r−ν/3 .

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 7.6: Graphs of cylindrical collision for (a) t = 0, (b) t = 30, (c) t = 60, (d) t = 90, (e) t = 120, and (f) t = 150.

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When the physical parameters and solitons’ positions are the same, the phase shifts in spherical and cylindrical geometry are different. But, when ν = 0, the phase shifts represented by Equation (7.64) reduce to the planar case. We have plotted all the figures by taking the negative dust charge (i.e., j = 1) and have discussed accordingly. If we consider the positive dust charge (i.e., j = −1), we will obtain some interesting behavior of the phase shift.

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7.3 Oblique Collision Now, we are going to explain the oblique collision of soliton. The oblique collisions between solitary waves are interesting nonlinear phenomena in plasmas. When two solitary waves approach closely, they interact, exchange their energies and positions with each other, and then separate off, regaining their original waveforms. During the collision, the solitary waves are remarkably stable entities, preserving their identities through collisions. The unique effect due to the collisions is their phase shift and the trajectories. Let us consider two small but finite amplitude solitary excitations, which propagate obliquely in the XY plane at an arbitrary angle with different velocities. Their interaction occurs at some event altering the trajectories of colliding waves, and then phase shifts appear. Realization of an acceptable physical profile of elastic interaction between waves through asymptotic expansion of plasma variables is possible. It is done with appropriate coordinates using stretching. The stretching process admits successful separation of variables to the desired nonlinear evolution and interaction profile by eliminating the secular terms with the help of an extended PLK method. The stretching coordinates are taken as ⎫ ξ = ε(k1 x + l1 y − c1 t) + ε2 P0 (η, τ ) + · · · ⎪ ⎬ (7.65) η = ε(k2 x + l2 y − c2 t) + ε2 Q0 (ξ, τ ) + · · · ⎪ ⎭ 3 τ =ε t where the functions Pj and Qj (j = 0, 1, 2, . . .) give the phase description of the interacting solitary waves in space–time evolution. These types of stretched coordinates were used in fluid dynamics also. Furthermore, we assume that before the collision takes place, the waves initially travel at directions given by r1 = (k1 , l1 ) and r2 = (k2 , l2 ). Therefore, they collide through an angle θ defined by cos θ = μ/(λ1 λ2 ), where μ = (k1 k2 + l1 l2 ),

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  λ1 = (k12 + l12 ), and λ2 = (k22 + l22 ). Here, λ1 and λ2 represent the normalized wavenumbers. If a wave that is slowly varying in a reference frame moving with the basic waves speed at an angle θ with respect to the x-axis may be described by φ = F (ξ, τ ), where ξ = n · x − λt, n = (cos θ, sin θ), τ = αt, and F satisfies the KdV-like equation 2αFτ + 1 3 2 2 2 αFξ + 2 βFξξξ + O(α ) = 0 in the first approximation. We proceed on the hypothesis that the interaction between two such waves may be described by φ(ξ1 , ξ2 , τ ) = F1 (ξ1 , τ ) + F2 (ξ2 , τ ) + αF12 (ξ1 , ξ2 , τ ), where ξn (n = 1, 2) is given by the above equation. The general problem is characterized by three parameters, which comprise the amplitudes of the two basic waves and the relative inclination of their normals. Let cn be the wave speed of the nth wave (n = 1, 2) and 2θ the angle between the wave normals n1 and n2 , then α, ε(c2 − c1 )/(c2 + c1 ), and k = sin2 θ are suitable measures of mean strength, relative strength, and obliquity, respectively. It follows from the perturbation equations that interaction is weak if k  α or strong if k = 0(α) as α → 0 (the adjective strong is used here in the sense of scattering theory, but it should be emphasized that the actual nonlinearity remains weak in the sense that the perturbation equations are valid). It also follows from the perturbation equations that ε = 14 (α2 − α1 ) + O(α2 ), where α1,2 d (d = quiescent depth) are the maximum amplitudes of the incoming waves. The more general case of solitary interaction involves two dimensions for the collision angle θ, which can be an arbitrary angle in the range 0 ≤ θ ≤ π, where the one-dimensional collisions are only special cases. 7.3.1 Oblique collision of DIASWs in quantum plasmas In a three-dimensional system, the angle θ between two wave propagation directions of the two solitary waves lies between 0 < θ < π. The head-on collision and the overtaking collisions are only two special cases for θ = π and θ = 0, respectively. In general, for the oblique interaction between two solitary waves in 3D, we must search for the evolution of the solitary waves propagating in two different directions. Hence, we need to employ a suitable asymptotic expansion to solve the original problem. However, when the included angle between the directions of propagation of impinging solitary waves is less than 120◦ , the effect of oblique interaction is stronger than that of the head-on one, but when the angle concerned is greater than 120◦ , the former is slightly weaker than the latter. The reality cannot be ignored that the one-dimensional geometry may not be a realistic situation

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in a laboratory device and space. However, the oblique collision of solitary waves in a two-dimensional geometry is more realistic in a magnetized dusty electronegative plasma, especially of the D and F regions of the Earth’s ionosphere. Now, we are going to investigate the oblique collision of two DIAWs in three-species quantum plasma. Like the previous problems, the extended PLK method is adopted to obtain a two-sided KdV equation, and Hirota’s method is employed to investigate the two soliton solutions. Quantum effects play a major role in collective nonlinear wave interactions in dense plasmas. The importance of quantum effects has also been recognized in the context of collective interactions in semiconductors [12]. There has been a growing interest in investigating new aspects of dense quantum plasmas by developing the quantum hydrodynamic (QHD) [13] and quantum kinetic equations [14] by incorporating the quantum force associated with the Bohm potential [15]. The Winger–Poisson (W-P) model [16] has been used to derive a set of QHD equations for a dense electron plasma. The QHD equations are composed of electron continuity, nonrelativistic electron momentum, and Poisson equations. The quantum nature appears in the electron momentum equations through the quantum statistical pressure, which requires the knowledge of the Winger distribution for a quantum mixture of electron wave functions, each characterized by the occupation probability. The quantum part of the electron pressure is repre√ √ sented as a quantum force −∇φB , where ∇φB = −(2 /2me ne )∇2 ne ,  is the Planck constant divided by 2π, me is the electron mass, and ne is the electron number density. In dense plasmas, quantum mechanical effects are important, since the de Broglie length of the charge carriers is comparable to the interparticle spacing. Let us study the two-dimensional obliquely propagating DIAWs in a three-species quantum dusty plasmas, whose constituents are dust particles, inertial ions with a background stationary dust of constant charge while the electrons are inertialess. DIAW in such a quantum plasma system is described by the normalized 2D basic equations as follows: ∂ ∂ ∂ni + (ni u) + (ni v) = 0 ∂t ∂x ∂y

(7.66)

∂u ∂u ∂u ∂φ +u +v + =0 ∂t ∂x ∂y ∂x

(7.67)

∂v ∂v ∂v ∂φ +u +v + =0 ∂t ∂x ∂y ∂x

(7.68)

Waves and Wave Interactions in Plasmas

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290

∂2φ ∂2φ + 2 = βne − αNd0 − ni ∂x2 ∂y

(7.69)

√ H2 φ − ne + 1 + √e ∇2 ne = 0 2 ne

(7.70)

where nj (j = i, and e, stands for ion, and electron, respectively) and φ are the number density and the electrostatic potential, respectively. u and v are the velocities of ions in the x and y directions, respectively. Normalization ni → v→

ni , no

ne →

ne , neo

φ→

eL , KB Te

x→

x , L

t → ω0 t,

u→

u , Lω0

v Lω0

  where L = (KB Te )/(4ΠZe2 n0 ) and ω0 = (4ΠZ 2 e2 n0 )/(mi ). e, KB , and Te are the electron charge, the Boltzmann constant, and the Fermi temperature, respectively. α = ±1 for positive/negative dust particles, i0 is the density ratio parameter with β = 1 + αNd0 , where β = nne0 zd0 nd0 Nd0 = ni0 , and He is the quantum diffraction parameter. The terms proportional to me /mi have been disregarded in Equation (7.70) in the limit me /mi 1. Now, let us consider two small but finite-amplitude wave-like perturbations that propagate obliquely at some angle θ in the XY plane with different velocities. They approach and interact after some time. Due to this interaction, they depart from each other, leaving a collision signature on waves which is nothing but a phase shift in the trajectories. Here, we use asymptotic expansions of plasma variables around the thermodynamics equilibrium state in a stretched coordinate that includes the phase variables. Here, the PLK method is used, and the following new oblique coordinates are (7.65). The directions of initial wave velocities are given by vectors r1 = (l1 , l1 ) and r2 = (l2 , l2 ), and the collision that acts through an angle θ is defined by cos θ = λ1μλ2 , where μ = (l1 l2 + l1 l2 ), λ1 = (l12 + l12 )1/2 , and λ2 = (l22 + l22 )1/2 . λ1 and λ2 represent the normalized wavenumbers. Here, we must know that θ = 0, i.e., cos θ = 1, otherwise the perturbation method used would not be valid. We take the asymptotic expansion of plasma variables away from thermodynamic equilibrium in the

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following way: nj = 1 +

∞ 

(k)

2k nj

+ ··· ,

u=

k=1

φ=

∞ 

∞ 

2k u1 + · · · ,

v=

k=1

∞ 

2k v1 + · · · ,

k=1

2k φ(k) + · · · .

k=1

Proceeding the same way as the previous problem, we have (1)

∂φξ

∂τ

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(1)

(1) (1) ∂φξ

+ Aφξ

∂ξ

(1)

+B

∂ 3 φξ

∂ξ 3

(1)

=0

(1)

∂φη ∂ 3 φη + B1 =0 ∂η ∂η 3  c3 βH 2 where A = 32 βc1 , B = 21 1 − 4 e , A1 = 32 βc2 , and B1 = Also (l1 l2 + l1 l2 ) = βc1 c2 cos θ. Thus, we obtain ∂φη ∂τ

+ A1 φ(1) η

∂P (0) = Cφ(1) η , ∂η where C =

βc1 (1+2 cos θ) c2 (3−2 cos θ)

and D =

(7.71) (7.72) c32 2

∂Q(0) (1) = Dφξ ∂ξ



1−

βHe2 4

.

(7.73)

βc2 (1+2 cos θ) c1 (3−2 cos θ) .

Equations (7.71) and (7.72) are two KdV equations. These waves are traveling in the ξ direction and η direction, respectively. To obtain a multisoliton solution of the KdV equation, let us employ Hirota’s direct method to (7.71) and (7.72). Accordingly, (1)

φξ =

12B ∂ 2 (log f ) A ∂ξ 2

(7.74)

where f = 1 + eθ1 and θ1 = kB −1/3 ξ − k 3 τ + δ. Using (7.74) in Equation (7.73), we get Q(0) (ξ, τ ) =

12BD ∂ 12B 2/3 D keθ1 (log f ) = . AC ∂ξ AC 1 + eθ1

(7.75)

Therefore, the corresponding phase shift is Q(0) = (l2 x + l2 y − c2 t)|ξ=−∞,η=0 − (l1 x + l1 y − c1 t)|ξ=∞,η=0 = 2 Q(0) (∞, τ ) − 2 Q(0) (−∞, τ ) =

12 2 DB 2/3 k. AC

(7.76)

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Waves and Wave Interactions in Plasmas

Similarly, the other phase shift is P (0) = −

12 2DB 2/3 k. AC

(7.77)

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Phase shifts in Equations (7.76) and (7.77) are similar to the investigations of different plasma models, but the approaches are different. Two soliton: Each of the KdV equations given by (7.71) and (7.72) has many soliton solutions. We consider here two-soliton solutions of each of the KdV equations. The two solitons for a particular KdV equation move in the same directions. The fast-moving soliton eventually overtakes the slower one. The two soliton solutions of (7.71) and (7.72) are propagated from the opposite directions. Initially, they are far from each other, but after some time, they come together, and the head-on collision will take place, and the solitons will finally depart. Using Hirota’s method, the twosoliton solution of the KdV equation (7.71) is given by (1)

φξ =

12B ∂ 2 (log g) A ∂ξ 2

(7.78)

where g = 1 + eθ1 + eθ2 + a12 eθ1 +θ2 , θi = ki B −1/3 ξ − ki3 τ + δi (i = 1, 2), and a12 = (k1 − k2 )2 /(k1 + k2 )2 . Using (7.78) in Equation (7.73), we obtain the corresponding phase shifts as Q(0) =

12 2 DB 2/3 (k1 + k2 ), AC

P (0) = −

12 2 DB 2/3 (k1 + k2 ). (7.79) AC

Three soliton: The three-soliton solution of (7.71) has the form (1)

φξ =

12B ∂ 2 (log h) A ∂ξ 2

(7.80)

where h = 1+eθ1 +eθ2 +eθ3 +a212 eθ1 +θ2 +a213 eθ3 +θ1 +a223 eθ2 +θ3 +a2 eθ1 +θ2 +θ3 ,  2 m θi = ki B −1/3 ξ − ki3 τ + δi (i = 1, 2, 3) and a2lm = kkll −k ; l, m = +km  3 1, 2, 3, l < m. a2 = l,m=1,l 1), λF h = 2kB TF h /4πnh0 e2 , ωpc = me /4πn0 e2 , TF h is the Fermi temperature of hot electron, KB is the Boltzmann constant, and e is the electron charge. The Fermi temperature of hot electrons is given by the relation me vF2 h /2 = kB TF h . Now, we derive the KdV equation from Equations (7.82)–(7.86) employing the RPT. The independent variables are stretched as ξ = (x − λt) and τ = 3 t and the dependent variables are expanded as n=1+

∞ 

k n(k) + · · · ,

k=1

nh = 1 +

∞  k=1

u=

∞ 

k u(k) + · · · ,

k=1 (k)

k nh

+ ··· ,

φ=

∞ 

(7.88) k (k)

φ

+ ···

k=1

where is a small nonzero parameter proportional to the amplitude of the perturbation. Now, substituting (7.88) into (7.82)–(7.86) and taking

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the terms in different power of , we obtain in the lowest order of the dispersion relation as λ2 = 1. In the next higher order of , we eliminate the second-order perturbed quantities from a set of equations to obtain the required KdV equation. Since this is a standard procedure, we skip the details and write the KdV equation ∂φ(1) ∂φ(1) ∂ 3 φ(1) − Aφ(1) +B =0 (7.89) ∂ξ ∂ξ 3 ∂τ  2 where A = λ2 (3α − 1) and B = λ2 1 − H4 . The amplitude and width of  4B a single soliton are respectively φm = − 3M A and W = M , where M is the normalized constant speed of the wave frame. We get the two solitons’ solution of the KdV equation (7.89) as φ(1) = −12 ×

B 1/3 A

k12 eθ1 + k22 eθ2 + a12 eθ1 +θ2 (k22 eθ1 + k12 eθ2 ) + 2(k1 − k2 )2 eθ1 +θ2 (1 + eθ1 + eθ2 + a12 eθ1 +θ2 )2 (7.90)

i ξ − ki3 τ + αi and a12 = (k1 − k2 )2 /(k1 + k2 )2 . a12 determines with θi = Bk1/3 the phase shifts of the respective solitons after overtaking takes place. When τ  1, the solution of Equation (7.89) asymptotically transforms into a superposition of two single soliton solutions     3B 1/3 k12 k1 (1) 2 1/3 2 sech ξ − B k1 τ − Δ1 φ ≈− A 2 2B 1/3    k22 k2 2 1/3 2 + sech ξ − B k2 τ − Δ2 (7.91) 2 2B 1/3 1/3 √ where Δi = ± 2BKi ln| a12 | (i = 1, 2). It is to be noted that the phase shifts Δ1 and Δ2 are of the same sign, and both of them are proportional to B 1/3 and amplitude. It is to be noted that B depends on H. The phase shifts will also depend on the parameter H. Here, the coefficient A is independent of H, but the coefficient B depends on H. The dispersion coefficient B vanishes at H = 2. This critical value of H destroys the KdV equation (Equation (7.89)), and no soliton can occur in this case. No soliton solution is possible for H > 2. However, we find that the H < 2 formation of soliton structure is possible.

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Fig. 7.9: Variation of the two soliton profiles φ(1) (Equation (7.90)) for different values of τ with H = 1.5, α = 1.1.

