Handbook of Exact Solutions to the Nonlinear Schrödinger Equations 0750324287, 9780750324281

Table of contents :
PRELIMS.pdf
Preface
Acknowledgments
Author Biographies
Usama Al Khawaja
Laila Al Sakkaf
Notation
CH001.pdf
Chapter 1 Introduction
References
CH002.pdf
Chapter 2 Fundamental Nonlinear Schrödinger Equation
A Glance at Chapter 2
A Statistical View of Chapter 2
2.1 NLSE with Cubic Nonlinearity
2.1.1 Real Dispersion and Nonlinearity Coefficients
2.2 Summary of Subsection 2.1.1
2.2.1 Complex Dispersion and Nonlinearity Coefficients
2.3 Summary of Subsection 2.2.1
References
CH003.pdf
Chapter 3 Nonlinear Schrödinger Equation with Power Law and Dual Power Law Nonlinearities
A Glance at Chapter 3
A Statistical View of Chapter 3
3.1 NLSE with Power Law Nonlinearity
3.1.1 Reduction to the Fundamental NLSE
3.2 Summary of Section 3.1
3.3 NLSE with Dual Power Law Nonlinearity
3.4 Summary of Section 3.3
References
CH004.pdf
Chapter 4 Nonlinear Schrödinger Equation with Higher Order Terms
A Glance at Chapter 4
A Statistical View of Chapter 4
4.1 NLSE with Third Order Dispersion, Self-Steepening, and Self-Frequency Shift
4.2 Summary of Section 4.1
4.3 Special Cases of Equation (4.1)
4.3.1 Case I: Hirota Equation (HE)
4.3.2 Case II: Sasa–Satsuma Equation (SSE)
4.4 NLSE with First and Third Order Dispersions, Self-Steepening, Self-Frequency Shift, and Potential
4.5 Summary of Section 4.4
4.6 NLSE with Fourth Order Dispersion
4.7 Summary of Section 4.6
4.8 NLSE with Fourth Order Dispersion and Power Law Nonlinearity
4.9 Summary of Section 4.8
4.10 NLSE with Third and Fourth Order Dispersions and Cubic and Quintic Nonlinearities
4.11 Summary of Section 4.10
4.12 NLSE with Third and Fourth Order Dispersions, Self-Steepening, Self-Frequency Shift, and Cubic and Quintic Nonlinearities
4.13 Summary of Section 4.12
4.14 NLSE with ∣ψ∣2-Dependent Dispersion
4.15 Infinite Hierarchy of Integrable NLSEs with Higher Order Terms
4.15.1 Constant Coefficients
4.15.2 Function Coefficients
4.16 Summary of Section 4.15
References
CH005.pdf
Chapter 5 Scaling Transformations
A Glance at Chapter 5
A Statistical View of Chapter 5
5.1 Fundamental NLSE to Fundamental NLSE with Different Constant Coefficients
5.2 Defocusing (Focusing) NLSE to Focusing (Defocusing) NLSE
5.3 Galilean Transformation (Movable Solutions)
5.4 Function Coefficients
5.4.1 Constant Dispersion and Complex Potential
5.4.2 Constant Dispersion and Real Quadratic Potential
5.4.3 Constant Dispersion and Real Linear Potential
5.4.4 Constant Nonlinearity and Complex Potential
5.4.5 Constant Nonlinearity and Real Quadratic Potential
5.4.6 Constant Nonlinearity and Real Linear Potential
5.5 Solution-Dependent Transformation
5.5.1 Special Case I: Stationary Solution, Constant Dispersion and Nonlinearity Coefficients
5.5.2 Special Case II: PT-Symmetric Potential
5.5.3 Special Case III: Stationary Solution, Constant Dispersion and Nonlinearity Coefficients, and Real Potential
5.6 Summary of Sections 5.1–5.5
5.7 Other Equations: NLSE with Periodic Potentials
5.7.1 General Case: sn2(x,m) Potential
5.7.2 Specific Case: sin2(x) Potential
5.8 Summary of Section 5.7
Reference
CH006.pdf
Chapter 6 Nonlinear Schrödinger Equation in (N + 1)-Dimensions
A Glance at Chapter 6
A Statistical View of Chapter 6
6.1 (N + 1)-Dimensional NLSE with Cubic Nonlinearity
6.2 (N + 1)-Dimensional NLSE with Power Law Nonlinearity
6.3 (N + 1)-Dimensional NLSE with Dual Power Law Nonlinearity
6.4 Galilean Transformation in (N + 1)-Dimensions (Movable Solutions)
6.5 NLSE in (2 + 1)-Dimensions with Φx1x2 Term
6.6 Summary of Sections 6.1–6.5
6.7 (N + 1)-Dimensional Isotropic NLSE with Cubic Nonlinearity in Polar Coordinate System
6.7.1 Angular Dependence
6.7.2 Constant Dispersion and Real Potential
6.8 Summary of Section 6.7
6.9 Power Series Solutions to (2 + 1)-Dimensional NLSE with Cubic Nonlinearity in a Polar Coordinate System
6.9.1 Family of Infinite Number of Localized Solutions
References
CH007.pdf
Chapter 7 Coupled Nonlinear Schrödinger Equations
A Glance at Chapter 7
A Statistical View of Chapter 7
7.1 Fundamental Coupled NLSE Manakov System
7.2 Summary of Section 7.1
7.3 Symmetry Reductions
7.3.1 Symmetry Reduction I
7.3.2 Symmetry Reduction II
7.3.3 Symmetry Reduction III
7.3.4 Symmetry Reduction IV
7.3.5 Symmetry Reduction V
7.4 Scaling Transformations
7.4.1 Linear and Nonlinear Coupling
7.4.2 Complex Coupling
7.4.3 Function Coefficients
7.5 Summary of Sections 7.3–7.4
7.6 (N + 1)-Dimensional Coupled NLSE
7.6.1 Reduction to 1D Manakov System
7.7 Symmetry Reductions of (N + 1)-Dimensional CNLSE to Scalar NLSE
7.7.1 Symmetry Reduction I
7.7.2 Symmetry Reduction II
7.7.3 Symmetry Reduction III
7.8 (N + 1)-Dimensional Scaling Transformations
7.8.1 Linear and Nonlinear Coupling
7.8.2 Complex Coupling
7.9 Summary of Sections 7.7–7.8
References
CH008.pdf
Chapter 8 Discrete Nonlinear Schrödinger Equation
A Glance at Chapter 8
A Statistical View of Chapter 8
8.1 Discrete NLSE with Saturable Nonlinearity
8.1.1 Nonstaggered Solutions
8.1.2 Staggered Solutions
8.2 Summary of Section 8.1
8.3 Short-period Solutions with General, Kerr, and Saturable Nonlinearities
8.4 Ablowitz–Ladik Equation
8.5 Summary of Section 8.4
8.6 Cubic-quintic Discrete NLSE
8.7 Summary of Section 8.6
8.8 Generalized Discrete NLSE
8.9 Summary of Section 8.8
8.10 Coupled Salerno Equations
8.11 Summary of Section 8.10
8.12 Coupled Ablowitz–Ladik Equation
8.13 Summary of Section 8.12
8.14 Coupled Saturable Discrete NLSE
8.15 Summary of Section 8.14
References
CH009.pdf
Chapter 9 Nonlocal Nonlinear Schrödinger Equation
A Glance at Chapter 9
A Statistical View of Chapter 9
9.1 Nonlocal NLSE
9.2 Nonlocal Coupled NLSE
9.3 Symmetry Reductions to Scalar Nonlocal NLSE
9.3.1 Symmetry Reduction I
9.3.2 Symmetry Reduction II
9.3.3 Symmetry Reduction III
9.4 Scaling Transformations
9.4.1 Linear and Nonlinear Coupling
9.4.2 Complex Coupling
9.5 Nonlocal Discrete NLSE with Saturable Nonlinearity
9.5.1 Nonstaggered Solutions
9.5.2 Staggered Solutions
9.6 Nonlocal Ablowitz–Ladik Equation
9.7 Nonlocal Cubic-Quintic Discrete NLSE
9.8 Summary of Chapter 9
APP1.pdf
Chapter
A.1 Derivation of Some Solutions of Section 2.1
A.1.1 Schematic Representation
A.1.2 Detailed Derivations
A.2 Derivation of Some Solutions of Section 3.1
A.2.1 Schematic Representation
A.2.2 Detailed Derivations
A.3 Derivation of Some Solutions of Section 3.3
A.3.1 Schematic Representation
A.3.2 Detailed Derivations
APP2.pdf
Chapter
B.1 Darboux Transformation
Link between NLSE and LP
Seed Solution
Darboux Transformation
Symmetry Reduction
B.1.1 Bright Soliton Solution: Zero Seed
B.1.2 Generalized Breather Solution for Focusing and Defocusing Nonlinearity: CW Seed
APP3.pdf
Chapter
C.1 Function Coefficients
C.2 Solution-Dependent Transformation
C.3 Similarity Transformation for the NLSE in (N + 1)-Dimensions

Citation preview

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations Usama Al Khawaja and Laila Al Sakkaf Department of Physics, UAE University, Al-Ain, UAE

IOP Publishing, Bristol, UK

ª IOP Publishing Ltd 2020 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publisher, or as expressly permitted by law or under terms agreed with the appropriate rights organization. Multiple copying is permitted in accordance with the terms of licences issued by the Copyright Licensing Agency, the Copyright Clearance Centre and other reproduction rights organizations. Permission to make use of IOP Publishing content other than as set out above may be sought at [email protected]. Usama Al Khawaja and Laila Al Sakkaf have asserted their right to be identified as the authors of this work in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988. Multimedia to accompany this book can be downloaded from https://iopscience.iop.org/book/9780-7503-2428-1. ISBN ISBN ISBN

978-0-7503-2428-1 (ebook) 978-0-7503-2426-7 (print) 978-0-7503-2427-4 (mobi)

DOI 10.1088/978-0-7503-2428-1 Version: 20191101 IOP ebooks British Library Cataloguing-in-Publication Data: A catalogue record for this book is available from the British Library. Published by IOP Publishing, wholly owned by The Institute of Physics, London IOP Publishing, Temple Circus, Temple Way, Bristol, BS1 6HG, UK US Office: IOP Publishing, Inc., 190 North Independence Mall West, Suite 601, Philadelphia, PA 19106, USA

Contents Preface

x

Acknowledgments

xii

Author Biographies

xiii

Notation

xiv

1

Introduction

1-1

References

1-6

2

Fundamental Nonlinear Schrödinger Equation

2-1

2.1

NLSE with Cubic Nonlinearity 2.1.1 Real Dispersion and Nonlinearity Coefficients Summary of Subsection 2.1.1 2.2.1 Complex Dispersion and Nonlinearity Coefficients Summary of Subsection 2.2.1 References

2.2 2.3

3

Nonlinear Schrödinger Equation with Power Law and Dual Power Law Nonlinearities

3.1

NLSE with Power Law Nonlinearity 3.1.1 Reduction to the Fundamental NLSE Summary of Section 3.1 NLSE with Dual Power Law Nonlinearity Summary of Section 3.3 References

3.2 3.3 3.4

2-1 2-2 2-33 2-40 2-43 2-45 3-1 3-1 3-2 3-6 3-8 3-14 3-17

4

Nonlinear Schrödinger Equation with Higher Order Terms

4-1

4.1

NLSE with Third Order Dispersion, Self-Steepening, and Self-Frequency Shift Summary of Section 4.1 Special Cases of Equation (4.1) 4.3.1 Case I: Hirota Equation (HE) 4.3.2 Case II: Sasa–Satsuma Equation (SSE) NLSE with First and Third Order Dispersions, Self-Steepening, Self-Frequency Shift, and Potential

4-3

4.2 4.3

4.4

v

4-9 4-13 4-13 4-13 4-13

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15

4.16

Summary of Section 4.4 NLSE with Fourth Order Dispersion Summary of Section 4.6 NLSE with Fourth Order Dispersion and Power Law Nonlinearity Summary of Section 4.8 NLSE with Third and Fourth Order Dispersions and Cubic and Quintic Nonlinearities Summary of Section 4.10 NLSE with Third and Fourth Order Dispersions, Self-Steepening, Self-Frequency Shift, and Cubic and Quintic Nonlinearities Summary of Section 4.12 NLSE with ∣ψ∣2-Dependent Dispersion Infinite Hierarchy of Integrable NLSEs with Higher Order Terms 4.15.1 Constant Coefficients 4.15.2 Function Coefficients Summary of Section 4.15 References

4-16 4-17 4-19 4-20 4-22 4-24 4-29 4-32 4-36 4-39 4-40 4-40 4-43 4-46 4-49

5

Scaling Transformations

5-1

5.1

Fundamental NLSE to Fundamental NLSE with Different Constant Coefficients Defocusing (Focusing) NLSE to Focusing (Defocusing) NLSE Galilean Transformation (Movable Solutions) Function Coefficients 5.4.1 Constant Dispersion and Complex Potential 5.4.2 Constant Dispersion and Real Quadratic Potential 5.4.3 Constant Dispersion and Real Linear Potential 5.4.4 Constant Nonlinearity and Complex Potential 5.4.5 Constant Nonlinearity and Real Quadratic Potential 5.4.6 Constant Nonlinearity and Real Linear Potential Solution-Dependent Transformation 5.5.1 Special Case I: Stationary Solution, Constant Dispersion and Nonlinearity Coefficients 5.5.2 Special Case II: PT-Symmetric Potential 5.5.3 Special Case III: Stationary Solution, Constant Dispersion and Nonlinearity Coefficients, and Real Potential Summary of Sections 5.1–5.5 Other Equations: NLSE with Periodic Potentials

5-4

5.2 5.3 5.4

5.5

5.6 5.7

vi

5-5 5-6 5-10 5-10 5-11 5-18 5-24 5-25 5-25 5-26 5-27 5-28 5-29 5-30 5-38

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

5.8

5.7.1 General Case: sn2(x, m) Potential 5.7.2 Specific Case: sin2(x) Potential Summary of Section 5.7 Reference

6

Nonlinear Schrödinger Equation in (N + 1)-Dimensions

6.1 6.2 6.3 6.4 6.5 6.6 6.7

(N + 1)-Dimensional NLSE with Cubic Nonlinearity (N + 1)-Dimensional NLSE with Power Law Nonlinearity (N + 1)-Dimensional NLSE with Dual Power Law Nonlinearity Galilean Transformation in (N + 1)-Dimensions (Movable Solutions) NLSE in (2 + 1)-Dimensions with Φx1x2 Term Summary of Sections 6.1–6.5 (N + 1)-Dimensional Isotropic NLSE with Cubic Nonlinearity in Polar Coordinate System 6.7.1 Angular Dependence 6.7.2 Constant Dispersion and Real Potential Summary of Section 6.7 Power Series Solutions to (2 + 1)-Dimensional NLSE with Cubic Nonlinearity in a Polar Coordinate System 6.9.1 Family of Infinite Number of Localized Solutions References

6.8 6.9

7

Coupled Nonlinear Schrödinger Equations

7.1 7.2 7.3

Fundamental Coupled NLSE Manakov System Summary of Section 7.1 Symmetry Reductions 7.3.1 Symmetry Reduction I From Manakov System to Fundamental NLSE 7.3.2 Symmetry Reduction II From Manakov System to Fundamental NLSE 7.3.3 Symmetry Reduction III From Vector NLSE to Fundamental NLSE 7.3.4 Symmetry Reduction IV From Three Coupled NLSEs to Manakov System 7.3.5 Symmetry Reduction V From Vector NLSE to Manakov System Scaling Transformations 7.4.1 Linear and Nonlinear Coupling 7.4.2 Complex Coupling

7.4

vii

5-38 5-39 5-40 5-40 6-1 6-4 6-11 6-12 6-16 6-22 6-24 6-33 6-34 6-35 6-38 6-41 6-42 6-42 7-1 7-4 7-13 7-17 7-17 7-17 7-18 7-19 7-22 7-22 7-22 7-25

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

7.5 7.6

7.7

7.8

7.9

7.4.3 Function Coefficients Summary of Sections 7.3–7.4 (N + 1)-Dimensional Coupled NLSE (N + 1)-Dimensional Manakov System 7.6.1 Reduction to 1D Manakov System Symmetry Reductions of (N + 1)-Dimensional CNLSE to Scalar NLSE 7.7.1 Symmetry Reduction I From (N + 1)-Dimensional Manakov System to (N + 1)-Dimensional Fundamental NLSE 7.7.2 Symmetry Reduction II From (N + 1)-Dimensional Manakov System to (N + 1)-Dimensional Fundamental NLSE 7.7.3 Symmetry Reduction III From (N + 1)-Dimensional Vector NLSE to (N + 1)-Dimensional Fundamental NLSE (N + 1)-Dimensional Scaling Transformations 7.8.1 Linear and Nonlinear Coupling 7.8.2 Complex Coupling Summary of Sections 7.7–7.8 References

8

Discrete Nonlinear Schrödinger Equation

8.1

Discrete NLSE with Saturable Nonlinearity 8.1.1 Nonstaggered Solutions 8.1.2 Staggered Solutions Summary of Section 8.1 Short-period Solutions with General, Kerr, and Saturable Nonlinearities Ablowitz–Ladik Equation Summary of Section 8.4 Cubic-quintic Discrete NLSE Summary of Section 8.6 Generalized Discrete NLSE Summary of Section 8.8 Coupled Salerno Equations Summary of Section 8.10 Coupled Ablowitz–Ladik Equation Summary of Section 8.12 Coupled Saturable Discrete NLSE Summary of Section 8.14 References

8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13 8.14 8.15

viii

7-26 7-30 7-32 7-32 7-34 7-34 7-35 7-36 7-37 7-37 7-39 7-40 7-42 8-1 8-2 8-2 8-9 8-16 8-22 8-22 8-30 8-33 8-37 8-39 8-47 8-48 8-55 8-58 8-67 8-71 8-73 8-74

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

9

Nonlocal Nonlinear Schrödinger Equation

9-1

9.1 9.2 9.3

Nonlocal NLSE Nonlocal Coupled NLSE Symmetry Reductions to Scalar Nonlocal NLSE 9.3.1 Symmetry Reduction I From Nonlocal Manakov System to Scalar Nonlocal NLSE 9.3.2 Symmetry Reduction II From Nonlocal Manakov System to Scalar Nonlocal NLSE 9.3.3 Symmetry Reduction III From Nonlocal Vector NLSE to Scalar Nonlocal NLSE Scaling Transformations 9.4.1 Linear and Nonlinear Coupling 9.4.2 Complex Coupling Nonlocal Discrete NLSE with Saturable Nonlinearity 9.5.1 Nonstaggered Solutions 9.5.2 Staggered Solutions Nonlocal Ablowitz–Ladik Equation Nonlocal Cubic-Quintic Discrete NLSE Summary of Chapter 9

9-3 9-4 9-7 9-7

9.4

9.5

9.6 9.7 9.8

9-8 9-9 9-10 9-10 9-12 9-13 9-13 9-15 9-15 9-17 9-21

Appendices A Derivation of Some Solutions of Chapters 2 and 3

A-1

B Darboux Transformation Single Soliton and Breather Solutions

B-1

C Derivation of the Similarity Transformations in Chapter 5

C-1

ix

Preface We have been involved for the past two decades with the nonlinear Schrödinger equation (NLSE) and its many variations including NLSE with higher-order terms, two- and three-dimensional NLSE, discrete NLSE, nonlocal NLSE, and coupled NLSEs. We noticed that, throughout the long history of NLSE, a large number of exact analytical solutions have been found and the number is still increasing as new solutions are being sought and discovered. As the basis for theoretical models of various research fields such as Bose–Einstein condensates of ultracold gases, nonlinear optics, and deep water waves, this equation is a subject of interest by the scientific communities in all three areas. Its known solutions are scattered in the literature of the different fields. For a beginner as well as for an expert researcher, it is difficult to keep track of the large number of known solutions. It is important for a researcher or a reviewer to know if a certain solution is a new solution, belongs to an existing class of solutions, or can be trivially obtained from another solution by a transformation. This book is the result of our effort towards serving the research communities involved with the NLSE in the different fields by collecting all known solutions in one document. In addition, the book organizes the solutions by classifying and grouping them based on the aspects and symmetries they have. Although most of the solutions presented in this book have been derived elsewhere using various methods, we attempt here to present a systematic derivation of many solutions and even have derived some new ones. We have also presented symmetries and reductions that connect different solutions through transformations and enable classifying new solutions into known classes. For the user to verify that the presented solutions do satisfy the NLSE, we provide Mathematica note books (available online at https://iopscience.iop.org/ book/978-0-7503-2428-1) containing all solutions in a one-to-one correspondence with the solutions in the text. The reader can run the Mathematica cell and see for themselves that it indeed satisfies the NLSE. This is also an efficient method for detecting and avoiding possible typo mistakes in the text. A large number of figures and animations are included to help visualize solutions and their dynamics. We have applied the following rules while collecting the solutions from the literature: (1) It has to be NLSE-related. As a result of this restriction, some interesting equations have been excluded such as the nonlinear Dirac equation, which we may consider in a future edition. (2) It has to satisfy the NLSE. We attempted to fix typo errors whenever it was obvious, but normally we did not invest much time in discovering what is wrong in the solution that does not satisfy the NLSE. (3) It has to be analytical. This excludes all numerical solutions. Some NLSEs do not admit analytical solutions but do have important stable numerical solutions such as in the case of two-dimensional and discrete NLSEs. Nonetheless, and in order to set a well-defined scope, we restrict our book to analytical solutions. We made our best effort to cite the reference where the solution appeared first. Quite often, however, solutions were either rederived in a later reference and put in a

x

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

different form, or were derived under different conditions. We cite the reference that we copied the solution from although a different variation of the solution may have appeared earlier. We do not claim any credit for the solutions collected, and we apologize for any citation mistake and for missing any solution. It should be mentioned that the solutions presented in this book have been the subject of study by a very large number of references. However, and as mentioned above, we cite only references that we copied the solutions from giving priority to any reference containing a large number of solutions. Due to the fact that we employed scaling transformations and symmetry reductions, many solutions published in the literature were not presented; the reader can reproduce those solutions using the transformations and symmetry reductions presented here. We give numerous examples on such cases. We should be grateful, however, if readers would draw our attention to any missing solutions. We also welcome criticism and comments hoping that they will lead to an enhanced second edition.

xi

Acknowledgments We are grateful for the substantial support of UAE University, including internal grant funding that supported this work. Fruitful discussions with our colleagues and collaborators, Lincoln Carr, Abdulaziz Alhaidari, Bakhtiyor Baizakov, Hocine Bahlouli, Saeed Al-Marzoug, Yuri Kivshar, Nail Akhmedeiv, Andrey Sokhorokov, Fathulla Abdullaev and Majid Taki are acknowledged. The process of collecting and deriving solutions and writing this book took about two years. We are grateful to our families for their support and patience during this time.

xii

Author Biographies Usama Al Khawaja Usama Al Khawaja obtained his Bachelorʼs degree in Physics from the University of Jordan in 1992 and Masterʼs degree in Physics with thesis research on two-dimensional neutral Fermi systems from the University of Jordan in 1996. He earned his PhD degree in theoretical Physics with dissertation research on Bose–Einstein condensation from the University of Copenhagen in 1999. Afterwards, he spent three years of postdoctoral research at Utrecht University in the Netherlands before joining the United Arab Emirates University in 2002 as an assistant professor. He is currently a full professor and Chairman of the Physics department at the United Arab Emirates University. His main areas of research are Bose–Einstein condensation, nonlinear and quantum optics, integrability, and exact solutions. His main achievements in integrability and exact solutions include developing a systematic search method of finding Lax pairs of a given nonlinear partial differential equation. He also developed a convergent power series method for solving nonlinear differential equations. He has authored more than 70 papers and obtained one patent on applying discrete solitons in all-optical operations.

Laila Al Sakkaf Laila Al Sakkaf obtained her Bachelorʼs degree in Physics from the United Arab Emirates University in 2015. Then she obtained her Masterʼs degree in Physics from the United Arab Emirates University in 2018 with thesis research on the iterative power series method for solving nonlinear differential equations. She is currently a research assistant and a PhD student at the Physics department of the United Arab Emirates University. Her current research focus is on integrability and exact solutions of differential equations modeling nonlinear physical phenomena.

xiii

Notation NLSE DNLSE CNLSE HONLSE 1D 2D 3D ND (N + 1)-D CW DW SW GN SN KN IPS IST PT AL LP DT HE SSE sn, cn, dn, nd, cd, sd, cs, ds, dc, ns

Nonlinear Schrödinger equation Discrete nonlinear Schrödinger equation Coupled nonlinear Schrödinger equation Nonlinear Schrödinger equation with higher-order terms One-dimensional/dimension Two-dimensional/dimensions Three-dimensional/dimensions N-dimensional/dimensions N dimensions in space, 1 refers to time Continuous wave Decaying wave Solitary wave General nonlinearity Saturable nonlinearity Kerr nonlinearity Iterative power series Inverse scattering transform Parity-Time Ablowitz–Ladik Lax-Pair Darboux transformation Hirota equation Sasa–Satsuma equation Jacobi elliptic functions

xiv

IOP Publishing

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations Usama Al Khawaja and Laila Al Sakkaf

Chapter 1 Introduction

The nonlinear Schrödinger equation is known in the literature, most commonly, with the following dimensionless form

iψt +

1 ψ + σ∣ψ ∣2 ψ = 0, 2 xx

(1.1)

where σ is a real constant, ψ = ψ (x , t ) is a complex function, and the subscripts are partial derivatives in terms of its two independent variables, x and t. It is the basis for theoretical models describing three major fields, namely: Bose–Einstein condensates of ultracold gases [1], nonlinear optics in fibers and waveguide arrays [2], and deep water waves [3]. In Bose–Einstein condensates, which is a quantum system, the nonlinear Schrödinger equation is the classical field limit of the analogous quantized field equation. The function ψ (x , t ) is the wave function of the macroscopic manyparticle system. To realize Bose–Einstein condensation, a confining (trapping) magnetic and optical potential is needed. This is accounted for by adding a potential term, V (x )ψ (x , t ), to the NLSE that then becomes the Gross–Pitaevskii equation [4, 5]. The nonlinear term corresponds to the interatomic interaction known as the Hartree–Fock energy with σ being proportional to the s-wave scattering length. The sign of σ can be both positive and negative, corresponding to attractive or repulsive interatomic interactions, respectively. The dispersion term corresponds to the kinetic energy pressure [1]. In nonlinear optics, the NLSE describes the propagation of pulses in nonlinear media such as optical fibers, photonic crystals, or waveguide arrays. It can be derived from Maxwell’s equations with ψ (x , t ) corresponding to the envelope of modulated electrical (or magnetic) field strength of the propagating pulse [6, 7]. The nonlinear term corresponds to the modulation of the refractive index of the medium as a response to the propagating light pulse, which is known in the nonlinear optics

doi:10.1088/978-0-7503-2428-1ch1

1-1

ª IOP Publishing Ltd 2020

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

community as the Kerr nonlinearity. The constant σ represents, in this case, the strength of the Kerr nonlinearity, which can also be positive or negative, leading to the focusing or defocusing NLSE, respectively. Here, the term ψxx corresponds to the dispersion of the pulse [2]. The NLSE describes also surface water waves where ψ (x , t ) corresponds to the intensity and phase of the waves. This description is restricted to deep water waves with a wavelength much smaller than the water depth. Shallow water waves are not described by the NLSE. The nonlinearity originates from the Bernoulli equation, its strength depends on the water depth, and it is always negative for deep water waves [3]. The above three examples suggest that the NLSE is a universal equation that describes the propagation of wave modulations in media with dispersion and nonlinearity. The NLSE is integrable and admits, in principle, an infinite number of independent solutions [8]. It was first solved by Zakharov and Shabat using the Inverse Scattering Transform (IST), which relies on associating the NLSE to a linear system of differential equations [9]. The system has been known since then as the Zakharov–Shabat system and the method was adopted to find other solutions of the NLSE and its variations. In general, the linear system is given in terms of a pair of matrices, known as the Lax pair, acting on an auxiliary field. The existence of a Lax pair establishes the integrability of a differential equation, at least within the Lax pair sense [10]. The IST is a powerful method for finding solutions of nonlinear differential equations [11]. It is distinguished among other methods by generating classes of an infinite hierarchy of solutions. It can also be used to exactly solve the nonlinear initial value problem for a given nonlinear differential equation. Many other methods of solving nonlinear differential equations have been applied to the NLSE. For the systematic derivations of solutions we present in this book, we use mainly the IST and separation of variables methods. There are many variations of the NLSE including NLSE with higher-order terms, NLSE in higher dimensions, NLSE with function coefficients and potential terms, coupled NLSEs, discrete NLSE, and nonlocal NLSE. It should be noted that we often refer to the NLSE and its variations simply by NLSE. Many of these variations turn out to be integrable and many others turn out to be related to the fundamental NLSE via some scaling transformations. All of these variations will be considered in this book. The book begins in chapter 2 with the fundamental NLSE. In this chapter we prefer to present solutions of NLSE with arbitrary constant coefficients a1 and a2 for dispersion and nonlinearity, respectively (see section 2.1). One may argue that this is not the ‘fundamental’ NLSE since with a simple scaling transformation, as shown in chapter 5, it transforms into another NLSE with no coefficients (iψt + ψxx + ∣ψ ∣2 ψ = 0), which may be more accurately denoted as the fundamental NLSE. This is indeed the case when a2 is real, but the scaling transformation does not work when a2 is complex. Therefore, by keeping the coefficients a1 and a2 explicitly in the NLSE, we will be able to consider solutions when the coefficients are complex. This chapter contains the largest number of solutions collected. The

1-2

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

solutions of this chapter can be used as a seed for transformations generating many solutions of other NLSEs in the subsequent chapters. The solutions can be categorized as: (i) stationary solutions of the form ψ (x , t ) = u(x )e iϕt , where u(x ) is a real function that can be localized or oscillatory and ϕ is a real constant, (ii) a class of breathers family, (iii) class of N-bright solitons, (iv) rational solutions that are fundamentally different from the breathers class. In chapter 3, we consider the NLSE with power law and dual power law nonlinearities. Here, the cubic nonlinearity of the fundamental NLSE is replaced first by a nonlinearity with general power, n, that is not restricted to integers. Then we consider an NLSE with two nonlinear terms; one with power n and another with power m, where again n and m are arbitrary real constants that do not have to be integers. In the first case we show, at the beginning of the chapter, that with a scaling transformation applied to stationary solutions, the NLSE with power law nonlinearity reduces to the fundamental NLSE, and thus all stationary solutions of chapter 2 lead to stationary solutions to the NLSE with power law nonlinearity. Many examples have been worked out explicitly. We could not find a similar transformation for the dual power law nonlinearity, but we have found 14 solutions for this case. In chapter 4, we consider the NLSE with higher-order terms. These include, following the nonlinear optics terminology: third-order dispersion, fourth-order dispersion, self-steepening, self-frequency shift, and power law nonlinearity. At the end, we consider an infinite hierarchy of integrable NLSEs. The first member of the hierarchy being the fundamental NLSE, the higher-order members turn out to comprise most of the known higher-order variations of the NLSE. Solutions to all member equations of the infinite hierarchy have been presented. In chapter 5, we present scaling transformations that reduce many variations of the NLSE to the fundamental one. These include transforming the NLSE with arbitrary constant coefficients to the one with no coefficients, transforming the NLSE with focusing (defocusing) nonlinearity to the NLSE with defocusing (focusing) nonlinearity, Galilean transformation to obtain movable solutions from static ones, transforming the NLSE with function coefficients and complex potential to the fundamental NLSE, and finally introducing a solution-dependent transformation where a seed solution is used to construct the transformation operator. The latter allowed for more possibilities including transforming an NLSE with constant coefficients and PT-symmetric potential to the fundamental NLSE. In chapter 6, we consider NLSE in higher dimensions. We start with a scaling transformation showing that an (N + 1)-dimensional NLSE, with N denoting the spatial dimensions and 1 denoting the temporal dimension, can be reduced to the onedimensional NLSE in terms of a reduced spatial variable. A Galilean transformation is then shown to apply where movable solutions of the (N + 1)-dimensional NLSE are obtained from the static solutions of the one-dimensional NLSE. An NLSE with mixed derivatives is also considered. Then, solutions of the (N + 1)-dimensional NLSE with power law and dual power law nonlinearities are presented. A scaling transformation of the (N + 1)-dimensional NLSE in polar coordinates is also worked out allowing one to consider solutions with cylindrical and spherical symmetries in

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Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

two- and three-dimensional geometries, respectively. At the end, we present our iterative power series method of obtaining convergent power series representations of the solutions. The method is applied here for nonintegrable cases such as some twoand three-dimensional NLSEs. In chapter 7, we consider the coupled NLSE. At first, we consider the fundamental coupled NLSE, known as the Manakov system. Being an integrable system, many solutions have been found and presented. Then, we show three simple symmetry reductions that transform the coupled NLSE to the scalar NLSE and symmetry reduction that transforms the vector NLSE (N-coupled NLSEs) to the Manakov system. Some interesting examples have been shown explicitly. We consider also a coupled system with additional linear coupling terms. A scaling transformation is performed to reduce this system to the fundamental Manakov system. Here, we found that, as a special case, one may obtain the solutions of a Manakov system from those of another Manakov system that differs in the values of the constant coefficients. Furthermore, one may obtain a solution of the Manakov system from another solution of the same system, which invokes the superposition principle known for linear differential equations. It is interesting to find such a principle applying for nonlinear differential equations. We have worked out explicitly a nontrivial example in this case. We have also considered a coupled system with additional complex coupling terms, which again with a scaling transformation was reduced to the fundamental Manakov system. Then, we consider the (N + 1)-dimensional coupled NLSEs. First, we show that, with a proper scaling transformation, this system reduces to the one-dimensional Manakov system. Then, we consider scaling transformations that reduce this system to the (N + 1)-dimensional scalar NLSE. Scaling transformations were then found for linear and nonlinear coupling, as mentioned above, but here generalized for (N + 1) dimensions. In chapter 8, we consider the discrete NLSE. The chapter starts with the discrete NLSE with saturable nonlinearity, which is integrable. Here, the solutions are classified into staggered and nonstaggered solutions. We show the transformation that links the two kinds of solutions. Then, we consider other types of nonlinearity including a general form of nonlinearity. Then, we considered the Ablowitz–Ladik equation, which is also integrable. Then, we consider an integrable discrete NLSE with cubic and quintic nonlinearities. A generalized discrete NLSE was then considered with many solutions satisfying the equation under some integrability conditions. Then, we consider coupled discrete NLSEs including the coupled Salerno equations and the coupled Ablowitz–Ladik equations. In chapter 9, we consider the nonlocal NLSE. We start with transformations that reduce the nonlocal NLSE to the fundamental NLSE. This is possible only for even or odd solutions in the variable x. Then, we consider coupled nonlocal NLSEs and reduce them to the local Manakov system. Then, we consider nonlocal coupled NLSEs with linear, nonlinear, and complex coupling. With scaling transformation, they are reduced to their local counterparts. Then, we consider the nonlocal discrete NLSE, nonlocal discrete coupled NLSEs, nonlocal Ablowitz–Ladik coupled system, and nonlocal NLSE with cubic and quintic nonlinearities. 1-4

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Chapters 2 and 3 are supplemented by appendix A and appendix B, where we lay out detailed derivations of most of the solutions presented in these chapters. In appendix A, we follow the standard methods of solving differential equations in a systematic manner in order to account for all possible solutions. Appendix B explains the Lax pair and Darboux transformation method and gives a detailed derivation of the bright soliton and breather solutions. Chapter 5 is supplemented with appendix C where we show the detailed derivations of the scaling transformation. All chapters are started by a ‘glance’; a summary that helps the reader to easily navigate through the text. All of the solutions presented in the text are rewritten together with the NLSE they satisfy in a Mathematica notebook (available online at https://iopscience.iop. org/book/978-0-7503-2428-1). The reader can run the cells in the Mathematica notebook in order to verify that the solutions do indeed satisfy the NLSE. This minimizes any possible typo errors in the text. It is also convenient to have the solution typed in for those readers who want to use the solutions in their calculations. Verification of the solutions was possible at different levels of accuracy. Some solutions satisfy the NLSE with all variables and parameters unspecified. For some other solutions, Mathematica could not verify the solution in a reasonable time, therefore we set arbitrary values for the variables and parameters. Here we used only integers or ratios of integers. The verification is then obtained with the integer ‘0’ as a result of substituting the solution in the NLSE. This result is numerical but with infinite accuracy. For the rest of the solutions, Mathematica could not verify the solutions with infinite accuracy. In this case, we set numerical values for the parameters and variables with a chosen number of digits. The verification leads to a numerical zero with accuracy increasing when the number of digits is increased. Some readers may still want to verify the last two cases with unspecified values of the variables. In such cases they will need to wait longer. For example, with a typical personal computer one may have to wait 15 minutes to verify the two-bright-solitons solution with unspecified variables. There are many solutions that can be obtained from other solutions for some specific values of the parameters. For the reader to have quick and easy access to the solutions that they might be interested in, we present such solutions explicitly. This book can be considered as the nucleus of a growing collection of solutions to the NLSE and its various variations. The search for new solutions is an ongoing effort by many researchers. We believe that new solutions will continue to appear. Our book will be a suitable host and help one to keep track of the new solutions and classify them into their proper classes, if any. It will be also a useful reference to judge what is claimed a new solution. In addition, the scaling transformations presented in this book will be an efficient tool to reveal whether a new solution can be obtained from an existing solution with a simple transformation. They will be helpful to derive new solutions for some specific setups. We aim at monitoring the literature for the appearance of new solutions of the NLSE and plan to update the book frequently with future editions. Researchers and users of this book are invited to contribute by suggesting and pointing out new or even missing solutions so that we can incorporate them into future editions. 1-5

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

References [1] Pethick C J and Smith H 2001 Bose-Einstein Condensation in Dilute Gases (Cambridge: Cambridge University Press) [2] See for instance: Hasegawa A and Kodama Y 1995 Solitons in Optical Communications (New York: Oxford University Press) Mollenauer L F and Gordon J P 2006 Solitons in Optical Fibers (Boston: Academic Press) Agrawal G P 2001 Nonlinear Fiber Optics 3rd ed (San Diego: Academic) Akhmediev N N and Ankiexicz A 1997 Solitons: Nonlinear Pulses and Beams (London: Chapman and Hall) Akhmediev N N and Ankiewicz A 2008 Dissipative Solitons: From Optics to Biology and Medicine (Berlin: Springer) Taylor J (ed) 1992 Optical Solitons: Theory and Experiment Cambridge Studies in Modern Optics pp I–Vi (Cambridge: Cambridge University Press) Kivshar Y S and Agrawal G P 2003 Optical Solitons (Burlington: Academic) Butcher P and Cotter D 1990 The Elements of Nonlinear Optics (Cambridge Studies in Modern Optics) (Cambridge: Cambridge University Press) Newell A C and Moloney J V 1992 Nonlinear Optics (Redwood City: Addison-Wesley) Taylor J R (ed) 1992 Optical Solitons-Theory and Experiment (Cambridge: Cambridge University Press) [3] Kharif C, Pelinovsky E and Slunyaev A 2009 Rogue waves in the ocean Advances in Geophysical and Environmental Mechanics and Mathematics (Berlin: Springer) [4] Gross E P 1961 Structure of a quantized vortex in boson systems Il Nuovo Cimento (1955–1965) 20 454–77 [5] Pitaevskii L P 1961 Vortex lines in an imperfect Bose gas Sov. Phys. JETP 13 451–4 [6] Hasegawa A and Tappert F 1973 Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion Appl. Phys. Lett. 23 142–4 [7] Hasegawa A and Tappert F 1973 Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion. Appl. Phys. Lett. 23 171–2 [8] Zakharov V E E and Manakov S V 1974 On the complete integrability of a nonlinear Schrödinger equation Theor. Math. Phys. 19 332–43 [9] Shabat A and Zakharov V 1972 Exact theory of two-dimensional self-focusing and onedimensional self-modulation of waves in nonlinear media Sov. Phys. JETP 34 62–9 [10] Al Khawaja U 2010 A comparative analysis of Painlevé, Lax pair, and similarity transformation methods in obtaining the integrability conditions of nonlinear Schrödinger equations J. Math. Phys. 51 053506–11 [11] Ablowitz M J and Clarkson P A 1991 Solitons, Nonlinear Evolution Equations and Inverse Scattering 149 (Cambridge: Cambridge University Press)

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IOP Publishing

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations Usama Al Khawaja and Laila Al Sakkaf

Chapter 2 Fundamental Nonlinear Schrödinger Equation

A Glance at Chapter 2

A Statistical View of Chapter 2 Equation 1 2 Total

Solutions

i ψt + a1 ψxx + a2 ∣ψ ∣2 ψ = 0 i ψt + (a1r + i a1i ) ψxx + (a2r + i a2i ) ∣ψ ∣2 ψ = 0 2

49 10 59

2.1 NLSE with Cubic Nonlinearity Equation:

i ψt + a1 ψxx + a2 ∣ψ ∣2 ψ = 0,

(2.1)

where ψ = ψ (x , t ) is the complex function profile, x and t are its two independent variables, a1 and a2 are arbitrary constants.

doi:10.1088/978-0-7503-2428-1ch2

2-1

ª IOP Publishing Ltd 2020

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Solutions: 2.1.1 Real Dispersion and Nonlinearity Coefficients Solution 1. Constant Amplitude I continuous wave (CW), t-dependent phase ⎡

2

ψ (x , t ) = A0 e i⎣a2 A0

(t − t0) + ϕ0⎤⎦

(2.2)

,

where A0, t0, and ϕ0 are arbitrary real constants. • Reference: [1]. Solution 2. Constant Amplitude II CW, x-dependent phase ⎡ i ±A 0

ψ (x , t ) = A0 e ⎢⎣

⎤ a2 a1 (x − x0) + ϕ0⎥⎦

(2.3)

,

where a1 a2 > 0, A0, x0, and ϕ0 are arbitrary real constants. • Derived in appendix A. Solution 3. Constant Amplitude III CW, t- and x-dependent phase ⎡

ψ (x , t ) = A0 e i⎣A1 (x − x0) + ( A0

2

a2 − A12 a1 (t − t0) + ϕ0⎤⎦

)

(2.4)

,

where A0, A1, x0, t0, and ϕ0 are arbitrary real constants. • Reference: [1]. Solution 4. Rational Solution I decaying wave (DW) ⎧ [a A + x − x ] 2

ψ (x , t ) =

⎫

1 2 0 + a2 A02 ln[A1+ t − t0 ] + ϕ0⎬ A0 i⎨ ⎭, e ⎩ 4 a1 [A1+ t − t0] A1 + t − t0

(2.5)

where A0, A1, A2, t0, x0, and ϕ0 are arbitrary real constants. • Reference: [1]. Solution 5. Rational Solution II

ψ (x , t ) =

1 −2 a1 e i ϕ0, a2 x − x0

2-2

(2.6)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

where a1 a2 < 0, x0 and ϕ0 are arbitrary real constants. • Reference: [2]. Solution 6. Rational Solution III higher order of (2.6)

ψ (x , t ) =

q1(x , t ) 1 , −a2 q2(x , t )

(2.7)

where

q1(x , t ) = 16

×

x12

⎛ ⎡ ⎞ ⎛ 2 2 λ* ⎜⎜ ⎢ 2 + 2 ⎢2 A1 x0 ⎜ x1 x + λ − λ*⎟ −A0 λ e ⎝ a1 ⎠ ⎝ ⎣

⎞ x − i λ* t⎟⎤ ⎟⎥ 2 a1 ⎠

⎥ ⎦

⎛ ⎞ 2 λ ⎜⎜ x + i λ t⎟⎟ 2 a ⎝ ⎠, 1 e ⎛

λ2 A0 2 λ* ⎜⎜⎝ e q2(x , t ) = 4 x0 A1

⎞ x − i λ* t⎟ ⎟ 2 a1 ⎠

⎛ ⎡ ⎞ 2 λ* ⎜⎜ ⎛ 2 ⎢2 x A ⎝ + ⎢ 0 1 + A0 ⎜⎝ a1 x1 x − x0⎟⎠ e ⎣ ⎞ ⎛ 2 x1 x + x0⎟ , − 8 λ2 ⎜ ⎠ ⎝ a1 ⎛ ⎞ 2 λ ⎜⎜ x + i λ t⎟⎟ 2 a ⎠ e ⎝ 1

⎞ x − i λ* t⎟⎤ ⎟⎥ 2 a1 ⎠

⎥ ⎦

λ = λ r + i λi , c1 = c1r + i c1i , c2 = c2r + i c2i , x0 = λ + λ*, x1 = ∣λ∣2 ,

A0 =

∣ c2 ∣2 , 2 λ*2 ∣ c1 ∣2 c2 , c1

A1 = a1 > 0, a2 < 0, λr , λi , c1r , c1i , c2r , c2i , and ϕ0 are arbitrary real constants.

The * denotes the complex conjugate. • Derived by Khelifa Elhadj and U Al Khawaja using the Darboux transformation, unpublished.

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Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Solution 7. sec(x)

ψ (x , t ) = A0

2 −2 a1 ⎡ ⎤ sec[A0 (x − x0)] e−i ⎣a1 A0 (t − t0) + ϕ0⎦, a2

(2.8)

where a1 a2 < 0, A0, x0, t0, and ϕ0 are arbitrary real constants. • Reference: [3] with m = 1 and α = γ = λ = ν = 0. Solution 8. csc(x)

ψ (x , t ) = A0

2 −2 a1 ⎡ ⎤ csc[A0 (x − x0)] e−i⎣ a1 A0 (t − t0) + ϕ0⎦, a2

(2.9)

where a1 a2 < 0, A0, x0, t0, and ϕ0 are arbitrary real constants. • Reference: [3] with m = 1 and α = γ = λ = ν = 0. Solution 9. tan(x)

ψ (x , t ) = A0

2 −2 a1 ⎡ ⎤ tan[A0 (x − x0)] e i⎣2 a1 A0 (t − t0) + ϕ0⎦, a2

(2.10)

where a1 a2 < 0, A0, x0, t0, and ϕ0 are arbitrary real constants. • Reference: [3] with m = 1 and α = γ = λ = ν = 0. Solution 10. cot(x)

ψ (x , t ) = A0

2 −2 a1 ⎡ ⎤ cot[A0 (x − x0)] e i ⎣2 a1 A0 (t − t0) + ϕ0⎦, a2

where a1 a2 < 0, A0, x0, t0, and ϕ0 are arbitrary real constants. • Reference: [3] with m = 1 and α = γ = λ = ν = 0. Solution 11. sech(x) bright soliton (Figure 2.1)

2-4

(2.11)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Figure 2.1. Bright soliton (2.12), with a1 = 1, a2 = 1/2 , A0 = 1, and x0 = t0 = ϕ0 = 0 . (a) Absolute value of (2.12), (b) real part of (2.12), and (c) imaginary part of (2.12).

Figure 2.2. Dark soliton (2.14), with a1 = 1/2 , a2 = −1, A0 = 1, and x0 = t0 = ϕ0 = 0 . (a) Absolute value of (2.14), (b) real part of (2.14), and (c) imaginary part of (2.14).

ψ (x , t ) = A0

2 2 a1 ⎡ ⎤ sech[A0 (x − x0)] e i ⎣a1 A0 (t − t0) + ϕ0⎦, a2

(2.12)

where a1 a2 > 0, A0, x0, t0, and ϕ0 are arbitrary real constants. • Reference: [1]. Solution 12. csch(x)

ψ (x , t ) = A0

2 −2 a1 ⎡ ⎤ csch[A0 (x − x0)] e i ⎣a1 A0 (t − t0) + ϕ0⎦, a2

where a1 a2 < 0, A0, x0, t0, and ϕ0 are arbitrary real constants. • Reference: [3] with m = 1 and α = γ = λ = ν = 0. Solution 13. tanh(x) dark soliton (Figure 2.2)

2-5

(2.13)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

ψ (x , t ) = A0

2 −2 a1 ⎡ ⎤ tanh[A0 (x − x0)] e−i ⎣2 a1 A0 (t − t0) + ϕ0⎦, a2

(2.14)

where a1 a2 < 0, A0, x0, t0, and ϕ0 are arbitrary real constants. • Reference: [3] with m = 1 and α = γ = λ = ν = 0. Solution 14. coth(x)

ψ (x , t ) = A0

2 −2 a1 ⎡ ⎤ coth[A0 (x − x0)] e−i ⎣2 a1 A0 (t − t0) + ϕ0⎦, a2

(2.15)

where a1 a2 < 0, A0, x0, t0, and ϕ0 are arbitrary real constants. • Reference: [3] with m = 1 and α = γ = λ = ν = 0. Solution 15. (Figure 2.3)

⎛ ⎧ ⎡ −a2 ⎤⎫⎞ ⎪ ⎟ ( x x ) a A ( t t ) ⎢ ψ (x , t ) = ⎜⎜A0 + i A1 tan ⎨ A − + − 0 2 0 0 ⎥⎬⎟ 1 ⎪ 2 a ⎣ ⎦ 1 ⎭⎠ ⎩ ⎝ ⎪

⎪

⎡

(2.16)

⎤

× e i ⎣a2 ( A0 − A1 ) (t − t0) + ϕ0⎦, 2

2

where a1a2 < 0, A0, A1, x0, t0, and ϕ0 are arbitrary real constants. • Reference: [4].

Figure 2.3. Plot of solution (2.16) with a1 = A1 = 1/2 , a2 = −1, A3 = 3/4 , and x0 = t0 = ϕ0 = 0 .

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Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Solution 16.

−a1 {sec[A0 (x − x0)] + tan[A0 (x − x0)]} 2 a2

ψ (x , t ) = A0 ×

⎡ a1 A 2 ⎤ 0 i ⎢ 2 (t − t0) + ϕ0⎥ ⎦, e ⎣

(2.17)

where a1 a2 < 0, A0, x0, t0, and ϕ0 are arbitrary real constants. • Reference: [3]. Solution 17.

⎛ ⎧ ⎡ x − x0 ⎤⎫ ⎪ −2 ⎜ ⎨ cot 2 A ( t t ) ⎢ ⎥⎬ A − − 1 0 0 ⎪ a2 ⎜⎝ ⎦⎭ ⎩ ⎣ a1 ⎧ ⎡ x − x0 ⎤⎫⎞ ⎪ ⎨ − tan ⎪A0 ⎢ − 2 A1 (t − t0)⎥⎬⎟⎟ ⎦⎭⎠ ⎩ ⎣ a1

⎪

ψ (x , t ) = A0

⎪

⎪

(2.18)

⎪

×e

⎡ A1 ⎤ 2 2 i ⎢ a (x − x0) − A1 − 8 A0 (t − t0) + ϕ0⎥ ⎣ 1 ⎦,

(

)

where a1 > 0, a2 < 0, A0, A1, x0, t0, and ϕ0 are arbitrary real constants. • Reference: [4]. Solution 18. ⎡

ψ (x , t ) = A0

2

a A −a1 ⎧ cos[A0 (x − x0)] ⎫ i ⎢⎣ 1 2 0 ⎨ ⎬e 2 a2 ⎩ 1 − sin[A0 (x − x0)] ⎭

where a1 a2 < 0, A0, x0, t0, and ϕ0 are arbitrary real constants. • Reference: [3].

2-7

⎤ (t − t0) + ϕ0⎥ ⎦,

(2.19)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Figure 2.4. Plot of solution (2.20) with a1 = −1/2 , a2 = 1/2 , A0 = 1/20 , and x0 = t0 = ϕ0 = 0 .

Solution 19. (Figure 2.4)

−2 a1 ⎧ 1 + tanh2[A0 (x − x0)] ⎫ −i ⎡⎣8 a1 A02 (t − t0) + ϕ0⎤⎦ ⎨ ⎬e , a2 ⎩ tanh[A0 (x − x0)] ⎭

ψ (x , t ) = A0

(2.20)

where a1 a2 < 0, A0, x0, t0, and ϕ0 are arbitrary real constants. • Reference: [3]. Solution 20.

ψ (x , t ) = A0

×

⎞ ⎛ ⎧ ⎡x − x ⎤ ⎜ b0 tan ⎨A0 ⎢⎣ a 0 − 2 A1 (t − t0)⎥⎦ − b1 ⎟ 1 ⎩ −2 ⎜ ⎟ ⎧ ⎡ x − x0 a2 ⎜ ⎤ ⎟ ⎜ b0 + b1 tan ⎨A0 ⎣⎢ a1 − 2 A1 (t − t0)⎦⎥ ⎟ ⎩ ⎠ ⎝

}

}

(2.21)

⎡ A (x − x 0 ) ⎤ − A12 − 2 A02 (t − t0) + ϕ0⎥ i⎢ 1 a ⎣ ⎦, 1 e

(

)

where a1 > 0, a2 < 0, A0, A1, b0, b1, x0, t0, and ϕ0 are arbitrary real constants. • Reference: [4]. Solution 21. (Figure 2.5)

ψ (x , t ) =

⎡ A0 ⎤ (x − x0) − A02 (t − t0) + ϕ0⎥ a1 ⎦,

q1(x , t ) i ⎢⎣ e q2(x , t )

2-8

(2.22)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Figure 2.5. Plot of solution (2.22) with a1 = 1/2 , a2 = −1/4 , A0 = −2 , b0 = b1 = −1, and x0 = t0 = ϕ0 = 0 .

Figure 2.6. Solitary wave (2.23), with a1 = 1, a2 = −1, A0 = 1, x0 = t0 = ϕ0 = 0 , and m = 1/2 . (a) Absolute value of (2.23), (b) real part of (2.23), and (c) imaginary part of (2.23).

where

q1(x , t ) = − 2 b0 A0

−2 a2 + 2 b1 −

−2 a2 ( x − x0 ) a1

+ 2 A0

−2 a2 (t − t0), 4 b0 a2 A0 (x − x0) − 8 b0 a2 A02 (t − t0) q2(x , t ) = 4 b02 a2 A02 + 2 b12 + a1 4 a2 A0 a (x − x0) (t − t0) + 4 a2 A02 (t − t0)2 , + 2 ( x − x0 ) 2 − a1 a1 a1 > 0, a2 < 0, A0, b0, b1, x0, t0, and ϕ0 are arbitrary real constants.

• Reference: [4]. Solution 22. sn(x, m) solitary wave (SW) (Figure 2.6)

ψ (x , t ) = A0

⎡ ⎤ 2m A0 ⎡ 2 ⎤ sn ⎢ (x − x0), m⎥ e−i ⎣A0 (t − t0) + ϕ0⎦, (2.23) ⎢⎣ a1 (1 + m) ⎥⎦ −a2 (1 + m)

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Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

where a1 (1 + m ) > 0, a2 m (1 + m ) < 0, m ≠ −1, A0, t0, x0, and ϕ0 are arbitrary real constants. • Reference: [5]. Solution 23. sn(x, −1) SW

ψ (x , t ) = A0

2 a1 sn [A0 (x − x0), −1] e i ϕ0, a2

(2.24)

where a1 a2 > 0, A0, x0, and ϕ0 are arbitrary real constants. • Derived in appendix A. Solution 24. cn(x, m) SW (Figure 2.7)

⎡ ⎤ 2m A0 cn ⎢ (x − x0), m⎥ a2 (2 m − 1) ⎢⎣ a1 (2 m − 1) ⎥⎦

ψ (x , t ) = A0 ⎡

2

× e i ⎣A 0

(t − t0) + ϕ0⎤⎦

(2.25)

,

where a1 (2 m − 1) > 0, a2 m (2 m − 1) > 0, m ≠ 1/2, A0, t0, x0, and ϕ0 are arbitrary real constants. • Reference: [5].

Figure 2.7. Solitary wave (2.25), with a1 = 1, a2 = 1, A0 = 1, x0 = t0 = ϕ0 = 0 , and m = 7/10 . (a) Absolute value of (2.25), (b) real part of (2.25), and (c) imaginary part of (2.25).

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Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Solution 25. cn(x, 1/2) SW

ψ (x , t ) = A0

⎡ 1⎤ a1 cn ⎢A0 (x − x0), ⎥ e i ϕ0, ⎣ 2⎦ a2

(2.26)

where a1 a2 > 0, m = 1/2, A0, x0, and ϕ0 are arbitrary real constants. • Derived in appendix A. Solution 26. dn(x, m) SW (Figure 2.8)

ψ (x , t ) = A0

⎡ ⎤ 2 A0 ⎡ 2 ⎤ dn ⎢ (x − x0), m⎥ e i ⎣A0 (t − t0) + ϕ0⎦, (2.27) ⎢⎣ a1 (2 − m) ⎥⎦ a2 (2 − m)

where a1 (2 − m ) > 0, a2 (2 − m ) > 0, m ≠ 2, A0, t0, x0, and ϕ0 are arbitrary real constants. • Reference: [5]. Solution 27. dn(x, 2) SW

ψ (x , t ) = A0

2 a1 dn [A0 (x − x0), 2] e i ϕ0, a2

(2.28)

where a1 a2 > 0, A0, x0, and ϕ0 are arbitrary real constants. • Derived in appendix A.

Figure 2.8. Solitary wave (2.27), with a1 = 1, a2 = 1, A0 = 1, x0 = t0 = ϕ0 = 0 , and m = 1/2 . (a) Absolute value of (2.27), (b) real part of (2.27), and (c) imaginary part of (2.27).

2-11

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Solution 28. nd(x, m) SW

⎡ A ⎤ 1 − m nd⎢ 1 (x − x0), m⎥ e−i [A2 (t − t0) + ϕ0], ⎣ a1 ⎦

ψ (x , t ) = A0

(2.29)

where A2 = (m − 2) A12 , a1 > 0, a2 =

2 A12 A02

,

0 < m ⩽ 1, A0, A1, x0, t0, and ϕ0 are arbitrary real constants.

• Reference: [6], taken from the nonlocal case. Solution 29. sd(x, m) SW

ψ (x , t ) = A0

⎡ A ⎤ m (1 − m) sd⎢ 1 (x − x0), m⎥ e−i [A2 (t − t0) + ϕ0], ⎣ a1 ⎦

(2.30)

where A2 = (1 − 2 m ) A12 , a1 > 0, a2 =

2 A12 A02

,

0 < m ⩽ 1, A0, A1, x0, t0, and ϕ0 are arbitrary real constants.

• Reference: [6], taken from the nonlocal case. Solution 30. cd(x, m) SW

ψ (x , t ) = A0

⎡ A ⎤ m cd⎢ 1 (x − x0), m⎥ e−i [A2 (t − t0) + ϕ0], ⎣ a1 ⎦

where A2 = (m + 1) A12 , a1 > 0, a2 =

−2 A12 A02

,

m > 0, A0, A1, x0, t0, and ϕ0 are arbitrary real constants.

• Reference: [6], taken from the nonlocal case.

2-12

(2.31)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Solution 31. sn2(x, m) solitary wave on a finite background (Figure 2.9)

ψ (x , t ) = A(x ) e i [ϕ(x, t ) + ϕ0], where A(x ) =

R3 + (R2 − R3) sn2[A0 (x − x0 ), m ] ,

ϕ(x , t ) = m= A0 =

(2.32)

2 λ0 Π

{

R3 − R2 , R3

}

am[A0 (x − x0 ), m ], m dn[A0 (x − x0 ), m ]

2 R3 A0 1 − m

sn2[A

0

(x − x0 ), m ]

+ λ2 (t − t0 ),

R2 − R3 , R1 − R3 a2 (R3 − R1) 2 a1

,

a1 a2 (R3 − R1) > 0, Rj, j = 1, 2, 3 are the three roots of Y (x ) = 2 a1 λ 02 − 2 a1 λ1 x − 2 λ2 x 2 + a2 x 3, Π is the incomplete elliptic integral, am is the amplitude for Jacobi elliptic functions, x0, t0, ϕ0, λ 0, λ1, and λ2 are arbitrary real constants.

• Derived in appendix A. • This solution can be written in terms of cn2 or dn2 by using: sn2(x , m ) = 1 − cn2(x , m ) = [1 − dn2(x , m )]/m. Solution 32. dn(x, m) + cn(x, m) SW

⎧A ⎡ A ⎤ B m ⎡ A ⎤⎫ cn ⎢ 1 (x − x0), m⎥⎬ ψ (x , t ) = ⎨ 0 dn ⎢ 1 (x − x0), m⎥ + 0 2 ⎣ a1 ⎦ ⎣ a1 ⎦⎭ (2.33) ⎩ 2 ⎪

⎪

⎪

⎪

× e−i [A2 (t − t0) + ϕ0],

Figure 2.9. Solitary wave on a finite background (2.32), with a1 = 1, a2 = 1, x0 = t0 = ϕ0 = 0 , λ 0 = 1/10 , λ1 = 0 , and λ2 = 1/2 . (a) Absolute value of (2.32), (b) real part of (2.32), and (c) imaginary part of (2.32).

2-13

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

where −(m + 1) A12

A2 = 2 B0 = ±A0, a1 > 0, a2 =

2 A12 A02

,

,

0 < m ⩽ 1, A0, A1, x0, t0, and ϕ0 are arbitrary real constants.

• Reference: [6], taken from the nonlocal case. Solution 33. SW

ψ (x , t ) = A0 m

⎡ cn ⎣⎢

A1 a1

⎤ ⎡ (x − x0), m⎦⎥ sn ⎣⎢ ⎡ dn ⎢⎣

A1 a1

A1 a1

⎤ (x − x0), m⎦⎥

⎤ (x − x0), m⎥⎦

e−i [A2 (t − t0) + ϕ0], (2.34)

where A2 = 2 (2 − m ) A12 , a1 > 0, a2 = −

2 A12 A02

,

m > 0, A0, A1, x0, t0, and ϕ0 are arbitrary real constants.

• Reference: [6], taken from the nonlocal case; we corrected the denominator. Solution 34. N-Bright Solitons

ψ (x , t ) =

1 a2

N

∑ψj (x, t ),

(2.35)

j=1

where ψj (x , t ) are solutions of N

∑ Mj k⎡⎣γj−1(x, t ) + γk*(x, t )⎤⎦ ψk(x, t ) = 1, k=1

2-14

j = 0, 1, 2, … , N ,

(2.36)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

a1 > 0, a2 > 0, λj = αj + i νj ,

Mjk = 1/(λj + λ k*),

γ j (x , t ) = e

λj (x − x 0 j ) + i 2 a1

(λj2 (t − t0 )/2 + ϕ0j )

,

γ j−1(x ,

t ) are the complex conjugate and the inverse of γk and γj , γk*(x , t ) and respectively, λ k* is the complex conjugate of λk , αj , νj , x0j , t0, and ϕ0j are arbitrary real constants.

• Reference: [7]. Solution 35. Two Bright Solitons (Figure 2.10)

ψ (x , t ) =

1 [ψ1(x , t ) + ψ2(x , t )], a2

(2.37)

where ψ1(x , t ) = ψ2(x , t ) =

⎡ ⎤ ⎡ ⎤ M12 ⎣γ1−1(x , t ) + γ2*(x , t )⎦ − M22 ⎣γ2−1(x , t ) + γ2*(x , t )⎦ ⎡ ⎤⎡ ⎤ ⎡ ⎤⎡ ⎤ M12 M21 ⎣γ1*(x , t ) + γ2−1(x , t )⎦ ⎣γ1−1(x , t ) + γ2*(x , t )⎦ − M11 M22 ⎣γ1−1(x , t ) + γ1*(x , t )⎦ ⎣γ2−1(x , t ) + γ2*(x , t )⎦ ⎡ ⎤ ⎡ ⎤ −M11 ⎣γ1−1(x , t ) + γ1*(x , t )⎦ + M21 ⎣γ1*(x , t ) + γ2−1(x , t )⎦ ⎡ ⎤⎡ ⎤ ⎡ ⎤⎡ ⎤ M12 M21 ⎣γ1*(x , t ) + γ2−1(x , t )⎦ ⎣γ1−1(x , t ) + γ2*(x , t )⎦ − M11 M22 ⎣γ1−1(x , t ) + γ1*(x , t )⎦ ⎣γ2−1(x , t ) + γ2*(x , t )⎦

,

,

Figure 2.10. Two bright solitons (2.37), with a1 = 1/2 , a2 = 1, α1 = 1, α2 = 2 , and x01 = ν2 = ϕ01 = ϕ02 = 0 . (a) x02 = ν1 = 0 , (b) x02 = 2 and ν1 = 0 , and (c) x02 = 3 and ν1 = 1/2 . Animation available online at https:// iopscience.iop.org/book/978-0-7503-2428-1.

2-15

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Figure 2.11. Three bright solitons (2.38), with a1 = 1/2 , a2 = 1, α1 = 2 , α2 = 3, α3 = 1, and ϕ01 = ϕ02 = ϕ03= t0 = 0 . (a) x01 = x02 = x03 = 1 and ν1 = ν2 = ν3 = 0 , (b) x01 = 4 , x02 = 2 , x03 = −2 , and ν1 = ν2 = ν3 = 0 , and (c) x01 = 4 , x02 = 2 , x03 = −2 , ν1 = ν3 = 0 , and ν2 = −1/2 . Animation available online at https:// iopscience.iop.org/book/978-0-7503-2428-1.

a1 > 0, a2 > 0, Mjk = 1/(λj + λ k*), λj

(x − x ) + i [λ2 (t − t )/2 + ϕ ]

0j 0 j 0j , γj (x , t ) = e 2 a1 λj = αj + i νj , αj , νj , x0j , t0, and ϕ0j are arbitrary real constants, N = 2 in (2.35).