In Figure 7.9, the time evaluation of the interaction of two soliton solution φ(1) vs ξ is plotted for the several values of τ . At τ = −4, the larger amplitude soliton is behind the smaller amplitude solitary wave. When τ = −1, two solitons merge and become one soliton at τ = 0. But at τ = 1, they separate from each other, and then finally, each appears as a separate soliton acquiring their respective unchanged speed and shape. It can be seen from the exact two soliton solutions and asymptotical solution that the amplitude of the merged soliton is greater than the amplitude of the shorter soliton but less than the amplitude of the taller soliton. Three soliton solution: To construct three soliton solution, we use the Hirota perturbation and finally, we get the three solitons solution of the

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298

KdV equation (7.89) as φ(1) = −12

B 1/3 L1 A M1

(7.92)

where

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  L1 = eθ1 +θ2 2(k1 − k2 )2 + 2(k1 − k2 )2 a13 a23 e2θ3 + a12 k12 eθ2 + a12 k22 eθ1   + eθ1 +θ3 2(k1 − k3 )2 + 2(k1 − k3 )2 a12 a23 e2θ2 + a13 k12 eθ3 + a13 k32 eθ1   + eθ2 +θ3 2(k2 − k3 )2 + 2(k2 − k3 )2 a12 a13 e2θ1 + a23 k22 eθ3 + a23 k32 eθ2 + k12 eθ1 + k22 eθ2 + k32 eθ3 + B1 eθ1 +θ2 +θ3   × a12 k32 eθ1 +θ2 + a13 k22 eθ1 +θ3 + a23 k12 eθ2 +θ3  + eθ1 +θ2 +θ3 a12 (k12 + k22 + k32 + 2k1 k2 − 2k1 k3 − 2k2 k3 ) + a13 (k12 + k22 + k32 + 2k1 k3 − 2k1 k2 − 2k2 k3 ) + a23 (k12 + k22 + k32 + 2k2 k3 − 2k1 k2 − 2k1 k3 )  + B1 (k12 + k22 + k32 + 2k1 k2 + 2k1 k3 + 2k2 k3 )  M1 = 1 + eθ1 + eθ2 + eθ3 + a12 eθ1 +θ2 + a13 eθ1 +θ3 2 + a23 eθ2 +θ3 + B1 eθ1 +θ2 +θ3 B1 = a12 a13 a23 ,

θi =

ki ξ − ki3 τ + αi , B 1/3

i = 1, 2, 3.

For τ  1, the above solution is asymptotically transformed into a superposition of three single-soliton solutions as (1)

φ

≈−

3  i

Ai sech

2



  ki 2 1/3  ξ − ki B τ − Δi 2B 1/3

(7.93)

where Ai = 3B 1/3 ki2 /2A (i = 1, 2, 3) is the amplitudes, and Δ1 = 1/3 1/3 1/3 ± 2Bk1 ln| aB231 |, Δ2 = ± 2Bk2 ln| aB131 |, and Δ3 = ± 2Bk3 ln| aB121 | are the phase shifts of the solitons. In Figure 7.10, the time evaluation of the interaction of three-soliton solution φ(1) vs ξ is plotted for the several values of τ . At τ = −8, the larger amplitude soliton is behind the smaller amplitude solitary wave. When τ = −4, two solitons merge and become one soliton at τ = 0. But at τ = 1, they separate from each other, and then finally, each appears as a separate soliton acquiring their original speed and shape.

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Fig. 7.10: Variation of the three soliton profiles φ(1) (Equation (7.92)) for different values of τ .

Figure 7.11 shows the variation of phase shift for respective solitons against H when the values of the other parameters are kept fixed. The phase shifts decrease with the increase in H, and the value of B decreases with the increase in H. Here, we have investigated the nature of the nonlinear propagation of two and three-electron acoustic soliton solutions in an unmagnetized quantum plasma consisting of ions and both cold and hot electrons. The KdV equation is derived by using RPT. Standard KdV equation is obtained by suitable transformation. Using Hirota’s direct method, we have obtained a two-soliton solution of the KdV equation. The propagation of the two solitons and the three solitons has been discussed. We have observed that the larger soliton moves faster, approaches the smaller ones, and after overtaking collision, both resume their original shape and speed. Although head-on collision and overtaking collision are different phenomena, they are qualitatively consistent with each other. In two solitons’ solutions, ki Δi (i = 1, 2)

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Fig. 7.11: Variation of the phase shift for three solitons against the parameter H.

have the same values. However, in the three soliton case, ki Δi is not the same for (i = 1, 2, 3). Each of them is different from the others. 7.5 Soliton Interaction and Soliton Turbulence In weakly dispersive media, the soliton is an essential part of the nonlinear wavefield. Their deterministic dynamics in the framework of the KdV equation is thoroughly researched [19–21]. Several authors have also studied soliton and multisoliton solutions of the KdV and KdV-like equations and their mutual interactions [22–26]. When a huge number of interacting waves propagate in different directions with different velocities, their interactions lead to a fast change in the wave pattern. The characteristics of the wavefield are thus described within the framework of statistical theory. Such theory is termed the theory of wave turbulence [27, 28]. Zakharov [29, 30] has discussed the characteristics of wave turbulence in an integrable system. The soliton turbulence is specified by the kinetic equations describing the parameters of the associated scattering problem and is a specific part of wave turbulence. In 1971, Zakharov [29] first described the fundamental role of pairwise soliton collisions within the framework of the KdV equation. Later, the fact was confirmed in Refs. [31–33]. The soliton turbulence in the integrable system is slightly degenerative because of the conservation of solitons in the interacting process. In the turbulence theory, the wavefield distribution and the moments (mean, variance, skewness, and kurtosis) of the random wavefield are obtained from the measurements [34–37]. Pelinovsky et al. [38, 39] studied the two soliton interactions as an

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elementary act of soliton turbulence in the framework of the KdV and modified KdV equation. They have shown that the nonlinear interaction of two solitons leads to a decrease in third and fourth-order moments of the wavefield, whereas the first and second moments remain constant. Some works have been reported on soliton turbulence [40–44]. Shurgalina [45, 46] has studied different features of soliton turbulence in the framework of Gardner’s equation with negative and positive cubic nonlinearities. Let us consider an unmagnetized dusty plasma consisting of cold inertial ions, stationary negative dust charge, and inertialess κ distributed electrons. The normalized basic equations governing the DIA waves are given by ∂n ∂(nu) + = 0, ∂t ∂x ∂u ∂u ∂φ +u =− , ∂t ∂x ∂x

(7.94) (7.95)

 −κ+1/2 ∂2φ φ = (1 − μ) 1 − −n+μ ∂x2 κ − 3/2

(7.96)

where n is the ion number density normalized to n0 , ne is the number density ofthe electrons, u is the ion velocity normalized to ion fluid speed, Cs = KB Te /mi , mi is the mass of ion, Te is the temperature of electron, and KB is the Boltzmann constant. The electrostatic wave potential φ is normalized to KB Te /e. Space and time variables are normalized to elec tron Debye radius λD =  KB Te /4πne0 e2 and the inverse of cold electron −1 plasma frequency ωpi = mi /4πne0 e2 , respectively. Here, μ = Zdnn0d0 , with nd0 being the number density of dusts and Zd being the dust charge number. To derive the KdV equation, RPT is applied. According to RPT, the independent variables are stretched as ξ = 1/2 (x − vt),

τ = 3/2 t.

(7.97)

The dependent variables are expanded as n=1+

∞ 

k nk + · · · ,

k=1

φ=

∞  k=1

k φk + · · · .

u=

∞  k=1

k u 1 + · · · , (7.98)

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302

Using Equations (7.97) and (7.98) in Equations (7.94)–(7.96) and comparing coefficients of lowest powers of , we obtain the linear propagation speed in a low-frequency limit as v2 =

1 , a(1 − μ)

(7.99)

with a = κ−1/2 κ−3/2 . Taking the coefficients of the next higher order of , we obtain the following KdV equation:

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∂ 3 φ1 ∂φ1 ∂φ1 + Aφ1 +B =0 ∂τ ∂ξ ∂ξ 3 where A =



3−2b(1−μ)v 4 2v



,B=

v3 2 ,

(κ−1/2)(κ+1/2) . 2(κ−3/2)2 ∂φ1 term φ1 ∂ξ of the

and b =

The coefficient A of the nonlinear (7.100) becomes zero for κ =

(7.100)

(4−3μ) 2(2−3μ) .

KdV Equation

(4−3μ) 2(2−3μ) −1 1/3

So, assuming κ = (μ < 1) and making the transformations ξ = B 1/3 ξ¯, φ1 = 6A B φ¯1 , and τ = τ¯ to the KdV Equation (7.100), we obtain the following standard KdV equation: ∂φ¯1 ∂ 3 φ¯1 ∂φ¯1 + 6φ¯1 ¯ + ¯3 = 0. ∂τ¯ ∂ξ ∂ξ

(7.101)

Using Hirota’s bilinear method, the two-soliton solution of Equation (7.101) is obtained [47] as ∂2 φ¯1 = 2 ¯2 (ln f ) ∂ξ

(7.102)

¯ ¯ ¯ ¯ 2 ¯ ¯ where f (ξ, τ ) = 1 + eθ1 + eθ2 + A212 eθ1 +θ2 , A12 = ηη11 −η +η2 , and θi = −2(ηi ξ − 3 4ηi τ¯ − αi ) for i = 1, 2. Here, αi are the initial phase of the solitons. Thus, the two-soliton solution of the KdV Equation (7.100) is

φ1 =

12B ∂ 2 (ln f ) A ∂ξ 2

where f (ξ, τ ) = 1 + eθ1 + eθ2 + A212 eθ1 +θ2 , and θi = −2 for i = 1, 2.

(7.103) 

ηi ξ B 1/3

− 4ηi3 τ − αi



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From Equation (7.103), we have φ1 =

12B 1/3 N A D

(7.104)

where D = (1 + eθ1 + eθ2 + A212 eθ1 +θ2 )2 and N = η12 eθ1 + η22 eθ2 + A212 (η22 eθ1 + η12 eθ2 )eθ1 +θ2 + 2(η1 − η2 )2 eθ1 +θ2 . From Equation (7.104), we have φ1 =

12B 1/3 A ×

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=

η12 eθ1 + η22 eθ2 + A212 (η22 eθ1 + η12 eθ2 )eθ1 +θ2 + 2(η1 − η2 )2 eθ1 +θ2 , (1 + eθ1 + eθ2 + A212 eθ1 +θ2 )2

12B 1/3 A ×

η12 e−θ1 −2θ2 + η22 e−2θ1 −θ2 + A212 (η12 e−θ1 + η22 e−θ2 ) + 2(η1 − η2 )2 e−θ1 −θ2 . (e−θ1 −θ2 + e−θ1 + e−θ2 + A212 )2

(7.105) Assuming τ  1, Equation (7.105) is asymptotically (up to exponentially small terms) approximated as  2 2 −θ1  12B 1/3 η1 A12 e η22 A212 e−θ2 φ1 ≈ + A (e−θ1 + A212 )2 (e−θ2 + A212 )2   48A2 B 1/3 η12 η22 = + −θ1 /2 −θ2 /2 A (A12 eθ1 /2 + e A12 )2 (A12 eθ2 /2 + e A12 )2   12B 1/3 2 φ1 ≈ η1 sech2 (θ1 /2 + ln A12 ) + η22 sech2 (θ2 /2 + ln A12 ) . A (7.106) Therefore, for τ  1, the two-soliton solution (7.103) of the KdV Equation (7.100) is asymptotically transformed into a superposition of two single solitons [26] φ1 =

2  i=1

Ai sech2



  B 1/3 ηi 2 1/3 ξ − 4η B τ − α − Δ i i i ηi B 1/3

where the amplitudes of the solitons are given by Ai =

12B 1/3 ηi2 , A

i = 1, 2,

(7.107)

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and the phase shifts (Δi , i = 1, 2) of the solitons due to interaction are given by Δ1 =

B 1/3 ln(A12 ), η1

Δ2 =

B 1/3 ln(A12 ). η2

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7.6 Statistical Characteristics of the Wavefield In this section, we will discuss the propagation and mutual interaction of the two solitons of the KdV Equation (7.100). Due to the complete integrability of the KdV Equation (7.100), the interaction of solitons is elastic. After the interaction, they regain the properties of soliton [29, 31–33, 47]. Two solitons with different amplitudes propagate in time since the speed of the solitons is proportional to their amplitudes, and the nonlinear interaction of the two solitons takes place at a certain time and space. Choosing α1 = 1 2 − 2Bη1/3 Δ1 , α2 = − 2Bη1/3 Δ2 . The interaction of the soliton elements occurs at the origin, i.e., at the point ξ = 0, τ = 0 [48]. For the above choice of α1 and α2 and assuming η1 > η2 , the amplitude of the resulting peak is obtained as φ∗1 =

12B 1/3 2 (η1 − η22 ). A

(7.108)

The pulse shape at the soliton interaction is instantly determined by the following equation:   ∂ 2 φ1 24 4 2 2 4 (0, 0) = − η − 4η η + 3η (7.109) 1 2 2 . ∂ξ 2 AB 1/3 1 Equation (7.109) indicates the concavity of wave profile at the strongest interaction point. The negative value of Equation (7.109) indicates that the wave profile is concave downwards. Therefore, the wave profile may maintain a single peak status at the strongest interaction point. The positive value of Equation (7.109) implies that the wave profile is concave upwards, and it will maintain two peak statuses at ξ = 0, τ = 0. Equation (7.109) is positive or negative according to whether (1 − R)(1 − 3R) is negative η2 or positive, where R = η22 and η1 > η2 . Equation (7.109) is negative for 1

R < 13 and is positive for 13 < R < 1. Figure 7.12 shows the interaction process of two solitons of Equation (7.100) for R = 0.3, κ = 2, μ = 0.1, and η1 = 0.4. When R < 13 , the larger soliton grabs the smaller soliton and merge as a single soliton at ξ = 0, τ = 0 and overtakes the smaller soliton.

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Fig. 7.12: Interaction process of two solitons of Equation (7.100) for κ = 2, μ = 0.1, η1 = 0.4, and R = 0.3.

Figure 7.13 presents the interaction process of two solitons for R = 0.5, κ = 2, μ = 0.1, and η1 = 0.4. When 13 < R < 1, the bigger and smaller solitons exchange their energy. In this case, the solitons never merge into a single soliton. The interaction of a large number of propagating waves in conservative systems results in changing the wave patterns fast. In such a scenario, the statistical theory is suitable for describing the wavefield. Such a theory is called weak wave turbulence. Soliton turbulence is stronger than the ordinary, weak turbulent plasma description. The conservation laws are important aspects in the study of turbulence theory. For a scalar partial differential equation with two independent variables x and t, and a single dependent variable u, the conservation law can be written as ∂X ∂T + =0 ∂t ∂x

(7.110)

where T and X are conserved density and flux, respectively. Moreover, both are polynomials of the solution u and its derivatives with respect to the space variable x [49]. If both T and ∂X ∂x are integrable over the interval (−∞, ∞), then on the assumption that X −→ 0 as |X| −→ ∞,

Waves and Wave Interactions in Plasmas

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306

Fig. 7.13: Interaction process of two solitons of Equation (7.100) for κ = 2, μ = 0.1, η1 = 0.4, and R = 0.5.

Equation (7.110) can be integrated as d dt



∞

−∞



⇒

 T dx = 0,

∞ −∞

T dx = constant.

(7.111)

Equation (7.111) is invariant with time and is termed as the invariant of motion or constant of motion [50, 51]. The KdV equation forms a completely integrable Hamiltonian system. Hence, it possesses an infinite number of conserved quantities [19–21]. The first four conservation laws of the KdV Equation (7.100) are  I1 =

−∞

 I2 =

∞

−∞

 I3 =

∞

∞

−∞

φ1 (ξ, τ )dξ,

(7.112)

φ21 (ξ, τ )dξ,

(7.113)



φ31 (ξ, τ ) −

3B A



∂φ1 ∂ξ

2  dξ,

(7.114)

Collision of Solitons



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I4 =

∞

−∞



φ41 (ξ, τ )

 2  2  12B ∂φ1 36B 2 ∂ 2 φ1 − φ1 + dξ. A ∂ξ 5A2 ∂ξ 2

307

(7.115)

The first three integrals (7.112)–(7.114) correspond to the conservation of mass, momentum, and energy, respectively. These integrals are preserved in the process of the wavefield evolution, and using the noninteracting solitons (7.107), the analytical calculations of I1 , I2 , I3 , and I4 are presented as follows: √ 4 3B 1/2 1/2 (7.116) I1 = √ (A1 + A2 ), A √ 8 3B 3/2 3/2 I2 = √ (A1 + A2 ), (7.117) 3 A √ 8 3B 5/2 5/2 I3 = √ (A1 + A2 ), (7.118) 5 A √ 32 3B 7/2 √ (A1 + A7/2 I4 = (7.119) 2 ). 35 A From Equations (7.116)–(7.119), it is observed that values of the integrals I1 , I2 , I3 , and I4 increase as amplitudes of the interacting solitons increase. The dynamics of multisoliton is affected significantly by mutual interactions of the solitons [29, 30]. To understand the effects of two soliton interactions on the statistical moments of the random wavefield, we consider the following integrals:  ∞ μn = φn1 dξ, n = 1, 2, 3, ..... (7.120) −∞

The integrals (7.120) are related to the statistical moments of the wavefield. The first integral moment (μ1 ) and the second integral moment (μ2 ) represent the mean and variance of the random wavefield, respectively. The first and second integral moments μ1 and μ2 are similar to Kruskal’s integral I1 and I2 . Hence, they are conserved for the two-soliton solution (7.103) which agrees with (7.116) and (7.117) (see Figure 7.14). Thus, the nonlinear interactions of the two solitons do not affect the mean and the variance of the wavefield. The third and fourth integral moments are defined by μ3 (7.121) M3 (τ ) = 3/2 , μ2 μ4 M4 (τ ) = 2 . (7.122) μ2

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Waves and Wave Interactions in Plasmas

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Fig. 7.14: The time dependence of the integrals I1 and I2 in the two soliton interaction with κ = 2, μ = 0.1, R = 0.5, and η1 = 0.4.

Fig. 7.15: The time dependence of skewness (M3 (τ )) and kurtosis (M4 (τ )) in the twosoliton interaction for κ = 2, μ = 0.1, R = 0.5, and η1 = 0.4.

The third and fourth moments M3 (τ ) and M4 (τ ) characterize the skewness and kurtosis (the word ‘kurtosis’ refers to the normalized fourth moment and not its difference from the Gaussian value 3-‘excess kurtosis’) of the random wavefield. The statistical measure of the vertical asymmetry of the wavefield is given by the skewness (M3 (τ )) and the kurtosis that provides information on the probability of occurrence of extreme waves [52]. The structures of μ3 and μ4 are different from Kruskal’s integrals I3 and I4 . Therefore, the skewness (M3 (τ )) and kurtosis (M4 (τ )) will not be conserved in the dominant interaction region. Figure 7.15 represents the numerical evolution of M3 (τ ) and M4 (τ ) for the two-soliton solution (7.103) with κ = 2, μ = 0.1, R = 0.5, and η1 = 0.4. In Figure 7.15, the skewness and kurtosis decrease in the dominant interaction region of the random wavefield. If the solitons do not interact with each other, then the skewness (M30 ) and kurtosis (M40 ) can be calculated using the noninteracting

Collision of Solitons

309

solitons (7.107) √ 1/4 5/2 5/2 6A (A1 + A2 ) = , 5(3B)1/4 (A3/2 + A3/2 )3/2 1 2 √ 7/2 7/2 3 3A (A1 + A2 ) 0 M4 = √ . 3/2 35 B (A3/2 + A2 )2 1 M30

(7.123)

(7.124)

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Also, from Figure 7.15, it is observed that the skewness (M3 (τ )) is calculated using (7.103). It varies about 7.25% from M30 that is calculated using the noninteracting solitons (7.107). Hence, the kurtosis (M4 (τ )) deviates about 8.03% from M40 due to two soliton interactions. Here, the skewness and kurtosis are always positive as all the two solitons are positive. 7.7 Plasma Parameters on Soliton Turbulence The soliton amplitude increases as the parameters κ and μ increase, as observed earlier. To show the effects of plasma parameters κ and μ on soliton turbulence, the maximum deviation Mi∗ = Mimax − Mimin (i = 3, 4) M max −M min

and the relative deviation Mi∗∗ = i M max i (i = 3, 4) in third and fourth i moments as a function of κ and μ have been calculated (Figures 7.16–7.19). From Figure 7.16, it is observed that the increase in the value of κ decreases the maximum deviation in the third and fourth moments. Also, the maximum deviation in third and fourth moments due to the interaction of solitons decreases as the parameter μ increases (Figure 7.17). Therefore, the spectral index κ and the parameter μ (unperturbed dust to ion ratio) have a strong influence on the soliton turbulence for the DIA waves for dusty plasma systems.

Fig. 7.16: Plot of maximum deviation of the third and fourth moments due to interaction as a function of κ with μ = 0.1, R = 0.5, and η1 = 0.4.

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Waves and Wave Interactions in Plasmas

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Fig. 7.17: Plot of maximum deviation of the third and fourth moments due to interaction as a function of μ with κ = 2, R = 0.5, and η1 = 0.4.

Fig. 7.18: Relative deviation in third and fourth moments due to interaction as a function of κ with μ = 0.1, R = 0.5, and η1 = 0.4.

Fig. 7.19: Relative deviation in third and fourth moments due to interaction as a function of μ with κ = 2, R = 0.5, and η1 = 0.4.

References [1] H. Ikezi, R. Taylor, and D. Baker, Phys. Rev. Lett. 25, 11 (1970).

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[2] T. Maxworthy, J. Fluid Mech. 76, 177 (1976). [3] P. Chatterjee, K. Roy, G. Mandal, S. V. Muniandy, and S. L. Yap, Phys. Plasmas 16, 122112 (2009). [4] F. Verheest, M. A. Hellberg, and W. A. Hereman, Phys. Rev. E 86, 036402 (2012). [5] F. Verheest, M. A. Hellberg, and W. A. Hereman, Phys. Plasmas 19, 092302 (2012). [6] M. Marklund and G. Brodin, Phys. Rev. Lett. 98, 025001 (2007). [7] G. Brodin and M. Marklund, New J. Phys. 9, 277 (2007). [8] M. Marklund, B. Eliasson, and P. K. Shukla, Phys. Rev. E 76, 067401 (2007). [9] A. P. Misra and N. K. Ghosh, Phys. Lett. A 372, 6412 (2008). [10] A. Mushtaq and S. V. Vladimirov, Eur. Phys. J. D. 64, 419 (2011). [11] F. A. Asenjo, J. Zamanian, N. Marklund, G. Brodin, and P. Johansson, New J. Phys. 14, 073042 (2012). [12] M. Bonitz, Phys. Rev. E 49, 5535 (1994). [13] G. Manfredi and F. Hass, Phys. Rev. B 64, 075316 (2001). [14] D. Kremp, T. Bornath, M. Bonitz, and M. Schlanges, Phys. Rev. E 60, 4725 (1999). [15] C. L. Gardener and C. Ringhofer, Phys. Rev. E 53, 157 (1996). [16] E. P. Winger, Phys. Rev. 40, 749 (1932). [17] A. C. Scott, F. Y. F. Chu, and D. W. McLaughlin, Proc. IEEE. 61, 1443 (1973). [18] G. B. Whitham, Linear and Nonlinear Waves. Wiley (1974). [19] S. P. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov, The Theory of Solitons: The Inverse Scattering Method. New York: Consultants (1984). [20] A. C. Newell, Solitons in Mathematics and Physics. Philadelphia: SIAM (1985). [21] P. G. Drazin and R. S. Johnson, Solitons: An Introduction. Cambridge University Press (1993). [22] B. Sahu and R. Roychoudhury, Astrophys. Space Sci. 345, 91–98 (2013). [23] B. Sahu, EPL 101, 55002 (2013). [24] A. Saha and P. Chatterjee, Astrophys. Space Sci. 353, 169–177 (2014). [25] G. Mandal, K. Roy, A. Paul, A. Saha, and P. Chatterjee, Z. Naturforsch. 70, 703–711 (2015). [26] K. Roy, S. K. Ghosh, and P. Chatterjee, Pramana J. Phys. 86, 873–883 (2016). [27] V. E. Zakharov, V. S. L’vov, and G. Falkovich, Kolmogorov Spectra of Turbulence. Berlin: Springer (1992), p. 6. [28] S. Nazarenko, Wave turbulence. Lecture Notes in Physics, p. 279. Berlin: Springer (2011) Vol. 26. [29] V. E. Zakharov, Sov. Phys. JETP 33, 538 (1971). [30] V. E. Zakharov, Stud. Appl. Math. 122, 219 (2009). [31] G. A. El and A. M. Kamchatnov, Phys. Rev. Lett. 95, 204101 (2005). [32] G. A. El, A. L. Krylov, S. A. Molchanov, and S. Venakides, Physica D 152–153, 653–664 (2001).

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[33] G. A. El, A. M. Kamchatnov, M. V. Pavlov, and S. A. Zykov, J. Nonlinear Sci. 21, 151–191 (2011). [34] K. Hasselmann, J. Fluid Mech. 12, 481 (1962). [35] S. Dyachenko, A. C. Newell, and V. E. Zakharov, Physica D 57, 96 (1992). [36] V. E. Zakharov and V. S. L’vov, Radiophys. Quantum Electron. 18, 1084 (1975). [37] V. S. Lvov, Y. V. Lvov, A. C. Newell, and V. E. Zakharov, Phys. Rev. E 56, 390 (1997). [38] E. N. Pelinovsky, E. G. Shurgalina, A. V. Sergeeva, T. G. Talipova, G. A. El, and R. H. J. Grimshaw, Phys. Lett. A 377, 272 (2013). [39] E. N. Pelinovsky and E. G. Shurgalina, Radiophys. Quan. Electr. 57, 737 (2015). [40] D. Dutykh and E. N. Pelinovsky, Phys. Lett A 378, 3102–3110 (2014). [41] E. G. Shurgalina and E. N. Pelinovsky, Phys. Lett. A 380, 2049–2053 (2016). [42] E. N. Pelinovsky and E. G. Shurgalina, “KDV soliton gas: Interactions and turbulence,” in I. Aronson, N. Rulkov, A. Pikovsky, and L. Tsimring (eds.), Challenges in Complexity: Advances in Dynamics, Patterns, Cognition. Springer (2017), pp. 295–306. [43] E. G. Shurgalina, E. N. Pelinovsky, and K. A. Gorshkov, Moscow Univ. Phys. Bull. 72, 441–448 (2017). [44] A. V. Slunyaev and E. N. Pelinovsky, Phys. Rev. Lett. 117, 214501 (2016). [45] E. G. Shurgalina, Radiophys. Quan. Electr. 60, 703–708 (2018). [46] E. G. Shurgalina, Fluid Dyn. 53, 59–64 (2018). [47] R. Hirota, The Direct Method in the Soliton Theory. Cambridge University Press (2004). [48] T. P. Moloney and P. F. Hodnett, Proc. R. Irish Acad. Sec. A Math. Phys. Sci. 89A, 205–217 (1989). [49] P. G. Drazin and R. S. Johnson, Solitons: An Introduction. Cambridge University Press (1996). [50] F. Verheest and W. Hereman, Phys. Scripta 50, 611 (1994). [51] U. Goktas and W. Hereman, J. Symb. Comput. 11, 1 (1999). [52] N. Mori and P. A. E. M. Janssen, J. Phys. Ocean 36, 1471–1483 (2006).

Chapter 8

Sagdeev’s Pseudopotential Approach

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8.1 Nonperturbative Approach Nonlinear waves are classified as small-amplitude waves and largeamplitude waves. The major problems of nonlinear waves in plasma physics are treated by the perturbation method. However, the perturbation method is applicable only to study small-amplitude waves. The reason is that for large-amplitude waves, the infinite series of perturbed quantities will not converge to the stable or equilibrium position. To study large-amplitude solitary waves, perturbation technique will not work, and accordingly, several nonperturbation techniques have been developed. For these techniques, the total nonlinearity of the system is taken into account and then the wave solutions are obtained from this consideration. One such method is Sagdeev’s pseudopotential method. In 1966, famous scientist R. Z. Sagdeev [1] pioneered a technique that later came to be known as Sagdeev’s pseudopotential approach or Sagdeev’s energy integral method. This method is used to study large amplitude solitary waves and shock waves, largeamplitude double layers, speed and shape of the solitary waves, and similar features of the wave. 8.2 Sagdeev’s Pseudopotential Approach 2

In 1966, R. Z. Sagdeev introduced an equation ddξφ2 = − dψ dφ , where φ is the electric potential, the stretched co-ordinate ξ = x − vt, x is the position and t is the time, v is the velocity of the wave, and ψ is an unknown quantity. This equation is named after him. The analogy of this equation with that of the newton potential equation is shown in Figure 8.1. From Figure 8.1, it is seen that the unknown quantity ψ(φ) can be taken as the potential if one considers the potential φ as position and ξ(= x − vt) 313

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Fig. 8.1: Comparison between the Newton potential and Sagdeev potential.

as time, i.e., if φ behaves as pseudo particle and ξ as pseudo time, then ψ is the pseudopotential. This is the reason why ψ(φ) is called pseudopotential or Sagdeev’s potential. Sagdeev’s potential plays an important role to find the shape and size of the large-amplitude solitary waves in plasmas. Let us evaluate the pseudopotential as well as the large-amplitude solitary waves in a simple plasma model, given in the following: Basic equations: Let us consider an unmagnetized, collisionless e-i plasma, where ions are mobile and electrons obey the Maxwell distribution. The basic equations in normalized forms are written as follows: Equation of continuity: ∂ni ∂(ni ui ) + =0 ∂t ∂x

(8.1)

Equation of momentum balance: ∂ui ∂ui ∂φ + ui =− ∂t ∂x ∂x

(8.2)

∂ 2 ui = ne − ni ∂x2

(8.3)

and Poisson equation:

where ne = eφ . Energy integral and the Sagdeev potential: Let us assume that the wave moves with the velocity of M . We consider the Galilean-type transformation (a traveling wave transformation), where the wave frame moves

Sagdeev’s Pseudopotential Approach

315

with the wave velocity M as ξ = x − M t. By substitution

∂ ∂x

=

d dξ

and

−M

∂ ∂t

d = −M dξ , (8.1)–(8.3) reduce to

dni d(ni ui ) + =0 dξ dξ

−M

dui dφ dui + ui =− dξ dξ dξ d2 φ = e φ − ni . dξ 2

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(8.4)

(8.5) (8.6) (8.7)

We assume that the equilibrium state is reached at both infinities (ξ → ±∞). Accordingly, we integrate and apply the boundary condition ni = 1, ui = 0, and φ = 0 at ξ ± ∞. After integrating and using these initial conditions, from (8.5) and (8.6), one obtains ni = −M ui +

M M − ui

u2i = −φ. 2

(8.8) (8.9)

After some simple algebra, we get M − ui = M

 1/2 2φ 1− 2 . M

(8.10)

Using Equations (8.8) and (8.10) in Equation (8.7), we get −1/2  ∂2φ 2φ φ = e − 1− 2 . ∂ξ 2 M

(8.11)

Now, we introduce the pseudopotential ψ as d2 φ dψ =− . 2 dξ dφ Integrating the above equation, we get  2 1 d φ dφ = −eφ − M (M 2 − 2φ) 2 + c1 . ψ=− dξ 2

(8.12)

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Putting the condition as when |ξ| → ∞, then ψ → 0, φ → 0, we obtain c1 = 1 + M 2 . Therefore, ψ can be expressed as   1/2  2φ φ 2 ψ =1−e +M 1− 1− 2 = ψe + ψi (8.13) M where ψe = 1 − eφ ,

  1/2  2φ ψi = M 2 1 − 1 − 2 . M

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φ

2



 1/2  2φ 1− 1− 2 . M

(8.14)

The expression in the box is called the “Sagdeev’s pseudopotential”, and ψi and ψe are the contributions of Sagdeev’s potential for ions and electrons, respectively. 8.2.1 Physical interpretation of Sagdeev’s potential 2

Let us compare Equation (8.12) with Newton’s potential equation ddt2x = − dφ dx , where x is the position, t is the time, and φ is the potential. We can see that Equation (8.12) can be considered as a one-dimensional motion of a particle of unit mass, whose pseudo position is φ at pseudo time ξ with pseudo velocity dφ/dξ in a pseudopotential well ψ(φ)[2, 3]. If the potential φ is treated as pseudo position and ξ as pseudo time, then ψ(φ) can be treated as pseudopotential. If the first term in integral (8.12) is multiplied by 2dφ/dξ and integrated, it can be written as (dφ/dξ)2 +2ψ(φ) = constant. It can be considered as the kinetic energy of a particle with position coordinate φ. ψ(φ) can be considered as the potential energy at that instant. Since kinetic energy is always non-negative, ψ(φ) should be negative (i.e., ψ(φ) ≤ 0) for the entire motion and zero is the maximum value of ψ(φ). 2 Again, from Equation (8.12), it is seen that ddξφ2 + ψ  (φ) = 0, i.e., the force acting on the particle is ψ  (φ), where ψ  (φ) = dψ dφ . It can be shown that  ψ(0) = ψ (0) = 0. The particle is in equilibrium at φ = 0 because both the velocity and the force acting on the particle at φ = 0 are zero. Now, if φ = 0 is an unstable equilibrium, the integral shows the motion of an oscillatory particle if ψ(φm ) = 0 for φm = 0. If the particle is displaced slightly from the position of unstable equilibrium, it moves away from the position and continues its motion until its velocity reaches zero, at φ = φm .