• Reference: [7]. Solution 36. Three Bright Solitons (Figure 2.11)

ψ (x , t ) =

1 [ψ1(x , t ) + ψ2(x , t ) + ψ3(x , t )], a2

2-16

(2.38)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

where

⎡ ψ1(x , t ) = −⎢ ⎣

({M

13

M22 ⎡⎣γ2−1(x , t ) + γ2*(x , t )⎤⎦ ⎡⎣γ1−1(x , t ) + γ3*(x , t )⎤⎦

−M12 M23 ⎡⎣γ1−1(x , t ) + γ2*(x , t )⎤⎦ ⎡⎣γ2−1(x , t ) + γ3*(x , t )⎤⎦

}

× −M13 ⎡⎣γ1−1(x , t ) + γ3*(x , t ) + M33 ⎡⎣γ3−1(x , t ) + γ3*(x , t )⎤⎦⎤⎦

}

{

− −M13 ⎡⎣γ1−1(x , t ) + γ3*(x , t )⎤⎦ + M23 ⎡⎣γ2−1(x , t ) + γ3*(x , t )⎤⎦

{

}

× M13 M32 ⎡⎣γ3−1(x , t ) + γ2*(x , t )⎤⎦ ⎡⎣γ1−1(x , t ) + γ3*(x , t )⎤⎦

{

− M12 M33 ⎡⎣γ1−1(x , t ) + γ2*(x , t )⎤⎦ ⎡⎣γ3−1(x , t ) + γ3*(x , t )⎤⎦

})

÷

({M

13

M22 ⎡⎣γ2−1(x , t ) + γ2*(x , t )⎤⎦ ⎡⎣γ1−1(x , t ) + γ3*(x , t )⎤⎦

− M12 M23 ⎡⎣γ1−1(x , t ) + γ2*(x , t )⎤⎦ ⎡⎣γ2−1(x , t ) + γ3*(x , t )⎤⎦

}

× M13 M31 ⎡⎣γ3−1(x , t ) + γ1*(x , t )⎤⎦ ⎡⎣γ1−1(x , t ) + γ3*(x , t )⎤⎦

{

− M11 M33 ⎡⎣γ1−1(x , t ) + γ1*(x , t )⎤⎦ ⎡⎣γ3−1(x , t ) + γ3*(x , t )⎤⎦

}

− M13 M21 ⎡⎣γ2−1(x , t ) + γ1*(x , t )⎤⎦ ⎡⎣γ1−1(x , t ) + γ3*(x , t )⎤⎦

{

−M11 M23 ⎡⎣γ1−1(x , t ) + γ1*(x , t )⎤⎦ ⎡⎣γ2−1(x , t ) + γ3*(x , t )⎤⎦

}

× M13 M32 ⎡⎣γ3−1(x , t ) + γ2*(x , t )⎤⎦ ⎡⎣γ1−1(x , t ) + γ3*(x , t )⎤⎦

{

})⎤⎥⎦,

− M12 M33⎡⎣γ1−1(x , t ) + γ2*(x , t )⎤⎦ ⎡⎣γ3−1(x , t ) + γ3*(x , t )⎤⎦

2-17

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

ψ2(x, t ) = (M23 M31 − M21 M33) γ2−1(x, t ) γ3−1(x, t ) + ⎡⎣M21 (M13 − M33) γ2−1(x, t )

{

+ ( − M13 + M23) M31 γ3−1(x, t )⎤⎦ γ3*(x, t ) + γ1−1(x, t ) ⎡⎣ M13 (M21 − M31) × γ1*(x, t ) + (M13 M21 − M11 M23) γ2−1(x, t ) + ( − M13 M31 + M11 M33) × γ3−1 ,(x, t ) + M11 ( − M23 + M33) γ3*(x, t )⎤⎦ + γ1*(x, t ) [M23 ( − M11 + M31) × γ2−1(x, t ) + (M11 − M21) M33 γ3−1(x, t ) + (M13 M21 − M11 M23 − M13 M31 + M23 M31 + M11 M33 − M21 M33) γ3*(x, t )⎤⎦

}

(

÷ M12 (M23 M31 − M21 M33)γ2−1(x, t ) γ2*(x, t ) γ3−1(x, t ) +

{ (− M

13

M22

+ M12 M23) M31 γ2*(x, t ) γ3−1(x, t ) + γ2−1(x, t ) ⎡⎣ M21 (M13 M32 − M12 M33) × γ2* (x, t ) + M13 ( − M22 M31 + M21 M32 ) γ3−1(x, t )⎤⎦

} γ*(x, t) + γ 3

−1 (x, 1

× M22 ( − M13 M31 + M11 M33) γ2*(x, t ) γ3−1(x, t ) + ⎡⎣ M11 ( − M23 M32

{

+ M22 M33 ) γ2*(x, t ) + M23 (M12 M31 − M11 M32 ) γ3−1(x, t )⎤⎦γ3*(x, t ) + γ2−1(x, t )⎡⎣ (M13 M21 − M11 M23) M32 γ2*(x, t ) + ( − M13 M22 M31 + M12 M23 M31 + M13 M21 M32 − M11 M23 M32 − M12 M21 M33 + M11 M22 M33 ) γ3−1(x, t ) + ( − M12 M21 + M11 M22 ) M33 γ3*(x, t )⎤⎦ + γ1*(x, t ) ⎡⎣( − M13 M22 + M12 M23) M31 γ2−1(x, t ) + M13 ( − M22 M31 + M21 M32 ) γ2*(x, t ) + M21 (M13 M32 − M12 M33) γ3−1(x, t ) + M12 (M23 M31 − M21 M33) γ3*(x, t )⎤⎦

}

{

× γ1*(x, t ) (M13 M21 − M11 M23) M32 γ3−1(x, t ) γ3*(x, t ) + γ2−1(x, t ) ⎡⎣ M23 (M12 M31 − M11 M32 ) γ2*(x, t ) + M11 ( − M23 M32 + M22 M33) γ3−1(x, t ) + M22 ( − M13 M31 + M11 M33) γ3*(x, t )⎤⎦ + γ2*(x, t ) ⎡⎣( − M12 M21 + M11 M22 ) M33 γ3−1(x, t ) + − M13 M22 M31

(

+ M12 M23 M31 + M13 M21 M32 − M11 M23 M32 − M12 M21 M33 ⎞ + M11 M22 M33) γ3*(x, t )⎤⎦ ⎟ , ⎠

}

2-18

t)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

ψ3(x , t ) =

{M

21

(M12 − M32) γ2−1(x , t ) γ2*(x , t ) + ⎡⎣ (M22 M31 − M21 M32) γ2−1(x , t )

+ ( − M12 + M22) M31 γ2*(x , t )⎤⎦ γ3−1(x , t ) + γ1*(x , t ) ⎡⎣ M22 ( − M11 + M31) × γ2−1(x , t ) + (M12 M21 − M11 M22 − M12 M31 + M22 M31 + M11 M32 − M21 M32 ) γ2*(x , t ) + (M11 − M21) M32 γ3−1(x , t )⎤⎦ + γ1−1(x , t ) ⎡⎣ M12 (M21 − M31) γ1*(x , t ) + (M12 M21 − M11 M22) γ2−1(x , t ) + M11 ( − M22 + M32) × γ2*(x , t ) + ( − M12 M31 + M11 M32) γ3−1(x , t )⎤⎦

}

(

÷ M12 ( − M23 M31 + M21 M33) γ2−1(x , t ) γ2*(x , t ) γ3−1(x , t ) +

{ (M

13

M22

− M12 M23) M31 γ2*(x , t ) γ3−1(x , t ) + γ2−1(x , t ) ⎡⎣ M21 ( − M13 M32 + M12 M33) × γ2* (x , t ) + M13 (M22 M31 − M21 M32) γ3−1(x , t )⎤⎦ γ3*(x , t ) + γ1−1(x , t )

}

× M22 (M13 M31 − M11 M33) γ2*(x , t ) γ3−1(x , t ) + ⎡⎣ M11 (M23 M32 − M22 M33)

{

× γ2* (x , t ) + M23 ( − M12 M31 + M11 M32) γ3−1(x , t )⎤⎦ γ3*(x , t ) + γ2−1(x , t ) × ⎡⎣ ( − M13 M21 + M11 M23) M32 γ2*(x , t ) + (M13 M22 M31 − M12 M23 M31 − M13 M21 M32 + M11 M23 M32 + M12 M21 M33 − M11 M22 M33) γ3−1(x , t ) + (M12 M21 − M11 M22) M33 γ3*(x , t )⎤⎦ + γ1*(x , t ) ⎡⎣ (M13 M22 − M12 M23) M31 × γ2−1(x , t ) + M13 (M22 M31 − M21 M32) γ2*(x , t ) + M21 ( − M13 M32 + M12 M33) × γ3−1 (x , t ) + M12 ( − M23 M31 + M21 M33) γ3*(x , t )⎤⎦ + γ1*(x , t )

}

{ (−M13 M21

+ M11 M23) M32 γ3−1(x , t ) γ3*(x , t ) + γ2−1(x , t ) ⎡⎣ M23 ( − M12 M31 + M11 M32) × γ2*(x , t ) + M11 (M23 M32 − M22 M33) γ3−1(x , t ) + M22 (M13 M31 − M11 M33) × γ3*(x , t )⎤⎦ + γ2*(x , t ) ⎡⎣ (M12 M21 − M11 M22 ) M33 γ3−1(x , t ) + (M13 M22 M31 − M12 M23 M31 − M13 M21 M32 + M11 M23 M32 + M12 M21 M33 − M11 M22 M33) γ3*(x , t )⎤⎦

}),

a1 > 0, a2 > 0, Mjk = 1/(λj + λ k*), λj

(x − x ) + i [λ2 (t − t )/2 + ϕ ]

0j 0 j 0j , γj (x , t ) = e 2 a1 λj = αj + i νj , αj , νj , x0j , t0, and ϕ0j are arbitrary real constants, N = 3 in (2.35).

• Reference: [7].

2-19

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Solution 37. N-Dark Solitons −2 vn −2 ⎧ ⎨1 − 2 i∑ μn (t ) e a1 a2 ⎩ n ⎪

ψ (x , t ) =

⎪

⎫ ⎣ − Q12 n + (λn + i vn ) (Q11n − 1)⎤⎦⎬ ⎭ (2.39)

(x − x 0 ) ⎡

⎪ ⎪

× e−2 i [t − t0+ ϕ0],

where vj =

1 − λ j2 ,

μj (t ) = e 4 vj λj (t − t0 ), Q11n and Q12n are obtained by solving the linear algebraic equations: μ (t )

Q12j + ∑ v n+ v Q12n e n

Q11j +

n

n

n

μn (t ) (λn + i vn )

− ∑Q11n

vn + vj

n

∑ vμ +(tv) Q11n e n

−2 v n ( x − x 0 ) a1

j

j

−2 v n ( x − x 0 ) a1

e

μn (t )(−λn + i vn )

+ ∑Q12n

vn + vj

n

−2 v n ( x − x 0 ) a1

=∑

μn (t )(λn + i vn ) vn + vj

n

e

−2 v n ( x − x 0 ) a1

μ (t )

= ∑ v n+ v e n

n

e

−2 v n ( x − x 0 ) a1

−2 v n ( x − x 0 ) a1

,

,

j

a1 > 0, a2 < 0, −1 < λj < 1, x0, t0, and ϕ0 are arbitrary real constants.

• Reference: [8], we corrected the exponential prefactor. Solution 38. Two Dark Solitons (Figure 2.12)

⎧ ⎛ ⎞ 2i ⎡ 2 1 1 ψ ( x , t ) = ⎨1 − + ⎢ ⎜ ⎟ − (λ1 − i v1) q1(x , t ) p(x , t ) ⎣ v1 + v2 ⎝ λ1 + i v1 λ2 + i v2 ⎠ ⎩ (2.40) ⎤⎫ ⎪ − 2 −2 i [ t − t 0 + ϕ ] 0 , e − (λ2 − i v2 ) q2(x , t ) ⎥⎬ ⎪ ⎥⎦⎭ a2 ⎪

⎪

where v1 =

1 − λ12 ,

v2 =

1 − λ 22 ,

q1(x , t ) = q2(x , t ) =

1 v1 1 v2

+e

2 v1 a1

(x − x0 ) − 4 v1 λ1 (t − t0 )

+e

2 v2 a1

(x − x0 ) − 4 v2 λ2 (t − t0 )

, ,

Figure 2.12. Two dark solitons, (2.40), with a1 = 1/2 , a2 = −1, λ2 = 3/10 , and x0 = t0 = ϕ0 = 0 .

2-20

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

p(x , t ) = (λ1 − i v1) (λ2 − i v2 ) q1(x , t ) q2(x , t ) −

1 (v1 + v2 )2

(

1 λ1 + i v1

+

2 1 , λ2 + i v2

)

a1 > 0, a2 < 0, λ1 = −λ2 , −1 < λ2 < 1, x0, t0, and ϕ0 are arbitrary real constants.

• Reference: [8], we corrected the exponential prefactor. Solution 39. Generalized First-Order Breather (form I*)

ψ (x, t ) =

⎧ ⎫ ⎪ 1 ⎪ κ 2 cosh[δ (t − t0)] + 2 i κ ν sinh[δ (t − t0)] ⎨ − 1⎬ e i [t − t0+ ϕ0], (2.41) a2 ⎪ 2 cosh[δ (t − t )] − 2 ν cos ⎡ κ (x − x )⎤ ⎪ 0 0 ⎥ ⎣⎢ 2 a1 ⎦ ⎩ ⎭

where a1 > 0, a2 > 0, κ = 2 1 − ν2 , δ = κ ν, ν , x0, t0, and ϕ0 are arbitrary real constants. • Reference: [9]. Solution 40. Generalized First-Order Breather (form II*)

ψ (x , t ) =

cos(A0 ) cos[q1(x , t ) + 2 i A1] − cosh(A1) cosh[q2(x , t ) + 2 i A0 ] cos(A0 ) cos[q1(x , t )] − cosh(A1) cosh[q2(x , t )] (2.42) × e i [a2 (t − t0) + ϕ0],

where

⎡ q1(x , t ) = v3 ⎢ ⎣ ⎡ q2(x , t ) = v2 ⎢ ⎣ v0 = v1 =

2 a2 a1 2 a2 a1

⎤ (x − x0 ) − a2 v0 (t − t0 )⎥, ⎦ ⎤ (x − x0 ) − a2 v1 (t − t0 )⎥, ⎦

sinh(2 A1) cos(2 A0 ) , cosh(A1) sin(A0 ) cosh(2 A1) sin(2 A0 ) − sinh(A ) cos(A ) , 1 0

v2 = −sinh(A1) cos(A0 ), v3 = cosh(A1) sin(A0 ), a1 a2 > 0, x0, t0, A0, A1, and ϕ0 are arbitrary real constants.

• Reference: [10]. 2-21

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Solution 41. Generalized First-Order Breather (form III*)

ψ (x , t ) =

A0 ⎛ ⎜1 − a2 ⎝

⎞ ⎡ 2 8 λr ⎤ p(x , t )⎟ e i ⎣A0 (t − t0) + ϕ0⎦, A0 ⎠

(2.43)

where p(x , t ) =

(A02 + Γ2) cos[q1(x, t )] + i (A02 − Γ2) sin[q1(x, t )] + 2 A0 {Γr cosh[q2(x, t )] − i Γi sinh[q2(x, t )]} , 2 A0 Γr cos[q1(x, t )] + (Γ2 + A02 ) cosh[q2(x, t )]

⎡x−x ⎤ 2 ⎢⎣ a 0 Δi − 2 (t − t0 ) (Δi λi + Δr λr )⎥⎦, 1 ⎡x−x ⎤ q2(x , t ) = δr + 2 ⎢⎣ a 0 Δr − 2 (t − t0 ) Δr λi + 2 (t − t0 ) Δi λr ⎥⎦, 1 ⎡ ⎤ Δr = Re ⎣ 2 (λr − i λi )2 − A02 ⎦, ⎡ ⎤ Δi = Im ⎣ 2 (λr − i λi )2 − A02 ⎦, Γr = Δr + 2 λr , Γi = Δi − 2 λi , q1(x , t ) = δi +

Γ = Γ r2 + Γ i2 , a1 > 0, a2 > 0, A0, λr , λi , x0, t0, and ϕ0 are arbitrary real constants.

• Reference: [11]. *Remark: There are different forms of the breather in the literature, as given by Solutions 39, 40, 41. In appendix B.1.2 we derive a fourth expression valid for focusing and defocusing nonlinearities and the arbitrary constants are expressed in terms of physical parameters. Solution 42. Periodicity in t and Localization in x Kuznetsov–Ma breather (Figure 2.13)

ψ (x, t ) =

⎧ ⎫ ⎪ 1 ⎪ − p2 cos[ω (t − t0)] − 2 i p ν sin[ω (t − t0)] ⎨ − 1⎬e i [t − t0+ ϕ0], (2.44) a2 ⎪ 2 cos[ω (t − t )] − 2 ν cosh ⎡ p (x − x )⎤ ⎪ 0 0 ⎥ ⎢⎣ 2 a1 ⎦ ⎩ ⎭

where a1 > 0, a2 > 0, p = 2 ν2 − 1 , ω = p ν, ν > 1, x0, t0, and ϕ0 are arbitrary real constants. 2-22

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Figure 2.13. Kuznetsov–Ma breather (2.44), with a1 = a2 = 1, ν = 1.5, and x0 = t0 = ϕ0 = 0. (a) Absolute value of (2.44), (b) real part of (2.44), and (c) imaginary part of (2.44). Animation available online at https:// iopscience.iop.org/book/978-0-7503-2428-1.

Figure 2.14. Akhmediev breather (2.45), with a1 = 1/2, a2 = 1, ν = 0.5, and x0 = t0 = ϕ0 = 0. (a) Absolute value of (2.45), (b) real part of (2.45), and (c) imaginary part of (2.45). Animation available online at https:// iopscience.iop.org/book/978-0-7503-2428-1.

• Reference: [9]. • This solution can be generated from (2.41) with ν > 1. Solution 43. Periodicity in x and Localization in t Akhmediev breather (Figure 2.14)

ψ (x, t ) =

⎧ ⎫ ⎪ ⎪ 2 κ cosh[δ (t − t0)] + 2 i κ ν sinh[δ (t − t0)] 1 ⎨ − 1⎬e i [t − t0+ ϕ0], (2.45) a2 ⎪ 2 cosh[δ (t − t )] − 2 ν cos ⎡ κ (x − x )⎤ ⎪ 0 0 ⎥ ⎣⎢ 2 a1 ⎦ ⎩ ⎭

where a1 > 0, a2 > 0, κ = 2 1 − ν2 , δ = κν , ν < 1, x0, t0, and ϕ0 are arbitrary real constants. • Reference: [9].

2-23

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Solution 44. Localization in x and t Peregrine soliton (Figure 2.15)

ψ (x , t ) =

⎡ ⎤ 1 ⎢ 4 + i 8 (t − t0 ) − 1⎥ e i [t − t0+ ϕ0], 2 ⎢ ⎥ 2 2 a2 1 + 4 (t − t0) + (x − x0) ⎣ ⎦ a1

(2.46)

where a2 > 0, x0, t0, and ϕ0 are arbitrary real constants. • Reference: [9]. • This solution can be generated from (2.41) in the limits ν → 1 and κ → 0. Solution 45. Generalized Two-Breathers Solution (Figures 2.16–2.20)

ψ (x , t ) =

1 ⎡ α( x , t ) + i β ( x , t ) ⎤ i [ t − t 0 + ϕ ] 0 , ⎢1 + ⎥e γ (x , t ) ⎦ a2 ⎣

(2.47)

where

Figure 2.15. Peregrine soliton (2.46), with a1 = a2 = 1, and x0 = t0 = ϕ0 = 0. (a) Absolute value of (2.46), (b) real part of (2.46), and (c) imaginary part of (2.46). Animation available online at https://iopscience.iop.org/ book/978-0-7503-2428-1.

Figure 2.16. Plot of solution (2.47), with a1 = 1/2, a2 = 1, ν1 = 0.5, ν2 = 0.85, x01 = x02 = t0 = ϕ0 = 0 , t01 = 5, and t02 = −5. (a) Absolute value of (2.47), (b) real part of (2.47), and (c) imaginary part of (2.47).

2-24

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Figure 2.17. Plot of solution (2.47), with a1 = 1/2, a2 = 1, ν1 = 1.3, ν2 = 1.85, x01 = 5, x02 = −5, and t01 = t02 = t0 = ϕ0 = 0 . (a) Absolute value of (2.47), (b) real part of (2.47), and (c) imaginary part of (2.47).

Figure 2.18. Plot of solution (2.47), with a1 = 1/2, a2 = 1, ν1 = 1.3, ν2 = 1.85, and x01 = x02 = t01 = t02= t0 = ϕ0 = 0 . (a) Absolute value of (2.47), (b) real part of (2.47), and (c) imaginary part of (2.47).

Figure 2.19. Plot of solution (2.47), with a1 = 1/2, a2 = 1, ν1 = 0.5, ν2 = 0.85, and x01 = x02 = t01 = t02 = t0= ϕ0 = 0 . (a) Absolute value of (2.47), (b) real part of (2.47), and (c) imaginary part of (2.47).

Figure 2.20. Plot of solution (2.47), with a1 = 1/2 , a2 = 1, ν1 = 0.5, ν2 = 1.5, and x01 = x02 = t01 = t02 = t0= ϕ0 = 0 . (a) Absolute value of (2.47), (b) real part of (2.47), and (c) imaginary part of (2.47).

2-25

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

⎧ ⎡ κ2 ⎤ 2 ⎪ δ 2 κ1 cos ⎢⎣ 2 a1 (x − x02 )⎥⎦ cosh[δ1 (t − t01)] 2 2 α(x , t ) = κ2 − κ1 ⎨ κ2 ⎪ ⎩ ⎡ κ ⎤ δ1 κ22 cos ⎢ 1 (x − x01)⎥ cosh[δ 2 (t − t02 )] ⎣ 2 a1 ⎦ − κ1 ⎫ ⎪ − κ12 − κ22 cosh[δ1 (t − t01)] cosh[δ 2(t − t02 )] ⎬ , ⎪ ⎭ ⎧ ⎡ κ2 ⎤ ⎪ δ1 δ 2 cos ⎣⎢ 2 a1 (x − x02 )⎦⎥ sinh[δ1 (t − t01)] β(x , t ) = − 2 κ12 − κ22 ⎨ κ2 ⎪ ⎩ − δ1 cosh[δ 2 (t − t02 )] sinh[δ1 (t − t01)] + δ 2 cosh[δ1 (t − t01)] sinh[δ 2 (t − t02 )] ⎫ ⎡ κ ⎤ δ1 δ 2 cos ⎢ 1 (x − x01)⎥ sinh[δ 2 (t − t02 )] ⎪ ⎣ 2 a1 ⎦ ⎬, − κ1 ⎪ ⎭ ⎡ κ ⎤ ⎡ κ ⎤ 2 δ1 δ 2 κ12 + κ22 cos ⎢ 1 (x − x01)⎥ cos ⎢ 2 (x − x02 )⎥ ⎣ 2 a1 ⎦ ⎣ 2 a1 ⎦ γ (x , t ) = κ1 κ2

(

)

(

)

(

)

(

)

(

)

− 2 κ12 + 2 κ22 − κ12 κ22 cosh[δ1 (t − t01)] cosh[δ 2 (t − t02 )] ⎧ ⎡ κ2 ⎤ ⎪ δ 2 cos ⎢⎣ 2 a1 (x − x02 )⎥⎦ cosh[δ1 (t − t01)] − 2 κ12 − κ22 ⎨ − κ2 ⎪ ⎩ ⎫ ⎡ κ ⎤ δ1 cos ⎢ 1 (x − x01)⎥ cosh[δ 2 (t − t02 )] ⎪ ⎣ 2 a1 ⎦ ⎬ + κ1 ⎪ ⎭ ⎧ ⎡ κ ⎤ ⎡ κ ⎤ + 4 δ1 δ 2⎨ sin ⎢ 1 (x − x01)⎥ sin ⎢ 2 (x − x02 )⎥ ⎣ ⎦ ⎣ ⎦ 2 2 a a 1 1 ⎩ ⎫ + sinh[δ1 (t − t01)] sinh[δ 2 (t − t02 )] ⎬ , ⎭

(

)

a1 > 0, a2 > 0,

2-26

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

κj = 2

1 − ν j2 ,

κ

δj = 2j 4 − κ j2 , νj , x0j , t0j , t0, and ϕ0 are arbitrary real constants, j = 1, 2.

• Reference: [9]. Solution 46. Specific Two-Breather Solution I Nonlinear superposition of Kuznetsov– Ma or Akhmediev breather with a Peregrine soliton (Figures 2.21–2.24)

ψ (x , t ) =

1 ⎡ α( x , t ) + i β ( x , t ) ⎤ i [ t − t 0 + ϕ ] 0 , ⎢1 + ⎥e γ (x , t ) ⎦ a2 ⎣

(2.48)

Figure 2.21. Plot of solution (2.48), with a1 = 1/2, a2 = 1, ν = 1.2 , x01 = 5, x02 = −5, and t01 = t02 = t0 = ϕ0 = 0 . (a) Absolute value of (2.48), (b) real part of (2.48), and (c) imaginary part of (2.48).

Figure 2.22. Plot of solution (2.48), with a1 = 1/2, a2 = 1, ν = 0.5, t01 = 5, t02 = −5, and x01 = x02 = t0 = ϕ0 = 0 . (a) Absolute value of (2.48), (b) real part of (2.48), and (c) imaginary part of (2.48).

Figure 2.23. Plot of solution (2.48), with a1 = 1/2, a2 = 1, ν = 1.2 , and x01 = x02 = t01 = t02 = t0 = ϕ0 = 0 . (a) Absolute value of (2.48), (b) real part of (2.48), and (c) imaginary part of (2.48).

2-27

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Figure 2.24. Plot of solution (2.48), with a1 = 1/2, a2 = 1, ν = 0.5, x01 = x02 = t01 = t02 = t0 = ϕ0 = 0 . (a) Absolute value of (2.48), (b) real part of (2.48), and (c) imaginary part of (2.48).

where

α( x , t ) =

κ 8

⎛ ⎡ ⎤ ⎜⎜8 δ cos ⎢ κ (x − x01)⎥ ⎣ 2 a1 ⎦ ⎝

⎞ ⎧ ⎡ ⎤ ⎫ 2 + κ ⎨ −8 + ⎢1 + 4 (t − t02 )2 + (x − x02 )2 ⎥ κ 2⎬ cosh[δ (t − t01)]⎟⎟ , ⎣ ⎦ ⎭ a1 ⎩ ⎠ ⎛ ⎫ ⎧ ⎡ ⎤ ⎪ ⎪ ⎜8 (t − t02 ) ⎨δ cos ⎢ κ (x − x01)⎥ − κ cosh[δ (t − t01)]⎬ ⎪ ⎪ ⎜ ⎣ 2 a1 ⎦ ⎭ ⎩ ⎝ ⎞ ⎡ ⎤ 2 + ⎢1 + 4 (t − t02 )2 + (x − x02 )2 ⎥ δ κ sinh[δ (t − t01)]⎟ , ⎣ ⎦ a1 ⎠ ⎡ ⎤ ⎫ 2 1 ⎡ ⎧ ⎢δ⎨ −16 + ⎢1 + 4 (t − t02 )2 + γ (x , t ) = − (x − x02 )2 ⎥ κ 2⎬ ⎣ ⎦ ⎭ 4 κ ⎢⎣ ⎩ a1

κ β (x , t ) = 4

⎛⎧ ⎤ ⎡ κ (x − x01)⎥ + κ ⎜⎜⎨ 16 + [ −3 + 4 (t − t02 )2 × cos ⎢ ⎣ 2 a1 ⎦ ⎝⎩ ⎪

⎪

+

⎧ 1 ⎫ 2 (x − x02 ) (x − x02 )2 ] κ 2⎬ cosh[δ (t − t01)] − 16 δ ⎨ ⎭ a1 ⎩ 2a1

⎞⎤ ⎫ ⎤ ⎡ κ ⎪ ⎟⎥ , (x − x01)⎥ + (t − t02 ) sinh[δ (t − t01)]⎬ × sin ⎢ ⎪⎟⎥ ⎦ ⎣ 2 a1 ⎭⎠⎦ a1 > 0, a2 > 0, κ = 2 1 − ν2 , κ δ = 2 4 − κ2, ν , x0j , t0j , t0, and ϕ0 are arbitrary real constants, j = 1, 2.

• Reference: [9]. • This solution can be generated from (2.47) in the limit κ2 → 0 with κ1 ≠ 0.

2-28

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Solution 47. Specific Two-Breather Solution II (Figures 2.25, 2.26) 1 ⎡ α( x , t ) + i β ( x , t ) ⎤ i [ t − t 0 + ϕ ] 0 , ψ (x , t ) = ⎢1 + ⎥e γ (x , t ) ⎦ a2 ⎣

(2.49)

where

α( x , t ) = −

⎧ ⎛ ⎡ 2κ ⎪ ⎨cosh[δ (t − t0)]⎜⎜ (δ 2 + κ 2 ) cos ⎢κ δ ⎪ ⎣ ⎝ ⎩

⎤ 1 ( x − x0 ) ⎥ 2a1 ⎦

⎤⎞ 1 (x − x0)⎥⎟⎟ 2a1 ⎦⎠ ⎫ ⎤ ⎪ 1 (x − x0)⎥ sinh[δ (t − t0)]⎬ , ⎪ 2a1 ⎦ ⎭ ⎛ ⎡ ⎤ 1 ( x − x0 ) ⎥ t0) ⎜⎜ −κ + δ cos ⎢κ 2a1 ⎣ ⎦ ⎝

− 2 δ κ cosh[δ (t − t0)] + δ 2 κ

⎡ 1 (x − x0)sin ⎢κ 2a1 ⎣

⎡ + δ (2 δ 2 − κ 2 ) (t − t0)cos ⎢κ ⎣ ⎧ 1 ⎪ ⎨8 δ (2 δ 2 − κ 2 ) (t − β (x , t ) = − 2δ κ ⎪ ⎩ ⎛ ⎡ ⎤ 1 1 ( x − x0 ) ⎥ + κ × cosh[δ (t − t0)]) + 8 δ 3 ⎜⎜cos ⎢κ ( x − x0 ) 2 2 a a1 ⎣ ⎦ 1 ⎝ ⎤⎞ ⎡ 1 × sin ⎢κ (x − x0)⎥⎟⎟ sinh[δ (t − t0)] 2a1 ⎦⎠ ⎣ ⎫ ⎪ 4 2 + κ (κ − 4 δ ) sinh[2 δ (t − t0 )] ⎬ , ⎪ ⎭ ⎧ ⎪ 1 ⎨ 32 δ 4 (δ 2 − κ 2 ) (t − t0)2 + κ 4 (δ 2 + κ 2 ) γ (x , t ) = − 4 δ2 κ2 ⎪ ⎩ ⎛ 1 ⎞ + 8 δ 2 κ 2 ⎜δ 2 (x − x0)2 + κ 2 (t − t0)2 ⎟ ⎝ 2a1 ⎠ ⎛ ⎡ ⎤⎞ 1 (x − x0)⎥⎟⎟ + 4 ⎜⎜κ 4 cosh[2 δ (t − t0 )] − δ 4 cos ⎢2 κ 2a1 ⎣ ⎦⎠ ⎝ ⎛ ⎡ ⎤ 1 ( x − x0 ) ⎥ − 4 δ κ 2 cosh[δ (t − t0)] ⎜⎜κ 3 cos ⎢κ 2a1 ⎣ ⎦ ⎝ ⎞ ⎡ ⎤ 1 1 (x − x0)⎥⎟⎟ (x − x0) sin ⎢κ + 4 δ2 2a1 2a1 ⎣ ⎦⎠ ⎫ ⎡ ⎤ ⎪ 1 2 2 2 (x − x0)⎥ sinh[δ (t − t0)]⎬ , − 16 δ κ (2 δ − κ ) (t − t0) cos ⎢κ ⎪ 2a1 ⎣ ⎦ ⎭

2-29

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Figure 2.25. Plot of solution (2.49), with a1 = 1/2, a2 = 1, ν = 1.5, and x0 = t0 = ϕ0 = 0 . (a) Absolute value of (2.49), (b) real part of (2.49), and (c) imaginary part of (2.49).

Figure 2.26. Plot of solution (2.49), with a1 = 1/2, a2 = 1, ν = 0.8, and x0 = t0 = 0 . (a) Absolute value of (2.49), (b) real part of (2.49), and (c) imaginary part of (2.49).

Figure 2.27. Second-order Peregrine soliton (2.50), with a1 = 1/2, a2 = 1, and x0 = t0 = ϕ0 = 0 . (a) Absolute value of (2.50), (b) real part of (2.50), and (c) imaginary part of (2.50). Animation available online at https:// iopscience.iop.org/book/978-0-7503-2428-1.

a1 > 0, a2 > 0, κ = 2 1 − ν2 , κ δ = 2 4 − κ2, ν , x0, t0, and ϕ0 are arbitrary real constants.

• Reference: [9]. • This solution can be generated from (2.47) in the limit κ2 → κ1, x01 = x02 = x0, and t01 = t02 = t0.

2-30

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Solution 48. Second-Order Peregrine Soliton (Figure 2.27)

ψ (x , t ) =

1 ⎡ α( x , t ) + i β ( x , t ) ⎤ i [ t − t 0 + ϕ ] 0 , ⎢1 + ⎥e γ (x , t ) ⎦ a2 ⎣

(2.50)

where 1 ⎡ 12 (x − x 0 ) 2 ⎢ − 3 + 72 (t − t0 )2 + 80 (t − t0 )4 + 96 ⎣ a1 ⎤ 48 4 (x − x 0 ) 2 (t − t0 ) 2 + 2 (x − x 0 ) 4 ⎥ , + a1 a1 ⎦ ⎡ 1 12 (t − t0 ) ⎢ − 15 + 8 (t − t0 )2 + 16 (t − t0 )4 − (x − x 0 ) 2 β (x , t ) = ⎣ 48 a1 ⎤ 16 4 (x − x 0 ) 2 (t − t0 ) 2 + 2 (x − x 0 ) 4 ⎥ , + a1 a1 ⎦ 54 −1 ⎡ (x − x 0 ) 2 γ (x , t ) = ⎢9 + 396 (t − t0 )2 + 432 (t − t0 )4 + 64 (t − t0 )6 + ⎣ 1152 a1 144 96 12 (x − x 0 ) 2 (t − t0 ) 2 + (x − x 0 ) 2 (t − t0 ) 4 + 2 (x − x 0 ) 4 − a1 a1 a1 ⎤ 48 8 + 2 (x − x 0 )4 (t − t0 )2 + 3 (x − x 0 )6⎥ , a1 a1 ⎦

α (x , t ) =

a2 > 0, x0, t0, and ϕ0 are arbitrary real constants.

• Reference: [9]. • This solution can be generated from (2.49) in the limit κ → 0 and δ = 0. Solution 49. Rogue Wave Triplet (Figure 2.28)

Figure 2.28. Rogue wave triplet (2.51), with a1 = 1/2, a2 = 1, xd = −15, td = −10 , and x0 = t0 = ϕ0 = 0 . (a) Absolute value of (2.51), (b) real part of (2.51), and (c) imaginary part of (2.51).