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Sagdeev’s Pseudopotential Approach

317

The force acting on the particle at φ = φm is −ψ  (φm ). If φm < 0, then the force acting on the particle at the point φ = φm is directed towards the point φ = 0 (if −ψ  (φm ) > 0, i.e., if ψ  (φm ) < 0). If φm > 0, then the force acting on the particle at the point φ = φm is directed towards the point φ = 0 (if −ψ  (φm ) < 0, i.e. if ψ  (φm ) > 0). As for the positive potential side ψ  (φm ) > 0 and for the negative potential side ψ  (φm ) < 0, the particle reflects back again to φ = 0 in both cases. Due to initial conditions assumed for ψ(φ), φ = 0 is a double root. Due to the above observations, the conditions for the existence of soliton solutions are as follows: Hence, the existence conditions for solitary waves can be written mathematically as (i) ψ(φ) = 0 at φ = 0 and φ = φm

(8.15)

(ii)  dψ(φ)  =0 dφ φ=0

and

 dψ(φ)  = 0. dφ φ=φm

(8.16)

There is also an additional requirement that the double root at φ = 0 corresponds to a local maximum at φ = 0. Hence, another condition is (iii)  d2 ψ(φ)  ≤0 dφ2 φ=0 (should it be < 0, otherwise = 0, will not give maximum, the (iv) ψ(φ) < 0

when φ lies between 0 and φm .

(8.17) d4 ψ dφ4

< 0, etc.)

(8.18)

If φm < φ < 0, rarefactive solitary waves exist, and if 0 < φ < φm , compressive solitary waves exist. Here, φm is the amplitude of the solitary wave. ψ(φ)|φ=0 = dψ dφ |φ=0 are already satisfied by the equilibrium charge neutrality condition and by the boundary condition chosen for the integrating constant.

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318

8.2.2 Determination of the range of Mach number  1/2   2φ We have ψ(φ) = 1 − eφ + M 2 1 − 1 − M = ψe + ψi . From Equation 2 (8.13), it is clear that ψe is negative for φ > 0. It is clear from Equation 2 (8.14) that for φ > M2 , then ψi becomes complex. But physically a complex 2 ion number density is not allowed. If 0 < φ ≤ M2 , then 0 < ψ ≤ M 2 . So 2 φ = M2 , then ψi is maximum and is equal to M 2 . The maximum value of 2 2 φ is allowed as φm = M2 . So, for that maximum value of φm = M2 , we get solitary wave ψ(φm ) = 0, i.e., ψ(M 2 /2) = 0

 1/2  2 M2 or 1 − e =0 +M 1− 1− 2 M 2  2 2  2 3  M M M2 1 1 ⇒1− 1+ + + + M 2 = 0. 2 2 2! 2 3!

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M 2 /2

2



So, neglecting higher-order terms, we get 1−1−

M2 < M4 M6 − − + M 2 = 0 ⇒ M 2 (24 − 6M 2 − M 4 ) = 0. 2 8 48

This implies that either M 2 = 0 or 24 − 6M 2 − M 4 = 0 √ −6 ± 132 2 ⇒ M = 0 or M = . 2

√ √ Taking positive sign, we have M 2 = −3 + 33 ⇒ M = −3 + 33 ≈ 1.66. Therefore, the range of Mach number for the existence of solitary waves is 0 < M < 1.66. 8.2.3 Shape of the solitary waves Multiplying both sides of Equation (8.12) by 2 dψ dξ and integrating with respect to ξ, we get 

dφ dξ

2 = −2ψ(φ) + C.

Sagdeev’s Pseudopotential Approach

Using the boundary conditions |ξ| → ∞, ψ → 0, and we get

319 dφ dξ

→ 0, ultimately

dφ = ± −2ψ. dξ Integrating, we obtain  ±ξ =

φ0

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φ

1

dφ. −2ψ(φ)

(8.19)

The shape of the soliton solution can be obtained from Equation (8.19). When φ0 is the value of φ, where ψ(φ) crosses the φ axis from below, i.e., φ0 is the amplitude of the soliton. From the above discussions, it is clear that soliton solution will exist if ψ(φ) is negative throughout the interval of integration, and ψ(φ) crosses φ axis from below at some value of φ (say φ0 ). Thus, φ0 is the amplitude of the wave. 8.2.4 Physical interpretation of double layers The double layer is a nonlinear potential structure in a plasma consisting of two adjacent layers with opposite electric charges, in which the existing potentials’ jump creates an electric field. Ions and electrons which enter the double layers are accelerated, decelerated, or reflected by the electric field. The double layer in plasma is the region of self-consistent potential drop, maintained by the appropriate distributions of accelerated and reflected particles. We classify double layers into two types: weak and strong double layers. It is based on the ratio of the potential drop in comparison with the plasma’s equivalent thermal potential. A double layer is said to be strong if the potential drop across the layer is greater than the equivalent thermal potential of the plasma’s component. It may be useful to note that the condition ψ(0) = ψ(φm ) = 0 is required to satisfy global charge neutrality and to balance the pressure on both sides of the double layers (known as Langmuir condition). Moreover, the boundary condition ψ  (0) = ψ  (φm ) = 0 must be fulfilled to ensure that the local charge is neutral and the electric field is zero at the boundaries. The conditions ψ  (0) < 0 and ψ  (φm ) < 0 are the generalized Bohm criteria. These have a simple physical interpretation of the net charges inside the double layers, and they must have the correct sign that is consistent with the electric field. Existence conditions for double layers: For a double layer, one needs successive double roots. It is because, in the parlance of classical mechanics,

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φ can transit from one value to another without coming back, with a rather sharp and shock, or kink-like transition between the two. Also, the double roots correspond to local maxima. The conditions on the Sagdeev potential now are (i) ψ(φ) = ψ  (φ) = 0

and ψ  (φ) < 0 at φ = 0

(8.20)

(ii) ψ(φm ) = ψ  (φm ) = 0

and ψ  (φm ) < 0 (φm = 0)

(8.21)

(iii)

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ψ(φ) is negative in the interval (0, φm ).

(8.22)

If φm > 0, then the double layer is called a compressive double layer, and if φm < 0, then the double layer is called a rarefractive double layer, where φm is some extremum value of the potential and is called the amplitude of the double layers. 8.2.5 Small amplitude approximation We shall now obtain the small amplitude approximation of ψ(φ). Expanding ψ(φ) about φ = 0, we get ψ(φ) = A + A0 φ + A1 φ2 + A2 φ3 + A3 φ4 + O(φ5 ) where

  A = ψ(φ)  A0 =  A1 =  A2 =  A3 =

=0 φ=0

dψ(φ) dφ



=0 φ=0

d2 ψ(φ) dφ2

d3 ψ(φ) dφ3 4

d ψ(φ) dφ4



= φ=0

 

=

3 − M4 6

=

15 − M 6 . 24M 6

φ=0

φ=0

1 − M2 2

(8.23)

Sagdeev’s Pseudopotential Approach

Expanding up to φ3 : Using the boundary condition ψ → 0 and as φ → 0, and if we consider up to φ3 terms, we get

321 dψ dφ

→0

ψ(φ) = A1 φ2 + A2 φ3 . 1 It is evident that the pseudopotential is reflected back at φ = − 3A A2 , which is the maximum potential φ0 of the solitary waves. The small-amplitude results arevalid if φ0 is small, i.e., if A2 is small. Hence, it is valid near the 2 . boundary ∂∂φψ2

Therefore,

φ=0

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dφ

= dξ. −2(A1 φ2 + A2 φ3 ) Hence, the KdV-type soliton solution is written as   ξ φ = φ0 sech2 δ 1 where ψ0 = − 3A A2 is the amplitude of the solitary wave and δ = the width of the solitary wave.

(8.24) √ 2 −A1

is

Expanding up to φ4 : If we consider up to φ4 term, we get ψ(φ) = A1 φ2 + A2 φ3 + A3 φ4 . Therefore, dφ

= dξ. 2 −2(A1 φ + A2 φ3 + A3 φ4 ) Integrating, we get    −1

A2 A22 A3 − − φ= − cosh A1 ξ . 3A1 9A21 2A1

(8.25)

Equation (8.25) is the modified KdV soliton solution provided A1 > 0. Again, if A22 = 9A12A3 , the above soliton solution would not exist. In that case, a shock wave solution is obtained from Equation (8.25), which is written as 3A1 φ=− [1 + tanh α(ξ + ξ0 )] (8.26) 2A2 2 where ξ0 is the integration constant and α = − 3√A2A . Equation (8.25) 3 is the small-amplitude cnoidal wave solution, and (8.26) is the smallamplitude shock wave solution.

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322

8.3 Effect of Finite Ion Temperature In Section 8.2, we have discussed the large-amplitude solitary waves in unmagnetized plasma neglecting the ion temperature. But there are real situations where the temperature of ions is considerable with the temperature of electrons. Incorporating that phenomena we may include the temperature ration σ = TTei , where Te and Ti are electron and ion temperature, respectively. Also, considering the electron inertia along with the ion continuity equation (8.1) and Poisson equation (8.3), we use the modified ion momentum equation as ∂ui ∂ui σ ∂pi ∂φ + ui + =− . ∂t ∂x ni ∂x ∂x

(8.27)

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Supplementing the energy equation by ∂pi ∂pi ∂pi + ui + 3pi = 0. ∂t ∂x ∂x

(8.28)

The electron continuity equation ∂ne ∂(ne ue ) + = 0. ∂t ∂x

(8.29)

The electron momentum equation   ∂ue ∂ue 1 ∂φ 1 ∂ne − + ue = . ∂t ∂x μ ∂x ne ∂x

(8.30)

1 e Here, pi is the ion pressure and μ = m Mi = 1836 . mi and me are ion mass and electron mass, respectively. To obtain a solitary wave solution, we make all the dependent variables depend on a single independent variable ξ = x−M t. Thus, Equations (8.1), (8.27)–(8.30) are written as

−M

d(ni ui ) dni + =0 dξ dξ

(8.31)

−M

dui dui σ dpi dφ + ui + =− dξ dξ ni dξ dξ

(8.32)

−M

dpi dpi dui + ui + 3pi =0 dξ dξ dξ

(8.33)

dne d(ne ue ) + =0 dξ dξ

(8.34)

−M

Sagdeev’s Pseudopotential Approach

−M

323

  due due 1 dφ 1 due − + ue = dξ dξ μ dξ ne dξ

(8.35)

d2 φ = ne − ni . dξ 2

(8.36)

Integrating Equations (8.31)–(8.36) and using the boundary conditions for localized perturbations, namely φ → 0, ni → 1, ne → 1, ui → 1, ue → 0, and pi → 1 as ξ → ±∞, we obtain from Equations (8.31) and (8.33) ni =

M M − ui

(8.37)

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pi = n3i . Replacing n1 in (8.27) by 1 − times (8.28) from it, we get −2M

ui M

(8.38)

, multiplying it by 2, and subtracting

dui dui dpi 3σ dui pi dφ − + 2ui + 3σ = −2 . dξ dξ dξ M dξ dξ

σ M

(8.39)

Integrating Equation (8.39) and using the boundary conditions, we get −2M ui + 2u2i + 3σpi − 3σ −

3σui pi = −2φ. M

(8.40)

Using Equations (8.37), (8.38), and (8.40), we get 

 1/2 1/2 (M 2 + 3σ − 2φ) − (M 2 + 3σ − 2φ)2 − 12σM 2 n= . 6σ

(8.41)

Similarly, integrating Equations (8.34) and (8.35) and using the boundary condition, we get M M − ue    M 1 φ = log + (M − ue )2 − M 2 . M − ue μ

ne =

(8.42) (8.43)

The pseudopotential ψ may be defined as dφ dψ =− dξ dφ

(8.44)

Waves and Wave Interactions in Plasmas

324

where ψ = ψi + ψe



ψi = −(3σM 6 )1/4 (e 2 − e θ

θ0 2

1 −3θ ) + (e 2 3

M + μM ue M − ue  2  M + 3σ − 2φ √ θ = cosh−1 , 12σM 2

(8.45)

 −3θ0 −e 2 ) ,

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ψe = 1 −

 2  M + 3σ θ0 = cosh−1 √ . 12σM 2

Soliton solution: The form of the pseudopotential ψ would determine whether a soliton (8.45) will exist or not. From  like solution of Equation 1 1 (8.17), we get − 1−μM 2 + M 2 −3σ < 0. Also, from another condition ψ(φm ) ≥ 0, when φm = min(φm1 , φm2 ) and φm1 =

M 2 −3σ , 2

φm2 = 12 M 2 .

8.4 Large-amplitude DASWs To study the effect of nonthermal ions on large-amplitude solitary waves, let us assume a four-component unmagnetized collisionless plasma consisting of negatively charged dust grains, electrons, and ions. The above nonthermal distribution for ion density has been discussed in Chapter 1. In this model, two types of dust grains are considered: smaller size positively and larger size negatively charged dust grains. Most of the researchers have considered the negatively charged dust only. The consideration of negatively charged dust is valid when the dust charging process by a collection of plasma particles (viz. electrons and ions) is much more important than other charging processes. But the dust grains can be positively charged by other charging processes also. The dust grains can be positively charged by three principal mechanisms. These are (i) photoemission in the presence of a flux of ultraviolet photons, (ii) thermionic emission induced by radiative heating, and (iii) secondary emission of electrons from the surface of the dust grains. Here, the electrons and ions obey the Boltzmann and nonthermal distribution, respectively. The basic normalized equations are ∂ ∂n1 + (n1 u1 ) = 0, ∂t ∂x ∂u1 ∂u1 ∂φ + u1 = , ∂t ∂x ∂x

(8.46) (8.47)

Sagdeev’s Pseudopotential Approach

325

∂n2 ∂ + (n2 u2 ) = 0, ∂t ∂x ∂u2 ∂u2 ∂φ + u2 = −αβ , ∂t ∂x ∂x

(8.48) (8.49)

∂2φ = n1 − (1 − μi + μe )n2 + μe eσφ ∂x2 − μi (1 + β1 φ + β1 φ2 )e−φ .

(8.50)

Here, all the notations and normalization are the same as in Section 5.3.

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Energy integral and the Sagdeev potential: By using the traveling wave transformation (8.4), Equations (8.46)–(8.50) are reduced to −M

dn1 d(n1 u1 ) + = 0, dξ dξ

−M −M

(8.51)

du1 du1 dφ + u1 = , dξ dξ dξ

(8.52)

dn2 d(n2 u2 ) + = 0, dξ dξ

−M

(8.53)

du2 dφ du2 + u2 = −αβ , dξ dξ dξ

(8.54)

d2 φ = n1 − (1 − μi + μe )n2 + μe eσφ − μi (1 + β1 φ + β1 φ2 )e−φ . dξ 2

(8.55)

Now, integrating with respect to ξ and using the boundary conditions ψ, u1 , u2 → 0, n1 , and n2 → 1 as |ξ| → ∞. From Equation (8.51)–(8.54), we get n1 =

M , M − u1

n2 =

M , M − u2

φ = −M u1 +

u21 , 2

u22 . 2 (8.56)

αβφ = M u2 −

Now, using (8.56) in (8.55) and introducing Sagdeev pseudopotential ψ(φ), we get Sagdeev’s equation as d2 φ dψ(φ) =− . dξ 2 dφ

(8.57)

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326

Sagdeev’s potential can be obtained by the similar procedure as discussed in the earlier section

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  1   1  2αβφ 2 2φ 2 M2 ψ(φ) = M 2 1 − 1 + 2 + (1 − μi + μe ) 1 − 1 − M αβ M2 μe + (1 − eσφ ) + μi [1 + 3β1 − (1 + 3β1 + 3β1 φ + β1 φ2 )e−φ ]. σ (8.58) Lower bounds of Mach number for the existence of solitary waves: Using soliton conditions (8.15)–(8.18), we can get an analytical condition to find a range of Mach number M for the existence of solitary waves. It can be easily checked that the condition ψ(φ) = 0 and ψ  (φ) = 0 at φ = 0 are satisfied. The condition ψ  (φ) < 0 at φ = 0 gives rise to the condition M > Mc , where Mc2 =

1 + αβ(1 − μi + μe ) . μe σ + (1 − β1 )μi

Small-amplitude approximation: To study the small-amplitude approximation of ψ(φ), let us expand ψ(φ) about φ = 0. Using the bound3 ary condition ψ → 0 and dψ dφ → 0 as φ → 0 and keeping up to φ terms, we get ψ(φ) = A1 1 M2 + 2 2 μe ) 3αM β4

where A1 =

φ2 φ3 + A2 2 6

(8.59)

αβ 3 (1 − μi + μe ) M 2 − σμe + (β1 − 1)μi and A2 = − M 4 +

(1 − μi + − σ 2 μe − (1 + 4β1 )μi . Hence, the KdV-type soliton solution is   2 ξ ψ = ψ0 sech δ

(8.60)

1 √2 where ψ0 = − 3A A2 is the amplitude of the solitary wave and δ = −A1 is the width of the solitary wave. Neglecting the nonthermal effect of ions (i.e., putting β1 = 0), (8.59) will become

ψ(φ) = A1

φ2 φ3 + A2 2 6

(8.61)

Sagdeev’s Pseudopotential Approach

where A1 = 2

2

1 M2 2

327

αβ 3 + (1 − μi + μe ) M 2 − σμe − μi and A2 = − M 4 + (1 − μi +

μe ) 3αM β4 − σ μe − μi .

Comparison between small- and large-amplitude solitary waves: Sayed and Mamun [4] studied this model for small-amplitude solitary waves using the RPT, and they obtained the KdV equation as ∂ψ1 ∂ψ1 ∂ 3 ψ1 + Aψ1 +B =0 ∂τ ∂ξ ∂ξ 3 where A =

− μi + μe )3α2 β 2 − 3 − V0 4 (μe σ 2 − μi )] and

B=

phase speed of the DAW, is given by

1 2V0 [(1−μi +μe )αβ] [(1 3 V0 2[1+(1+μe −μi )αβ) . V0 , the

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(8.62)

1 + (1 + μe − μi )αβ . σμe + μi

(8.63)

To get the steady-state solution, they used the transformation ξ = η − U0 τ and the usual boundary conditions, where U0 is the velocity of the frame of the transformed coordinate. Using the usual technique, one obtains   d2 ψ1 2 ψ12 ∂V1 = (8.64) U0 ψ1 − =− dξ 2 B 2 ∂ψ1 where V1 (ψ1 ) = −

U0 2 1 3 ψ1 + ψ . B 3B 1

(8.65)

Now, to compare the small-amplitude approximation of ψ(φ) of (8.61) with the values of V1 (ψ1 ) of (8.65) obtained by the RPT [4], we first replace M by V0 + U0 , when U0 is small. Then, keeping only first-order terms (in U0 ), it can easily be verified that ψ(φ) in (8.61) reduces to V1 (ψ1 ) given in (8.65). Hence, V1 (ψ1 ) obtained by the RPT in Ref. [4] is nothing but a small-amplitude approximation of ψ(φ) of (8.61). It is obvious from Equation (8.57) that ψ(φ) satisfies the existence conditions (8.15)–(8.17) for solitary waves. It can be shown that ψ(φ) crosses the φ axis for φ < 0 for 1.60 ≤ M ≤ 1.88. So, the rarefractive solitary waves exist. For M = 1.88, ψ(φ) crosses the φ axis at φ = −1.765. Hence, |φm | = ψ0 = 1.765 is the amplitude of the rarefractive solitary waves. It is also seen that from Figure 8.2 that the amplitude of the solitary waves increases with the increase in the Mach number, i.e., phase velocity of the waves.

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Waves and Wave Interactions in Plasmas

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Fig. 8.2: Plot of ψ(φ) vs. φ for M = 1.6, 1.88, and 2.1. The other parameters are α = 0.01, β = 40, μi = 0.5, μe = 0.2, σ = 0.5, α1 = 0.03.

Fig. 8.3: Plot of ψ(φ) vs. φ for M = 1.9, 2.2, and 2.5. The other parameters are α = 0.02, β = 40, μi = 0.4, μe = 0.3, σ = 0.6, α1 = 0.04.

It is clear from Figure 8.3 that ψ(φ) crosses the φ axis for positive values of ψ for 1.9 ≤ M ≤ 2.2. So, the compressive solitary waves exist. For M = 2.2, ψ(φ) crosses the φ axis at φ = 2.82. Hence, |ψm | = ψ0 = 2.82 is the amplitude of the compressive solitary waves. It is also seen from this figure that the amplitude of the solitary waves increases with the increase in velocity. To see the effect of α1 on the amplitude of rarefractive solitary waves, Figure 8.4 is drawn. Here, the amplitude of the solitary wave |ψ0 | is plotted against α1 . It is seen that the amplitude of the rarefractive solitary waves increases very slowly with the increase in α1 . Hence, α1 has a significant effect on the speed and shape of solitary waves in four-component plasmas. Figure 8.5 shows that if the value of α1 is between 0.0 and 0.2, the width of the solitary waves starts increasing. Again, when the value of α1 > 0, then the width of the solitary waves starts decreasing.

Sagdeev’s Pseudopotential Approach

329

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Fig. 8.4: Plot of |ψ0 | against α1 with the parameters of Figure 8.2.

Fig. 8.5: Plot of δ vs. α1 with parameters of Figure 8.2.

8.5 Large-amplitude Double Layers In Chapter 1, we have discussed dusty plasma. Here, we consider a dusty plasma model consisting of massive, micron-sized, negatively charged inertial dust grains, high- and low-temperature isothermal ions, and nonthermal electrons having Tsallis distribution. The presence of energetic particles in a plasma can change the astrophysical plasma environment, and the distribution functions are considered highly nonthermal. In Chapter 1, we have already discussed the above q distribution. The nonextensive nonthermal velocity of electron distribution is physically meaningful because the distribution shoulders and high energy states are more prominent in the case of the extensive nonthermal electron. Accordingly, we have studied the effect of the nonthermal electron nonextensivity on DA double layers. However, expressions for electron number density are different for the values of q > 1

330

Waves and Wave Interactions in Plasmas

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and −1 < q < 1. It is surprising that the previous analysis [5, 6], involving the pure Tsallis-type distribution (i.e., for α = 0), uses the same electron number density distribution for both the regions. In this study, we have restricted the values of q in a limited range 0.6 < q < 1 (arguments for selecting this range are discussed in Ref. [7]). For simplicity, we assume the dust grains’ charge qd = −zd e, where zd is the number of charges residing on the dust grain. Charge neutrality at equilibrium requires nih0 + nic0 = ne0 + zd nd0 , where nj0 is the equilibrium particle density and nj (j = d, e, ih, and ic stands for the negative dust particle, electron, high-temperature isothermal ion, and low-temperature isothermal ion, respectively) is the density. The electrons are assumed to follow a nonextensive nonthermal velocity distribution function given by   1/(q−1) vx4 vx2 fe (vx ) = Cq,α 1 + α 4 1 − (q − 1) 2 vte 2vte

(8.66)

where vte = (Te /me )1/2 , Te , and me are the electron thermal velocity, the electron temperature, and the electron mass, respectively. ⎧  1 Γ( 1−q )(1 − q)5/2 ⎪ me n , e0 1 1 1 ⎪ 2πTe Γ( 1−q − 52 )[3α + ( 1−q − 32 )( 1−q − 52 )(1 − q)2 ] ⎪ −1 < q < 1 ⎨ Cq,α = ⎪  1 1 1 Γ( q−1 + 32 )(q − 1)5/2 ( q−1 + 32 )( q−1 + 52 ) me ⎪ ne0 , 1 1 1 2πTe Γ( q−1 + 1)[3α + (q − 1)2 ( q−1 + 32 )( q−1 + 52 )] ⎪ ⎩ q>1 (8.67) is the constant of normalization. Here, α, q, and Γ are the number of nonthermal electrons, the strength of nonextensivity, and the Gamma function, respectively. For q > 1, the distribution function (8.66) represents a thermal cut-off on the maximum value of the velocity of the electrons, given by  2Te , (8.68) vmax = me (q − 1) beyond which no probable states exist. For q = 1, the distribution [8] is obtained. High energy states are more probable than in the extensive case, when q < 1. Integrating Equation (8.66) overall velocity space, we get the

Sagdeev’s Pseudopotential Approach

electron density as ⎧  +∞ ⎪ fe (vx )dvx ⎨ −∞ ne (φ) =  +v max ⎪ ⎩ fe (vx )dvx −vmax



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= ne0

eφ 1 + (q − 1) Te

331

for −1 < q < 1, for q > 1

(q+1)/2(q−1)     2  eφ eφ 1+A +B Te Te (8.69)

where A = −16qα/(3 − 14q + 15q 2 + 12α) and B = 16(2q − 1)qα/(3 − 14q + 15q 2 + 12α). In the extensive limiting case (q → 1), density (8.69) reduces to the nonthermal electron density,     2    4α eφ 4α eφ eφ ne (φ) = ne0 1 − + exp . (8.70) 1 + 3α Te 1 + 3α Te Te The dynamics of low phase velocity DA oscillations, whose phase speed is much less than the electron and ion thermal velocities, is governed by the following normalized equations: ∂nd ∂(nd ud ) + = 0, ∂t ∂x ∂ud σ ∂pd ∂ud ∂φ + ud + = zd , ∂t ∂x nd ∂x ∂x

(8.71) (8.72)

∂p ∂p ∂ud + ud + 3p = 0, ∂t ∂x ∂x

(8.73)

∂2φ = zd nd + ne − nic − nih . ∂x2

(8.74)

j = d, e, ih, and ic stands for negative dust particle, high-temperature isothermal ion, low-temperature isothermal ion, and electron, respectively. nj is the number density, uj is the charged velocity, φ is the electric potential, and pj is the charge fluid pressure. Normalization: Here, the normalization is taken as x→

x , λD

pd →

t → ωpd t,

pd , zd nd0 Td

φ→

nj → eφ Tef f

nj , nj0

ud →

ud , Cd

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Waves and Wave Interactions in Plasmas

 12 1 T −1 where λD = 4πzdefe2fnd0 , ωpd = (md /4πzd2 e2 nd0 ) 2 , pd0 = zd nd0 Td , and

Cd = zd Tef f /md . mj , Tj , and e are mass, temperatures, and elementary charge. Moreover, we define Tef f = (f /Te + μ/Tic + ν/Tih )−1 (the effective temperature), σ = Td /Tef f , f = ne0 /(zd nd0 ), σe = Tef f /Te , μ = nic /(zd nd0 ), and ν = nih /(zd nd0 ). Here, μ + ν = 1. The densities of nonthermal electrons feature Tsallis distribution [9] and the thermal two-temperature ions are given, respectively, by

 (q+1)/2(q−1) ne = f 1 + Aσe φ + Bσe2 φ2 1 + (q − 1)σe φ   φ nic = μ exp − , μ + νβ1   β1 φ nih = ν exp − μ + νβ1 where β1 = Tc /Th = (low-temperature)/(high-temperature), A = −16qα/ (3 − 14q + 15q 2 + 12α), B = 16(2q − 1)qα/(3 − 14q + 15q 2 + 12α), and α is the nonthermal parameter. In the extensive limiting case (q → 1), density (8.70) reduces to the nonthermal electron density [8]   4α 4α 2 2 σe φ + σ φ exp(σe φ). (8.75) ne = f 1 − 1 + 3α 1 + 3α e In contrast, for α = 0, density (8.70) reduces to the nonextensive electron density ne = f [1 + (q − 1)σe φ](q+1)/2(q−1) [10]. Sagdeev potential: Using transformations (8.4) in Equations (8.71)– (8.74) and then integrating the transformed equation with the help of boundary condition φ → 0, ud → ud0 (the equilibrium dust drift speed), nd → 1 as |ξ| → ∞, we get M − ud0 , M − ud  2 u2d 3σ M − ud0 A zd φ = −M ud + + + 2 2 M − ud 2 nd =

(8.76) (8.77)

where A = 2M ud0 − u2d0 − 3σ. Equation (8.77) can be solved to find ud as an explicit function of φ  12 

1 ud = M − √ M 2 + 2zd φ − A + (M 2 + 2zd φ − A)2 − 12σ(M − ud0 )2 . 2 (8.78)

Sagdeev’s Pseudopotential Approach

333

Integrating Equation (8.74) and using the above given boundary conditions, we get d2 φ ∂ψ(φ) =− , 2 dξ ∂φ

(8.79)

where the pseudopotential is 

3  M − ud0 M − ud 3q−1    2(q−1) 2(1 + Aσe φ + Bσe2 φ2 ) +f − 1 + (q − 1)σe φ (3q − 1)σe 5q−3   2(q−1) 4(A + 2Bσe φ) + 1 + (q − 1)σe φ (3q − 1)(5q − 3)σe 7q−5     2(q−1) 16B + 1 − 1 + (q − 1)σe φ (3q − 1)(5q − 3)(7q − 5)σe   2 2A + 1− (3q − 1)σe (5q − 3)   φ + μ(μ + νβ1 ) 1 − e− μ+νβ1



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ψ(φ) =

− (M − ud0 )ud0 + σ + (M − ud0 )ud − σ

  β1 φ ν(μ + νβ1 ) − μ+νβ 1 + 1−e . β1

(8.80)

When we consider only dust particles and two-temperature isothermal ions without electron (ne = 0), our result agrees with Tagare’s result [11]. For the double layers, the Sagdeev potential ψ(φ) should be negative between φ = 0 and φm , where φm is some double root of the pseudopotential corresponding to the maximum wave amplitude. Therefore, for the formation of double layers, ψ(φ) must satisfy conditions (8.20)–(8.22). To find the region of existence of the double layer, one has to study the nature of ψ(φ). In Figure 8.6, ψ(φ) is plotted against φ for q = 0.61 (solid line) and 0.65 (dashed line). It is seen that for q = 0.61, conditions (8.20)–(8.22) are both satisfied, and hence, a double layer exists. However, it is found that for q = 0.65, condition (8.20) is satisfied but ψ(φ) = 0 at φ = φm , i.e., V (φ) does not cross the φ axis at any point other than 0. Hence, for q = 0.62, a double layer could not exist. It is, therefore, found that q has a significant role in the formation of the double layer.

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Waves and Wave Interactions in Plasmas

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Fig. 8.6: The pseudopotential ψ(φ) is plotted against φ for q = 0.61 (solid line) and 0.65 (dashed line). Other parameters are M = 1.5, σ = 0.02, μ = 0.282, β = 0.0224, ud0 = 0.001, σe = 0.05, zd = 825, f = 0.0317.

Fig. 8.7: The effect of the finite dust temperature on the pseudopotential ψ(φ) for σ = 0.02 (solid line) and 0.001 (dashed line). Other parameters are the same as Figure 8.6.