2-31

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

ψ (x , t ) =

1 ⎡ α( x , t ) + i β ( x , t ) ⎤ i [ t − t 0 + ϕ ] 0 , ⎢1 + ⎥e γ (x , t ) ⎦ a2 ⎣

(2.51)

where ⎧ 4 α (x , t ) = − 12 ⎨ − 3 + 2 (x − x 0 )4 − 128 td a ⎩ 1 12 (x − x 0 )2 [1 + 4 (t − t0 )2 ] + a1

1 (x − x 0 ) 2a1

⎫ + 8 [9 + 10 (t − t0 )2 ] (t − t0 )2 − 128 xd (t − t0 ) ⎬ , ⎭ ⎛ ⎧ 1 4 td (x − x 0 ) (t − t0 ) + (t − t0 ) ⎨ − 15 + 2 (x − x 0 )4 β (x , t ) = − 24 ⎜⎜ − 128 a1 2 a1 ⎩ ⎝ ⎫ 4 (x − x 0 )2 [ − 3 + 4 (t − t0 )2 ]⎬ + 8 (t − t0 )2 [1 + 2 (t − t0 )2 ] + ⎭ a1 ⎞ ⎡ ⎤ 2 ( x − x 0 ) 2 − 4 ( t − t 0 ) 2 ⎥x d ⎟ , + 16 ⎢1 + ⎣ ⎦ ⎠ a1 8 γ (x , t ) = 9 + 3 (x − x 0 )6 + 396 (t − t0 )2 + 432 (t − t0 )4 + 64 (t − t0 )6 a1 12 6 (x − x 0 ) 2 + 2 (x − x 0 )4 [1 + 4 (t − t0 )2 ] + a1 a1 × {9 + 8 (t − t0 )2 [ − 3 + 2 (t − t0 )2 ]} ⎛ ⎧ ⎛ 1 ⎞3/2 + 128 ⎜⎜8 td2 + td ⎨4 ⎜ ⎟ (x − x 0 )3 ⎩ ⎝ 2 a1 ⎠ ⎝ ⎫ 1 (x − x 0 ) [ − 1 − 4 (t − t0 ) 2 ] ⎬ +3 2 a1 ⎭ ⎧ ⎫⎞ ⎡ ⎤ 6 ( x − x 0 ) 2 − 4 ( t − t 0 ) 2 ⎥ + 8 x d ⎬⎟ , + x d ⎨( t − t 0 ) ⎢ − 9 + ⎣ ⎦ a1 ⎩ ⎭⎠ ⎪ ⎪

⎪ ⎪

a1 > 0, a2 > 0, xd, td, x0, t0, and ϕ0 are arbitrary real constants.

• Reference: [12], solution (8) with δ = 0.

2-32

⎡

a2 A02

a2 a1

2-33

ψ (x , t ) =

e

i

−2 a1 a2

[a1 A2 + x − x0 ]2 +a2 4 a1 [A1+ t − t0 ]

e i ϕ0

{

⎡

2

}

(t−t0 )+ϕ0⎤⎦

A02 ln[A1+t−t0 ]+ϕ0

a2−A12 a1) (t−t0 )+ϕ0⎤⎦

sec[A0 (x − x0 )] e−i ⎣a1 A0

q1(x, t ) 1 −a2 q2(x, t )

−2 a1 1 a2 x − x0

A0 A1 + t − t0

7. ψ (x , t ) = A0

6. ψ (x , t ) =

5. ψ (x , t ) =

4.

2

⎤ (x−x0 )+ϕ0⎦

(t−t0 )+ϕ0⎤⎦

3. ψ (x , t ) = A0 e i ⎡⎣A1 (x−x0)+(A0

2. ψ (x , t ) = A e i ⎣±A0 0

⎡

1. ψ (x , t ) = A0 e i ⎣

# Solution

continuous wave, x-dependent phase

continuous wave, t-dependent phase

Name

(2.3)

(2.2)

Eq. #

a1 a2 < 0 , A0, x0, t0, and ϕ0 are arbitrary real constants

See text

a1 a2 < 0 , x0, and ϕ0 are arbitrary real constants

A0, A1, A2, t0, x0, and ϕ0 are arbitrary real constants

—

(2.8)

(2.7)

(2.6)

(2.5)

(Continued)

of Solution 5

higher order

—

decaying wave

A0, A1, x0, t0, and ϕ0 are arbitrary continuous wave, (2.4) real constants t- and xdependent phase

a1 a2 > 0 , A0, x0, and ϕ0 are arbitrary real constants

A0, t0, and ϕ0 are arbitrary real constants

Conditions

i ψt + a1 ψxx + a2 ∣ψ ∣2 ψ = 0

Equation

Note: For lengthy conditions, the reader is referred to the solutions in subsection 2.1.1.

2.2 Summary of Subsection 2.1.1

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

2-34

−2 a1 a2

2 a1 a2

−2 a1 a2

−2 a1 a2

−2 a1 a2

10. ψ (x , t ) = A0

11. ψ (x , t ) = A0

12. ψ (x , t ) = A0

13. ψ (x , t ) = A0

14. ψ (x , t ) = A0

2

×e

⎡

2

2

2

⎡

2

⎤

⎤⎫⎞ (x − x0 ) + a2 A0 (t − t0 )⎥⎬⎟ ⎦⎭⎠

(t−t0 )+ϕ0⎤⎦

Equation

– —

a1 a2 < 0 , A0, x0, t0, and ϕ0 are arbitrary real constants a1 a2 < 0 , A0, x0, t0, and ϕ0 are arbitrary real constants

–

dark soliton

–

a1a2 < 0 , A0, x0, t0, and ϕ0 are arbitrary real constants

–

a1a2 < 0 , A0, A1, x0, t0, and ϕ0 are – arbitrary real constants

a1 a2 < 0 , A0, x0, t0, and ϕ0 are arbitrary real constants

a1 a2 < 0 , A0, x0, t0, and ϕ0 are arbitrary real constants

a1 a2 < 0 , A0, x0, t0, and ϕ0 are arbitrary real constants

bright soliton

–

a1 a2 < 0 , A0, x0, t0, and ϕ0 are arbitrary real constants

a1 a2 > 0 , A0, x0, t0, and ϕ0 are arbitrary real constants

Name

Conditions

i ψt + a1 ψxx + a2 ∣ψ ∣2 ψ = 0

{sec[A0 (x − x0 )] + tan[A0 (x − x0 )]}

−a2 2 a1

coth[A0 (x − x0 )] e−i ⎣2 a1 A0

(t−t0 )+ϕ0⎤⎦

(t−t0 )+ϕ0⎤⎦

⎡

2

tanh[A0 (x − x0 )] e−i ⎣2 a1 A0

⎡

(t−t0 )+ϕ0⎤⎦

(t−t0 )+ϕ0⎤⎦

(t−t0 )+ϕ0⎤⎦

(t−t0 )+ϕ0⎤⎦

csch[A0 (x − x0 )] e i ⎣a1 A0

⎡ a1 A 2 ⎤ i ⎢ 2 0 (t−t0 )+ϕ0⎥ ⎣ ⎦

−a1 2 a2

2

⎡

cot[A0 (x − x0 )] e i ⎣2 a1 A0

× e i ⎣a2 (A0 −A1 ) (t−t0)+ϕ0⎦

⎡

2

⎡

2

tan[A0 (x − x0 )] e i ⎣2 a1 A0

⎛ ⎧ ⎡ ψ (x , t ) = ⎜A0 + i A1 tan ⎨A1 ⎢ ⎩ ⎣ ⎝

16. ψ (x , t ) = A0

15.

−2 a1 a2

9. ψ (x , t ) = A0

⎡

csc[A0 (x − x0 )] e−i ⎣ a1 A0

sech[A0 (x − x0 )] e i ⎣a1 A0

−2 a1 a2

8. ψ (x , t ) = A0

# Solution

(Continued )

(2.17)

(2.16)

(2.15)

(2.14)

(2.13)

(2.12)

(2.11)

(2.10)

(2.9)

Eq. # Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

ψ (x , t ) = A0

2-35

i

} ⎡

⎪

⎪

2m a2 (2 m − 1)

24. ψ (x , t ) = A 0

A0 a1 (1 + m )

⎡ cn ⎢ ⎣

⎤ (x−x0 )−(A12 −8 A02 ) (t−t0 )+ϕ0⎦

(t−t0 )+ϕ0⎤⎦

⎤ ⎡ 2 ⎤ (x − x0 ), m⎥e−i ⎣A0 (t−t0)+ϕ0⎦ ⎦

A0 a1 (2 m − 1)

⎤ ⎡ 2 ⎤ (x − x0 ), m⎥e i ⎣A0 (t−t0)+ϕ0⎦ ⎦

sn [A0 (x − x0 ), − 1] e i ϕ0

2 a1 a2

⎡ sn ⎢ ⎣

⎤ (x−x0 )−A02 (t−t0 )+ϕ0⎦

23. ψ (x , t ) = A0

A0 a1

2m −a2 (1 + m )

⎡ i⎣

⎤ A02 ) (t−t0 )+ϕ0⎦

⎪

⎪

2

e−i ⎣8 a1 A0

⎧ ⎛ ⎞ ⎡ x − x0 ⎤⎫ ⎜ b0 tan ⎨⎩A0 ⎢⎣ a1 − 2 A1 (t − t0)⎥⎦⎬⎭ − b1 ⎟ ⎜ ⎟ ⎧ ⎫ ⎜ b0 + b1 tan ⎨A0 ⎡⎢ x − x0 − 2 A1 (t − t0)⎤⎥⎬ ⎟ a ⎣ ⎦ ⎭⎠ 1 ⎩ ⎝

22. ψ (x , t ) = A 0

e

}

A1 a1

⎡ a1 A 2 ⎤ i ⎢ 2 0 (t−t0 )+ϕ0⎥ ⎦ e ⎣

1 + tanh2[A0 (x − x0)] tanh[A0 (x − x0)]

⎡ A1 (x − x0 ) −(A12 −2 a1 ⎣

−2 a2

{

{

cos[A0 (x − x0)] 1 − sin[A0 (x − x0)]

}

}

⎛ ⎧ ⎡x − x ⎤ ⎜cot ⎨A0 ⎣⎢ a 0 − 2 A1 (t − t0 )⎥⎦ 1 ⎩ ⎝

−2 a1 a2

−a1 2 a2

q1(x, t ) q2(x, t )

×e

ψ (x , t ) = A0

21. ψ (x , t ) =

20.

−2 a2

⎧ ⎡x − x ⎤ ⎞ i⎡ − tan ⎨A0 ⎢ a 0 − 2 A1 (t − t0 )⎥ ⎟ e ⎣ ⎦ ⎠ ⎩ ⎣ 1

ψ (x , t ) = A0

19. ψ (x , t ) = A 0

18.

17.

a1 (2 m − 1) > 0 , a2 m (2 m − 1) > 0 , m ≠ 1/2 , A0, t0, x0, and ϕ0 are arbitrary real constants

a1 a2 > 0 , A0, x0, and ϕ0 are arbitrary real constants

a1 (1 + m ) > 0, a2 m (1 + m ) < 0, m ≠ − 1, A0, t0, x0, and ϕ0 are arbitrary real constants

See text

solitary wave

solitary wave

solitary wave

–

–

–

a1 a2 < 0 , A0, x0, t0, and ϕ0 are arbitrary real constants a1 > 0 , a2 < 0, A0, A1, b0, b1, x0, t0, and ϕ0 are arbitrary real constants

–

–

a1 a2 < 0 , A0, x0, t0, and ϕ0 are arbitrary real constants

a1 > 0 , a2 < 0, A0, A1, x0, t0, and ϕ0 are arbitrary real constants

(Continued)

(2.25)

(2.24)

(2.23)

(2.22)

(2.21)

(2.20)

(2.19)

(2.18)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

A0 a1 (2 − m )

2-36

A1 a1

⎤ (x − x0 ), m⎥ e−i [A2 (t−t0)+ϕ0 ] ⎦

⎤ (x − x0 ), m⎥ e−i [A2 (t−t0)+ϕ0 ] ⎦

⎤ (x − x0 ), m⎥ e−i [A2 (t−t0)+ϕ0 ] ⎦

⎡ m cd⎢ ⎣

30. ψ (x , t ) = A 0

A1 a1

⎡ m (1 − m ) sd⎢ ⎣

29. ψ (x , t ) = A 0

A1 a1

⎡ 1 − m nd⎢ ⎣

28. ψ (x , t ) = A 0

Equation

A02

2 A12

A02

, 0 < m ⩽ 1, A0, A1,

A02

, m > 0 , A0, A1, x0, t0, and ϕ0 are arbitrary real constants

a2 =

−2 A12

A2 = (m + 1) A12 , a1 > 0 ,

x0, t0, and ϕ0 are arbitrary real constants

a2 =

2 A12

A2 = (1 − 2 m ) A12 , a1 > 0 ,

solitary wave

solitary wave

solitary wave

Name

solitary wave

solitary wave

, solitary wave

0 < m ⩽ 1, A0, A1, x0, t0, and ϕ0 are arbitrary real constants

A2 = (m − 2) A12 , a1 > 0 , a2 =

a1 a2 > 0 , A0, x0, and ϕ0 are arbitrary real constants

a1 (2 − m ) > 0, a2 (2 − m ) > 0, m ≠ 2 , A0, t0, x0, and ϕ0 are arbitrary real constants

a1 a2 > 0 , m = 1/2, A0, x0, and ϕ0 are arbitrary real constants

Conditions

i ψt + a1 ψxx + a2 ∣ψ ∣2 ψ = 0

⎤ ⎡ 2 ⎤ (x − x0 ), m⎥e i ⎣A0 (t−t0)+ϕ0⎦ ⎦

dn [A0 (x − x0 ), 2] e i ϕ0

2 a1 a2

27. ψ (x , t ) = A0

⎡ dn ⎢ ⎣

1 cn ⎡⎣A0 (x − x0 ), 2 ⎤⎦ e i ϕ0

2 a2 (2 − m )

a1 a2

26. ψ (x , t ) = A 0

25. ψ (x , t ) = A0

# Solution

(Continued )

(2.31)

(2.30)

(2.29)

(2.28)

(2.27)

(2.26)

Eq. #

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

2-37

A0 2

B0 m 2

N

j=1

∑ψj (x, t ),

A1 a1

[ψ1(x , t ) + ψ2(x , t )]

[ψ1(x , t ) + ψ2(x , t ) + ψ3(x , t )]

1 a2

36. ψ (x , t ) =

k=1

0 < m ⩽ 1, A0, A1,

A02

, m > 0 , A0, A1, x0,

See text

See text

See text

t0, and ϕ0 are arbitrary real constants

a2 = −

2 A12

A2 = 2 (2 − m ) A12 , a1 > 0 ,

(2.35)

(2.34)

three bright solitons

(Continued)

(2.38)

two bright solitons (2.37)

N-bright solitons

solitary wave

(2.33)

solitary wave on a (2.32) finite background

, B0 = ± A0 , a1 > 0 , solitary wave

x0, t0, and ϕ0 are arbitrary real constants

a2 =

2 2 A12 , A02

−(m + 1) A12

See text

e−i [A2 (t−t0)+ϕ0 ] A2 =

j = 0, 1, 2, … , N

}

⎤ (x − x0 ), m⎥ ⎦

+ λ2 (t − t0 )

e−i [A2 (t−t0)+ϕ0 ]

⎡ cn ⎢ ⎣

× ∑ Mj k ⎡⎣γ j−1(x , t ) + γk*(x , t )⎤⎦ ψk (x , t ) = 1,

N

1 a2

⎤ (x − x0 ), m⎥ + ⎦

1 a2

ψ (x , t ) =

A1 a1

⎡ A ⎤ ⎡ A ⎤ cn ⎢ 1 (x − x0), m⎥ sn ⎢ 1 (x − x0), m⎥ ⎣ a1 ⎦ ⎣ a1 ⎦ ⎡ A1 ⎤ dn ⎢ (x − x0), m⎥ ⎣ a1 ⎦

⎡ dn ⎢ ⎣

ψ (x , t ) = A0 m

{

2 R3 A0 1 − m sn2[A0 (x − x0), m ]

35. ψ (x , t ) =

34.

33.

32. ψ (x , t ) =

ϕ (x , t ) =

⎧R − R ⎫ 2 λ0 Π⎨ 3 2 , am[A0 (x − x0), m ], m⎬ dn[A0 (x − x0), m ] ⎩ R3 ⎭

A(x ) = R3 + (R2 − R3) sn2[A0 (x − x0 ), m ] ,

31. ψ (x , t ) = A(x ) e i [ϕ(x, t )+ϕ0 ],

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

2-38

ψ (x , t ) =

ψ (x , t ) =

ψ (x , t ) =

42.

ψ (x , t ) =

41. ψ (x , t ) =

40. ψ (x , t ) =

39.

38.

37.

# Solution

(Continued )

⎪

⎪

⎧ ⎨1 − 2 i ⎩

⎧ ⎨1 − ⎩

2i ⎡ 2 ⎢ v1 + v2 p (x , t ) ⎣

n

(

∑ μ n (t ) e

+

(x − x 0 )

1 λ1 + i v1

−2 v n a1

1 λ2 + i v2

)

⎫ ⎧ ⎪ ⎪ κ 2 cosh[δ (t − t0)] + 2 i κ ν sinh[δ (t − t0)] ⎨ − 1⎬ e i [t−t0+ϕ0 ] ⎡ κ ⎤ ⎪ ⎪ 2 cosh[δ (t − t0)] − 2 ν cos ⎢ (x − x0)⎥ ⎣ 2 a1 ⎦ ⎭ ⎩

1 a2

A0 a2

8 λr A0

)

⎡

2

p(x , t ) e i ⎣A0

(t−t0 )+ϕ0⎤⎦

⎫ ⎧ ⎪ ⎪ −p2 cos[ω (t − t0)] − 2 i p ν sin[ω (t − t0)] ⎨ − 1⎬ e i [t−t0+ϕ0 ] ⎡ p ⎤ ⎪ ⎪ 2 cos[ω (t − t0)] − 2 ν cosh ⎢ (x − x0)⎥ ⎣ 2 a1 ⎦ ⎭ ⎩

(1 −

× e i [a2 (t−t0)+ϕ0 ]

cos(A0 ) cos[q1(x, t ) + 2 i A1] − cosh(A1) cosh[q2(x, t ) + 2 i A0 ] cos(A0 ) cos[q1(x, t )] − cosh(A1) cosh[q2(x, t )]

1 a2

⎪

⎪

⎫ [ −Q12 n + (λn + i vn ) (Q11n − 1)]⎬ ⎭

− (λ1 − i v1) q1(x , t ) − (λ2 − i v2 ) q2(x , t )]} e−2 i [t−t0+ϕ0 ]

−2 a2

× e−2 i [t−t0+ϕ0 ]

−2 a2

generalized first-order breather III

generalized first-order breather II

generalized first-order breather I

two dark solitons

N-dark solitons

Name

Kuznetsov–Ma a1 > 0 , a2 > 0, p = 2 ν 2 − 1 , breather ω = p ν , ν > 1, x0, t0, and ϕ0 are arbitrary real constants

See text

See text

a1 > 0 , a2 > 0, κ = 2 1 − ν 2 , δ = κ ν , ν , x0, t0, and ϕ0 are arbitrary real constants

See text

See text

Conditions

i ψt + a1 ψxx + a2 ∣ψ ∣2 ψ = 0

Equation

(2.44)

(2.43)

(2.42)

(2.41)

(2.40)

(2.39)

Eq. # Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

α (x , t ) + i β ( x , t ) ⎤ ⎦ γ (x , t )

α (x , t ) + i β ( x , t ) ⎤ ⎦ γ (x , t )

⎡ ⎣1 +

⎡ ⎣1 +

1 a2

1 a2

49. ψ (x , t ) =

α (x , t ) + i β ( x , t ) ⎤ ⎦ γ (x , t )

⎡ ⎣1 +

1 a2

47. ψ (x , t ) =

48. ψ (x , t ) =

α (x , t ) + i β ( x , t ) ⎤ ⎦ γ (x , t )

⎡ ⎣1 +

1 a2

46. ψ (x , t ) =

2-39 e i [t−t0+ϕ0 ]

e i [t−t0+ϕ0 ]

e i [t−t0+ϕ0 ]

e i [t−t0+ϕ0 ]

e i [t−t0+ϕ0 ]

See text

See text

See text

See text

rogue wave triplet

second-order Peregrine soliton

specific twobreathers II

specific twobreathers I

generalized two-breathers

See text

α (x , t ) + i β ( x , t ) ⎤ ⎦ γ (x , t )

a2 > 0, x0, t0, and ϕ0 are arbitrary Peregrine soliton real constants

Akhmediev a1 > 0 , a2 > 0, κ = 2 1 − ν 2 , breather δ = κ ν , ν < 1, x0, t0, and ϕ0 are arbitrary real constants

⎡ ⎤ 4 + i 8 (t − t 0 ) ⎢ ⎥ e i [t−t0+ϕ0 ] − 1 2 ⎢⎣ 1 + 4 (t − t0)2 + a (x − x0)2 ⎥⎦ 1

⎡ ⎣1 +

1 a2

⎫ ⎧ ⎪ ⎪ κ 2 cosh[δ (t − t0)] + 2 i κ ν sinh[δ (t − t0)] ⎨ − 1⎬ e i [t−t0+ϕ0 ] ⎡ κ ⎤ ⎪ ⎪ 2 cosh[δ (t − t0)] − 2 ν cos ⎢ (x − x0)⎥ ⎣ 2 a1 ⎦ ⎭ ⎩

1 a2

ψ (x , t ) =

ψ (x , t ) =

1 a2

45. ψ (x , t ) =

44.

43.

(2.51)

(2.50)

(2.49)

(2.48)

(2.47)

(2.46)

(2.45)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

2.2.1 Complex Dispersion and Nonlinearity Coefficients Here, a1 = a1r + i a1i , a2 = a2r + i a2i , where a1r , a1i , a2r , and a2i are real constants. Solution 1. Constant Amplitude I CW, t-dependent phase ⎡

2

ψ (x , t ) = A0 e i⎣ a2r A0

(t − t0) + ϕ0⎤⎦

(2.52)

,

where a2i = 0, A0, t0, and ϕ0 are arbitrary real constants. • Derived in appendix A. Solution 2. Constant Amplitude II CW, x-dependent phase ⎡ i ±A 0

ψ (x , t ) = A0 e ⎢⎣

⎤ a2r a1r (x − x0) + ϕ0⎥⎦

(2.53)

,

where a1r a2r > 0, a a a1i = 1ar 2i , 2r A0, x0, and ϕ0 are arbitrary real constants. • Derived in appendix A. Solution 3. Rational Solution I DW

ψ (x , t ) =

1 e i ϕ0, 2 a2i (t − t0)

(2.54)

where a2i > 0, a2r = 0, t0 and ϕ0 are arbitrary real constants. • Derived in appendix A. Solution 4. Rational Solution II DW

ψ (x , t ) =

1 i e 2 a2i (t − t0)

{ A (x − x ) − a 0

0

2-40

1r

A02 (t − t0) +

},

a2r ln[2 a2i (t − t0)] + ϕ0 2 a2i

(2.55)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

where a2i > 0, a1i = 0, A0, t0, x0, and ϕ0 are arbitrary real constants. • Derived in appendix A. Solution 5.

ψ (x , t ) = A0

×

a1r 2 2 a A e 1 r 0 (t − t 0 )

− a2i

(2.56)

⎧ ⎫ a (a + a ) A 2 a 2 i ⎨A0 (x − x0) − 1r 2ia 2r 0 (t − t0) + 2r ln⎡⎣−a2i + e 2 a1r A0 (t − t0)⎤⎦ + ϕ0⎬ i 2 2 a 2i ⎭, e ⎩

where a1i = −a1r , A0, t0, x0, and ϕ0 are arbitrary real constants. • Derived in appendix A. Solution 6. 2

ψ (x , t ) = A0

×

a1i e a1i A0 2

a2i e 2 a1i A0

(t − t 0 )

(t − t 0 )

−1

(2.57)

⎛ a i ⎜A0 (x − x0) − a1r A02 (t − t0) + 2r ln 2 a2i e ⎝

{

2 e 2 a1i A0 A1

⎡a e 2 a1i A02 (t − t0)− 1⎤ + ϕ ⎞⎟ 0 ⎣ 2i ⎦ ⎠

}

,

where A0, A1, t0, x0, and ϕ0 are arbitrary real constants. • Derived in appendix A. Solution 7.

ψ (x , t ) = A0 e

−a1r A12

⎡ ⎤ a A2 2 i ⎢A1 (x − x0) − a1r A12 (t − t0) − 2r 0 2 e−2 a1r A1 (t − t0)+ ϕ0⎥ (t − t 0 ) e ⎣ 2 a1r A1 ⎦,

(2.58)

where a1i = −a1r , a2i = 0, A0, A1, t0, x0, and ϕ0 are arbitrary real constants. • Derived in appendix A. Solution 8. sn(x, −1) SW

ψ (x , t ) = A0

2 a1r sn [A0 (x − x0), −1] e i ϕ0, a2r

2-41

(2.59)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

where a1r a2r > 0, a a a1i = 1ar 2i , 2r A0, x0, and ϕ0 are arbitrary real constants. • Derived in appendix A. Solution 9. cn(x, 1/2) SW

ψ (x , t ) = A0

⎡ 1⎤ a1r cn ⎢A0 (x − x0), ⎥ e i ϕ0, ⎣ 2⎦ a2r

(2.60)

where a1r a2r > 0, a a a1i = 1ar 2i , 2r A0, x0, and ϕ0 are arbitrary real constants. • Derived in appendix A. Solution 10. dn(x, 2) SW

ψ (x , t ) = A0

2 a1r dn [A0 (x − x0), 2] e i ϕ0, a2r

where a1r a2r > 0, a a a1i = 1ar 2i , 2r A0, x0, and ϕ0 are arbitrary real constants. • Derived in appendix A.

2-42

(2.61)

2-43

7.

6.

1

a

2i

(t−t0 )+ 2 a2r ln[2 a2i (t−t0 )]+ϕ0

}

ψ (x , t ) = A0

×e

(t−t0 ) − 1

2 e−a1r A1 (t−t0)

{e 2 a1i A02 A1

}

⎡a e 2 a1i A02 (t−t0 )−1⎤ +ϕ ⎞⎟ 0 ⎣ 2i ⎦ ⎠

⎡ ⎤ a2r A02 2 i ⎢A1 (x−x0 )−a1r A12 (t−t0 )− e−2 a1r A1 (t−t0 )+ϕ0⎥ 2 a1r A12 ⎣ ⎦ e

⎛ a i ⎜A0 (x−x0 )−a1r A02 (t−t0 )+ 2 a2r ln 2i ⎝

a2i e 2

a1i A02

2 a1i e a1i A0 (t−t0 )

⎧ ⎫ a1r (a2i + a2r ) A02 2 a (t−t0 )+ 2 a2r ln⎡⎣−a2i +e 2 a1r A0 (t−t0 )⎤⎦+ϕ0⎬ i ⎨A0 ( x − x 0 ) − a 2i 2i ⎩ ⎭ e

–

a1i = − a1r , a2i = 0 , A0, A1, t0, x0, and ϕ0 are – arbitrary real constants

A0, A1, t0, x0, and ϕ0 are arbitrary real constants

a1i = − a1r , A0, t0, x0, and ϕ0 are arbitrary – real constants

decaying wave

a2i > 0 , a1i = 0, A0, t0, x0, and ϕ0 are arbitrary real constants

2

e i {A0 (x−x0)−a1r A0

a

a2i > 0 , a2r = 0 , t0 and ϕ0 are arbitrary real decaying wave constants

a

continuous wave, t-dependent phase

Name

(2.58)

(2.57)

(2.56)

(2.55)

(2.54)

(2.53)

(2.52)

Eq. #

(Continued)

a1r a2r > 0 , a1i = 1ar 2i , A0, x0, and ϕ0 are continuous wave, 2r x-dependent phase arbitrary real constants

a2i = 0 , A0, t0, and ϕ0 are arbitrary real constants

Conditions

i ψt + (a1r + i a1i ) ψxx + (a2r + i a2i ) ∣ψ ∣2 ψ = 0

Equation

e i ϕ0

⎤ (x−x0 )+ϕ0⎦

a1r 2 e 2 a1r A0 (t−t0 ) − a2i

2 a2i (t − t0)

ψ (x , t ) = A0

×

1

2 a2i (t − t0)

5. ψ (x , t ) = A0

4. ψ (x , t ) =

3. ψ (x , t ) =

2. ψ (x , t ) = A e i ⎡⎣±A0 0

(t−t0 )+ϕ0⎤⎦

a 2r a1r

2

1. ψ (x , t ) = A0 e i ⎡⎣ a2r A0

# Solution

2.3 Summary of Subsection 2.2.1

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

10. ψ (x , t ) = A0

9. ψ (x , t ) = A0

8. ψ (x , t ) = A0

# Solution

(Continued )

2 a1r a2r

a1r a2r

2 a1r a 2r

dn [A0 (x − x0 ), 2] e i ϕ0

1 cn ⎡⎣A0 (x − x0 ), 2 ⎤⎦ e i ϕ0

sn [A0 (x − x0 ), − 1] e i ϕ0

a1r a2i , a 2r

Name

(2.61)

a

a1r a2r > 0 , a1i = 1ar 2i , A0, x0, and ϕ0 are solitary wave 2r arbitrary real constants

a

(2.60)

a

(2.59)

Eq. #

a1r a2r > 0 , a1i = 1ar 2i , A0, x0, and ϕ0 are solitary wave 2r arbitrary real constants

a

A0, x0, and ϕ0 are solitary wave a1r a2r > 0 , a1i = arbitrary real constants

Conditions

i ψt + (a1r + i a1i ) ψxx + (a2r + i a2i ) ∣ψ ∣2 ψ = 0

Equation

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

2-44

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

References [1] Zaitsev V F and Polyanin A D 2003 Handbook of Nonlinear Partial Differential Equations (New York: Chapman and Hall) [2] He B and Meng Q 2016 Qualitative analysis and explicit exact solitary, kink and anti-kink wave solutions of the generalized nonlinear Schrödinger equation with parabolic law nonlinearity Commun. Theor. Phys. 65 1–10 [3] Zayed E M and Al-Nowehy A G 2017 Exact solutions for the perturbed nonlinear Schrödinger equation with power law nonlinearity and Hamiltonian perturbed terms Optik 139 123–44 [4] Ma W X and Chen M 2009 Direct search for exact solutions to the nonlinear Schrödinger equation Appl. Math. Comput. 215 2835–42 [5] Bronski J C, Carr L D, Deconinck B and Kutz J N 2001 Bose–Einstein condensates in standing waves: The cubic nonlinear Schrödinger equation with a periodic potential Phys. Rev. Lett. 86 1402–5 [6] Khare A and Saxena A 2015 Periodic and hyperbolic soliton solutions of a number of nonlocal nonlinear equations J. Math. Phys. 56 032104–27 [7] Gordon J P 1983 Interaction forces among solitons in optical fibers Opt. Lett. 8 596–8 [8] Blow K J and Doran N J 1985 Multiple dark soliton solutions of the nonlinear Schrödinger equation Phys. Lett. A 107 55–8 [9] Kedziora D J, Ankiewicz A and Akhmediev N 2012 Second-order nonlinear Schrödinger equation breather solutions in the degenerate and rogue wave limits Phys. Rev. E 85 066601–9 [10] Kharif C, Pelinovsky E and Slunyaev A 2010 Rogue waves in the ocean Advances in Geophysical and Environmental Mechanics and Mathematics (Berlin: Springer) [11] Al Khawaja U and Taki M 2013 Rogue waves management by external potentials Phys. Lett. A 37 2944–9 [12] Chowdury A, Kedziora D J, Ankiewicz A and Akhmediev N 2015 Breather solutions of the integrable quintic nonlinear Schrödinger equation and their interactions Phys. Rev. E 91 022919–11

2-45

IOP Publishing

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations Usama Al Khawaja and Laila Al Sakkaf

Chapter 3 Nonlinear Schrödinger Equation with Power Law and Dual Power Law Nonlinearities

A Glance at Chapter 3

A Statistical View of Chapter 3 Equation 1 2 Total

Solutions

∣ψ ∣n

i ψt + a1 ψxx + a2 ψ=0 i ψt + a1 ψxx + a2 ∣ψ ∣n ψ + a3 ∣ψ ∣m ψ = 0 2

13 14 27

3.1 NLSE with Power Law Nonlinearity Equation:

i ψt + a1 ψxx + a2 ∣ψ ∣n ψ = 0,

(3.1)

where ψ = ψ (x , t ) is the complex function profile, x and t are its two independent variables, n, a1, and a2 are arbitrary real constants.

doi:10.1088/978-0-7503-2428-1ch3

3-1

ª IOP Publishing Ltd 2020

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

3.1.1 Reduction to the Fundamental NLSE Case I: CW profile If ⎡

a2 − A12 a1 (t − t0) + ϕ0⎤⎦

ψ (x , t ) = A0 e i ⎣A1(x − x0) + ( A0

)

2

(3.2)

is a stationary solution of the fundamental NLSE with cubic nonlinearity, (2.1), then ⎡

ψ (x , t ) = A02/ n e i ⎣A1(x − x0) + ( A0

2

a2 − A12 a1 (t − t0) + ϕ0⎤⎦

)

(3.3)

is a stationary solution to the NLSE with power law nonlinearity, (3.1). Case II: x-dependent profile If

ϕ(x , t ) = u(x ) e iλt

(3.4)

is a stationary solution of the fundamental NLSE with cubic nonlinearity, (2.1), then

⎛ n + 2 ⎞1/ n i ψ (x , t ) = ⎜ 2 ⎟ u(x )2/ n e ⎝ n ⎠

4λ t n2

(3.5)

is a stationary solution to the NLSE with power law nonlinearity, (3.1). Solutions: Solution 1. Constant Amplitude I continuous wave (CW), t-dependent phase ⎡

2

ψ (x , t ) = A02/ n e i ⎣A0

a2 (t − t0) + ϕ0⎤⎦

(3.6)

,

where A0, t0, and ϕ0 are arbitrary real constants. • Reference: [1]. Solution 2. Constant Amplitude II CW, x-dependent phase

ψ (x , t ) = A02/ n e

⎡ i ⎢A 0 ⎣

⎤ a2 a1 (x − x0) + ϕ0⎥⎦

(3.7)

,

where a1 a2 > 0, A0, x0, and ϕ0 are arbitrary real constants. • Derived in appendix A. Solution 3. Constant Amplitude III CW, t- and x-dependent phase ⎡

ψ (x , t ) = A02/ n e i ⎣A1 (x − x0) + ( A0

2

a2 − A12 a1 (t − t0) + ϕ0⎤⎦

)

where A0, A1, x0, t0, and ϕ0 are arbitrary real constants. • Reference: [1]. 3-2

,

(3.8)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Solution 4. Rational Solution I decaying wave (DW)

ψ (x , t ) =

2 −n ⎡ ⎤ n 2 ⎢ 2 a2 ∣ A0 ∣ (t − t0) 2 + (x − x0) + ϕ ⎥ 0 2−n 4 a1 (t − t0) ⎥⎦ ,

i A0 ⎢ e ⎣ t − t0

(3.9)

where A0, x0, t0, and ϕ0 are arbitrary real constants. • Reference: [1]. Solution 5. Rational Solution II DW

⎡ ⎢ ψ (x , t ) = ⎢ ⎢⎣ n

2

⎤n ⎥ iϕ ±1 ⎥ e , −a2 (x − x0) ⎥⎦ 2 a1 (2 + n )

(3.10)

where a1 a2 (2 + n ) < 0, A0, x0, and ϕ0 are arbitrary real constants. • Reference: [2]. Solution 6. sec(x) ⎡

1

⎧ −2 A 2 a1 (n + 2) ⎫ n −i ⎢ 4 a1 2A0 0 2 ψ (x , t ) = ⎨ sec [A0 (x − x0)]⎬ e ⎣ n a2n2 ⎩ ⎭

2

⎤ (t − t0) + ϕ0⎥ ⎦,

(3.11)

⎤ (t − t0) + ϕ0⎥ ⎦,

(3.12)

where a1 a2 (n + 2) < 0, A0, x0, t0, and ϕ0 are arbitrary real constants. • Reference: [3] with α = γ = λ = ν = 0. Solution 7. csc(x) ⎡

1

⎧ −2 A 2 a1 (n + 2) ⎫ n −i ⎢ 4 a1 2A0 0 2 ⎨ ψ (x , t ) = csc [A0 (x − x0)]⎬ e ⎣ n a2n2 ⎩ ⎭

2

where a1 a2 (n + 2) < 0, A0, x0, t0, and ϕ0 are arbitrary real constants. • Reference: [3] with α = γ = λ = ν = 0. Solution 8. sech(x) bright soliton 1

⎡

⎧ 2 A 2 a1 (n + 2) ⎫ n i ⎢ 4 a1 A0 0 2[A (x − x )]⎬ e ⎣ n 2 ψ (x , t ) = ⎨ sech 0 0 a2n2 ⎩ ⎭ where 3-3

2

⎤ (t − t0) + ϕ0⎥ ⎦,

(3.13)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

a1 a2 (n + 2) > 0, A0, x0, t0, and ϕ0 are arbitrary real constants.