In Figure 8.7, ψ(φ) is plotted against φ, for σ = 0.02 (solid line) and σ = 0.001 (dashed line). It can be seen that a double layer exists for σ = 0.02. So, it is found that a finite dust temperature has a significant role in the formation of the double layer. 8.6 Effect of Ion Kinematic Viscosity The presence of dissipation effects in plasma plays an important role in the dynamics of nonlinear waves. The dissipation arises due to the Landau damping, kinematic viscosity, wave–particle interaction, and so on. Experimentally, the effects of dissipation caused by the kinematic viscosity on the propagation of solitary wave structures were observed and discussed in

Sagdeev’s Pseudopotential Approach

335

Ref. [12]. Here, we consider the nonextensivity of electrons, which has been discussed in Chapter 1. The basic normalized equations are ∂ni ∂(ni ui ) + = 0, ∂t ∂x

(8.81)

∂ui ∂ui ∂φ ∂ 2 ui −η 2 , + ui =− ∂t ∂x ∂x ∂x

(8.82)

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∂2φ = (ne − ni δ1 + zd δ2 ) ∂x2

(8.83)

where nj , uj , φ, Tj , η, Cs , and λde are the number density, the charged velocity, the plasma potential, the temperature, the kinematic viscosity, the ion acoustic speed, and the Debye length, respectively. Here, j = i, and e, which stands for ions, and electrons, respec−1 Te /mi , ωpi = (mi /4πni0 e2 )1/2 , λi = tively. η = 1/Cs λDe , Cs = q+1

(Te /4πni0 e2 )1/2 δ1 = ni0 /ne0 , δ2 = nd0 /ne0 , and ne = (1 + (q − 1)φ) 2(q−1) . zd is the dust charge number and the dust grains are assumed to be massive, immobile, and negatively charged. Normalization: Here, the normalizations are taken as ni →

ni , ni0

ne →

ne , ne0

ui →

ui , Cs

φ→

eφ , Te

t → ωpi t,

x→

x . λi

By using transformation (8.4), Equations (8.81)–(8.83) are reduced to −M

d(ni ui ) dni + = 0, dξ dξ

−M

dui dφ d2 ui dui −η 2 , + ui =− dξ dξ dξ dξ d2 φ = n e − n i δ1 + z d δ2 . dξ 2

(8.84) (8.85) (8.86)

Using boundary conditions ni → 1, ui → 0, when ξ → ±∞, we obtain from Equations (8.84)–(8.85) ni =

M M − ui

(M − ui )2 M2 dui = −φ−η . 2 dξ 2

(8.87) (8.88)

Waves and Wave Interactions in Plasmas

336

Let  F (φ) =

φ

0

ni dφ.

(8.89)

Now using (8.87)–(8.89) in (8.86), we get d2 φ ∂ψ = 2 dξ ∂φ

(8.90)

where

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ψ(φ) =

(3q−1) 2 2 [1 + (q − 1)φ] 2(q−1) − δ1 F (φ) + zd δ2 φ − . (3q − 1) (3q − 1) (8.91)

Small-amplitude approximation: To get the explicit expression of pseudopotential, one has to get the explicit expression for F (φ). From Equations (8.87) and (8.89), we have F  (φ) =

M . M − ui

(8.92)

So, dui M dφ =  F  (φ) . dξ [F (φ)]2 dξ

(8.93)

w = f  (φ) = ni

(8.94)

Taking

and using the results of Equations (8.88) and (8.93), we get 

w =

ηM



w2 M 2 2 4 3q−1 [1

− φw2 −

M2 2

3q−1

+ (q − 1)φ] 2(q−1) − 2δ1 F  (φ) + 2zd δ2 φ −

4 3q−1

. (8.95)

It is seen from Equation (8.95) that the equation even holds when η is zero, provided one takes the appropriate limit. For η = 0, w2 can be obtained from the following equation: w2 M 2 M2 − φw2 − = 0. 2 2

(8.96)

Sagdeev’s Pseudopotential Approach

337

So, M ni = w =

. M 2 − 2φ

(8.97)

For nonzero but small η, we consider terms up to O(η) and get from Equation (8.95) ηM dφ M dξ + . w=

M 2 − 2φ (M 2 − 2φ)2

(8.98)

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Using (8.86) in Equation (8.98), we have M δ1 η dφ (q+1) d2 φ δ1 M dξ 2(q−1) −

= [1 + (q − 1)φ] − + z d δ2 . dξ 2 M 2 − 2φ (M 2 − 2φ)2

(8.99)

Considering the terms up to O( Mφ2 ) in Equation (8.99) and differentiating with respect to ξ, we get   d3 φ dφ q + 1 δ1 (q + 1)(q − 3) dφ δ1 η d2 φ − − 3 2. = + φ 3 2 dξ dξ 2 M 2 dξ M dξ

(8.100)

Equation (8.100) is the well-known KdVB equation written in terms of the single variable ξ = x − M t. Using the boundary conditions φ = 0, dφ dξ = 0 and a particular solution of the above KdVB equation [13] is obtained as follows: φ=

A2 g 2 − Bg)2

(e−Aξ/2

(8.101)

 (q+1)(q−3) 2ηδ1 where A = − 5M and B = ± . g is the integration constant 3 6 that depends on the initial condition and M is determined from the following equation: 25(q + 1)M 6 − 50δ1 M 4 + 12δ12 η 2 = 0.

(8.102)

When Bg = −1, the above solution is the same as the solution obtained by W. Malfliet [14] who used the tanh method. However, if η is not too small, one cannot derive Equation (8.101). For arbitrary η, one can get Sagdeev’s potential up to any order in φ by the suitable expansion of F (φ). Now,

Waves and Wave Interactions in Plasmas

338

considering F (φ) up to φ4 terms, we have F (φ) = φ + bφ2 + cφ3 + dφ4

(8.103)

where b is determined from the following equation:  (q + 1) 2 2bM − 1 − 2bηM − 2δ1 b = 0. 2 c and d are given by c=

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and

4b − 2b2 M 2 − 2ηM b(q + 1)(q − 3)

[12(1 + q) − 48δ1 b][3M 2 − 6ηM (q + 1)/2 − 2δ1 b]

   q + 1 − 4δ1 b 4δ1 ηM b d 12ηM − 4M 2 −√ √ 2 2 q + 1 − 4δ1 b = 6bcV 2 − (4b2 + 6c) + d1

where



  (q + 1)(q − 3) + 24δ1 c ηM b {(q + 1)(q − 3) + 24δ1 c}2 √ √ d1 = cηM + √ 144(q + 1 − 4δ1 b)3/2 2 2 q + 1 − 4δ1 b 2  (q + 1)(q − 3)(3q − 5) √ − . 48 q + 1 − 4δ1 b

One can also write ψ=−

 2 1 dφ . 2 dξ

(8.104)

We get from our previous equation ψ=−

3q−1 2 2 [1 + (q − 1)φ] 2(q−1) + δ1 F (φ) − δ2 zd φ + . (8.105) (3q − 1) 3q − 1

Keeping terms up to φ2 , we get d2 φ = A1 φ + A2 φ2 dξ 2

(8.106)

where A1 = −(q + 1)/2 + 2bδ1 and A2 = (q − 3)(q + 1)/8 + 3cδ1 . Integrating Equation (8.106), with the same boundary conditions, we obtain √  3A1 A1 ξ 2 φ=− sech . (8.107) 2A2 2

Sagdeev’s Pseudopotential Approach

339

Now, keeping terms up to φ3 of Equation (8.105), we have d2 φ = A1 φ + A2 φ2 + A3 φ3 dξ 2

(8.108)

where A3 = −(q − 3)(q + 1)(3q − 5)/48 + 4δ1 d. Solving Equation (8.108), we obtain   −1 

A2 A22 A3 − φ= − + cosh( A1 ξ) . (8.109) 3A1 9A21 2A1 If A22 = 92 A1 A3 , then solution (8.109) would not be valid and a shock wave solution is obtained, which is given by

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φ = P (1 + tanh α(ξ + ξ0 ))

(8.110)

3A1 2 where ξ0 is the integration constant and P = − 2A , α = 3√A2A , (A0 > 0). 2 3 If δ1 b = (q + 1)/4 and δ1 c = (3 − q)(q + 1)/24, it is seen from Equations (8.107) and (8.109) that the solution does not exist. For these values of parameters, one must take a higher-order term. When A1 < 0, but δ1 c = q(3−q)(q = 1)/24, both these solutions will show periodic behavior. Hence, for solitary wave solution, we assume A1 > 0 and A3 = 0. Figure 8.8 shows the plot of ψ vs. φ for η = 0.03 (dotted line) and 0.05 (bold line) when δ1 = 2.235, q = 1.1. It is seen that as η increases, the amplitude of the soliton increases. To show the effect of the nonextensivity of electrons, we use the q distribution function fe (v) = Cq {1 + (q − 1)[me v 2 /2Te − eφ/Te ]}1/(q−1) , where

Fig. 8.8: The plot of ψ vs. φ for η = 0.03 (dotted line) and 0.05 (solid line).

340

Waves and Wave Interactions in Plasmas

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Fig. 8.9: The plot of ψ vs. φ for q = 1.2 (dotted line) and 1.5 (solid line).

fe (v) is the particular distribution that maximizes the Tsallis entropy and Cq is the normalization constant. To see the effect of nonextensive parameter q in Figure 8.9, ψ is plotted against φ for q = 1.2 (dotted line) and 1.5 (solid line). Other parameters are the same as those in Figure 8.8. We find from this figure that the amplitude of the soliton increases highly as q increases. Hence, both q and η play significant roles in the formation and shape of solitary waves in plasma. For large-amplitude DIASWs, one has either to solve the coupled Equations (8.94) and (8.95) or has to include higher-order terms in the expansion. However, the solution given in Equation (8.109) is already a higher-order solution that cannot be obtained easily using the RPT method. For small amplitude, one can neglect the term of the order φ4 in the pseudopotential, while for a shock wave solution, one has to include higher-order terms. 8.7 DIASWs in Magnetized Plasma In a strong magnetic field, ion pressure has a significant effect on the formation of solitary waves. If the magnetized plasma is in a collisionless state and the magnetic field is strong, the pressure becomes anisotropic. The plasma then behaves differently in each of the parallel and perpendicular directions. The ion pressure obeys two different equations of state in both directions in the magnetic field. Chew–Goldberger–Low (CGl) theory [15] or a double adiabatic theory can be used when there is no coupling between the perpendicular and parallel pressure. Anisotropic pressure has a significant effect on the solitary waves. In anisotropic plasmas, the pressure tensor

Sagdeev’s Pseudopotential Approach

341

can be written in a matrix form as ⎛ ⎞ p⊥ 0 0 P = ⎝ 0 p⊥ 0 ⎠ . 0 0 p If p⊥ = p , this becomes isotropic pressure. According to the CGL theory, the scalar pressures p⊥ and p can be written as  p⊥ = p⊥0

ni ni0



 and p = p0

ni ni0

3

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where p⊥0 and p0 are the perpendicular and parallel pressures at equilibrium. Subindex 0 denotes the value at the equilibrium. The Boltzmann relation gives the density of electrons as eφ

ne = ne0 e Te .

(8.111)

The basic normalized equations are

∂vx ∂t ∂vy ∂t ∂vz ∂t

∂n ∂nvx ∂nvz + + = 0, ∂t ∂x ∂z   p⊥0 1 ∂n ∂ ∂ ∂φ + vx + vz + vy − , vx = − Te ni0 n ∂x ∂x ∂z ∂x   ∂ ∂ + vx + vz vy = −vx , ∂x ∂z   3p0 ∂n ∂ ∂ ∂φ − + vx + vz n , vz = − Te ni0 ∂z ∂x ∂z ∂z  2  ∂ ∂2 + 2 φ = β[eφ − δ1 n + δ2 ] ∂x2 ∂z

(8.112) (8.113) (8.114) (8.115) (8.116)

where n, φ, and zd are the number density, the electrostatic potential, and the dust charge number, respectively. The charge of the dust is written as qd = −ezd , where e is the elementary charge. vi and mi are the velocity and the mass of ions. We assume that the wave is propagating in the xz r2

i0 plane. β = λg2 , δ1 = nne0 , and δ2 = nnde0zd . rg = CΩs is the ion gyroradius and e  1/2 e λe = 4πnTe0 is the electron Debye length. 2 e

Normalizations: Normalizations are taken as Ωt → t,

(Cs /Ω)∇ → ∇,

vi /Cs → v,

ni /ni0 → n,

eφ/Te → φ

Waves and Wave Interactions in Plasmas

342

where Cs = (Te /mi )1/2 is the ion acoustic velocity, Ω = gyrofrequency, and nj0 are equilibrium densities.

eB0 mi c

is the ion

Sagdeev potential: To obtain the solutions for Equations (8.112)– (8.116), we introduce a variable defined in a moving coordinate given by ξ = lx x + lz z − M t

(8.117)

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where lx and lz are directional cosines and M is the Mach number of the localized wave. Equations (8.112)–(8.116) can then be reduced to ordinary differential equations in terms of ξ. M (1 − n) + n(lx vx + lz vz ) = 0, (8.118)   p⊥0 1 dn d d dφ dvx −M + vx lx + vz lz + vy − lx vx = −lx Te ni0 n dξ, dξ dξ dξ dξ (8.119)   dvy d d −M + vx lx + vz lz (8.120) vy = −vx , dξ dξ dξ   3p0 dvz d d dφ dn − −M + vx lx + vz lz nlz , (8.121) vz = −lz Te ni0 dξ dξ dξ dξ dξ (lx2 + lz2 )

d2 φ = β[eφ − δ1 n + δ2 ]. dξ 2

Using (8.118) in (8.119), we get   p⊥0 1 dn M dvx dφ = lx + − vy . Te ni0 n dξ n dξ dξ

(8.122)

(8.123)

Using (8.119), we get M dvy = vx . n dξ

(8.124)

  3p0 dn M dvz dφ = lz + n . Te ni0 dξ n dξ dξ

(8.125)

Using (8.121), we have

Considering  F (φ) =

0

φ

ndφ,

(8.126)

Sagdeev’s Pseudopotential Approach

343

  p0 3 lz n . F (φ) + Te ni0 M

(8.127)

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we have vz =

Using (8.126) and (8.127) in (8.118), we obtain     l2 1 l2 p0 lx vx + z F (φ) + z (n3 − 1) = 1 −  M. M M Te ni0 F (φ)

(8.128)

From (8.122), we have (lx2

+



dφ 2 dξ

1 lz2 )

2 = β[eφ − 1 − δ1 F (φ) + δ2 φ].

(8.129)

As can be seen in (8.129), Equation (8.122) can be integrated to the form  2 1 dφ + ψ(φ) = 0 (8.130) 2 dξ where ψ(φ) is the Sagdeev potential. In a small-amplitude approximation, F (φ) can be expanded in terms of φ such that F (φ) = φ + a1 φ2 + a2 φ3 + a3 φ4 + a4 φ5 + · · ·

(8.131)

and from (8.128), we can get vx = Aφ + Bφ2 + Cφ3 + Dφ4 where A= B= C= D=

 ! p l2 − Mz 1 + 6a1 Te0 + 2a M 1 ni0  l2 − Mz a1 +  l2 − Mz a2 +

 l2 − Mz a3 +

lx p0 Te ni0 (9a2

,

! + 12a21 ) + 3a2 M − 4a21 M lx

p0 3 Te ni0 (8a1

(8.132)

,

! + 12a3 + 36a1 a2 ) + 4a3 M − 12a1 a2 M + 8a31 M lx

p0 Te ni0 (15a4

+ 27a22 + 48a1 a3 ) + 36a21 a2



lx +

5a4 M −

9a22 M

− 16a1 a3 M + 36a21 a2 M − 16a41 M . lx

,

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Waves and Wave Interactions in Plasmas

From (8.129), we get  2      dφ 1 1 2β 2 = 2 (1 + δ2 − δ1 )φ + − δ1 a1 φ + − δ1 a2 φ3 dξ lx + lz2 2 6    1 + − δ1 a3 φ4 . (8.133) 24 Keeping terms up to φ2 order, we can get Sagdeev’s equation as

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d2 φ dψ = A1 φ + A2 φ2 = − (8.134) dξ 2 dφ 1

β β where A1 = l2 +l 2 (1 − 2δ1 a1 ) and A2 = l2 +l2 2 − 3δ1 a2 . From Equation x z x z (8.124), we get M

dvy dφ = vx F  (φ), dφ dξ

and from Equation (8.123), we get   p⊥0 1 dn M dvx dφ − vy = lx + lx . Te ni0 F  (φ) dφ F  (φ) dφ dξ

(8.135)

(8.136)

Equating the terms of φ with equal powers in Equations (8.135) and (8.136), the values of a1 and a2 can be decided by the following equations: A = Y1 (1 − 2δ1 a1 )   p⊥0 (X2 − 1) 3M X2 a2 + 3Y1 δ1 − 3 Te ni0 lx M

(8.137)

  Y1 lx = + Z1 X2 − 2a1 A + (1 − 2δ1 a1 )a1 X1 − 2Y1 . (8.138) 2 2M

The expressions for X1 , X2 , Y1 , and Z1 are given as  lx M β lx + 2a1 pT⊥0 − M A n 4M 2 β e i0 X1 = 2 , Y1 = , lx + lz2 lx2 + lz2 2 lz a1 2 M + 4a 1 M 2M 2 β(1 − 2δ1 a1 ) Z1 = . , X2 = 1 + lx lx2 + lz2 The solution for (8.133) is given by 3(1 − 2δ1 a1 ) φ=− sech2 1 − 6δ1 a2

√  A1 ξ . 2

(8.139)

Sagdeev’s Pseudopotential Approach

345

Again, keeping terms up to φ3 , we have

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d2 φ = A1 φ + A2 φ2 + A3 φ3 (8.140) dξ 2 1

β where A3 = l2 +l 2 6 − 4δ1 a3 . a3 is obtained from the following equation: x z     p⊥0 lx M 4M a3 4Y1 δ1 + 3X1 (1 − 2δ1 a1 ) − + lx Te ni0 M lx  2   p0 1 lz = + 12a1 a2 M − 8a31 M a2 + Te ni0 lx M Y1 − 4Aa21 − 4Ba1 − 6Aa2 + 6    p⊥0 lx a1 l x 3X1 X1 1 + − 3δ1 a2 −B+3 a2 + (1 − 2δ1 a1 ) 2 2 Te ni0 M 4lx M  2  lz2 p0 l a2 l 2 a2 × x + z + + 12a1 a2 M − 8a31 M . M M M Te ni0 Solution for Equation (8.134) is given by   −1

A2 A22 A3 − − φ= − cosh( A ξ) . 1 3A1 9A21 2A1

(8.141)

If A22 = 92 A1 A3 , then solution (8.141) would not be valid and a shock wave (double-layer) solution is obtained, which is given by φ=−

3A1 [1 + tanh α(ξ + ξ0 )] 2A2

(8.142)

2 where ξ0 is the integration constant and α = − 3√A2A , A3 > 0. 3 In Equation (8.139), it can be seen that if 2δ1 a1 = 1, then the amplitude of the solitary wave will be 0 and if 6δ1 a2 = 1, then the amplitude will be infinite. Thus, the expansion of V (φ) up to O(φ3 ) is valid, only if δ1 a1 = 12 or δ1 a2 = 16 . If δ1 a1 = 12 or δ1 a2 = 16 , then the expansion should be made up to O(φ4 ) and in that case, the soliton solution will be given by Equation (8.141). Again, if A22 = 92 A1 A3 , then Equation (8.141) no longer admits a solitary solution. For that particular set of parameters, we get a shock wave solution, which is given by Equation (8.142). Furthermore, from Equations (8.139) and (8.141), one more condition is required, A1 > 0, for the existence of soliton solution. From the discussions so far, it is clear that the amplitude of the solitary waves depends on a1 , a2 , a3 , which are

346

Waves and Wave Interactions in Plasmas

functions of β. This means that the amplitude of the solitary wave changes with the change of the external magnetic field.

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8.8 Solitary Kinetic Alfven Waves We have already discussed Alfven waves in Chapter 1. We have seen that there are different types of Alfven waves. The kinetic Alfven waves grow from the perpendicular effects. These waves follow an electric field and propagate perpendicularly to the ambient magnetic field. It can be stimulated in plasma either by drift wave instability or resonant mode conversion of a surface magnetohydrodynamic wave. KAW involves an exact solution that springs from the balance between the perpendicular dispersion and the compressible nonlinearity of the waves. The electrons tend to follow the magnetic lines of force due to their small Larmor radii. It produces a charge separation and leads to what is known as the kinetic Alfven waves (KAWs). Ion drift, for example, contributes to the kinetic properties of Alfven waves. The KAW develops a longitudinal parallel electrostatic field due to the finite-Larmor radius (FLR) effect. They also have dispersive characteristics. Solitary kinetic Alfven wave (SKAW) exists due to the interplay between the nonlinear steepening and their dispersive character [16]. They play an important role in the study of the coupling of the ionosphere and magnetosphere. The strong electromagnetic solitary spikes dominate the auroral low-frequency turbulence, according to the observation of the Freja satellite [17, 18]. These structures could be interpreted as hump-type and dip-type SKAWs. A recent observation from Freja and Fast satellites [19] showed the presence of SKAWs in Earth’s ionosphere. We will discuss the effect of ion temperature on the existence of largeamplitude solitary waves and double layers in the plasma model. Here, we consider a two-fluid homogeneous plasma model in a uniform ambient magnetic field. ∂ne vez ∂ne + =0 ∂t ∂z   ∂vez ∂vez 1 ∂ne + vez = α − Ez − ∂t ∂z ne ∂z ∂ni vix ∂ni viz ∂ni + + =0 ∂t ∂x ∂z

(8.143) (8.144) (8.145)

Sagdeev’s Pseudopotential Approach

347

∂vix ∂vix ∂vix σ ∂ni + vix + viz = αQEx + viy − ∂t ∂x ∂z ni ∂x ∂viy ∂viy ∂viy + vix + viz = −vix ∂t ∂x ∂z ∂viz ∂viz ∂viz σ ∂ni + vix + viz = αQEz − ∂t ∂x ∂z ni ∂z   ∂ 3 Ex ∂ 3 Ez 1 ∂ 2 ne ∂ 2 (ni viz ) − 2 + = + . ∂z ∂x ∂x2 ∂z αQ ∂t2 ∂t∂z

(8.146) (8.147) (8.148) (8.149)

nj (j = i and e stand for ions and electrons, respectively) and ωpi are e the density and the ion plasma frequency, respectively. Q = m mi , Ex = ∂φ2 β ∂φ1 cB0 Te − ∂x , Ez = − ∂z , VA = 1 , σ = T , and α = 2Q . Two potentials i (4πn0 mi ) 2

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φ1 and φ2 are included to justify a low-β plasma model. Normalization: Here, normalization is taken as follows: nj →

nj , n0

t → Ωt,

Electric fields are normalized by by B0 .

vi → VA ,

Te Ωci evA ,

x→

cx . ωpi

and magnetic field is normalized

Energy integral and the Sagdeev potential: We introduce a stationary independent variable η = xkx + zkz − ωt, where ω = VVA is the phase velocity of the wave in the unit of the Alfven velocity VA and kx2 + kz2 = 1. Here, it is assumed that a uniform ambient magnetic field B0 acts along the z-direction on the plasma. Equations (8.143)–(8.149) can be expressed in the stationary frame, and the resulting ordinary differential equations are solved by taking the following boundary conditions: vix = viz = vez = 0 at ni = ne = 1 when η → ∞. The electron velocity along the z-direction is   ω 1 vez = 1− . (8.150) kz ne The electron density is 1

ne = e− kz where A =

ω2 2αkz2 .



 Ez dη

  1 exp A 1 − 2 ne

The following relationship   1 kx vix + kz viz = ω 1 − ni

(8.151)

(8.152)

348

Waves and Wave Interactions in Plasmas

is obtained from Equation (8.145) and will be used later to obtain vix and viy . Maxwell Equation (8.149) in stationary frame is given by   3 2 d3 Ex 1 d2 (ni viz ) 2 d Ez 2 d ne −kx kz2 + k k = − ωk . (8.153) ω z x z dη 3 dη 3 αQ dη 2 dη 2

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Applying the charge neutrality condition ni = ne = n and the boundary conditions as mentioned above, the electric field in the z-direction is calculated as   2A 1 dn . (8.154) Ez = kz − n3 n dη The components of the ion velocity are given as follows:     kz 2A viz = − n + 2A + 1 − σ(n − 1) , αQ − ω n       ω 1 2A kz2 vix = − 1− αQ + n − 2A − 1 + σ(n − 1) , kx n ωkx n  2    1 n 1 n2 k2 n3 viy = − z2 αQ 2An + −n+ − An2 − −A+ kx 2 ω 2 3 2  3  n n2 1 dη − × + . 3 2 6 dn

(8.155) (8.156) 1 6

 +σ

(8.157)

By substituting vix , viy in Equation (8.146) expressed in stationary frame, we have        ω2 kz2 1 dn 2A σ kx σ Ex = − + αQ − 3 + + + αQkx n3 αQkx n n n nαQ dη  2     1 1 1 n k2 n3 n2 − z2 αQ 2An + − −n+ − An2 − −A+ αQkx 2 ω 6 2 3 2  3  n n2 1 dη − + . (8.158) +σ 3 2 6 dn After successive integration ofEquation (8.153) and using Equation (8.155) → 0 and n → 1, we get  dEx dEz 1 1 −kx kz2 + kx2 kz = [ω 2 n − ωkz nviz ] + c = ω 2 (n − 1) dη dη αQ αQ     2A − kz2 n αQ + n − 2A − 1 + σ(n − 1) . n (8.159)

and the boundary conditions

dn dη

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349

Now, by substituting Ex and Ez from Equations (8.154) and (8.158) into Equation (8.159) and after some algebraic manipulations, we get  2 4 !2 kz n2 σ 1 dn ω2 2 − (n − 2A) − 2 dη n6 αQ αQ  2 k kz2 1 − ωz2   ω2σ k2 σ n2 = − 2 2 (n − log n) + z − 2An − n 2A log n + α Q αQ 2     k 2 σ 2 n2 ω4 1 1 + 2z 2 −n + 2 2 − + 2 α Q α Q n 2n 2   2 2 k ω 2A 1 A − 2 + log n + − z + αQ n n n   2 2 2 k σω 1 ω − − z2 2 log n + (n − log n) α Q n αQ     n2 kz2 σ n2 2 + kz 2A log n + − 2An − n + −n αQ 2 2     2Aω 2 1 2A 1 1 A 2 − + 2 − 2Akz − 2 + log n + + + αQ n 2n n n n   2Akz2 σ 1 − log n + +C (8.160) αQ n ! 2 k2 kz2 σ ω2 where C = 2z (1 + 2A)2 + 4A + α2σQ2 + αQ (4A + 1) + αQ (A + 1)(1 + ! 2 2 k σ ω σ z kz2 ) + αQ . + 2αQ + αQ Finally, we get  2 1 dn + ψ(n, α, Q, ω, kz , σ) = 0 2 dη where Sagdeev’s potential ψ(n, α, Q, ω, kz , σ) is given as ψ(n, α, Q, ω, kz , σ)  k2  2  n4 1 − ωz2 kz kz2 σ kz2 σ 2 =− + + n4 !2 2 Q2 2 2 2 αQ 2α σn ω kz2 αQ − (n2 − 2A) − αQ     ω2 kz2 σ 2 3 σ kz2 σ 2 − kz (2A + 1) + 2(A + 1) + 2 2 n 1+ + αQ αQ αQ α Q

(8.161)

Waves and Wave Interactions in Plasmas

350

  ω2 kz2 σ σ 2 + + 1 − kz − n2 log n αQ αQ αQ  2  k σ2 k2 σ + z (1 + 2A)2 + 4A + 2 2 + z (4A + 1) 2 α Q αQ   ω2 σ ω2 k2 σ + + + z (A + 1)(1 + kz2 ) + n2 αQ αQ 2αQ αQ  2 2    ω ω kz2 σ σ 2 2 − + 2A + (2A + 1)kz + + 2Akz 1 + 2A + n αQ αQ αQ αQ  2 2   ω ω 2 2 2 + + A(1 + kz ) + 2A kz . αQ 2αQ Downloaded from www.worldscientific.com



Soliton and other types of solutions may be obtained similarly as discussed in the previous section. 8.9 Collapse of EA Solitary Waves To study the speed and shape of electron acoustic solitary waves, we consider the basic equations in an e-i plasma, where ions and hot electrons are mobile in the back ground of Maxwell distribution. ∂(nj uj ) ∂nj + = 0, ∂t ∂x ∂uj ∂uj μj ∂pj ∂φ + uj + = −zj μj , ∂t ∂x nj ∂x ∂x

(8.162) (8.163)

∂pj ∂pj ∂uj + uj + 3pj = 0, ∂t ∂x ∂x

(8.164)

∂2φ = n0h eφ − Σj zj nj ∂x2

(8.165)

where μj = mj /mi , zj = qj /e, and j = i, c represent ions and cold electrons, respectively. The subscript h denotes the hot electron. Normalizations: The normalizations are taken as  nj me pj Tj , uj → u j , pj → , Tj → , nj → n0e n0e Th Th Th x x→ , t → ωi t. λD

φ→

eφ , Th

Sagdeev’s Pseudopotential Approach

351

nj , pj , Tj , and φ are the number density, the pressure, the temperatures, and electric potential, respectively. me is the mass of the electron. Also, λD and and the ion plasma frequency, respectively.  ωi are the Debye length λD =

KB Th 4πn0e e2

and ωi =

4πne0 e2 . mi

Sagdeev potential and collapse of solitons: By using the traveling wave transformation (8.4), Equations (8.162)–(8.165) are reduced to −M

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−M

dnj d(nj uj ) + = 0, dξ dξ

duj duj μj dpj dφ + uj + =− , dξ dξ nj dξ dξ

−M

dpj dpj du + uj + 3pj = 0, dξ dξ dξ d2 φ = n0h eφ + nc − ni . dξ 2

(8.166) (8.167) (8.168) (8.169)

Integrating the above equation and using the boundary condition φ, dφ dξ , ui , uc , uh → 0, ni → 1, nc → n0c , pi → Ti , pc → n0c Tc at |ξ| → ∞, we get nc =

M n0c . M − uc

(8.170)

Now, eliminating φ, nc , ni , pc , pi in terms of uc and keeping terms up to O(μ), we get ∂ψ d2 uc = dξ 2 ∂u

(8.171)

ψ(uc ) = g(uc )[ψh (uc ) + ψc (uc ) + ψi (uc )].

(8.172)

where

 1 2 , ψh (uc ) = n0h (1 − ev1 ), ψc (uc ) = Here, g(uc ) = −  3Tc M 2 −1 (M−uc ) (M 4 −uc )  ! μv 2 M3 i v1 n0c M uc + Tc 1 − (M−u , ψi (uc ) = v1 + 2M12 + 3μT 3 4M 2 , and v1 = c)   u2 3Tc −M uc + 2c 1 − (M−u . ψh (uc ), ψc (uc ), and ψi (uc ) manifest the 2 c) effects of nh , nc , and ni , respectively. All ψh (uc ), ψc (uc ), and ψi (uc ) are obtained from Equation (8.169) by using Equation (8.167). Calculations

Waves and Wave Interactions in Plasmas

352

are done neglecting the terms of O(μ2 ). Thus,   d2 uc n0c M μv1 3μTi v1 − 1 − + = g (u ) n e − 1 c 0h M2 dξ 2 M − uc 4M 2     M3 + g2 (uc ) n0h (1 − ev1 ) + n0c M uc + Tc 1 − (M − uc )3  μv1 2 3μTi v1 + v1 + + (8.173) 2M 2 4M 2 9

2

5

c ) +9Tc M (M−uc ) , where g1 (uc ) = (M−uc )−3 [3Tc1M 2 −(M−uc )4 ] , g2 (uc ) = 2 (M−u [3Tc M 2 −(M−uc )4 ]3

(u )2

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and ψ(uc ) = 2c . To find the region of existence of solitary waves, one has to study the nature of the function ψ(uc ), and φ1 (uc ) is defined as ψ(uc ) =

(uc )2 2

(8.174)

where ∂ψ ∂u = φ1 (uc ).

uc =

(8.175)

φ1 (uc ) has two roots: one being at uc = 0 and the other at some point u = uc1 (≥ 0). Also, φ1 (uc ) should be positive on the interval (0, uc1 ) and negative on (uc1 , umax ). Here, umax is obtained from the nonzero root of ψ(uc ). To get the shape of the solitary wave, we have solved the differential equation numerically uc = ψ1 (uc ) with u0c = 0.423062, u0c = 0, and Figure 8.10 depicts the soliton solution uc (ξ) plotted against ξ. It is seen that u0c = 0.423062 is the critical value for uc . For u > u0c , the soliton solution ceases to exist as can be seen in Figure 8.11. In this figure, u0c is

Fig. 8.10: The soliton solution uc (ξ) plotted against ξ.

Sagdeev’s Pseudopotential Approach

353

Fig. 8.11: The collapse of soliton uc (ξ) plotted against ξ.

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taken as 0.423063 (all other parameters are the same as in Figure 8.10). Hence, it is seen that for an extremely small change of the value of u0c , the periodic behavior of the wave as well as solitary wave is destroyed. 8.10 Collapse of DASWs in Presence of Trapped Ions The ion and electron distribution functions are significantly modified in the presence of large-amplitude solitary waves, which are excited by the two-stream instability [19]. In this context, one may consider a vortex-like distribution for ions in the plasma. Accordingly, we consider the trapped or vortex-like ion distribution fi = fif + fit , where 2 1 1 fif = √ e− 2 (vi +2φ) 2π 2 1 1 fit = √ e− 2 σit (vi +2φ) 2π

f or |vi | >

−2φ

(8.176)

f or |vi | ≤

−2φ.