• Reference: [3] with α = γ = λ = ν = 0. Solution 9. csch(x) 1

⎡

⎧ −2 A 2 a1 (n + 2) ⎫ n i ⎢ 4 a1 A0 0 2[A (x − x )]⎬ e ⎣ n 2 ψ (x , t ) = ⎨ csch 0 0 a2n2 ⎩ ⎭

2

⎤ (t − t0) + ϕ0⎥ ⎦,

(3.14)

where a1 a2 (n + 2) < 0, A0, x0, t0, and ϕ0 are arbitrary real constants. • Reference: [3] with α = γ = λ = ν = 0. Solution 10. Generalized Oscillatory Solution

ψ (x , t ) = A(x ) e i ϕ0,

(3.15)

where A(x ) = Y −1(x − x0 ), A(x ) ⎡ 1 1 n + 3 2 a2 An + 2 (x ) ⎤ Y [A(x )] = 2F1⎣ 2 , n + 2 , n + 2 , a A (n + 2) ⎦, A0 1 0 −1 is the inverse operator of the 2F1 is the hypergeometric function and Y function Y [A(x )], A0 > 0, x0 and ϕ0 are arbitrary real constants. • Derived in appendix A. Solution 11. sn(x, m) for n = 1 solitary wave (SW)

ψ (x , t ) = { R3 + (R2 − R3) sn2[A0 (x − x0), m ]} e i [λ1 (t − t0) + ϕ0], where m= A0 =

R2 − R3 , R1 − R3

Rj, j = 1, 2, 3, are the three roots of Y (x ) = 3 λ 0 +

a2 (R3 − R1) 6 a1

,

a1 a2 (R3 − R1) > 0, n = 1, λ 0 , λ1, x0, t0, and ϕ0 are arbitrary real constants.

• Derived in appendix A.

3-4

3 λ1 a1

(3.16)

x2 −

2a2 a1

x 3,

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Solution 12. sn(x, m) for n = 4 SW

R1 sn[A0 (x − x0), m ]

ψ (x , t ) =

where m=

R1 − R3 R3

+

sn2[A0

e i [λ1 (t − t0) + ϕ0],

(3.17)

(x − x0), m ]

R3 (R1 − R2 ) , R2 (R1 − R3)

Rj, j = 1, 2, 3, are the three roots of Y (x ) = 3 λ 0 + a2 R2 (R1 − R3) 3 a1

A0 =

3 λ1 a1

x−

a2 a1

x 3,

,

R1 > 0, a1 a2 R2 (R1 − R3) > 0, n = 4, λ 0 , λ1, x0, t0, and ϕ0 are arbitrary real constants.

• Derived in appendix A. Solution 13. sn2(x, m) for n = 4 SW on a finite background

ψ (x , t ) =

R1 (R2 − R 4 ) + R2 (R 4 − R1) sn2[A0 (x − x0), m ] −R2 + R 4 + (R1 −

R 4 ) sn2[A0

e i ϕ(x, t ),

(3.18)

(x − x0), m ]

where

λ0 ϕ(x , t ) =

(

(R2 − R1) A0

{

Π

R2 (R1 − R 4) , R1 (R2 − R 4)

R1 R2 +

}

1−

2

(R2 − R3) (R1 − R 4) sn [A0 (x − x0), m ] (R1 − R3) (R2 − R 4)

λ0 (x − x0) + λ2 (t − t0) + ϕ0 , R2

(R − R ) (R − R )

m = (R2 − R3) (R1 − R 4 ) , 1 3 2 4 Rj, j = 1, 2, 3, 4, are the four roots of Y (x ) = 3 a1 λ 02 − 3 a1 λ1 x − 3 λ2 x 2 + a2 x 4 , A0 =

−a2 (R1 − R3) (R2 − R 4 ) 3 a1

)

am[A0 (x − x0), m ], m dn[A0 (x − x0), m ]

,

Π is the incomplete elliptic integral, am is the amplitude for Jacobi elliptic functions, a1 a2 (R1 − R3) (R2 − R 4 ) < 0, n = 4, λ 0 , λ1, λ2 , x0, t0, and ϕ0 are arbitrary real constants.

• Derived in appendix A.

3-5

⎡

⎡

2

3-6

2

8.

7.

2

⎡

1

⎡

2

⎧ 2 A 2 a1 (n + 2) ⎫ n i ⎢ 4 a1 A0 ψ (x , t ) = ⎨ 0 a n 2 sech2[A0 (x − x0 )]⎬ e ⎣ n2 2 ⎩ ⎭

1

a1 a2 (n + 2) < 0 , A0, x0, t0, and ϕ0 are arbitrary real constants

⎤ (t − t0 ) + ϕ0⎥ ⎦

a1 a2 (n + 2) > 0 , A0, x0, t0, and ϕ0 are arbitrary real constants

a1 a2 (n + 2) < 0 , A0, x0, t0, and ϕ0 are arbitrary real constants

(3.7)

(3.6)

Eq. #

decaying wave

bright soliton

–

–

(3.13)

(3.12)

(3.11)

(3.10)

(3.9)

continuous wave, t- and (3.8) x-dependent phase

continuous wave, x-dependent phase

continuous wave, t-dependent phase

Name

a1 a2 (2 + n ) < 0 , A0, x0, and ϕ0 are arbitrary – real constants

A0, x0, t0, and ϕ0 are arbitrary real constants

⎤ (t − t0 ) + ϕ0⎥ ⎦

⎤ (t − t0 ) + ϕ0⎥ ⎦

⎧ −2 A02 a1 (n + 2) ⎫ n −i ⎢ 4 a1 A0 2[A (x − x )]⎬ e ⎣ n2 ψ (x , t ) = ⎨ csc 0 0 2 a2 n ⎩ ⎭

2

⎡

1

⎧ −2 A02 a1 (n + 2) ⎫ n −i ⎢ 4 a1 A0 2[A (x − x )]⎬ e ⎣ n2 ψ (x , t ) = ⎨ sec 0 0 2 a2 n ⎩ ⎭

⎤n ⎥ eiϕ (x − x 0 ) ⎥ ⎦

6.

−a 2 2 a1 (2 + n )

±1

2−n ⎤ ⎡ 2 a ∣ A ∣n (t − t ) 2 (x − x ) 2 i ⎢ 2 0 2−n 0 + 4 a (t −0 t ) + ϕ0⎥ ⎥ ⎢ 1 0 ⎦ e ⎣

⎡ ψ (x , t ) = ⎢ ⎢⎣ n

ψ (x , t ) =

A0 t − t0

A0, A1, x0, t0, and ϕ0 are arbitrary real constants

a1 a2 > 0 , A0, x0, and ϕ0 are arbitrary real constants

A0, t0, and ϕ0 are arbitrary real constants

Conditions

Equation: i ψt + a1 ψxx + a2 ∣ψ ∣n ψ = 0

5.

4.

a2 − A12 a1) (t − t0 ) + ϕ0⎤⎦

⎤ (x − x0 ) + ϕ0⎥⎦

3. ψ (x , t ) = A02/n ei ⎣A1 (x − x0) + (A0

a2 a1

⎡ 2 ⎤ A02/ n ei ⎣A0 a2 (t − t0) + ϕ0⎦

2. ψ (x , t ) = A 2/n ei ⎢⎣A0 0

1. ψ (x , t ) =

# Solution

3.2 Summary of Section 3.1

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

1

⎡

2

⎧ −2 A02 a1 (n + 2) ⎫ n i ⎢ 4 a1 A0 2[A (x − x )]⎬ e ⎣ n2 ψ (x , t ) = ⎨ csch 0 0 a2 n2 ⎩ ⎭

⎤ (t − t0 ) + ϕ0⎥ ⎦

3-7

ϕ (x , t ) =

13. ψ (x , t ) =

12. ψ (x , t ) =

+ sn2[A0 (x − x0 ), m ]

ei [λ1 (t − t0) + ϕ0]

+

λ0

λ0 R2

0

0

⎟

} dn[A (x − x ), m]⎞⎠

eiϕ(x, t ),

(R2 − R3) (R1 − R 4 ) sn2[A0 (x − x0 ), m ] (R1 − R3) (R2 − R 4 )

(x − x0 ), m ], m

(x − x0 ), m ]

(x − x0 ) + λ2 (t − t0 ) + ϕ0

1−

R2 (R1 − R 4 ) , am[A0 R1 (R2 − R 4 )

R1 R2

{

⎛ (R2 − R1) ⎜ Π ⎝ A0

−R2 + R 4 + (R1 − R 4 ) sn2 [A0

R1 (R2 − R 4 ) + R2 (R 4 − R1) sn2 [A0 (x − x0 ), m ]

R1 − R3 R3

R1 sn[A0 (x − x0 ), m ]

11. ψ (x , t ) = {R3 + (R2 − R3) sn2[A0 (x − x0 ), m ]} ei [λ1 (t − t0) + ϕ0]

10. ψ (x , t ) = A(x ) ei ϕ0 , A(x ) = Y −1(x − x0 )

9.

⎡ 1 1 n + 3 2 a2 An + 2 (x ) ⎤ 2F1⎢ ⎣ 2 , n + 2 , n + 2 , a1 A0 (n + 2) ⎥⎦, hypergeometric function and Y −1

A0

A (x )

− R1) > 0, n = 1,

x 3,

R3 (R1 − R2 ) , R2 (R1 − R3)

a2 R2 (R1 − R3) 3 a1

x 3,

, n = 4, R1 > 0 ,

x−

a2 a1

solitary wave

2 1 3 2 4 , Π is the A0 = 3 a1 incomplete elliptic integral, am is the amplitude for Jacobi elliptic functions, a1 a2 (R1 − R3) (R2 − R 4 ) < 0, n = 4, λ 0 , λ1, λ2 , x0, t0, and ϕ0 are arbitrary real constants

−a (R − R ) (R − R )

(R − R ) ( R − R ) m = (R2 − R 3) (R1 − R 4 ) , Rj, j = 1, 2, 3, 4 are the solitary wave 1 3 2 4 four roots of Y (x ) = 3 a1 λ 02 − 3 a1 λ1 x − 3 λ2 x 2 + a2 x 4 ,

a1 a2 R2 (R1 − R3) > 0, λ 0 , λ1, x0, t0, and ϕ0 are arbitrary real constants

A0 =

3 λ1 a1

Rj, j = 1, 2, 3 are the three

roots of Y (x ) = 3 λ 0 +

m=

λ 0 , λ1, x0, t0, and ϕ0 are arbitrary real constants

A0 =

generalized oscillatory solution

–

Rj, j = 1, 2, 3 are the three roots of solitary wave

3λ 2a 3 λ0 + a 1 x2 − a 2 1 1 a2 (R3 − R1) , a1 a2 (R3 6 a1

R2 − R3 , R1 − R3

Y (x ) =

m=

is 2F1 is the the inverse operator of the function Y [A(x )], A0 > 0, x0 and ϕ0 are arbitrary real constants

Y [A(x )] =

a1 a2 (n + 2) < 0 , A0, x0, t0, and ϕ0 are arbitrary real constants

(3.18)

(3.17)

(3.16)

(3.15)

(3.14)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

3.3 NLSE with Dual Power Law Nonlinearity Equation:

i ψt + a1 ψxx + a2 ∣ψ ∣n ψ + a3 ∣ψ ∣m ψ = 0,

(3.19)

where ψ = ψ (x , t ) is the complex function profile, x and t are its two independent variables, n, m, a1, a2, and a3 are arbitrary real constants. Solutions: Solution 1. Constant Amplitude I CW, t-dependent phase

ψ (x , t ) = A0 e i [(∣A0 ∣

n

a2+∣ A0 ∣m a3) (t − t0) + ϕ0 ],

(3.20)

where A0, t0, and ϕ0 are arbitrary real constants. • Derived in appendix A. Solution 2. Constant Amplitude II CW, x-dependent phase

ψ (x , t ) = A0

⎡ ∣ A ∣n a +∣ A ∣m a ⎤ 0 2 0 3 i⎢ (x − x0) + ϕ0⎥ a1 ⎣ ⎦, e

(3.21)

where A0, x0, and ϕ0 are arbitrary real constants. • Derived in appendix A. Solution 3. Constant Amplitude III CW, t- and x-dependent phase ⎡

ψ (x , t ) = A0 e i ⎣A1 (x − x0) + (∣A0 ∣

n

a2+∣ A0 ∣m a3 − a1 A12) (t − t0) + ϕ0⎤⎦

,

(3.22)

where A0, A1, x0, t0, and ϕ0 are arbitrary real constants. • Derived in appendix A. Solution 4. Rational Solution I DW

ψ (x , t ) =

A0 t − t0

2 −m 2 −n ⎧ ⎫ ⎪ ⎪ 2 ∣ A0 ∣m a3 (t − t0) 2 2 ∣ A0 ∣n a2(t − t0) 2 [2 a1 A1+ (x − x0)] 2 i ⎨− − + + ϕ0⎬ ⎪ ⎪ m−2 n−2 4 a1(t − t0) ⎭, e⎩

where

3-8

(3.23)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

n ≠ 2, m ≠ 2, A0, A1, x0, t0, and ϕ0 are arbitrary real constants.

• Derived in appendix A. Solution 5. Rational Solution II 1

⎡ ⎤n −2 a1 a2 (n + 1) (n + 2) ψ (x , t ) = ⎢ ⎥ e i ϕ0, ⎣ a1 a3 (n + 2)2 + a22 n2 (1 + n ) (x − x0)2 ⎦

(3.24)

where a1 a2 < 0, a1 a3 > 0, m = 2 n, x0 and ϕ0 are arbitrary real constants. • Reference: [2]. Solution 6.

ψ (x , t ) =

−a23 2 a3

x − x0 12 a1 a3 + a22 (x − x0)2

⎡ a2 ⎤ −i ⎢ 2 (t − t0) + ϕ0⎥ ⎦, e ⎣ 4 a3

(3.25)

where n = 2, m = 4, a2 a3 < 0, a1 a3 > 0, x0, t0, and ϕ0 are arbitrary real constants. • Reference: [2] with m = 1. Solution 7.

⎧ n+1 ψ (x , t ) = ⎨ ⎩ (n + 2) [1 + 2 e where A0 =

1

⎫n ⎬ e i [A1 (t − t0) + ϕ0], A 0 (x − x 0 ) ] ⎭

a2 n2 (n + 1) , a1 (n + 2)2 a1 A0 , n2

A1 = m = 2 n, a3 = −a2 ,

3-9

(3.26)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

a1 a2 (n + 1) > 0, x0, t0, and ϕ0 are arbitrary real constants.

• Reference: [2]. Solution 8. sech(x) flat-top soliton (Figure 3.1)

⎛ ⎜ ψ (x , t ) = ⎜ ⎜ ⎜ a2 ⎝

1

A0 A1 (n ⎧ ⎡ ⎨A0 + 2 cosh⎢n ⎣ ⎩

⎞n ⎟ + 2) ⎟ e i [A1 (t − t0) + ϕ0], ⎤ ⎟ A1 ( x − x0 ) ⎥ ⎟ a1 ⎦ ⎠

(3.27)

}

where A0 =

4 a 22 (n + 1) A1 δ (n + 2)2

δ = a3 +

,

a 22

(n + 1) , A1 (n + 2)2

m = 2 n, a1 A1 > 0, A1 δ (n + 1) > 0, x0, t0, A1, and ϕ0 are arbitrary real constants.

• Reference: [4]. Solution 9. tanh(x) dark soliton (Figure 3.2) 1

⎡

⎛ 2 a1 A 2 (n + 2) ⎞ n i⎢ 4 a1 2A0 0 ψ (x , t ) = ⎜ {1 − tanh[A0 (x − x0)]}⎟ e ⎣ n a2 n2 ⎠ ⎝

2

⎤ (t − t0) + ϕ0⎥ ⎦,

(3.28)

where A0 =

−a 22 n2 (1 + n ) 4 a1 a3 (2 + n )2

,

Figure 3.1. Plot of solution (3.27). (a) Flat-top soliton with a3 = −0.069444444444 , (b) bright soliton with a3 = 0.03055555555. For the other parameters: a1 = A1 = 2 , a2 = 1, n = 4, and x0 = t0 = ϕ0 = 0 .

3-10

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Figure 3.2. Dark soliton (3.28) with a1 = a2 = 1, a3 = −1, n = 2, and x0 = t0 = ϕ0 = 0 .

m = 2 n, a1 a2 (n + 2) > 0, a1 a3 (n + 1) < 0, x0, t0, and ϕ0 are arbitrary real constants.

• Reference: [5]. Solution 10. coth(x) 1

⎡

⎛ 2 a1 A 2 (n + 2) ⎞ n i⎢ 4 a1 2A0 0 ψ (x , t ) = ⎜ {1 − coth[A0 (x − x0)]}⎟ e ⎣ n a2 n2 ⎝ ⎠

2

⎤ (t − t0) + ϕ0⎥ ⎦,

(3.29)

where −a 22 n2 (1 + n ) 4 a1 a3 (2 + n )2

A0 =

,

m = 2 n, a1 a2 (n + 2) > 0, a1 a3 (n + 1) < 0, x0, t0, and ϕ0 are arbitrary real constants.

• Reference: [6]. Solution 11. sinh(x)

ψ (x , t ) = ±

μ1 μ2

(

)

μ12 + μ12 + 4 μ22 sinh2[A0 (x − x0)]

where 3 a2 a3

μ1 = μ2 =

A0 =

1 2

9 a 22

+

−3 a2 a3 a3 μ12 μ22 12 a1

a32

+

+ 9 a 22 a32

4 A1 a3

+

, 4 A1 a3

,

,

3-11

⎡A2 A a ⎤ i ⎢ 0 21 21 (t − t0) + ϕ0⎥ ⎢⎣ a3 μ1 μ 2 ⎥⎦ e ,

(3.30)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

3 a2 a3


0, n = 2, m = 4, x0, t0, A1, and ϕ0 are arbitrary real constants.

• Reference: [2] with m = 1. Solution 12. cosh(x) 1

⎛ n + 1 ⎞2 n ψ (x , t ) = ⎜ ⎟ ⎝ 2 a2n a3 n2 ⎠ ⎧ ⎪ ×⎨ ⎪ ⎩ ×

1

2

2 a2 n (n + 1) a3 (n + 2) 2

⎡ + A02 cosh⎢ ⎣

A02 A0 2 a1

⎤ ( x − x0 ) ⎥ + ⎦

2

2 a2 n (n + 1) a3 (n + 2) 2

⎫n ⎪ ⎬ ⎪ ⎭

(3.31)

⎡ A2 ⎤ i ⎢ 0 2 (t − t0) + ϕ0⎥ n 2 ⎣ ⎦, e

where a1 > 0, a2 = 1, a3 (n + 1) > 0, m = 2 n, x0, t0, A0, and ϕ0 are arbitrary real constants. • Reference: [7] with A2 = B2 = 0. Solution 13. sin(x) 1

⎛ −1 ⎞ n ψ (x , t ) = ⎜ ⎟ ⎝ 2 ⎠ ⎧ ⎪ ×⎨ ⎪ ⎩ ×

1

⎡ 2 (n + 1) ⎤ 2 n ⎢ n ⎥ ⎣ a 2 a3 n2 ⎦ 1

2 a2 n 2 (n + 1) a3 (n + 2) 2

⎡ − A02 sin⎢ ⎣

A02 A0 2 a1

⎡ −A 2 ⎤ 0 i⎢ (t − t0) + ϕ0⎥ 2 n 2 ⎦, e ⎣

3-12

⎤ ( x − x0 ) ⎥ + ⎦

2 a2 n 2 (n + 1) a3 (n + 2) 2

⎫n ⎪ ⎬ ⎪ ⎭

(3.32)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

where a1 > 0, a2 = 1, a3 (n + 1) > 0, m = 2 n, x0, t0, A0, and ϕ0 are arbitrary real constants. • Reference: [7] with A2 = B2 = 0. Solution 14. cos(x) 1

⎛ −1 ⎞ n ψ (x , t ) = ⎜ ⎟ ⎝ 2 ⎠ ⎧ ⎪ ×⎨ ⎪ ⎩ ×

1

⎡ 2 (n + 1) ⎤ 2 n ⎢ n ⎥ ⎣ a 2 a3 n2 ⎦ 1

2 a2 n 2 (n + 1) a3 (n + 2) 2

⎡ − A02 cos⎢ ⎣

A02 A0 2 a1

⎤ ( x − x0 ) ⎥ + ⎦

⎡ −A 2 ⎤ 0 i⎢ (t − t0) + ϕ0⎥ 2 n 2 ⎦, e ⎣

where a1 > 0, a2 = 1, a3 (n + 1) > 0, m = 2 n, x0, t0, A0, and ϕ0 are arbitrary real constants. • Reference: [7] with A2 = B2 = 0.

3-13

2 a2 n 2 (n + 1) a3 (n + 2) 2

⎫n ⎪ ⎬ ⎪ ⎭

(3.33)

n

ψ (x , t ) = A0 e

∣ A0 ∣n a2+∣ A0 ∣m a3 a1

n

⎡ i⎢ ⎣

⎡

3-14

6.

5.

× 1

2−m 2−n ⎫ ⎧ ⎪ ⎪ −2 A0m a3 (t − t0 ) 2 2 A n a 2 (t − t 0 ) 2 [2 a A + (x − x )]2 i⎨ − 0 n−2 + 14 a1 (t − t )0 + ϕ0⎬ m−2 1 0 ⎪ ⎪ ⎭ ⎩ e

A0 t − t0

ψ (x , t ) =

−a 23 2 a3

12

a1 a3 + a 22

(x − x 0

x − x0 )2

e

⎡ a2 ⎤ −i ⎢ 4 a2 (t − t0 ) + ϕ0⎥ ⎣ 3 ⎦

⎡ ⎤ n i ϕ0 −2 a1 a2 (n + 1) (n + 2) ψ (x , t ) = ⎢⎣ ⎦ e a1 a3 (n + 2)2 + a 22 n2 (1 + n ) (x − x0 )2 ⎥

4. ψ (x , t ) =

Eq. #

(3.25)

n = 2, m = 4, a2 a3 < 0 , a1 a3 > 0 , x0, t0, and ϕ0 are arbitrary – real constants

(3.23)

(3.24)

decaying wave

continuous wave, (3.22) t- and xdependent phase

continuous wave, (3.21) x-dependent phase

continuous wave, (3.20) t-dependent phase

Name

–

a1 a2 < 0 , a1 a3 > 0, m = 2 n , x0 and ϕ0 are arbitrary real constants

n ≠ 2 , m ≠ 2 , A0, A1, x0, t0, and ϕ0 are arbitrary real constants

A0, A1, x0, t0, and ϕ0 are arbitrary real constants

A0, x0, and ϕ0 are arbitrary real constants

A0, t0, and ϕ0 are arbitrary real constants

Conditions

Equation: i ψt + a1 ψxx + a2 ∣ψ ∣n ψ + a3 ∣ψ ∣m ψ = 0

a2+∣ A0 ∣m a3 − a1 A12 ) (t − t0 ) + ϕ0⎤⎦

⎤ (x − x 0 ) + ϕ 0 ⎥ ⎦

a2+∣ A0 ∣m a3) (t − t0 ) + ϕ0 ]

3. ψ (x , t ) = A0 ei ⎣A1 (x − x0) + (∣ A0 ∣

2.

1. ψ (x , t ) = A0 ei [(∣ A0 ∣

# Solution

3.4 Summary of Section 3.3

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

3-15

12.

1

ψ (x , t ) =

ψ (x , t ) =

ψ (x , t ) =

)

1 2n

⎡ + A02 cosh⎢ ⎣

⎡ A2 ⎤ i ⎢ 02 (t − t0 ) + ϕ0⎥ ⎣2 n ⎦

2 a2 n2 (n + 1) a3 (n + 2)2

n+1 2 a 2n a3 n2

×e

μ1 μ2

2 a1

A0

A02 ⎤ (x − x 0 )⎥ + ⎦

μ12 + (μ12 + 4 μ22 ) sinh2[A0 (x − x0 )]

⎡ A 2 A1 a1 ⎤ i ⎢ 0 2 2 (t − t0 ) + ϕ0⎥ ⎣ a3 μ1 μ2 ⎦

⎧ ⎪ ×⎨ ⎪ ⎩

(

×e

)

1 n

1 n

2 a2 n2 (n + 1) a3 (n + 2)2

{1 − coth[A0 (x − x0 )]}

⎡ 4 a1 A 2 ⎤ 0 i⎢ (t − t0 ) + ϕ0⎥ ⎦ e ⎣ n2

2 a1 A02 (n + 2) a2 n2

×

(

)

{1 − tanh[A0 (x − x0 )]}

⎡ 4 a1 A 2 ⎤ 0 i⎢ (t − t0 ) + ϕ0⎥ ⎦ e ⎣ n2

2 a1 A02 (n + 2) a2 n2

×

(

⎛ ⎞n ⎜ ⎟ i [A (t − t ) + ϕ ] A0 A1 (n + 2) ψ (x , t ) = ⎜ ⎧ e 1 0 0 ⎡ ⎤⎫ ⎟ A1 ⎜ a2 ⎨A0 + 2 cosh⎢n a (x − x0)⎥⎬ ⎟ ⎣ ⎦⎭ ⎠ 1 ⎝ ⎩

1

⎧ ⎫n n+1 ⎬ ei [A1 (t − t0) + ϕ0] ψ (x , t ) = ⎨ ⎡ ⎤ ⎩ (n + 2) ⎣1 + 2 e A0 (x − x0)⎦ ⎭

11. ψ (x , t ) = ±

10.

9.

8.

7.

1

⎫n ⎪ ⎬ ⎪ ⎭

a2 n2 (n + 1) , a1 (n + 2)2

A1 =

a1 A0 , n2

m = 2 n , a3 = − a2 ,

4 a 22 (n + 1) A1 δ (n + 2)2

, δ = a3 +

a 22 (n + 1) , A1 (n + 2)2

m = 2 n , a1 A1 > 0 ,

−a 22 n2 (1 + n ) 4 a1 a3 (2 + n )2

, m = 2 n , a1 a2 (n + 2) > 0 ,

−a 22 n2 (1 + n ) 4 a1 a3 (2 + n )2

, m = 2 n , a1 a2 (n + 2) > 0 ,

3 a2 a3

12 a1 3

+

4 A1 a3

, 3 a2 < a

a32

9 a 22

a3 μ12 μ22

+ a32

9 a 22

+

, μ2 =

4 A1 a3

1 2

+

a32

9 a 22

+

4 A1 a3

, , a1 a3 > 0 , n = 2, m = 4,

−3 a2 a3

–

–

dark soliton

(3.31)

(3.30)

(3.29)

(3.28)

(3.27)

(3.26)

(Continued)

flat-top soliton

–

a1 > 0 , a2 = 1, a3 (n + 1) > 0 , m = 2 n , x0, t0, A0, and ϕ0 are – arbitrary real constants

x0, t0, A1, and ϕ0 are arbitrary real constants

A0 =

μ1 =

a1 a3 (n + 1) < 0 , x0, t0, and ϕ0 are arbitrary real constants

A0 =

a1 a3 (n + 1) < 0 , x0, t0, and ϕ0 are arbitrary real constants

A0 =

A1 δ (n + 1) > 0 , x0, t0, A1, and ϕ0 are arbitrary real constants

A0 =

a1 a2 (n + 1) > 0 , x0, t0, and ϕ0 are arbitrary real constants

A0 =

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

14.

13.

ψ (x , t ) =

ψ (x , t ) =

# Solution

(Continued )

3-16

⎡ − A02 sin⎢ ⎣

1

⎡ − A02 cos⎢ ⎣

⎡ 2 (n + 1) ⎤ 2 n ⎢⎣ a2n a3 n2 ⎥⎦

2 a2 n2 (n + 1) a3 (n + 2)2

1 n

⎡ −A 2 ⎤ i ⎢ 02 (t − t0 ) + ϕ0⎥ ⎣2n ⎦

⎧ ⎪ ×⎨ ⎪ ⎩

−1 2

×e

2 a2 n2 (n + 1) a3 (n + 2)2

⎡ 2 (n + 1) ⎤ ⎣⎢ a2n a3 n2 ⎦⎥

1 2n

⎡ −A 2 ⎤ i ⎢ 02 (t − t0 ) + ϕ0⎥ ⎣2n ⎦

( )

×e

⎧ ⎪ ×⎨ ⎪ ⎩

−1 2

( )

1 n

2 a1

A0

A02

2 a1

A0

A02

⎤ ( x − x 0 )⎥ + ⎦

⎤ ( x − x 0 )⎥ + ⎦

2 a2 n2 (n + 1) a3 (n + 2)2

2 a2 n2 (n + 1) a3 (n + 2)2

1

⎫n ⎪ ⎬ ⎪ ⎭

1

⎫n ⎪ ⎬ ⎪ ⎭

Name

a1 > 0 , a2 = 1, a3 (n + 1) > 0 , m = 2 n , x0, t0, A0, and ϕ0 are arbitrary real constants

–

a1 > 0 , a2 = 1, a3 (n + 1) > 0 , m = 2 n , x0, t0, A0, and ϕ0 are – arbitrary real constants

Conditions

Equation: i ψt + a1 ψxx + a2 ∣ψ ∣n ψ + a3 ∣ψ ∣m ψ = 0

(3.33)

(3.32)

Eq. #

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

References [1] Zaitsev V F and Polyanin A D 2003 Handbook of Nonlinear Partial Differential Equations (New York: Chapman and Hall) [2] He B and Meng Q 2016 Qualitative analysis and explicit exact solitary, kink and anti-kink wave solutions of the generalized nonlinear Schrödinger equation with parabolic law nonlinearity Commun. Theor. Phys. 65 1–10 [3] Zayed E M and Al-Nowehy A G 2017 Exact solutions for the perturbed nonlinear Schrödinger equation with power law nonlinearity and Hamiltonian perturbed terms Optik 139 123–44 [4] Al Khawaja U and Bahlouli H 2019 Integrability conditions and solitonic solutions of the nonlinear Schrödinger equation with generalized dual-power nonlinearities, PT-symmetric potentials, and space-and time-dependent coefficients Commun. Nonlinear Sci. Numer. Simul. 69 248–60 [5] Triki H and Biswas A 2011 Dark solitons for a generalized nonlinear Schrödinger equation with parabolic law and dual-power law nonlinearities Math. Methods Appl. Sci. 34 958–62 [6] Mirzazadeh M, Eslami M, Milovic D and Biswas A 2014 Topological solitons of resonant nonlinear Schödinger’sequation with dual-power law nonlinearity by GG-expansion technique Optik 125 5480–9 [7] Zhang L H and Si J G 2010 New soliton and periodic solutions of (1 + 2)-dimensional nonlinear Schrödinger equation with dual-power law nonlinearity Commun. Nonlinear Sci. Numer. Simul. 15 2747–54

3-17

IOP Publishing

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations Usama Al Khawaja and Laila Al Sakkaf

Chapter 4 Nonlinear Schrödinger Equation with Higher Order Terms

A Glance at Chapter 4

doi:10.1088/978-0-7503-2428-1ch4

4-1

ª IOP Publishing Ltd 2020

4-2

9 2

i ψt + a1 ψxx + a2 ∣ψ ∣2 ψ + a3 ∣ψ ∣4 ψ + i a 4 ψxxx + a5 ψxxxx + i a 6 (∣ψ ∣2 ψ )x + i a7 (∣ψ ∣2 )x ψ = 0

i ψt + (1 − ∣μ∣ ∣ψ ∣2 ) ψxx + 2 (1 − ∣μ∣) ∣ψ ∣2 ψ = 0

i ψt + a2 k2 − i a3 k3 + a 4 k4 − i a5 k5 + … m = 0

i ψt + a2(t ) k2 − i a3(t ) k3 + a 4(t ) k4 − i a5(t ) k5 + … m = 0

11

8

9

10

11

Total

57

2

8

9

i ψt + a1 ψxx + a2 ∣ψ ∣2 ψ + a3 ∣ψ ∣4 ψ + i a 4 ψxxx + a5 ψxxxx = 0

i ψt + a1 ψxx + a2 ∣ψ ∣2 ψ + a3 ψxxxx = 0

5

7

5

i ψt + i a1 ψx + a2 ψxx − i a3 ψxxx + a 4 ∣ψ ∣2 ψ − i a5 ∣ψ ∣2 ψx − i a 6 ψ 2 ψx* − a7 ψ = 0

4

5

4

i ψt + a1 ψxx + a2 ∣ψ ∣2 ψ + i a 4 [ψxxx + (∣ψ ∣2 )x ψ + ∣ψ ∣2 ψx ] = 0

3

i ψt + a1 ψxx + a2 ∣ψ ∣2 n ψ + a3 ψxxxx = 0

0

i ψt + a1 ψxx + a2 ∣ψ ∣2 ψ + i a3 ψxxx + i a 4 ∣ψ ∣2 ψx = 0

2

6

0

i ψt + a1 ψxx + a2 ∣ψ ∣2 ψ + i a3 ψxxx + i a 4 (∣ψ ∣2 ψ )x + i a5 (∣ψ ∣2 )x ψ = 0

13

Solutions

1

Equation

A Statistical View of Chapter 4

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

4.1 NLSE with Third Order Dispersion, Self-Steepening, and Self-Frequency Shift Equation:

i ψt + a1 ψxx + a2 ∣ψ ∣2 ψ + i a3 ψxxx + i a 4 (∣ψ ∣2 ψ )x + i a5 (∣ψ ∣2 )x ψ = 0,

(4.1)

where ψ = ψ (x , t ) is the complex function profile, x and t are its two independent variables, aj are arbitrary real constants, j = 1, 2, 3, 4, 5. Solutions: Solution 1. Constant Amplitude continuous wave (CW), t- and x-dependent phase

ψ ( x , t ) = c e i [ c (x − x 0 ) + c

2

(−a1+ a2 + c a3 − c a 4) (t − t0) + ϕ0 ],

(4.2)

where x0, t0, c, and ϕ0 are arbitrary real constants. Solution 2. sec(x,t)

ψ (x , t ) = ± −μ1 sec

{

}

(4.3)

}

(4.4)

μ2 [x − x0 + c1 (t − t0)] e i [c2 (x − x0) + c3 (t − t0) + ϕ0],

where 6 (c1 + 2 a1 c2 − 3 a3 c22 ) < 0, 3 a 4 + 2 a5 2 c + 2 a1 c2 − 3 a3 c2 μ2 = 1 > 0, a3 −3 a2 a3 + a1 (3 a 4 + 2 a5 ) , c2 = 6 a3 (a 4 + a5 ) a c +2a2 c c3 = 8 a1 c22 − 8 a3 c23 − 1 1 a 1 2 3

μ1 =

+ 3 c1 c2 , x0, t0, c1, and ϕ0 are arbitrary real constants.