(8.177)

The ion distribution function is continuous in velocity space and satisfies the regularity requirements for an admissible BGK solution. Here, the ion velocity vi in Equations (8.176) and (8.177) is normalized by the ion thermal speed vT i ; σit = TTiti , where Ti is the free ion temperature and Tit is the trapped ion temperature. Integrating the ion distributions over velocity space, we get ion number density as

1 ni = I(−φ) + √ e−σit φ erf ( −σit φ), σit

1 ni = I(−φ) +

WD ( σit φ) π|σit |

f or σit > 0

(8.178)

f or σit < 0

(8.179)

Waves and Wave Interactions in Plasmas

354

where √ I(x) = [1 − erf ( x)]ex ,  x 2 2 erf (x) = √ e−y dy, π 0  x 2 2 WD (x) = e−x ey dy. 0

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We consider a three-component dusty plasma system consisting of massive, micron-sized, negatively charged, inertial dust grains, Boltzmann distributed electron, and vortex-like ion distribution. The basic equations are ∂nd ∂(nd ud ) + = 0, ∂t ∂x ∂ud ∂φ ∂ud + ud = , ∂t ∂x ∂x

(8.180) (8.181)

∂2φ = nd − ni + ne (8.182) ∂x2 √ √ where ni = e−φ erfc −φ + σ1it e−σit φ erf −σit φ and ne = μ0 eαφ . nj (j = i, e, and d denote ion, electron, and dust particles, respectively), ud , and φ are the number density, the dust fluid velocity, and the electrostatic potential, respectively. Ti , Md , and e are the ion temperature, the dust mass, and the electron charge, respectively. Normalizations: The normalizations are taken as nj →

nj , nj0

ud →

ud , Cd

φ→

eφ , Ti

x→

x , λD

t → ωpd t.

λD and ωpd are the Debye length and plasma frequency, respec the2 dust  4πzd nd0 e2 β Ti 1 tively. λD = , μ1 = 1−β , μ0 = 1−β , 4πzd nd0 e2 , ωpd = md β=

ne0 ni0 ,

and α =

Ti Te .

Te is the electron temperature.

Sagdeev potential and collapse of solitons: By using the traveling wave transformation (8.4), Equations (8.180)–(8.182) are reduced to −M

dnd d(nd ud ) + = 0, dξ dξ

−M

dud dud dφ + ud = , dξ dξ dξ d2 φ = nd − ni + ne . dξ 2

(8.183) (8.184) (8.185)

Sagdeev’s Pseudopotential Approach

355

Integrating the above equation and using the boundary conditions φ, 0, nd → 1, ud → ud0 at |ξ| → ∞, we obtain 1 nd =  1+

dφ dξ

→

.

(8.186)

∂ 2 ud ∂ψ(ud ) = ∂ξ 2 ∂ud

(8.187)

2φ M2

Using (8.186) in Equation (8.185), we get

where ψi (ud ) + ψe (ud ) + ψd (ud ) , ψd (ud ) = M ud, (M − ud )2      u2 u2 u2 u2 1 d d √ eMud − 2 erf M ud − d ψi (ud ) = − eMud − 2 erfc M ud − d + 2 2 σit π  u2 1 + √ M ud − d (σit − 1)] + 1, 2 σit π

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ψ(ud ) = −

  u2 −Mud + 2d

α μ0  ψe (u) = 1−e α

.

Expanding erf and erfc functions and neglecting much higher-order terms O(φ4 ), Equation (6.18) can be written as  ψi (u) = −μ1 1 + vud −

u2d 2



−

u2d

4(1 − σit ) √ vud − 3 π 2



 32 +

3  u2d  5 −vu + 2 2 d 2 8(1 − u √ − vud − d − 2 6 15 π 4   u2d  7 −vu + 2 2 3 d 2 u 16(1 − σit ) √ − + . vud − d 105 π 2 24

−vud +

u2d 2

2

2

2 σit )

(8.188)

2

Hence, ψ(u) and ddξu2 can be obtained up to O(φ4 ) from Equations (8.187) and (8.188). We can also write ψ(ud ) =

(ud )2 . 2

(8.189)

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Waves and Wave Interactions in Plasmas

To find the region of existence of solitary waves, one has to study the nature of the functions ψ(ud ), and φ1 (ud ) is defined by

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ud =

∂ψd = φ1 (ud ). ∂ud

For solitary wave, φ1 (ud ) has two roots: one being at ud = 0 and other at some point ud = ud1 (≥ 0). Also, φ1 (ud ) should be positive on the interval (0, ud1 ) and negative on (ud1 , udmax ). Here, udmax is obtained from the nonzero root of ψd (ud ). To get the shape of the traveling solitary wave, one has to solve φ1 (ud ) = ud numerically with suitable boundary conditions. To get the shape of the solitary wave, we have solved numerically ud = φ1 (ud ) with ud = 0.324728, ud = 0, and Figure 8.12 depicts the soliton solution ud (ξ) plotted against ξ. ud1 = 0.324728 is the critical value for ud . For ud > ud1 , the soliton solution ceases to exist, and it is shown in Figure 8.13. In this figure, ud1 is taken as 0.324729. Hence, it is seen that a small change of the value of ud can destroy the periodic behavior of the wave as well as solitary wave. The divergent part of the wave for the negative

Fig. 8.12: The soliton solution u d (ξ) plotted against ξ for ud1 = 0.324728.

Fig. 8.13: The collapse of soliton u d (ξ) plotted against ξ for ud1 = 0.324728.

Sagdeev’s Pseudopotential Approach

357

value of ud cannot be shown because of the presence of the square root of ud in the differential equation.

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References [1] R. Z. Sagdeev in, In M. A. Leontovich (eds.), Reviews of Plasma Physics. Vol. 4 , New York: Consultants Bureau (1966), p. 23. [2] M. Remoissenet, Waves called Solitons. 3rd ed. Burlin: Springer verlag (1999). [3] A. Das, A. Bandyopadhyay, and K. P. Das, J. Plasma Phys. 28, 149 (2012). [4] F. Sayed and A. A. Mamun, Phys. Plasmas 14, 014501 (2007). [5] K. Roy, T. K. Maji, M. K. Ghorui, P. Chatterjee, and R. Roychowdhury, Astrophys. Space Sci. 352, 151 (2014). [6] U. N. Ghosh, P. Chatterjee, and M. Tribeche, Phys. Plasmas 79, 789 (2013). [7] G. Williams, I. Kourakis, F. Verheest, and M. A. Hellberg, Phys. Rev. E 88, 023103 (2013). [8] R. A. Cairns, A. A. Mamun, R. Bingham, R. Bostron, R. O. Dendy, C. M. C. Nair, and P. K. Shukla, Geophys. Res. Lett. 22, 2709 (1995). [9] M. Tribeche, R. Amour, and P. K. Shukla, Phys. Rev. E 85, 037401 (2012). [10] M. Tribeche, L. Djebani, and R. Amour, Phys. Plasmas 17, 042114 (2010). [11] S. G. Tagare, Phys. Plasmas 4, 3167 (1977). [12] Y. Nakamura and A. Sarma, Phys. Plasmas 8, 3921 (2001). [13] S. Maitra and R. Roychoudhury, Phys. Plasmas 12, 054502 (2005). [14] W. Malfliet, Am. J. Phys. 60, 650 (1992). [15] G. F. Chew, M. L. Goldberger, and F. E. Low, Proc. R. Soc. London, Ser. A 236, 112 (1956). [16] D. J. Wu, G. L. Huang, D. Y. Wang, and C. G. Falthammar, Phys. Plasmas 3, 2879 (1966). [17] J. E. Wahlund, P. Louran, T. Chust, H. De Feraudy, A. Roux, B. Holback, P. O. Dovner, and G. Holmgren, Geophys. Res. Lett. 21, 1831. Doi: 10.1029/94GL01289. [18] P. Louran, J. E. Wahlund, T. Chust, H. De Feraudy, A. Roux, B. Holback, P. O. Dovner, and G. Holmgren, Geophys. Res. Lett. 21, 1847. Doi: 10.1029/94GL00882. [19] J. S. Makela, A. Malkki, H. Koskinen, B. Holback, and L. Elliason, J. Geophys. Res. [Space Phys.] 103, 9391 (1998). Doi: 10.1029/98JA00212; K. Stasiewicz, G. Holmgren, and L. Zanetti, ibid 103, 4251 (1998). Doi: 10.1029/97JA02007.

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Chapter 9

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Conclusion and Future Scopes

In this book, we have studied the linear and nonlinear waves in different plasma environments (classical plasma, dusty plasma, and quantum plasma). Moreover, the interaction of waves in various plasma models is also investigated. We have considered the fluid approach for this study. To study linear waves, the fluid equations are linearized in the neighborhood of an equilibrium point, and the first-order perturbed quantities are considered wavy (proportional to ekx−iωt , where k is the wavenumber and w is the natural frequency). The relationship between the frequency and the wave number is obtained and is called the dispersion relation. This relation gives different wave modes in plasma. Several wave modes and their features are investigated in various unmagnetized and magnetized plasmas. Nonlinear evolution equations (NLEEs) are becoming significant in modern research because of their wide range of applications. Today, the research of NLEEs is becoming very popular. It is applied in various fields such as mathematical physics, nonlinear mechanics, particle physics, plasma physics, nonlinear optics, marine science, atmospheric science, automation, and others. The trend originates from the fact that NLEEs may explain a wide range of natural phenomena that linear equations cannot do. Wave phenomena are significant in understanding natural phenomena, as it maintains a good relationship between the theory and experiment or observation. Naturally, as almost all physical phenomena are nonlinear (do not satisfy the principle of superposition), the derivation of the nonlinear evolution equation for such nonlinear phenomena is a must. After explaining the basic properties of plasma theory, we have focused on the basic properties of nonlinear waves. We have explained these in the first two chapters. Chapter 4 discusses the derivation of nonlinear waves in a plasma medium by using the

359

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Waves and Wave Interactions in Plasmas

famous reductive perturbation technique (RPT). The reductive perturbation technique is a special type of multiscale perturbation technique where two different timescales are incorporated into the coordinate frame by considering two different stretched coordinates. The perturbation parameter  considered here depends upon ω and k, where ω is the frequency and k is the wave vector. This technique gives different information for different values of . The lowest order of  gives the dispersion relation and phase speed. The next orders provide the desired evolution equation and evolution equation for dressed soliton, respectively. Chapter 3 discusses different techniques to obtain the traveling wave solutions of different nonlinear evolution equations like solitary waves, shock waves, and change of shape of soliton to shocks and vice versa. In the presence of damping or forcing terms in the evolution equation, we get the solution from the conservation principles of KdV or KdV-type equations as and when required. The plasma devices used in the plasma laboratory are spherical and cylindrical. So, the evolution equations required to model the situation should contain nonplaner terms. In Chapter 6, we derive evolution equations in nonplaner geometry. The equation contains a nonsingular term at t = 0. The solution of such an equation in the neighborhood of the singular point is obtained by numerical simulation. Sometimes, the results observed in the laboratory do not match the theoretical results obtained from KdV-type equations. In those cases, a higher-order correction of KdV or similar equations is necessary. Considering the next higher-order terms in RPT, one can easily obtain the equation whose solution improves the result obtained from solitons. Such improved solitons are called dressed solitons. In Chapter 5, we consider the properties of dressed solitons in different plasma environments also. Some nonperturbed methods are applied to understand the existence of solitons and to observe other important features like the dependence of parameters on the amplitude and width of a soliton. Sagdeev’s pseudopotential approach is one such method by which one can understand the effect of plasma parameters’ region where the solitons or shocks exist. It also explains how the plasma parameters affect the amplitude and width of the solitons. An improved technique has been developed to understand the speed and shape of solitons. It is also shown that the solitons are very sensitive to initial conditions, and the shape of the soliton is destroyed for a small change of initial speed.

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Conclusion and Future Scopes

361

Interactions of solitons are the main issue for soliton dynamics. Starting from the famous work of Zabusky and Kruskal, several theoretical and experimental investigations have been conducted to understand the interaction of solitary waves. Chapter 7 investigates all the interactions of solitons like head-on, overtaking, and oblique collision. Theoretically, the solitons should regain their shape after an infinite time. But it has been observed that the solitons regain their shape after a finite time. Moreover, in that finite time, the collision of multisoliton causes weak turbulence called soliton turbulence in the medium. Again, the head-on collision causes relatively strong turbulence than overtaking collision. The nonlinear medium plasmas show several nonlinear waves other than solitons. Those are cnoidal waves, envelop solitons, lump, breather, dromions, etc. Interactions of those waves may lead to new phenomena which may be discussed in the future and may be included in the next edition. Possible development may be done in near future: (1) Interaction between soliton and lump, lump–lump, lump–breather, and soliton–breather in plasmas can be investigated. (2) Effect of damping and forcing on soliton turbulence is another important area that may be given emphasis. (3) Chaos and hyperchaos in the presence of externally applied force may be investigated. (4) Effect of noise may be incorporated in place of externally applied periodic force considered in all problems.

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Index

Dust ion acoustic waves, 77 Dusty plasma, 32

Abel equation, 110 Alfven wave, 66 Average random kinetic energy, 22 Average velocity, 22

Electromagnetic wave, 52 Electron plasma wave, 72 Electrostatic ion cyclotron waves, 55 Envelope soliton, 224 Equation of energy, 32 Equation of motion, 16, 30 Extended PLK method, 267 Extraordinary wave, 61

Boltzmann constant, 2 Boltzmann equation, 28 Burgers’ equation, 95, 102, 166 Cnoidal wave, 90 Collision and coupling limit, 9 Conservation laws, 112 Continuity equation, 16, 29 Cyclotron frequency, 13

Fluid equation, 34 Forced Schamel KdV equation, 159 Further MKP equation, 183

Damped forced KdV equation, 153 Damped forced MKdV equation, 155 Damped KdVB equation, 175 Damped KP equation, 189 Damped ZK equation, 197 de Broglie wavelength, 37 Debye length, 4 Debye shielding, 4, 33 Debye sphere, 5 Density, 22 Dispersion relation, 46, 139 Distribution function, 21 Double layers, 319 Dressed soliton, 201 Dust acoustic waves, 76

Gardner’s equation, 148, 150 Gardner’s generalization, 115 Group velocity, 47 Gyrofrequency, 12 Gyroradius, 12 Head-on collision, 265 Higher-order approximation, 202 Hirota’s method, 121 Hydromagnetic wave, 66 Hyperbolic tangent method, 98 Ion acoustic wave, 73 363

364

Waves and Wave Interactions in Plasmas

KdV equation, 88, 100, 137 KdV Burgers’ equation, 103, 173 KP equation, 96, 178 KP Burgers’ equation, 109, 186

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Linear wave, 50 Longitudinal wave, 53 Lower hybrid frequency, 56 Mach number, 318 Macroscopic variables, 22 Magnetized plasma, 11 Magnetosonic wave, 69 Maxwellian distribution function, 22 Maxwell’s equations, 18 Miura transformation, 114 Modified Burgers’ equation, 170 Modified Gardner’s equation, 150 Modified KdV equation, 93, 101, 144 Modified KP equation, 97, 181 Multisoliton, 121 Neutrality, 32 Non-Maxwellian distribution function, 23 Nonthermal distribution, 23 Nonlinear Schrodinger equation, 225 Nonlinear wave, 78, 87 Nonplanar geometry, 240 Nonperturbative approach, 313 Oblique collision, 287 Ordinary wave, 60 Overtaking collision, 294 Parallel wave, 53 Particular solution, 206 Perpendicular wave, 53 Perturbation technique, 131

Phase velocity, 46 Plasma, 1 Plasma frequency, 8, 33 Plasma oscillation, 51 Plasma temperature, 2 PLK method, 266 Pressure equations, 17 Pressure tensor, 22 q-nonextensive distribution, 25 Quantum plasma, 36 Quasineutrality, 7 Renormalization, 203 Response time, 7 RPT, 135 Saha equation, 2 Sagdeev’s pseudopotential approach, 313 Schamel-type KdV equation, 94 Small amplitude approximation, 320 Solitary wave, 82 Solitary kinetic Alfven wave, 346 Soliton, 82 Soliton turbulence, 300 Standing wave, 46 Superthermal distribution, 24 Tanh-coth method, 105 Transverse wave, 53 Upper hybrid frequency, 53 Vlasov equation, 28 ZK equation, 192 ZK Burgers’ equation, 195