• Reference: [1]. Solution 3. csc(x,t)

ψ (x , t ) = ± −μ1 csc

{

μ2 [x − x0 + c1 (t − t0)] e i [c2 (x − x0) + c3 (t − t0) + ϕ0],

where 6 (c1 + 2 a1 c2 − 3 a3 c22 ) < 0, (3 a 4 + 2 a5 ) 2 c + 2 a1 c2 − 3 a3 c2 μ2 = 1 > 0, a3 −3 a2 a3 + a1 (3 a 4 + 2 a5 ) , c2 = 6 a3 (a 4 + a5 )

μ1 =

4-3

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

(a c + 2 a 2 c )

c3 = 8 a1 c22 − 8 a3 c23 − 1 1 a 1 2 + 3 c1 c2 , 3 x0, t0, c1, and ϕ0 are arbitrary real constants.

• Reference: [1]. Solution 4. tan(x,t)

ψ (x , t ) = ± μ1 tan

{

}

(4.5)

}

(4.6)

−μ2 [x − x0 + c1 (t − t0)] e i [c2 (x − x0) + c3 (t − t0) + ϕ0],

where 3 (c1 + 2 a1 c2 − 3 a3 c22 ) > 0, (3 a 4 + 2 a5 ) c + 2 a1 c2 − 3 a3 c22 μ2 = 1 < 0, 2 a3 −3 a2 a3 + a1 (3 a 4 + 2 a5 ) , c2 = 6 a3 (a 4 + a5 ) (a c + 2 a 2 c ) c3 = 8 a1 c22 − 8 a3 c23 − 1 1 a 1 2

μ1 =

+ 3 c1 c2 , x0, t0, c1, and ϕ0 are arbitrary real constants. 3

• Reference: [1]. Solution 5. cot(x,t)

ψ (x , t ) = ± μ1 cot

{

−μ2 [x − x0 + c1 (t − t0)] e i [c2 (x − x0) + c3 (t − t0) + ϕ0],

where 3 (c1 + 2 a1 c2 − 3 a3 c22 ) > 0, (3 a 4 + 2 a5 ) 2 c + 2 a1 c2 − 3 a3 c2 μ2 = 1 < 0, 2 a3 −3 a2 a3 + a1 (3 a 4 + 2 a5 ) , c2 = 6 a3 (a 4 + a5 ) (a c + 2 a 2 c ) c3 = 8 a1 c22 − 8 a3 c23 − 1 1 a 1 2 3

μ1 =

+ 3 c1 c2 , x0, t0, c1, and ϕ0 are arbitrary real constants.

• Reference: [1]. Solution 6. sech(x,t) bright soliton (Figure 4.1)

ψ (x , t ) = ± −μ1 sech

{

}

−μ2 [x − x0 + c1 (t − t0)] e i [c2 (x − x0) + c3 (t − t0) + ϕ0], (4.7)

where μ1 = μ2 =

6 (c1 + 2 a1 c2 − 3 a3 c22 ) < 0, (3 a 4 + 2 a5 ) 2 c1 + 2 a1 c2 − 3 a3 c2 < 0, a3

4-4

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Figure 4.1. Bright soliton (4.7) with a1 = 1/2 , a2 = 1, a3 = −1, a 4 = −3, a5 = 2 , c1 = 9/10 , and x0 = t0 = ϕ0 = 0 .

c2 =

−3 a2 a3 + a1 (3 a 4 + 2 a5 ) , 6 a3 (a 4 + a5 ) (a c + 2 a 2 c )

c3 = 8 a1 c22 − 8 a3 c23 − 1 1 a 1 2 + 3 c1 c2 , 3 x0, t0, c1, and ϕ0 are arbitrary real constants.

• Reference: [1]. Solution 7. csch(x,t)

ψ (x , t ) = ± μ1 csch

{

}

(4.8)

}

(4.9)

−μ2 [x − x0 + c1 (t − t0)] e i [c2 (x − x0) + c3 (t − t0) + ϕ0],

where 6 (c1 + 2 a1 c2 − 3 a3 c22 ) > 0, (3 a 4 + 2 a5 ) 2 c + 2 a1 c2 − 3 a3 c2 μ2 = 1 < 0, a3 −3 a2 a3 + a1 (3 a 4 + 2 a5 ) , c2 = 6 a3 (a 4 + a5 ) (a c + 2 a 2 c ) c3 = 8 a1 c22 − 8 a3 c23 − 1 1 a 1 2

μ1 =

+ 3 c1 c2 , x0, t0, c1, and ϕ0 are arbitrary real constants. 3

• Reference: [1]. Solution 8. tanh(x,t) dark soliton (Figure 4.2)

ψ (x , t ) = ± −μ1 tanh

{

μ2 [x − x0 + c1 (t − t0)] e i [c2 (x − x0) + c3 (t − t0) + ϕ0],

where 3 (c1 + 2 a1 c2 − 3 a3 c22 ) < 0, (3 a 4 + 2 a5 ) 2 c + 2 a1 c2 − 3 a3 c2 μ2 = 1 > 0, 2 a3 −3 a2 a3 + a1 (3 a 4 + 2 a5 ) , c2 = 6 a3 (a 4 + a5 )

μ1 =

4-5

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Figure 4.2. Dark soliton (4.9) with a1 = −4 , a2 = 1, a3 = −1, a 4 = −1, a5 = 2 , c1 = 9/10 , and x0 = t0 = ϕ0 = 0 .

(a c + 2 a 2 c )

c3 = 8 a1 c22 − 8 a3 c23 − 1 1 a 1 2 + 3 c1 c2 , 3 x0, t0, c1, and ϕ0 are arbitrary real constants.

• Reference: [1]. Solution 9. coth(x,t)

ψ (x , t ) = ± −μ1 coth

{

}

μ2 [x − x0 + c1 (t − t0)] e i [c2 (x − x0) + c3 (t − t0) + ϕ0],

(4.10)

where 3 (c1 + 2 a1 c2 − 3 a3 c22 ) < 0, (3 a 4 + 2 a5 ) 2 c + 2 a1 c2 − 3 a3 c2 μ2 = 1 > 0, 2 a3 −3 a2 a3 + a1 (3 a 4 + 2 a5 ) , c2 = 6 a3 (a 4 + a5 ) (a c + 2 a 2 c ) c3 = 8 a1 c22 − 8 a3 c23 − 1 1 a 1 2 3

μ1 =

+ 3 c1 c2 , x0, t0, c1, and ϕ0 are arbitrary real constants.

• Reference: [1]. Solution 10. Rational Solution decaying wave (DW)

⎫ ⎧ ⎪ ⎪ −6 a3 ⎬ ψ (x , t ) = ± ⎨ ⎪ 3 a 4 + 2 a5 ⎡⎣ x − x0 + 3 a3 c22 − 2 a1 c2 (t − t0) + c1⎤⎦ ⎪ (4.11) ⎭ ⎩

(

× e i [c2 (x − x0) + c3 (t − t0) + ϕ0], where c2 = c3 =

a1(3 a 4 + 2 a5 ) − 3 a2 a3 , 6 a3 (a 4 + a5 ) 3 2 a3 c2 − a1 c2 ,

4-6

)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

a3 < 0, x0, t0, c1, and ϕ0 are arbitrary real constants.

• Reference: [1]. Solution 11.

ψ (x , t ) = ±

⎛ μ1 ⎜ cot −2 μ1 [x − x0 − c1 (t − t0)] μ2 ⎜ csc −2 μ1 [x − x0 − c1 (t − t0)] + ⎝

{

}

{

}

⎞ ⎟ 1 ⎟⎠

(4.12)

× e i [c2 (x − x0) + c3 (t − t0) + ϕ0], where −c1 + 2 a1 c2 − 3 a3 c22 < 0, a3 3 a 4 + 2 a5 μ2 = 3 a < 0, 3 −3 a2 a3 + a1 (3 a 4 + 2 a5 ) , c2 = 6 a3 (a 4 + a5 ) (a c − 2 a 2 c ) c3 = 8 a1 c22 − 8 a3 c23 + 1 1 a 1 2

μ1 =

− 3 c1 c2 , x0, t0, c1, and ϕ0 are arbitrary real constants. 3

• Reference: [1]. Solution 12. ψ (x , t ) ⎛ − 3 sec 2 μ [x − x − c (t − t )] + tan 2 μ [x − x − c (t − t )] 0 1 0 0 1 0 1 1 =± ⎜⎜ 2 sec 2 μ1 [x − x0 − c1 (t − t0)] + 1 ⎝ μ1 i [c (x − x ) + c (t − t ) + ϕ ] 0 3 0 0 , e 2 × μ2

{

}

{

{

}

where c1 − 2 a1 c2 + 3 a3 c22 > 0, a3 −3 a 4 − 2 a 5 μ2 = > 0, 3 a3 3 a a − a (3 a + 2 a ) c2 = − 2 63 a (1a + a4 ) 5 , 3 4 5

μ1 =

(a c − 2 a 2 c )

c3 = 8 a1 c22 − 8 a3 c23 + 1 1 a 1 2 − 3 c1 c2 , 3 x0, t0, c1, and ϕ0 are arbitrary real constants.

• Reference: [1].

4-7

} ⎞⎟

⎟(4.13) ⎠

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Solution 13. ψ (x, t ) ⎛ 5 csch 2 μ [x − x − c (t − t )] + coth 2 μ [x − x − c (t − t )] 0 1 0 0 1 0 1 1 =± ⎜ ⎜ 2 csch 2 μ1 [x − x 0 − c1 (t − t0 )] + 1 ⎝ μ × − 1 e i [c 2 (x − x0) + c3 (t − t0) + ϕ0 ], μ2

{

}

{

{

}

where −c1 + 2 a1 c2 − 3 a3 c22 > a3 3a +2a μ2 = 43 a 5 < 0, 3 a (3 a + 2 a ) − 3 a a c2 = 1 64a (a 5+ a ) 2 3 , 3 4 5

μ1 =

0,

(a c − 2 a 2 c )

c3 = 8 a1 c22 − 8 a3 c23 + 1 1 a 1 2 − 3 c1 c2 , 3 x0, t0, c1, and ϕ0 are arbitrary real constants.

• Reference: [1].

4-8

} ⎞⎟

⎟ (4.14) ⎠

2

(−a1+ c a3 + a2 − c a 4 ) (t − t0 ) + ϕ0 ]

4-9

4.

3. ψ (x , t ) = ± − μ1 csc{ μ2 [x − x0 + c1 (t − t0 )]} × ei [c2 (x − x0) + c3 (t − t0) + ϕ0]

2. ψ (x , t ) = ± − μ1 sec{ μ2 [x − x0 + c1 (t − t0 )]} × ei [c2 (x − x0) + c3 (t − t0) + ϕ0]

1. ψ (x , t ) = c ei [c (x − x0) + c

# Solution

Equation

4

5

Name

Eq. #

(a1 c1 + 2 a12 c2 ) a3

+ 3 c1 c2,

6 (c1 + 2 a1 c2 − 3 a3 c22 ) (3 a 4 + 2 a5)

< 0,

(a1 c1 + 2 a12 c2 ) a3

+ 3 c1 c2,

μ1 =

3 (c1 + 2 a1 c2 − 3 a3 c22 ) (3 a 4 + 2 a5)

> 0,

x0, t0, c1, and ϕ0 are arbitrary real constants

−

μ2 =

c1 + 2 a1 c2 − 3 a3 c22 > 0, a3 −3 a2 a3 + a1 (3 a 4 + 2 a5) , c2 = 6 a3 (a 4 + a5) 2 3 c3 = 8 a1 c2 − 8 a3 c2

μ1 =

x0, t0, c1, and ϕ0 are arbitrary real constants

−

μ2 =

c1 + 2 a1 c2 − 3 a3 c22 > 0, a3 −3 a2 a3 + a1 (3 a 4 + 2 a5) , c2 = 6 a3 (a 4 + a5) 2 3 c3 = 8 a1 c2 − 8 a3 c2

–

–

(Continued)

(4.5)

(4.4)

x0, t0, c, and ϕ0 are arbitrary real constants continuous wave, t- and (4.2) x- dependent phase – 6 (c1 + 2 a1 c2 − 3 a3 c22 ) (4.3) , μ1 = < 0 (3 a + 2 a )

Conditions

i ψt + a1 ψxx + a2 ∣ψ ∣2 ψ + i a3 ψxxx + i a 4 (∣ψ ∣2 ψ )x + i a5 (∣ψ ∣2 )x ψ = 0

4.2 Summary of Section 4.1

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

4-10

7. ψ (x , t ) = ± μ1 csch{ − μ2 [x − x0 + c1 (t − t0 )]} × ei [c2 (x − x0) + c3 (t − t0) + ϕ0]

6. ψ (x , t ) = ± − μ1 sech{ − μ2 [x − x0 + c1 (t − t0 )]} × ei [c2 (x − x0) + c3 (t − t0) + ϕ0]

5. ψ (x , t ) = ± μ1 cot{ − μ2 [x − x0 + c1 (t − t0 )]} × ei [c2 (x − x0) + c3 (t − t0) + ϕ0]

× ei [c2 (x − x0) + c3 (t − t0) + ϕ0]

ψ (x , t ) = ± μ1 tan{ − μ2 [x − x0 + c1 (t − t0 )]}

(a1 c1 + 2 a12 c2 ) a3

+ 3 c1 c2,

3 (c1 + 2 a1 c2 − 3 a3 c22 ) (3 a 4 + 2 a5)

> 0,

(a1 c1 + 2 a12 c2 ) a3

+ 3 c1 c2,

6 (c1 + 2 a1 c2 − 3 a3 c22 ) (3 a 4 + 2 a5)

< 0,

(a1 c1 + 2 a12 c2 ) a3

+ 3 c1 c2,

c2 =

μ2 =

μ1 =

> 0, c1 + 2 a1 c2 − 3 a3 c22 < 0, a3 −3 a2 a3 + a1 (3 a 4 + 2 a5) , 6 a3 (a 4 + a5)

6 (c1 + 2 a1 c2 − 3 a3 c22 ) (3 a 4 + 2 a5)

x0, t0, c1, and ϕ0 are arbitrary real constants

−

μ2 =

c1 + 2 a1 c2 − 3 a3 c22 < 0, a3 −3 a2 a3 + a1 (3 a 4 + 2 a5) , c2 = 6 a3 (a 4 + a5) 2 3 c3 = 8 a1 c2 − 8 a3 c2

μ1 =

x0, t0, c1, and ϕ0 are arbitrary real constants

−

μ2 =

c1 + 2 a1 c2 − 3 a3 c22 < 0, 2 a3 −3 a2 a3 + a1 (3 a 4 + 2 a5) , c2 = 6 a3 (a 4 + a5) 2 3 c3 = 8 a1 c2 − 8 a3 c2

μ1 =

x0, t0, c1, and ϕ0 are arbitrary real constants

−

c1 + 2 a1 c2 − 3 a3 c22 < 0, 2 a3 −3 a2 a3 + a1 (3 a 4 + 2 a5) , c2 = 6 a3 (a 4 + a5) 2 3 c3 = 8 a1 c2 − 8 a3 c2

μ2 =

–

bright soliton

–

(4.8)

(4.7)

(4.6)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

4-11

11. ψ (x , t ) = ±

10. ψ (x , t ) = ±

−6 a3

μ1 ⎛ cot{ −2 μ1 [x − x0 − c1 (t − t0 )]} ⎞ ⎜ ⎟ μ2 ⎝ csc{ −2 μ1 [x − x0 − c1 (t − t0 )]} + 1 ⎠

× ei [c2 (x − x0) + c3 (t − t0) + ϕ0]

3 a 4 + 2 a5 [x − x0 + (3 a3 c22 − 2 a1 c2 ) (t − t0 ) + c1]

× ei [c2 (x − x0) + c3 (t − t0) + ϕ0]

{

9. ψ (x , t ) = ± − μ1 coth{ μ2 [x − x0 + c1 (t − t0 )]} × ei [c2 (x − x0) + c3 (t − t0) + ϕ0]

8. ψ (x , t ) = ± − μ1 tanh{ μ2 [x − x0 + c1 (t − t0 )]} × ei [c2 (x − x0) + c3 (t − t0) + ϕ0]

}

(a1 c1 + 2 a12 c2 ) a3

+ 3 c1 c2,

3 (c1 + 2 a1 c2 − 3 a3 c22 ) (3 a 4 + 2 a5)

< 0,

(a1 c1 + 2 a12 c2 ) a3

+ 3 c1 c2,

3 (c1 + 2 a1 c2 − 3 a3 c22 ) (3 a 4 + 2 a5)

< 0,

(a1 c1 + 2 a12 c2 ) a3

+ 3 c1 c2,

a1(3 a 4 + 2 a5) − 3 a2 a3 , 6 a3 (a 4 + a5) 3 2 a3 c2 − a1 c2 , a3
0, 2 a3 −3 a2 a3 + a1 (3 a 4 + 2 a5) , c2 = 6 a3 (a 4 + a5) 2 3 c3 = 8 a1 c2 − 8 a3 c2

μ2 =

μ1 =

x0, t0, c1, and ϕ0 are arbitrary real constants

−

μ2 =

c1 + 2 a1 c2 − 3 a3 c22 > 0, 2 a3 −3 a2 a3 + a1 (3 a 4 + 2 a5) , c2 = 6 a3 (a 4 + a5) 2 3 c3 = 8 a1 c2 − 8 a3 c2

μ1 =

x0, t0, c1, and ϕ0 are arbitrary real constants

−

c3 = 8 a1 c22 − 8 a3 c23

–

decaying wave

–

dark soliton

(Continued)

(4.12)

(4.11)

(4.10)

(4.9) Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

4-12

⎛ ×⎜ ⎝

ei [c2 (x − x0) + c3 (t − t0) + ϕ0]

⎠

⎟

⎠

2 csch{ 2 μ1 [x − x0 − c1 (t − t0 )]} + 1

5 csch{ 2 μ1 [x − x0 − c1 (t − t0 )]} + coth{ 2 μ1 [x − x0 − c1 (t − t0 )]} ⎞

2

μ1 μ2

2 sec{ 2 μ1 [x − x0 − c1 (t − t0 )]} + 1

⎟

−3 sec{ 2 μ1 [x − x0 − c1 (t − t0 )]} + tan{ 2 μ1 [x − x0 − c1 (t − t0 )]} ⎞

13. ψ (x , t ) = ± − μ1 ei [c2 (x − x0) + c3 (t − t0) + ϕ0] μ

×

12. ψ (x , t ) = ± ⎛⎜ ⎝

+

(a1 c1 − 2 a12 c2 ) a3

− 3 c1 c2,

−3 a2 a3 + a1 (3 a 4 + 2 a5) , 6 a3 (a 4 + a5) 2 3 8 a1 c2 − 8 a3 c2

3 a2 a3 − a1 (3 a 4 + 2 a5) , 6 a3 (a 4 + a5)

> 0,

(a1 c1 − 2 a12 c2 ) a3

− 3 c1 c2,

−c1 + 2 a1 c2 − 3 a3 c22 a3

> 0,

(a1 c1 − 2 a12 c2 ) a3

− 3 c1 c2,

x0, t0, c1, and ϕ0 are arbitrary real constants

+

3 a 4 + 2 a5 < 0, 3 a3 a1 (3 a 4 + 2 a5) − 3 a2 a3 , c2 = 6 a3 (a 4 + a5) 2 3 c3 = 8 a1 c2 − 8 a3 c2

μ2 =

μ1 =

x0, t0, c1, and ϕ0 are arbitrary real constants

+

c3 = 8 a1 c22 − 8 a3 c23

c2 = −

−3 a 4 − 2 a5 3 a3

μ2 =

> 0,

c1 − 2 a1 c2 + 3 a3 c22 a3

μ1 =

x0, t0, c1, and ϕ0 are arbitrary real constants

c3 =

c2 =

–

–

(4.14)

(4.13)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

4.3 Special Cases of Equation (4.1) 4.3.1 Case I: Hirota Equation (HE)

i ψt + a1 ψxx + a2 ∣ψ ∣2 ψ + i a3 ψxxx + i a 4 ∣ψ ∣2 ψx = 0.

(4.15)

Solutions of (4.15) can be obtained from solutions of 4.1 for a5 = −a 4 . 4.3.2 Case II: Sasa–Satsuma Equation (SSE)

i ψt + a1 ψxx + a2 ∣ψ ∣2 ψ + i a 4 [ψxxx + (∣ψ ∣2 )x ψ + ∣ψ ∣2 ψx ] = 0.

(4.16)

Solutions of (4.16) can be obtained from solutions of 4.1 for a 4 = a3 and a5 = 0.

4.4 NLSE with First and Third Order Dispersions, Self-Steepening, Self-Frequency Shift, and Potential Equation: i ψt + i a1 ψx + a2 ψxx − i a3 ψxxx + a 4 ∣ψ ∣2 ψ − i a5 ∣ψ ∣2 ψx − i a6 ψ 2 ψx* − a7 ψ = 0,

(4.17)

where ψ = ψ (x , t ) is the complex function profile, x and t are its two independent variables, aj are real constants, j = 1, 2, … , 7. Solutions: Solution 1. Constant Amplitude CW, t- and x-dependent phase

ψ (x , t ) = A0 e i [A1 (x − x0) + A2 (t − t0) + ϕ0],

(4.18)

where A2 = A02 [a 4 + A1 (a5 − a6 )] − A1 [a1 + A1 (a2 + a3 A1)] − a7 , x0, t0, A0, A1, ϕ0, and aj are arbitrary real constants, j = 1, 2, … , 7. Solution 2. ψ (x , t ) = λ tanh{η [x − x 0 − χ (t − t0 )]} + i ρ sech{η [x − x 0 − χ (t − t0 )]}, (4.19)

where a1 = −2 α1 Ω + 3 a3 Ω2 , a2 = α1 − 3 a3 Ω, α α a3 = 31 α 3 , 2

4-13

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

a 4 = α2 − α3 Ω , a5 = 2 α3 + α4 , a6 = α3 + α4 , a7 = κ + α1 Ω2 − a3 Ω3, α α4 = − 23 , κ=− Ω=

2 α1 α22

3 α32 α2 , α3

,

χ = −(α1 Ω + α3 λ2 ) − a3 η 2 , α η = 3 a3 (ρ 2 − λ2 ) , 3

α2 α1

(ρ 2 − λ 2 ) > 0 , x0, t0, α1, α2 , α3, ρ, and λ are arbitrary real constants.

• Reference: [2]. Solution 3. ψ (x , t ) = i β + λ tanh{η [x − x0 − χ (t − t0)]} + i λ sech{η [x − x0 − χ (t − t0)]},

where a1 = −2 α1 Ω + 3 a3 Ω2 , a2 = α1 − 3 a3 Ω, a3 = 0, a 4 = α2 − α3 Ω , a5 = 2 α3 + α4 , a6 = α3 + α4 , a7 = κ + α1 Ω2 − a3 Ω3, α4 = −α3, κ = (α2 − α3 Ω) (λ2 + β 2 ) − α1 Ω2 , χ = −(2 α1 Ω + α3 β 2 ), α βλ η = − 3α , 1

λ=

2 α1 (α3 Ω − α2 ) α3

,

α1 (α3 Ω − α2 ) > 0, x0, t0, α1, α2 , α3, β, ρ, and Ω are arbitrary real constants.

• Reference: [2].

4-14

(4.20)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Figure 4.3. Plot of solution (4.21) with α2 = 1, α4 = 3, Ω = λ = 1/5, β = −3, and x0 = t0 = 0 .

Solution 4. (Figure 4.3)

ψ (x , t ) = i β + λ tanh[η (x − x0)] + i ρ sech[η (x − x0)], where a1 = −2 α1 Ω + 3 a3 Ω2 , a2 = α1 − 3 a3 Ω, a3 = 0, a 4 = α2 − α3 Ω , a5 = 2 α3 + α4 , a6 = α3 + α4 , a7 = κ + α1 Ω2 − a3 Ω3, α1 = 0, 3α α4 = − 2 3 , κ = (α2 − α3 Ω) (λ2 + β 2 ), β (α − α Ω) η = − λ (α2 + α3 ) , 3

4

ρ = λ2 + 2 β 2 , x0, t0, α2 , α3, β, λ , and Ω are arbitrary real constants.

• Reference: [2].

4-15

(4.21)

Equation

4-16 χ = − (α1 Ω + α3

λ2 ) − a3

η2 ,

η

,

3 α32 α = 3 a3 3

2 α1 α22

− λ2 ) > 0

x0, t0, α1, α2 , α3, ρ, and λ are arbitrary real constants

Ω=

α2 , α3

a7 = κ + α1 Ω2 − a3 Ω3, α4 = − 23 , κ = −

α

a 4 = α2 − α3 Ω , a5 = 2 α3 + α4 , a 6 = α3 +

(ρ 2 − λ 2 ) ,

α1 α3 , 3 α2 α2 α 4 , α (ρ 2 1

a1 = − 2 α1 Ω + 3 a3 Ω2 , a2 = α1 − 3 a3 Ω , a3 =

4. ψ (x , t ) = i β + λ tanh[η (x − x0 )] + i ρ sech[η (x − x0 )]

α3 β λ , α1

λ=

α3

2 α1 (α3 Ω − α2 )

, α1 (α3 Ω − α2 ) > 0 ,

β (α2 − α3 Ω) , λ (α3 + α4 )

ρ=

λ2 + 2 β 2 ,

x0, t0, α2 , α3, β , λ , and Ω are arbitrary real constants

κ = (α2 − α3 Ω) (λ2 + β 2 ), η = −

a1 = − 2 α1 Ω + 3 a3 Ω2 , a2 = α1 − 3 a3 Ω , a3 = 0 , a 4 = α2 − α3 Ω , a5 = 2 α3 + α4 , a 6 = α3 + α4 , a7 = κ + α1 Ω2 − a3 Ω3, 3α α1 = 0 , α4 = − 2 3 ,

x0, t0, α1, α2 , α3, β , ρ, and Ω are arbitrary real constants

χ = − (2 α1 Ω + α3 β 2 ), η = −

3. ψ (x , t ) = i β + λ tanh{η [x − x0 − χ (t − t0 )]} a1 = − 2 α1 Ω + 3 a3 Ω2 , a2 = α1 − 3 a3 Ω , a3 = 0 , + i λ sech{η [x − x0 − χ (t − t0 )]} a 4 = α2 − α3 Ω , a5 = 2 α3 + α4 , a 6 = α3 + α4 , α4 = − α3, a7 = κ + α1 Ω2 − a3 Ω3, κ = (α2 − α3 Ω) (λ2 + β 2 ) − α1 Ω2 ,

2. ψ (x , t ) = λ tanh{η [x − x0 − χ (t − t0 )]} + i ρ sech{η [x − x0 − χ (t − t0 )]}

continuous wave

A2 = A02 [a 4 + A1 (a5 − a 6 )] − A1 [a1 + A1 (a2 + a3 A1)] − a7 , x0,t0, A0, A1, ϕ0 , and aj are arbitrary real constants, j = 1, 2, … , 7

1. ψ (x , t ) = A0 ei [A1 (x − x0) + A2 (t − t0) + ϕ0]

–

–

–

t- and x-dependent phase

Name

Conditions

# Solution

i ψt + i a1 ψx + a2 ψxx − i a3 ψxxx + a 4 ∣ψ ∣2 ψ − i a5 ∣ψ ∣2 ψx − i a 6 ψ 2 ψx* − a7 ψ = 0

4.5 Summary of Section 4.4

(4.21)

(4.20)

(4.19)

(4.18)

Eq. #

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

4.6 NLSE with Fourth Order Dispersion Equation:

i ψt + a1 ψxx + a2 ∣ψ ∣2 ψ + a3 ψxxxx = 0,

(4.22)

where ψ = ψ (x , t ) is the complex function profile, x and t are its two independent variables, aj are arbitrary real constants, j = 1, 2, 3. Solutions: Solution 1. Constant Amplitude CW, t- and x-dependent phase

ψ (x , t ) = c e

⎡ a ⎤ i ⎢ a1 (x − x0) + c 2 a2 (t − t0) + ϕ0⎥ ⎣ 3 ⎦,

(4.23)

where a1 a3 > 0, x0, t0, c and ϕ0 are arbitrary real constants. Solution 2. sec(x) ⎡

ψ (x , t ) = a1

⎡ ⎤ i ⎢ −4 a1 −3 a1 sec2 ⎢ (x − x0)⎥e ⎣ 25 a3 10 a2 a3 ⎣ 20 a3 ⎦

2

⎤ (t − t0) + ϕ0⎥ ⎦,

(4.24)

⎤ (t − t0) + ϕ0⎥ ⎦,

(4.25)

where a1 a3 > 0, a2 a3 < 0, x0, t0, and ϕ0 are arbitrary real constants. • Reference: [3]. Solution 3. csc(x) ⎡

ψ (x , t ) = a1

⎡ ⎤ i ⎢ −4 a1 −3 a1 csc2⎢ (x − x0)⎥e ⎣ 25 a3 10 a2 a3 ⎣ 20 a3 ⎦

where a1 a3 > 0, a2 a3 < 0, x0, t0, and ϕ0 are arbitrary real constants. • Reference: [3].

4-17

2

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Solution 4. sech(x) bright soliton ⎡

ψ (x , t ) = a1

⎡ −a1 ⎤ i ⎢ −4 a1 −3 sech2 ⎢ (x − x0)⎥e ⎣ 25 a3 10 a2 a3 ⎣ 20 a3 ⎦

2

⎤ (t − t0) + ϕ0⎥ ⎦,

(4.26)

⎤ (t − t0) + ϕ0⎥ ⎦,

(4.27)

where a1 a3 < 0, a2 a3 < 0, x0, t0, and ϕ0 are arbitrary real constants. • Reference: [3]. Solution 5. csch(x) ⎡

ψ (x , t ) = a1

⎡ −a1 ⎤ i ⎢ −4 a1 −3 2 csch ⎢ (x − x0)⎥e ⎣ 25 a3 10 a2 a3 ⎣ 20 a3 ⎦

where a1 a3 < 0, a2 a3 < 0, x0, t0, and ϕ0 are arbitrary real constants. • Reference: [3].

4-18

2

4-19

5.

4.

3.

2.

1.

#

ψ (x , t ) = a1

ψ (x , t ) = a1

ψ (x , t ) = a1

ψ (x , t ) = a1

ψ (x , t ) = c e

Solution

a1 a3

−3 10 a2 a3

−3 10 a2 a3

−3 10 a2 a3

⎢⎣

⎡ csch2⎣⎢

⎡ sech2

⎡ csc 2⎣⎢

⎡ sec 2 ⎣⎢

−a1 20 a3

⎤ (t − t0 ) + ϕ0⎥ ⎦

⎡ −4 a 2

⎤ i⎢ 1 (x − x0 )⎥⎦e ⎣ 25 a3

⎤ i⎢ 1 (x − x0 )⎥⎦e ⎣ 25 a3

⎤ (t − t0 ) + ϕ0⎥ ⎦

⎤ (t − t0 ) + ϕ0⎥ ⎦

⎤ (t − t0 ) + ϕ0⎥ ⎦

⎡ −4 a 2

⎡ −4 a 2

Equation

x0, t0, and ϕ0 are arbitrary real constants

a1 a3 < 0 , a2 a3 < 0 ,

x0, t0, and ϕ0 are arbitrary real constants

a1 a3 < 0 , a2 a3 < 0 ,

x0, t0, and ϕ0 are arbitrary real constants

a1 a3 > 0 , a2 a3 < 0 ,

x0, t0, and ϕ0 are arbitrary real constants

–

bright soliton

–

–

t- and x-dependent phase

x0, t0, c and ϕ0 are arbitrary real constants

a1 a3 > 0 , a2 a3 < 0 ,

continuous wave,

Name

a1 a3 > 0 ,

Conditions

i ψt + a1 ψxx + a2 ∣ψ ∣2 ψ + a3 ψxxxx = 0

⎡ −4 a 2

⎤ i⎢ 1 (x − x0 )⎥⎦e ⎣ 25 a3

⎤ i⎢ 1 (x − x0 )⎥⎦e ⎣ 25 a3

−a1 20 a3

a1 20 a3

a1 20 a3

⎤ (x − x0 ) + c 2 a2 (t − t0 ) + ϕ0⎦⎥

−3 10 a2 a3

⎡ i ⎢⎣

4.7 Summary of Section 4.6

(4.27)

(4.26)

(4.25)

(4.24)

(4.23)

Eq. #

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

4.8 NLSE with Fourth Order Dispersion and Power Law Nonlinearity Equation:

i ψt + a1 ψxx + a2 ∣ψ ∣2 n ψ + a3 ψxxxx = 0,

(4.28)

where ψ = ψ (x , t ) is the complex function profile, x and t are its two independent variables, n and aj are arbitrary real constants, j = 1, 2, 3. Solutions: Solution 1. Constant Amplitude CW, t- and x-dependent phase

ψ (x , t ) = c e

⎡ a ⎤ i ⎢ a1 (x − x0) + c 2 n a2 (t − t0) + ϕ0⎥ ⎣ 3 ⎦,

(4.29)

where a1 a3 > 0, x0, t0, c, and ϕ0 are arbitrary real constants. Solution 2. sec(x)

ψ (x , t ) =

{

1 n

−μ1 sec2 ⎡⎣ μ2 (x − x0)⎤⎦

}

e−i ⎡⎣μ3 (t − t0) + ϕ0⎤⎦,

(4.30)

where μ1 = μ2 = μ3 =

a12 (n + 1) (n + 2) (3 n + 2) 4 a3 a2 (n2 + 2 n + 2)2 a1 n2 > 0, 4 a3 (n2 + 2 n + 2) 2 2 a1 (n + 1) , a3 (n2 + 2 n + 2)2

< 0,

x0, t0, and ϕ0 are arbitrary real constants. • Reference: [3]. Solution 3. csc(x)

ψ (x , t ) =

{

−μ1 csc2⎡⎣ μ2 (x − x0)⎤⎦

where μ1 = μ2 =

a12 (n + 1) (n + 2) (3 n + 2) 4 a3 a2 (n2 + 2 n + 2)2 a1 n2 > 0, 4 a3 (n2 + 2 n + 2)

1 n −i [μ3 (t − t0) + ϕ0 ] ,

}e

< 0,

4-20

(4.31)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

μ3 =

a12 (n + 1)2 , a3 (n2 + 2 n + 2)2

x0, t0, and ϕ0 are arbitrary real constants. • Reference: [3]. Solution 4. sech(x) bright soliton

ψ (x , t ) =

{

−μ1 sech2 ⎡⎣ −μ2 (x − x0)⎤⎦

1 n

}

e−i [μ3 (t − t0) + ϕ0],

(4.32)

1 n −i [μ3 (t − t0) + ϕ0 ] ,

(4.33)

where μ1 = μ2 = μ3 =

a12 (n + 1) (n + 2) (3 n + 2) 4 a3 a2 (n2 + 2 n + 2)2 a1 n2 < 0, 4 a3 (n2 + 2 n + 2) 2 a1 (n + 1)2 , a3 (n2 + 2 n + 2)2

< 0,

x0, t0, and ϕ0 are arbitrary real constants. • Reference: [3]. Solution 5. csch(x)

ψ (x , t ) =

{

−μ1 csch2⎡⎣ −μ2 (x − x0)⎤⎦

}e

where μ1 = μ2 = μ3 =

a12 (n + 1) (n + 2) (3 n + 2) 4 a3 a2 (n2 + 2 n + 2)2 a1 n2 < 0, 4 a3 (n2 + 2 n + 2) 2 2 a1 (n + 1) , a3 (n2 + 2 n + 2)2

< 0,

x0, t0, and ϕ0 are arbitrary real constants. • Reference: [3].

4-21

Solution

4-22

4. ψ (x , t ) =

3. ψ (x , t ) =

2. ψ (x , t ) =

a1 a3

(x − x 0 ) + c 2 n

1 n

1 n

}

− μ1 sech2 ⎡⎣ − μ2 (x − x0 )⎤⎦

× e−i [μ3 (t − t0) + ϕ0]

{

1 n

}

}

− μ1 csc 2⎡⎣ μ2 (x − x0 )⎤⎦

× e−i [μ3 (t − t0) + ϕ0]

{

⎤ a2 (t − t0 ) + ϕ0⎦⎥

− μ1 sec 2 ⎡⎣ μ2 (x − x0 )⎤⎦

⎡ i ⎣⎢

× e−i [μ3 (t − t0) + ϕ0]

{

1. ψ (x , t ) = c e

#

4.9 Summary of Section 4.8

a3

a12 (n 2 (n + 1)2 , + 2 n + 2)2

a1 4 a3 (n2 + 2 n + 2)

n2

> 0,

a12 (n + 1) (n + 2) (3 n + 2) 4 a3 a2 (n2 + 2 n + 2)2

< 0,

a3

a12 (n 2 + 2 n + 2)2

(n + 1)2

,

a1 4 a3 (n2 + 2 n + 2)

n2

> 0,

a12 (n + 1) (n + 2) (3 n + 2) 4 a3 a2 (n2 + 2 n + 2)2

< 0,

a1 n2 4 a3 (n2 + 2 n + 2)

< 0,

< 0, μ3 =

a12 (n + 1) (n + 2) (3 n + 2) 4 a3 a2 (n2 + 2 n + 2)2

a12 (n + 1)2 , a3 (n2 + 2 n + 2)2

x0, t0, and ϕ0 are arbitrary real constants

μ2 =

μ1 =

x0, t0, and ϕ0 are arbitrary real constants

μ3 =

μ2 =

μ1 =

x0, t0, and ϕ0 are arbitrary real constants

μ3 =

μ2 =

μ1 =

x0, t0, c, and ϕ0 are arbitrary real constants

a1 a3 > 0 ,

Conditions

i ψt + a1 ψxx + a2 ∣ψ ∣2 n ψ + a3 ψxxxx = 0

Equation

Eq. #

bright soliton

–

–

(4.32)

(4.31)

(4.30)

continuous wave, t- and x-dependent phase (4.29)

Name

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

5. ψ (x , t ) =

1 n

}

− μ1 csch2⎡⎣ − μ2 (x − x0 )⎤⎦

× e−i [μ3 (t − t0) + ϕ0]

{ a12 (n + 1)2 , a3 (n2 + 2 n + 2)2

μ3 =

< 0,

< 0,

x0, t0, and ϕ0 are arbitrary real constants

a1 n2 4 a3 (n2 + 2 n + 2)

a12 (n + 1) (n + 2) (3 n + 2) 4 a3 a2 (n2 + 2 n + 2)2

μ2 =

μ1 =

–

(4.33)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

4-23

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

4.10 NLSE with Third and Fourth Order Dispersions and Cubic and Quintic Nonlinearities Equation:

i ψt + a1 ψxx + a2 ∣ψ ∣2 ψ + a3 ∣ψ ∣4 ψ + i a 4 ψxxx + a5 ψxxxx = 0,

(4.34)

where ψ = ψ (x , t ) is the complex function profile, x and t are its two independent variables, aj are arbitrary real constants, j = 1, 2, 3, 4, 5. Solutions: Solution 1. Constant Amplitude CW, t- and x-dependent phase 2 2 2 ⎡ 2 ⎤ ψ (x , t ) = c1 e i {c2 (x − x0) + ⎣c1 (a2+ c1 a3) + c2 (−a1+ c2 a 4+ c2 a5)⎦ (t − t0) + ϕ0},

(4.35)

where x0, t0, c1, c2, and ϕ0 are arbitrary real constants. Solution 2. sec(x,t)

ψ (x , t ) = ± 2 c 1

−6 a5 sec[c1 (x − x0) + c2 (t − t0)] e i [c3 (x − x0) + c 4 (t − t0) + ϕ0], (4.36) a3

where c1 = −

⎛ 3 a 42 + 8 a5 ⎜a1 + a2 ⎝

c2 = −c1 c3 =

−6 a5 a3

80 a52

(

a 43 + 4 a1 a 4 a5 192 a53

⎞ ⎟ ⎠

,

),

−a 4 , 4 a5

c4 = a5 c14 −

(8 a5 a1 + 3 a 42 ) 8 a5

c12 −

3 a 44 + 16 a1 a 42 a5 256 a53

,

a3 a5 < 0,

(

3 a 42 + 8 a5 a1 + a2

−6 a 5 a3

) > 0,

x0, t0, and ϕ0 are arbitrary real constants. • Reference: [4]. Solution 3. csc(x,t)

ψ (x , t ) = ± 2 c 1

−6 a5 csc[c1 (x − x0) + c2 (t − t0)] e i [c3 (x − x0) + c 4 (t − t0) + ϕ0], (4.37) a3

4-24

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

where c1 = −

⎛ 3 a 42 + 8 a5 ⎜a1 + a2 ⎝

⎞ ⎟ ⎠

80 a52

(

c2 = −c1 c3 =

−6 a5 a3

a 43 + 4 a1 a 4 a5 192 a53

,

),

−a 4 , 4 a5 (8 a5 a1 + 3 a 42 ) 8 a5

c4 = a5 c14 −

3 a 44 + 16 a1 a 42 a5

c12 −

256 a53

,

a3 a5 < 0,

(

−6 a 5 a3

3 a 42 + 8 a5 a1 + a2

) > 0,

x0, t0, and ϕ0 are arbitrary real constants. • Reference: [4]. Solution 4. tan(x,t)

ψ (x , t ) = ± 2 c 1

−6 a5 tan[c1 (x − x0) + c2 (t − t0)] e i [c3 (x − x0) + c 4 (t − t0) + ϕ0], (4.38) a3

where c1 = −

⎛ −3 a 42 − 8 a5 ⎜a1 + a2 ⎝

⎞ ⎟ ⎠

160 a52

(

c2 = −c1 c3 =

−6 a5 a3

a 43 + 4 a1 a 4 a5 192 a53

,

),

−a 4 , 4 a5

c4 = 16 a5 c14 +

(8 a5 a1 + 3 a 42 ) 4 a5

c12 −

3 a 44 + 16 a1 a 42 a5 256 a53

,

a3 a5 < 0,

(

−6 a 5 a3

3 a 42 + 8 a5 a1 + a2

) < 0,

x0, t0, and ϕ0 are arbitrary real constants. • Reference: [4]. Solution 5. cot(x,t)

ψ (x , t ) = ± 2 c 1

−6 a5 cot[c1 (x − x0) + c2 (t − t0)] e i [c3 (x − x0) + c 4 (t − t0) + ϕ0], (4.39) a3

where c1 = − c2 = −c1

⎛ −3 a 42 − 8 a5 ⎜a1 + a2 ⎝ 160 a52

(

a 43 + 4 a1 a 4 a5 192 a53

−6 a5 a3

⎞ ⎟ ⎠

,

), 4-25

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

c3 =

−a 4 , 4 a5

c4 = 16 a5 c14 +

(8 a5 a1 + 3 a 42 ) 4 a5

c12 −

3 a 44 + 16 a1 a 42 a5 256 a53

,

a3 a5 < 0,

(

−6 a 5 a3

3 a 42 + 8 a5 a1 + a2

) < 0,

x0, t0, and ϕ0 are arbitrary real constants. • Reference: [4]. Solution 6. csch(x,t) ψ (x , t ) = ± 2 c1

− 6 a5 csch[c1 (x − x 0 ) + c 2 (t − t0 )] e i [c3 (x− x0) + c 4 (t − t0) + ϕ0], a3

(4.40)

where c1 = −

⎛ −3 a 42 − 4 a5 ⎜2 a1 + a2 ⎝

⎞ ⎟ ⎠

80 a52

(

c2 = −c1 c3 =

−24 a5 a3

a 43 + 4 a1 a 4 a5 192 a53

,

),

−a 4 , 4 a5

c4 = a5 c14 +

2

(8 a5 a1 + 3 a 4 ) 8 a5

3 a 44 + 16 a1 a 42 a5

c12 −

256 a53

,

a3 a5 < 0,

(

−3 a 42 − 4 a5 2 a1 + a2

−24 a5 a3

) > 0,

x0, t0, and ϕ0 are arbitrary real constants. • Reference: [4], we corrected the constant prefactor. Solution 7. sech(x,t) bright soliton ψ (x , t ) = ± 2 c1

− 6 a5 sech[c1 (x − x 0 ) + c 2 (t − t0 )] e i [c3 (x− x0) + c 4 (t − t0) + ϕ0], a3

where c1 = −

⎛ −3 a 42 − 8 a5 ⎜a1 − a2 ⎝

c2 = −c1 c3 =

80 a52

(

a 43 + 4 a1 a 4 a5 192 a53

−6 a5 a3

⎞ ⎟ ⎠

,

),

−a 4 , 4 a5

4-26

(4.41)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

c4 = a5 c14 +

(8 a5 a1 + 3 a 42 ) 8 a5

c12 −

(

−6 a 5 a3

3 a 44 + 16 a1 a 42 a5 256 a53

,

a3 a5 < 0, −3 a 42 − 8 a5 a1 − a2

) > 0,

x0, t0, and ϕ0 are arbitrary real constants. • Reference: [4]. Solution 8. tanh(x,t) dark soliton

ψ (x , t ) = ± 2 c 1

−6 a5 tanh[c1 (x − x0) + c2 (t − t0)] a3

(4.42)

e i [c3 (x − x0) + c 4 (t − t0) + ϕ0], where ⎛ 3 a 42 + 8 a5 ⎜a1 + a2 ⎝

c1 = −

⎞ ⎟ ⎠

160 a52

(

c2 = −c1 c3 =

−6 a5 a3

a 43 +

4 a1 a 4 a5

192 a53

,

),

−a 4 , 4 a5

c4 = 16 a5 c14 −

2

(8 a5 a1 + 3 a 4 ) 4 a5

c12 −

3 a 44 + 16 a1 a 42 a5 256 a53

,

a3 a5 < 0,

(

−6 a 5 a3

3 a 42 + 8 a5 a1 + a2

) > 0,

x0, t0, and ϕ0 are arbitrary real constants. • Reference: [4]. Solution 9. coth(x,t)

ψ (x , t ) = ± 2 c 1

−6 a5 coth[c1 (x − x0) + c2 (t − t0)] a3

e i [c3 (x − x0) + c 4 (t − t0) + ϕ0], where c1 = − c2 = −c1

⎛ 3 a 42 + 8 a5 ⎜a1 + a2 ⎝ 160 a52

(

a 43 + 4 a1 a 4 a5 192 a53

−6 a5 a3

⎞ ⎟ ⎠

,

),

4-27

(4.43)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

c3 =

−a 4 , 4 a5

c4 = 16 a5 c14 −

(8 a5 a1 + 3 a 42 ) 4 a5

c12 −

3 a 44 + 16 a1 a 42 a5 256 a53

a3 a5 < 0,

(

3 a 42 + 8 a5 a1 + a2

−6 a 5 a3

) > 0,

x0, t0, and ϕ0 are arbitrary real constants. • Reference: [4].

4-28

,

sec[c1 (x − x0 ) + c2 (t − t0 )]

4-29

−6 a5 a3

csc[c1 (x − x0 ) + c2 (t − t0 )]

× ei [c3 (x − x0) + c4 (t − t0) + ϕ0]

3. ψ (x , t ) = ± 2 c 1

× ei [c3 (x − x0) + c4 (t − t0) + ϕ0]

−6 a5 a3

Equation

(

80

a52

−6 a5 a3

) > 0,

(

80

a52

a3 a5 < 0 , 3 a 42 + 8 a5 (a1 + a2

−6 a5 a3

) > 0,

256 a53

3 a 44 + 16 a1 a 42 a5

−a 4 , 4 a5

,

c12 −

3

), c =

−6 a5 ⎞ ⎟ a3 ⎠

(8 a5 a1 + 3 a 42 ) 8 a5

192 a53

a 43 + 4 a1 a 4 a5

c4 = a5 c14 −

c2 = − c1

c1 = −

⎛ 3 a 42 + 8 a5 ⎜a1 + a2 ⎝

x0, t0, and ϕ0 are arbitrary real constants

a3 a5 < 0 , 3 a 42 + 8 a5 (a1 + a2

256 a53

3 a 44 + 16 a1 a 42 a5

−a 4 , 4 a5

,

c12 −

3

), c =

−6 a5 ⎞ ⎟ a3 ⎠

(8 a5 a1 + 3 a 42 ) 8 a5

192 a53

a 43 + 4 a1 a 4 a5

c4 = a5 c14 −

c2 = − c1

c1 = −

⎛ 3 a 42 + 8 a5 ⎜a1 + a2 ⎝

,

,

x0, t0, c1, c2, and ϕ0 are arbitrary real constants

Conditions

i ψt + a1 ψxx + a2 ∣ψ ∣2 ψ + a3 ∣ψ ∣4 ψ + i a 4 ψxxx + a5 ψxxxx = 0

2 2 2 2 ei {c2 (x − x0) + [c1 (a2 + c1 a3) + c2 (−a1+ c2 a4 + c2 a5)] (t − t0) + ϕ0}

2. ψ (x , t ) = ± 2 c 1

1. ψ (x , t ) = c1

# Solution

4.11 Summary of Section 4.10

–

–

(4.37)

(4.36)

(4.35)

Eq. #

(Continued)

continuous wave, t- and xdependent phase

Name

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

−6 a5 a3

tan[c1 (x − x0 ) + c2 (t − t0 )]

−6 a5 a3

cot[c1 (x − x0 ) + c2 (t − t0 )]

4-30

−6 a5 a3

csch[c1 (x − x0 ) + c2 (t − t0 )]

× ei [c3 (x − x0) + c4 (t − t0) + ϕ0]

6. ψ (x , t ) = ± 2 c1

× ei [c3 (x − x0) + c4 (t − t0) + ϕ0]

5. ψ (x , t ) = ± 2 c 1

× ei [c3 (x − x0) + c4 (t − t0) + ϕ0]

4. ψ (x , t ) = ± 2 c1

( a 42 + 8 a5 (a1 + a2

−6 a5 a3

) < 0,

256 a53

3 a 44 + 16 a1 a 42 a5

,

−6 a5 a3

) < 0,

( a 4 a5

3

c12 −

x0, t0, and ϕ0 are arbitrary real constants

−24 a5 a3

256 a53

,

) > 0,

3 a 44 + 16 a1 a 42 a5

a3 a5 < 0 , − 3 a 42 − 4 a5 (2 a1 + a2

(8 a5 a1 + 3 a 42 ) 8 a5

, −a 4 , 4 a5

−24 a5 ⎞ ⎟ a3 ⎠

), c =

80 a52 192 a53

a 43 + 4 a1

c4 = a5 c14 +

c2 = − c1

c1 = −

⎛ −3 a 42 − 4 a5 ⎜2 a1 + a2 ⎝

x0, t0, and ϕ0 are arbitrary real constants

a3 a5 < 0 , 3 a 42 + 8 a5 (a1 + a2

c4 = 16

160 a 43 + 4 a1 a 4 a5

a52

−6 a5 ⎞ ⎟ a3 ⎠

,

−a ), c3 = 4 a4 , 192 a53 5 3 a 44 + 16 a1 a 42 a5 (8 a5 a1 + 3 a 42 ) 2 , a5 c14 + c 1 − 4 a5 256 a53

c2 = − c1 (

c1 = −

⎛ −3 a 42 − 8 a5 ⎜a1 + a2 ⎝

x0, t0, and ϕ0 are arbitrary real constants

a3 a5 < 0 , 3

c12 −

192 a53 (8 a5 a1 + 3 a 42 ) 4 a5

,

−a 4 , 4 a5

3

−6 a5 ⎞ ⎟ a3 ⎠

), c =

a52

a 4 a5

160 a 43 + 4 a1

c4 = 16 a5 c14 +

c2 = − c1

c1 = −

⎛ −3 a 42 − 8 a5 ⎜a1 + a2 ⎝

x0, t0, and ϕ0 are arbitrary real constants

–

–

–

(4.40)

(4.39)

(4.38)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

−6 a5 a3

sech[c1 (x − x0 ) + c2 (t − t0 )]

−6 a5 a3

tanh[c1 (x − x0 ) + c2 (t − t0 )]

4-31

−6 a5 a3

coth[c1 (x − x0 ) + c2 (t − t0 )]

× ei [c3 (x − x0) + c4 (t − t0) + ϕ0]

9. ψ (x , t ) = ± 2 c 1

× ei [c3 (x − x0) + c4 (t − t0) + ϕ0]

8. ψ (x , t ) = ± 2 c 1

× ei [c3 (x − x0) + c4 (t − t0) + ϕ0]

7. ψ (x , t ) = ± 2 c 1

(

80

,

−6 a5 a3

(

(

) > 0,

256 a53

3 a 44 + 16 a1 a 42 a5

(

c12 −

192 a53

(8 a5 a1 + 3 a 42 ) 4 a5

, −a 4 , 4 a5

3

−6 a5 ⎞ ⎟ a3 ⎠

), c =

160

a52

a 43 + 4 a1 a 4 a5

−6 a5 a3

) > 0,

256 a53

3 a 44 + 16 a1 a 42 a5

x0, t0, and ϕ0 are arbitrary real constants

a3 a5 < 0 , 3 a 42 + 8 a5 (a1 + a2

c4 = 16 a5 c14 −

c2 = − c1

c1 = −

⎛ 3 a 42 + 8 a5 ⎜a1 + a2 ⎝

x0, t0, and ϕ0 are arbitrary real constants

3 a 42 + 8 a5 a1 + a2

−6 a5 a3

c12 −

(8 a5 a1 + 3 a 42 ) 4 a5

, −a 4 , 4 a5

3

−6 a5 ⎞ ⎟ a3 ⎠

), c =

a 4 a5

192 a53

a 43 + 4 a1

160 a52

⎛ 3 a 42 + 8 a5 ⎜a1 + a2 ⎝

c4 = 16 a5 c14 −

c2 = − c1

c1 = −

,

) > 0,

x0, t0, and ϕ0 are arbitrary real constants

a3 a5 < 0 , − 3 a 42 − 8 a5 (a1 − a2

256 a53

3 a 44 + 16 a1 a 42 a5

−a 4 , 4 a5

c12 −

3

−6 a5 ⎞ ⎟ a3 ⎠

), c =

(8 a5 a1 + 3 a 42 ) 8 a5

192 a53

a 43 + 4 a1 a 4 a5

a52

⎛ −3 a 42 − 8 a5 ⎜a1 − a2 ⎝

c4 = a5 c14 +

c2 = − c1

c1 = −

,

, a3 a5 < 0 ,

–

dark soliton

bright soliton

(4.43)

(4.42)

(4.41)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

4.12 NLSE with Third and Fourth Order Dispersions, Self-Steepening, Self-Frequency Shift, and Cubic and Quintic Nonlinearities Equation: i ψt + a1 ψxx + a2 ∣ψ ∣2 ψ + a3 ∣ψ ∣4 ψ + i a 4 ψxxx + a5 ψxxxx + i a6 (∣ψ ∣2 ψ )x + i a7 (∣ψ ∣2 )x ψ = 0,

(4.44)

where ψ = ψ (x , t ) is the complex function profile, x and t are its two independent variables, aj are arbitrary real constants, j = 1, 2, 3, 4, 5, 6, 7. Solutions: Solution 1. Constant Amplitude CW, t- and x-dependent phase ⎡

ψ (x , t ) = c1 e i ⎣c2 (x − x0) + (c1

2

a2 + c14 a3 (t − t0) + ϕ0⎤⎦

)

,

(4.45)

where a4 =

a 6 c12 + a1 c2 − a5 c23 c22

,

x0, t0, c1, c2, and ϕ0 are arbitrary real constants. Solution 2. sec(x,t) ψ (x , t ) =

− 6 (4 a5 c 3 + a 4 ) sec[x − x 0 + c1 (t − t0 )] e i [c3 (x− x0) + c 2 (t − t0) + ϕ0], 3 a 6 + 2 a7

(4.46)

where c1 = −2 a1 c3 + a 4 (3 c32 + 1) + 4 a5 (c33 + c3), c2 = −a1 (c32 + 1) + a 4 (c33 + 3 c3) + a5 (c34 + 6 c32 + 1), (4 a5 c3 + a 4 ) (3 a2 + 2 c3 a7 ) a1 = + 2 a 4 c3 + 2 a5 (c32 + 5), 3a +2a 6

a3 =

7

−2 a5 (3 a 6 + 2 a7 )2 , 3 (a 4 + 4 a5 c3)2

(4 a5 c3 + a 4 ) (3 a6 + 2 a7 ) < 0, x0, t0, c3, and ϕ0 are arbitrary real constants.

• Reference: [5]. Solution 3. csc(x,t) ψ (x , t ) =

− 6 (4 a5 c 3 + a 4 ) csc[x − x 0 + c1 (t − t0 )] e i [c3 (x− x0) + c 2 (t − t0) + ϕ0], 3 a 6 + 2 a7

4-32

(4.47)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

where c1 = −2 a1 c3 + a 4 (3 c32 + 1) + 4 a5 (c33 + c3), c2 = −a1 (c32 + 1) + a 4 (c33 + 3 c3) + a5 (c34 + 6 c32 + 1), (4 a5 c3 + a 4 ) (3 a2 + 2 c3 a7 ) a1 = + 2 a 4 c3 + 2 a5 (c32 + 5), 3a +2a 6

a3 =

7

−2 a5 (3 a 6 + 2 a7 )2 , 3 (a 4 + 4 a5 c3)2

(4 a5 c3 + a 4 ) (3 a6 + 2 a7 ) < 0, x0, t0, c3, and ϕ0 are arbitrary real constants.

• Reference: [5]. Solution 4. tan(x,t) ψ (x , t ) =

− 6 (4 a5 c 3 + a 4 ) tan[x − x 0 + c1 (t − t0 )] e i [c3 (x− x0) + c 2 (t − t0) + ϕ0], 3 a 6 + 2 a7

(4.48)

where c1 = −2 a1 c3 + a 4 (3 c32 − 2) + 4 a5 (c33 − 2 c3), c2 = a1 (2 − c32 ) + a 4 (c33 − 6 c3) + a5 (c34 − 12 c32 + 16), (4 a5 c3 + a 4 ) (3 a2 + 2 c3 a7 ) a1 = + 2 a 4 c3 + 2 a5 (c32 − 10), 3a +2a 6

a3 =

7

−2 a5 (3 a 6 + 2 a7 )2 , 3 (a 4 + 4 a5 c3)2

(4 a5 c3 + a 4 ) (3 a6 + 2 a7 ) < 0, x0, t0, c3, and ϕ0 are arbitrary real constants.

• Reference: [5]. Solution 5. cot(x,t) ψ (x , t ) =

− 6 (4 a5 c 3 + a 4 ) cot[x − x 0 + c1 (t − t0 )] e i [c3 (x− x0) + c 2 (t − t0) + ϕ0], 3 a 6 + 2 a7

where c1 = −2 a1 c3 + a 4 (3 c32 − 2) + 4 a5 (c33 − 2 c3), c2 = a1 (2 − c32 ) + a 4 (c33 − 6 c3) + a5 (c34 − 12 c32 + 16), (4 a5 c3 + a 4 ) (3 a2 + 2 c3 a7 ) a1 = + 2 a 4 c3 + 2 a5 (c32 − 10), 3a +2a 6

a3 =

7

−2 a5 (3 a 6 + 2 a7 )2 , 3 (a 4 + 4 a5 c3)2

(4 a5 c3 + a 4 ) (3 a6 + 2 a7 ) < 0, x0, t0, c3, and ϕ0 are arbitrary real constants.

• Reference: [5].

4-33

(4.49)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Solution 6. sech(x,t) bright soliton

ψ (x , t ) =

6 (4 a5 c3 + a 4 ) sech[x − x0 + c1 (t − t0)] e i [c3 (x − x0) + c2 (t − t0) + ϕ0], (4.50) 3 a6 + 2 a7

where c1 = −2 a1 c3 + a 4 (3 c32 − 1) + 4 a5 (c33 − c3), c2 = a1 (1 − c32 ) + a 4 (c33 − 3 c3) + a5 (c34 − 6 c32 + 1), (4 a5 c3 + a 4 ) (3 a2 + 2 c3 a7 ) a1 = + 2 a 4 c3 + 2 a5 (c32 − 5), 3a +2a 6

a3 =

7

−2 a5 (3 a 6 + 2 a7 )2 , 3 (a 4 + 4 a5 c3)2

(4 a5 c3 + a 4 ) (3 a6 + 2 a7 ) > 0, x0, t0, c3, and ϕ0 are arbitrary real constants.

• Reference: [5]. Solution 7. csch(x,t) ψ (x , t ) =

− 6 (4 a5 c 3 + a 4 ) csch[x − x 0 + c1 (t − t0 )] e i [c3 (x− x0) + c 2 (t − t0) + ϕ0], (4.51) 3 a 6 + 2 a7

where c1 = −2 a1 c3 + a 4 (3 c32 − 1) + 4 a5 (c33 − c3), c2 = a1 (1 − c32 ) + a 4 (c33 − 3 c3) + a5 (c34 − 6 c32 + 1), (4 a5 c3 + a 4 ) (3 a2 + 2 c3 a7 ) a1 = + 2 a 4 c3 + 2 a5 (c32 − 5), 3a +2a 6

a3 =

7

−2 a5 (3 a 6 + 2 a7 )2 , 3 (a 4 + 4 a5 c3)2

(4 a5 c3 + a 4 ) (3 a6 + 2 a7 ) < 0, x0, t0, c3, and ϕ0 are arbitrary real constants.

• Reference: [5]. Solution 8. tanh(x,t) dark soliton ψ (x , t ) =

− 6 (4 a5 c 3 + a 4 ) tanh[x − x 0 + c1 (t − t0 )] e i [c3 (x− x0) + c 2 (t − t0) + ϕ0], (4.52) 3 a 6 + 2 a7

where c1 = −2 a1 c3 + a 4 (3 c32 + 2) + 4 a5 (c33 + 2 c3), c2 = −a1 (c32 + 2) + a 4 (6 c3 + c33) + a5 (c34 + 12 c32 + 16), (4 a5 c3 + a 4 ) (3 a2 + 2 c3 a7 ) a1 = + 2 a 4 c3 + 2 a5 (c32 + 10), 3a +2a 6

a3 =

7

−2 a5 (3 a 6 + 2 a7 )2 , 3 (a 4 + 4 a5 c3)2

4-34

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

(4 a5 c3 + a 4 ) (3 a6 + 2 a7 ) < 0, x0, t0, c3, and ϕ0 are arbitrary real constants.

• Reference: [5]. Solution 9. coth(x,t) ψ (x , t ) =

− 6 (4 a5 c 3 + a 4 ) coth[x − x 0 + c1 (t − t0 )] e i [c3 (x− x0) + c 2 (t − t0) + ϕ0], (4.53) 3 a 6 + 2 a7

where c1 = −2 a1 c3 + a 4 (3 c32 + 2) + 4 a5 (c33 + 2 c3), c2 = −a1 (c32 + 2) + a 4 (6 c3 + c33) + a5 (c34 + 12 c32 + 16), (4 a5 c3 + a 4 ) (3 a2 + 2 c3 a7 ) a1 = + 2 a 4 c3 + 2 a5 (c32 + 10), 3a +2a 6

a3 =

7

−2 a5 (3 a 6 + 2 a7 )2 , 3 (a 4 + 4 a5 c3)2

(4 a5 c3 + a 4 ) (3 a6 + 2 a7 ) < 0, x0, t0, c3, and ϕ0 are arbitrary real constants.

• Reference: [5].

4-35

4-36

4.

3.

ψ (x , t ) =

ψ (x , t ) =

ψ (x , t ) =

csc[x − x0 + c1 (t − t0 )]

tan[x − x0 + c1 (t − t0 )]

× ei [c3 (x − x0) + c2 (t − t0) + ϕ0]

−6 (4 a5 c3 + a 4 ) 3 a6 + 2 a7

× ei [c3 (x − x0) + c2 (t − t0) + ϕ0]

−6 (4 a5 c3 + a 4 ) 3 a6 + 2 a7

× ei [c3 (x − x0) + c2 (t − t0) + ϕ0]

sec[x − x0 + c1 (t − t0 )]

4 2 ei [c2 (x − x0) + (c1 a2 + c1 a3) (t − t0) + ϕ0]

−6 (4 a5 c3 + a 4 ) 3 a6 + 2 a7

ψ (x , t ) = c1

1.

2.

Solution

#

Equation

−2 a5 (3 a6 + 2 a7 )2 , 3 (a 4 + 4 a5 c3)2

(4 a5 c3 + a 4 ) (3 a 6 + 2 a7 ) < 0 ,

+ 2 a 4 c3 + 2 a5 (c32 + 5),

−2 a5 (3 a6 + 2 a7 )2 , 3 (a 4 + 4 a5 c3)2

a3 =

+ 2 a 4 c3 + 2 a5 (c32 + 5),

−2 a5 (3 a6 + 2 a7 )2 , 3 (a 4 + 4 a5 c3)2

a3 =

+ 2 a 4 c3 + 2 a5 (c32 − 10), (4 a5 c3 + a 4 ) (3 a 6 + 2 a7 ) < 0 ,

(4 a5 c3 + a 4 ) (3 a2 + 2 c3 a7 ) 3 a6 + 2 a7

a1 =

c1 = − 2 a1 c3 + a 4 (3 c32 − 2) + 4 a5 (c33 − 2 c3), c2 = a1 (2 − c32 ) + a 4 (c33 − 6 c3) + a5 (c34 − 12 c32 + 16),

x0, t0, c3, and ϕ0 are arbitrary real constants

(4 a5 c3 + a 4 ) (3 a 6 + 2 a7 ) < 0 ,

(4 a5 c3 + a 4 ) (3 a2 + 2 c3 a7 ) 3 a6 + 2 a7

a1 =

c1 = − 2 a1 c3 + a 4 (3 c32 + 1) + 4 a5 (c33 + c3), c2 = − a1 (c32 + 1) + a 4 (c33 + 3 c3) + a5 (c34 + 6 c32 + 1),

x0, t0, c3, and ϕ0 are arbitrary real constants

a3 =

(4 a5 c3 + a 4 ) (3 a2 + 2 c3 a7 ) 3 a6 + 2 a7

–

–

–

c1 = − 2 a1 c3 + a 4 (3 c32 + 1) + 4 a5 (c33 + c3), c2 = − a1 (c32 + 1) + a 4 (c33 + 3 c3) + a5 (c34 + 6 c32 + 1), a1 =

continuous wave, t- and x-dependent phase

Name

, x0, t0, c1, c2, and ϕ0 a4 = c22 are arbitrary real constants

a6 c12 + a1 c2 − a5 c23

Conditions

i ψt + a1 ψxx + a2 ∣ψ ∣2 ψ + a3 ∣ψ ∣4 ψ + i a 4 ψxxx + a5 ψxxxx + i a 6 (∣ψ ∣2 ψ )x + i a7 (∣ψ ∣2 )x ψ = 0

4.13 Summary of Section 4.12

(4.48)

(4.47)

(4.46)

(4.45)

Eq.#

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

4-37

8.

7.

6.

5.

ψ (x , t ) =

ψ (x , t ) =

ψ (x , t ) =

ψ (x , t ) =

cot[x − x0 + c1 (t − t0 )]

sech[x − x0 + c1 (t − t0 )]

csch[x − x0 + c1 (t − t0 )]

tanh[x − x0 + c1 (t − t0 )]

× ei [c3 (x − x0) + c2 (t − t0) + ϕ0]

−6 (4 a5 c3 + a 4 ) 3 a6 + 2 a7

× ei [c3 (x − x0) + c2 (t − t0) + ϕ0]

−6 (4 a5 c3 + a 4 ) 3 a6 + 2 a7

× ei [c3 (x − x0) + c2 (t − t0) + ϕ0]

6 (4 a5 c3 + a 4 ) 3 a6 + 2 a7

× ei [c3 (x − x0) + c2 (t − t0) + ϕ0]

−6 (4 a5 c3 + a 4 ) 3 a6 + 2 a7

−2 a5 (3 a6 + 2 a7 )2 , 3 (a 4 + 4 a5 c3)2

a3 =

+ 2 a 4 c3 + 2 a5 (c32 − 10),

−2 a5 (3 a6 + 2 a7 )2 , 3 (a 4 + 4 a5 c3)2

a3 =

+ 2 a 4 c3 + 2 a5 (c32 − 5),

−2 a5 (3 a6 + 2 a7 )2 , 3 (a 4 + 4 a5 c3)2

a3 =

+ 2 a 4 c3 + 2 a5 (c32 − 5),

−2 a5 (3 a6 + 2 a7 )2 , 3 (a 4 + 4 a5 c3)2

a3 =

+ 2 a 4 c3 + 2 a5 (c32 + 10),

x0, t0, c3, and ϕ0 are arbitrary real constants

(4 a5 c3 + a 4 ) (3 a 6 + 2 a7 ) < 0 ,

(4 a5 c3 + a 4 ) (3 a2 + 2 c3 a7 ) 3 a6 + 2 a7

a1 =

c1 = − 2 a1 c3 + a 4 (3 c32 + 2) + 4 a5 (c33 + 2 c3), c2 = − a1 (c32 + 2) + a 4 (6 c3 + c33) + a5 (c34 + 12 c32 + 16),

x0, t0, c3, and ϕ0 are arbitrary real constants

(4 a5 c3 + a 4 ) (3 a 6 + 2 a7 ) < 0 ,

(4 a5 c3 + a 4 ) (3 a2 + 2 c3 a7 ) 3 a6 + 2 a7

a1 =

c1 = − 2 a1 c3 + a 4 (3 c32 − 1) + 4 a5 (c33 − c3), c2 = a1 (1 − c32 ) + a 4 (c33 − 3 c3) + a5 (c34 − 6 c32 + 1),

x0, t0, c3, and ϕ0 are arbitrary real constants

(4 a5 c3 + a 4 ) (3 a 6 + 2 a7 ) > 0 ,

(4 a5 c3 + a 4 ) (3 a2 + 2 c3 a7 ) 3 a6 + 2 a7

a1 =

c1 = − 2 a1 c3 + a 4 (3 c32 − 1) + 4 a5 (c33 − c3), c2 = a1 (1 − c32 ) + a 4 (c33 − 3 c3) + a5 (c34 − 6 c32 + 1),

x0, t0, c3, and ϕ0 are arbitrary real constants

(4 a5 c3 + a 4 ) (3 a 6 + 2 a7 ) < 0 ,

(4 a5 c3 + a 4 ) (3 a2 + 2 c3 a7 ) 3 a6 + 2 a7

a1 =

c1 = − 2 a1 c3 + a 4 (3 c32 − 2) + 4 a5 (c33 − 2 c3), c2 = a1 (2 − c32 ) + a 4 (c33 − 6 c3) + a5 (c34 − 12 c32 + 16),

x0, t0, c3, and ϕ0 are arbitrary real constants

dark soliton

–

bright soliton

–

(Continued)

(4.52)

(4.51)

(4.50)

(4.49)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

9.

#

ψ (x , t ) =

Solution

(Continued )

coth[x − x0 + c1 (t − t0 )]

× ei [c3 (x − x0) + c2 (t − t0) + ϕ0]

−6 (4 a5 c3 + a 4 ) 3 a6 + 2 a7

7

−2 a5 (3 a6 + 2 a7 )2 , 3 (a 4 + 4 a5 c3)2

(4 a5 c3 + a 4 ) (3 a 6 + 2 a7 ) < 0 ,

x0, t0, c3, and ϕ0 are arbitrary real constants

a3 =

6

(4.53)

–

c1 = − 2 a1 c3 + a 4 (3 c32 + 2) + 4 a5 (c33 + 2 c3), c2 = − a1 (c32 + 2) + a 4 (6 c3 + c33) + a5 (c34 + 12 c32 + 16), (4 a5 c3 + a 4 ) (3 a2 + 2 c3 a7 ) a1 = + 2 a 4 c3 + 2 a5 (c32 + 10), 3a +2a

Eq.#

Name

Conditions

i ψt + a1 ψxx + a2 ∣ψ ∣2 ψ + a3 ∣ψ ∣4 ψ + i a 4 ψxxx + a5 ψxxxx + i a 6 (∣ψ ∣2 ψ )x + i a7 (∣ψ ∣2 )x ψ = 0

Equation

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

4-38

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

4.14 NLSE with ∣ψ ∣2 -Dependent Dispersion Equation:

i ψt + (1 − ∣μ∣ ∣ψ ∣2 ) ψxx + 2 (1 − ∣μ∣) ∣ψ ∣2 ψ = 0,

(4.54)

where ψ = ψ (x , t ) is the complex function profile, x and t are its two independent variables, μ is an arbitrary real constant. Solutions: Solution 1. Constant Amplitude CW, t- and x-dependent phase

ψ (x , t ) = A0 e i [A1 (x − x0) + A2 (t − t0) + ϕ0],

(4.55)

where A2 = 2 A02 + A02 ∣μ∣ (A12 − 2) − A12 , x0, t0, A0, A1, μ, and ϕ0 are arbitrary real constants. Solution 2. Peakon (Figure 4.4)

ψ (x , t ) = A0 e

−

2 − 2 ∣μ ∣ ∣ x − x0 ∣ ∣μ ∣

e−i [A1 (t − t0) + ϕ0],

where 2 − 2 ∣μ∣ > 0, ∣μ∣ < 1, (for peakon solution) 2 A1 = 2 − ∣ μ ∣ , x0, t0, A0, and ϕ0 are arbitrary real constants. • Reference: [7], we corrected the prefactors of x and t.

Figure 4.4. Peakon solution (4.56) at t = 0, with A0 = 1, a2 = −1/2 , μ = 0.7 , and ϕ0 = 0 .

4-39

(4.56)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

4.15 Infinite Hierarchy of Integrable NLSEs with Higher Order Terms 4.15.1 Constant Coefficients Equation:

i ψt + a2 k2 − i a3 k3 + a 4 k 4 − i a5 k5 + … m = 0,

(4.57)

where ψ = ψ (x , t ) is the complex function profile, x and t are its two independent variables, aj are in general arbitrary real constants, j = 2, 3, 4, 5, …m, δ kj = ( −1) j δ ψ * ∫ pj + 1 dx , pj = ψ

∂ ∂x

( )+∑ pj − 1 ψ

p j1 + j2 = j − 1 j1

p j2 .

Starting with p1 = ∣ψ ∣2 , the next three pj terms are then p2 = ψ ψx*, p3 = ∣ψ ∣4 + ψ ψx*x, 2 p4 = ψ (ψx ψ * + 4 ∣ψ ∣2 ψx* + ψx*x x ), while the first four kj terms are k2 = ψx x + 2 ∣ψ ∣2 ψ , k3 = ψx x x + 6 ∣ψ ∣2 ψx,

k4 = ψx x x x + 8 ∣ψ ∣2 ψx x + 6 ∣ψ ∣4 ψ + 4 ∣ψx∣2 ψ + 6 ψx2 ψ *+2 ψ 2 ψx*x, k5 = ψx x x x x + 10 ∣ψ ∣2 ψx x x + 10(∣ψx∣2 ψ )x + 20 ψ * ψx ψx x + 30 ∣ψ ∣4 ψx.

Hierarchy of Integrable NLSEs: j = 2,

iψt + a2(ψxx + 2∣ψ ∣2 ψ ) = 0,

(4.58)

iψt + a2 (ψxx + 2∣ψ ∣2 ψ ) − i a3 (ψxxx + 6∣ψ ∣2 ψx ) = 0,

(4.59)

j = 3, … Solutions to all Equations in the Hierarchy:

Solution 1. Constant Amplitude CW, t-dependent phase

ψ (x , t ) = c

⎡ ∞ (2 n )! ⎤ i ⎢c 2 ∑ a2 n c 2 n − 2 (t − t0) + ϕ0⎥ 2 ⎥⎦ e ⎢⎣ n =1 (n !) ,

where x0, t0, c, and ϕ0 are arbitrary real constants. • Reference: [6].

4-40

(4.60)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Solution 2. sech(x,t) bright soliton

ψ (x , t ) = c sech[c (x − x0) + v (t − t0)] e i [ϕ1 (t − t0) + ϕ0], where

(4.61)

∞

ϕ1 =

∑a2 n c 2 n, n=1 ∞

v=

∑a2 n+ 1 c 2 n+ 1, n=1

x0, t0, c, and ϕ0 are arbitrary real constants. • Reference: [6]. Solution 3. Localization in x and t Peregrine soliton

⎫ ⎧ ⎡ 1 + 2 i b (t − t ) ⎤ 0 ψ ( x , t ) = c ⎨4 ⎢ ⎥ − 1⎬ e i [ϕ1 (t − t0) + ϕ0], ⎦ d (x , t ) ⎭ ⎩ ⎣ where

(4.62)

∞ (2 n )!

ϕ1 = c 2∑ (n !)2 a2 n c 2 n − 2 , n=1

d (x , t ) = 1 + 4 b 2 (t − t0 )2 + 4 [c (x − x0 ) + v (t − t0 )]2 , ∞

b=

∑ n((2n!)n)! a2 n c 2 n, 2

n=1 ∞

v=

∑ (2(nn +!) 1)! a2 n+ 1 c 2 n+ 1, 2

n=1

x0, t0, c, and ϕ0 are arbitrary real constants. • Reference: [6]. Solution 4. Rational Solution DW

⎧ ⎫ 4 ⎬ e i ϕ0, 1 ψ (x , t ) = c ⎨ − ⎩ 1 + 4[c (x − x0) + v (t − t0)]2 ⎭ where v=

∞

∑ (2(nn +!) 1)! a2 n+ 1 c 2 n+ 1, 2

n=1

x0, t0, c, and ϕ0 are arbitrary real constants. • Reference: [6].

4-41

(4.63)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Solution 5. Periodicity in x and Localization in t Akhmediev breather ψ (x , t ) = ⎧ ⎡ ⎤ ⎡ ⎤⎫ 2 2 κ 2 cosh⎢b κ 1 − κ4 (t − t0)⎥ + i κ 4 − κ 2 sinh⎢b κ 1 − κ4 (t − t0)⎥ ⎪ ⎪ ⎪ ⎣ ⎦ ⎣ ⎦⎪ ⎬ (4.64) c ⎨1 + ⎡ ⎤ κ2 ⎪ ⎪ 2 cos[κ [c (x − x ) + v (t − t )]] − 2 cosh b κ 4 1 ( t t ) − κ − − 0 0 0 ⎥ ⎪ ⎢⎣ ⎪ 4 ⎦⎭ ⎩ × e i [ϕ1 (t − t0) + ϕ0] ,

where

⎛

∞

v=

n

⎞

r 2r

∑ (2 nn+! 1)! a2 n+ 1 c 2 n+ 1⎜⎜∑ (n −(−r1))! κ(2 r r+!1)! ⎟⎟, ⎝ r=0

n=1 ∞

ϕ1 =

⎠

∑ (2(n n!))! a2 n c 2 n, 2

n=1 ∞

b=2

⎛

n

⎞

r 2r

∑ (2 nn+! 1)! a2 n+ 2 c 2 n+ 2⎜⎜∑ (n −(−r1))! κ(2 r r+!1)! ⎟⎟,

⎝ r=0 ⎠ n=0 0 < κ < 2, x0, t0, c, and ϕ0 are arbitrary real constants.

• Reference: [6]. Solution 6. Periodicity in t and Localization in x Kuznetsov–Ma breather

ψ (x , t ) = c ×

⎡ 2(1 − κ ) d (t ) − 2 κ d (x , t ) + 2 i 1 − 2 κ d (t ) ⎤ 1 3 2 ⎥ 2⎢ 2 d3(x , t ) − 2 κ d1(t ) ⎦ ⎣ e i [ϕ1 (t − t0) + ϕ0],

where d1(t ) = cos[2 1 − 2 κ b (t − t0 )], d2(t ) = sin[2 1 − 2 κ b (t − t0 )], d3(x , t ) = cosh[2 1 − 2 κ [c (x − x0 ) + v (t − t0 )], n ∞ ⎛ ⎞ (2 r − 1) !! κ r n 2 n 1 + ⎜ ⎟⎟, v = ∑ 4 a2 n + 1 c ⎜1 + ∑ r! ⎝ ⎠ n=1 r=1 ∞

ϕ1 =

∑ (2(n n!))! a2 n c 2 n (2 κ )n, 2

n=1 ∞

b=2

∑

4n

a2 n + 2

⎛ ⎜ + ⎝

c 2 n + 2 ⎜1

n

r

⎞

∑ (2 r −r1)! !! κ ⎟⎟,

⎠ n=0 r=1 0 < κ < 1/2, x0, t0, c, and ϕ0 are arbitrary real constants.

• Reference: [6].

4-42

(4.65)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Solution 7. dn(x,t) solitary wane (SW)

ψ (x , t ) = c dn[c (x − x0) + v (t − t0), m ] e i [ϕ1 (t − t0) + ϕ0], where

∞

v=

∑ a2 n+ 1 c 2 n+ 1 mn Pn( m2 n=1 ∞

ϕ1 =

∑ a2 n c 2 n mn Pn( m2

(4.66)

)

−1 ,

)

−1 ,

n=1

Pn is the set of orthogonal Legendre polynomials of the first kind, 0 < m < 1, x0, t0, c, and ϕ0 are arbitrary real constants. • Reference: [6]. Solution 8. cn(x,t) SW

ψ (x , t ) = where v=

⎤ c coth(κ ) ⎡ c (x − x0) + v (t − t0) , m⎥ e i [ϕ1 (t − t0) + ϕ0], cn⎢ sinh(κ ) ⎣ ⎦ 2

(4.67)

∞

∑ a2 n+ 1 c 2 n+ 1 sinh−2 n(κ ) Pn[sinh2(κ )], n=1 ∞

ϕ1 =

∑ a2 n c 2 n sinh−2 n(κ ) Pn[sinh2(κ )], n=1

Pn is the set of orthogonal Legendre polynomials of the first kind, 1 0 < [m = 2 cosh2(κ )] < 1 with κ real, example: κ = 1/2, x0, t0, c, and ϕ0 are arbitrary real constants. • Reference: [6].

4.15.2 Function Coefficients If ψ (x , t ) is a solution of an infinite hierarchy with constant coefficients

i ψt + a02 k2 − i a03 k3 + a0 4 k 4 − i a05 k5 + … m = 0,

(4.68)

then ψ (x , t ) will be also a solution of the same infinite hierarchy with t-dependent coefficients

i ψt + a2(t ) k2 − i a3(t ) k3 + a 4(t ) k 4 − i a5(t ) k5 + … m = 0,

4-43

(4.69)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

after making the following replacements in ψ : a 0 2 j (t − t0 ) → ∫ a2 j (t ) dt , a 0 2 j + 1 (t − t0 ) → ∫ a2 j + 1(t ) dt , j = 1, 2, 3, 4, …,

where a 0 n are in general arbitrary real constants, an(t ) are arbitrary real functions, n = 2, 3, 4, 5, ….

Example 1. sech(x,t) bright soliton Given ψ (x , t ) = c sech[c (x − x0 ) + v (t − t0 )] ei [ϕ1 (t − t0 ) + ϕ0 ]

is a solution of (4.68), where ∞

ϕ1 =

∑a02 n c 2 n, n=1 ∞

v=

∑a 02 n+ 1 c 2 n+ 1, then n=1

⎧ ψ (x , t ) = c sech⎨ c ( x − x0 ) + ⎪ ⎩ ⎪

⎛

∞

∑∫ n=1

∞

⎪

is a solution of (4.69), where x0, t0, c, and ϕ0 are arbitrary real constants. • Reference: [6]. Example 2. Localization in (x,t) Peregrine soliton Given ψ (x , t ) = c

{4 ⎡⎣

1 + 2 i b (t − t0 ) ⎤ ⎦ d (x , t )

−1

}e

i [ϕ1 (t − t0 ) + ϕ0 ]

is a solution of (4.68), where ∞

(2 n )!

ϕ1 = c 2∑ (n !)2 a 0 2 n c 2 n − 2 , n=1

d (x , t ) = 1 + 4 b 2 (t − t0 )2 + 4 [c (x − x0 ) + v (t − t0 )]2 , ∞

b=

∑ n((2n!)n)! a 02 n c 2 n, 2

n=1 ∞

v=

⎞

⎫ i ⎜⎜ ∑ ∫ a2 n(t ) dtc 2n+ ϕ0⎟⎟ ⎠ (4.70) a2 n + 1(t ) dtc 2n + 1⎬ e ⎝ n =1 ⎪ ⎭

∑ (2(nn +!) 1)! a 02 n+ 1 c 2 n+ 1, then 2

n=1

4-44

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

⎧ ⎡ ⎪ ⎢1 + 2 i ⎪ ⎢ ψ ( x , t ) = c ⎨4 ⎢ ⎪ ⎢ ⎪ ⎢ ⎩ ⎣ ×

⎛ ∞ (2 n )! i ⎜⎜c 2 ∑ 2 e ⎝ n = 1 ( n !)

⎫ ⎤ n(2 n )! ⎪ ∫ a2 n(t ) dtc 2 n ⎥ 2 ( n! ) ⎪ ⎥ n=1 − 1⎬ ⎥ d (x , t ) ⎪ ⎥ ⎪ ⎥⎦ ⎭ ∞

∑

(4.71)

⎞

∫ a2 n(t ) dtc 2 n −2+ ϕ0⎟⎟

⎠,

is a solution of (4.69), where ⎡∞ n (2 n )! d (x, t ) = 1 + 4 ⎢ ∑ ⎢⎣ n = 1 (n!) 2

∫

⎤2 ⎡ a2 n(t ) dtc 2 n⎥ + 4 ⎢c (x − x 0 ) + ⎥⎦ ⎣⎢

x0, t0, c, and ϕ0 are arbitrary real constants. • Reference: [6].

4-45

∞

⎤2

n=1

⎦

∑ (2(nn +!) 21)! ∫ a2 n + 1(t ) dtc 2 n + 1⎥⎥

,

4-46

⎡ ∞ ⎤ (2 n )! i ⎢c 2 ∑ a2 n c 2 n − 2 (t − t0 ) + ϕ0⎥ (n !) 2 ⎢⎣ ⎥⎦ n=1 e

q1(t ) =

+i κ

−1

4 − κ 2 sinh[b κ

1−

(t − t0 )] κ2 4

(t − t0 )],

}

i [ϕ1 (t − t0 ) + ϕ0 ]

− 1 e i ϕ0

}e

i [ϕ1 (t − t0 ) + ϕ0 ] ,

κ2 4

}e

1−

q1(t ) q2(x, t )

4 1 + 4[c (x − x0 ) + v (t − t0 )]2

1 + 2 i b (t − t0 ) ⎤ ⎦ d (x , t )

{1 +

{

{4 ⎡⎣

cosh[b κ

ψ (x , t ) = c

5.

κ2

ψ (x , t ) = c

ψ (x , t ) = c

ψ (x , t ) = c sech[c (x − x0 ) + v (t − t0 )] ei [ϕ1 (t − t0) + ϕ0]

ψ (x , t ) = c

Solution

4.

3.

2.

1.

#

Equation: i ψt + a2 k2 − i a3 k3 + a 4 k4 − i a5 k5 + … m = 0

4.16 Summary of Section 4.15

∞

∞

n=1

n=1

∑ a2 n c 2 n , v = ∑ a2 n + 1 c 2 n + 1,

(2 n )!

∞

2

n=1

2

∑ (2(nn +!) 1)! a2 n + 1 c 2 n + 1,

n=1 ∞

∑ n((2n!)n)! a2 n c 2 n,

∞ 2

n=1

∑ (2(nn +!) 1)! a2 n + 1 c 2 n + 1,

n=1

∑

n=1 ∞

∞

⎛

n

r 2r

⎞

(2 n )! (n ! ) 2

a2 n

c2 n,

⎝r=0

⎠

∑ (2 nn+! 1)! a2 n + 1 c 2 n + 1 ⎜⎜∑ (n −(−r1))! κ(2 r r+!1)! ⎟⎟ ϕ1 =

v=

x0, t0, c, and ϕ0 are arbitrary real constants

v=

x0, t0, c, and ϕ0 are arbitrary real constants

v=

b=

d (x , t ) = 1 + 4 b 2 (t − t0 )2 + 4 [c (x − x0 ) + v (t − t0 )]2 ,

n=1

ϕ1 = c 2 ∑ (n !)2 a2 n c 2 n − 2 ,

∞

x0, t0, c, and ϕ0 are arbitrary real constants

ϕ1 =

x0, t0, c, and ϕ0 are arbitrary real constants

Conditions

Constant Coefficients

Akhmediev breather

decaying wave

Peregrine soliton

bright soliton

continuous wave, t-dependent phase

Name

(4.64)

(4.63)

(4.62)

(4.61)

(4.60)

Eq. #

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

7.

6.

1−

κ2 4

(t − t0 )],

⎡ 2(1 − κ ) d1(t ) − 2 κ d3(x, t ) + 2 i 1 − 2 κ d2(t ) ⎤ 2 ⎢⎣ ⎥⎦ 2 d3(x, t ) − 2 κ d1(t )

ψ (x , t ) = c dn[c (x − x0 ) + v (t − t0 ), m ] ei [ϕ1 (t − t0) + ϕ0]

× ei [ϕ1 (t − t0) + ϕ0]

ψ (x , t ) = c

− 2 cosh[b κ

q2(x , t ) = 4 − κ 2 cos[κ [c (x − x0 ) + v (t − t0 )]]

∞

⎛

n

r 2r

⎞

n=0

⎝r=0

4-47

2

⎛

n

r

⎞

∞

− 1),

− 1),

Pn is the set of orthogonal Legendre polynomials of the first kind, 0 < m < 1, x0, t0, c, and ϕ0 are arbitrary real constants

n=1

∑ a2 n c 2 n mn Pn( m2

n=1 ∞

∑ a2 n + 1 c 2 n + 1 mn Pn( m2

ϕ1 =

v=

n=0

⎝ ⎠ r=1 x0, t0, c, and ϕ0 are arbitrary real constants

∑ 4n a2 n + 2 c 2 n + 2 ⎜⎜1 + ∑ (2 r −r1)! !! κ ⎟⎟,

n=1 ∞

∑ (2(n n!))! a2 n c 2 n (2 κ )n , 0 < κ < 1/2,

b=2

ϕ1 =

∞

1 − 2 κ b (t − t0 )],

d2(t ) = sin[2 1 − 2 κ b (t − t0 )], d3(x , t ) = cosh[2 1 − 2 κ [c (x − x0 ) + v (t − t0 )], ∞ n ⎛ ⎞ (2 r − 1) !! κ r ⎟⎟, v = ∑ 4n a2 n + 1 c 2 n + 1 ⎜⎜1 + ∑ r! ⎝ ⎠ n=1 r=1

d1(t ) = cos[2

⎠

∑ (2 nn+! 1)! a2 n + 2 c 2 n + 2 ⎜⎜∑ (n −(−r1))! κ(2 r r+!1)! ⎟⎟,

0 < κ < 2, x0, t0, c, and ϕ0 are arbitrary real constants

b=2

solitary wave

solitary wave

(4.66)

(4.65)

(Continued)

Kuznetsov–Ma breather

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

ψ (x , t ) =

8.

2

c coth(κ )

⎡ c (x − x0) + v (t − t0) ⎤ cn⎣ , m⎦ ei [ϕ1 (t − t0) + ϕ0] sinh(κ )

Function Coefficients

κ = 1/2

Pn is the set of orthogonal Legendre polynomials of the first kind, 1 0 < [m = 2 cosh2(κ )] < 1 with κ real, example:

n=1

∑ a2 n c 2 n sinh−2 n(κ ) Pn[sinh2(κ )],

n=1 ∞

∑ a2 n + 1 c 2 n + 1 sinh−2 n(κ ) Pn[sinh2(κ )],

ϕ1 =

v=

∞

Conditions

Constant Coefficients

4-48

2.

1.

#

⎪

∫

∞

×

⎛ ∞ i ⎜⎜c 2 ∑ e ⎝ n=1

n=1

⎞ a2 n(t ) dtc 2 n + ϕ0⎟⎟ ⎠

(2 n )! (n !) 2

∫

⎪

⎫

⎞ a2 n(t ) dtc 2 n − 2 + ϕ0⎟⎟ ⎠

⎭

⎪

∑ ∫ a2 n + 1(t ) dtc 2 n + 1⎬

∞ ⎫ ⎧ ⎡ ⎤ n(2 n )! ∫ a2 n(t ) dtc 2 n ⎪ ⎪ ⎢ 1 + 2 i n∑ ⎥ (n !) 2 =1 ψ (x , t ) = c ⎨4 ⎢ ⎥ − 1⎬ d (x , t ) ⎪ ⎪ ⎢ ⎥⎦ ⎭ ⎩ ⎣

×

⎛∞ i ⎜⎜ ∑ e ⎝ n=1

⎪

⎧ ψ (x , t ) = c sech⎨c (x − x0 ) + ⎩

Solution

(2 n + 1)! (n ! ) 2

⎤2

⎥⎦

⎦

∫ a2 n+1(t ) dtc2 n+1⎥⎥ ,

∫ a2 n(t )

⎤2 dtc 2 n⎥

x0, c, and ϕ0 are arbitrary real constants

n=1

+∑

∞

+ 4 [c ( x − x 0 )

⎡∞ n(2 n )! d ( x , t ) = 1 + 4 ⎢ ∑ (n ! ) 2 ⎢⎣ n = 1

x0, c, and ϕ0 are arbitrary real constants

Conditions

Equation: i ψt + a2(t ) k2 − i a3(t ) k3 + a 4(t ) k4 − i a5(t ) k5 + … m = 0

Solution

#

Equation: i ψt + a2 k2 − i a3 k3 + a 4 k4 − i a5 k5 + … m = 0

(Continued )

Peregrine soliton

bright soliton

Name

Name

(4.71)

(4.70)

Eq. #

(4.67)

Eq. # Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

References [1] Zayed E M and Al-Nowehy A G 2017 Exact solutions for the perturbed nonlinear Schrödinger equation with power law nonlinearity and Hamiltonian perturbed terms Optik 139 123–44 [2] Li Z, Li L, Tian H and Zhou G 2000 New types of solitary wave solutions for the higher order nonlinear Schrödinger equation Phys. Rev. Lett. 84 4096-9 [3] Wazwaz A M 2006 Exact solutions for the fourth order nonlinear Schrödinger equations with cubic and power law nonlinearities Math. Comput. Modelling 43 802–8 [4] Huang Y and Liu P 2014 New exact solutions for a class of high-order dispersive cubic-quintic nonlinear Schrödinger equation J. Math. Res. 6 104–8 [5] Zayed E M and Al-Nowehy A G 2017 Exact solutions and optical soliton solutions for the nonlinear Schrödinger equation with fourth-order dispersion and cubic-quintic nonlinearity Ric. Mat. 66 531–52 [6] Ankiewicz A, Kedziora D J, Chowdury A, Bandelow U and Akhmediev N 2016 Infinite hierarchy of nonlinear Schrödinger equations and their solutions Phys. Rev. E 93 012206 [7] Kevrekidis P G 2009 The Discrete Nonlinear Schrödinger Equation: Mathematical Analysis Numerical Computations and Physical Perspectives (Springer Tracts in Modern Physics vol 232) (Berlin: Springer)

4-49

IOP Publishing

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations Usama Al Khawaja and Laila Al Sakkaf

Chapter 5 Scaling Transformations Known also as similarity transformations

A Glance at Chapter 5

doi:10.1088/978-0-7503-2428-1ch5

5-1

ª IOP Publishing Ltd 2020

1 0

i Φt + a1 Φxx + a2 ∣Φ∣n Φ + a3 ∣Φ∣m Φ = 0

i Φt + b1(x , t ) Φxx + b2(x , t ) ∣Φ∣2 Φ + [b3r (x , t ) + i b3i (x , t )] Φ = 0

5

6

x2 Φ = 0

1

i Φt + a1 Φxx + a2 ∣Φ∣n Φ = 0

4

5-2

13

12

a2 b10 c5 a1 c22 c7

a2 b10 e γ t a1 c22

∣Φ∣2 Φ −

∣Φ∣2 Φ +

a1 c22

2 b10 c5

xΦ=0

x2 Φ = 0

c7 g4″(t )

γ2 4 b10

x2 Φ = 0

β γ 2 [3 β + β cos(2 γ t ) + 2 α sin(γ t )] 8 b10 [α + β sin(γ t )]2

4 b10 g52(t )

g5(t ) g5″(t ) − 2 g5′2(t )

∣Φ∣2 Φ +

∣Φ∣2 Φ −

a2 b10 [α + β sin(γ t )]

a1 c22

a2 b10 g5(t )

i Φt +

a1 b20 c22 a2 g5(t )

Φxx + b20 ∣Φ∣2 Φ +

4 a1 b20 c22 g5(t )

a2 [g5′2(t ) − g5(t ) g5″(t )]

x2 Φ = 0

i Φt + b1(x , t ) Φxx + b20 ∣Φ∣2 Φ + [b3r (x , t ) + i b3i (x , t )] Φ = 0

i Φt + b10 Φxx +

i Φt + b10 Φxx +

10

11

i Φt + b10 Φxx +

i Φt + b10 Φxx +

9

8

0

0

4

3

3

3

0

3

i Φt + a1 Φxx + a2 ∣Φ∣2 Φ = 0

3

i Φt + b10 Φxx + b2(x , t ) ∣Φ∣2 Φ + [b3r (x , t ) + i b3i (x , t )] Φ = 0

2

i Φt + a1 Φxx − a2 ∣Φ∣2 Φ = 0

2

7

2

i Φt + a11 Φxx + a22 ∣Φ∣2 Φ = 0

Solution

1

Equation

A Statistical View of Chapter 5

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

⎞⎤ + Bxx(x , t )⎟⎟⎥ Φ = 0 ⎥ ⎠⎦

i ψt +

i ψt +

20

20

Total

⎤ (x − x 0 )⎥ ⎦

ψxx − ∣ψ ∣2 ψ + V0 sin2(x ) ψ = 0

a1 b10

b10 g12(t )

1 2

⎧ ⎪ Φ+⎨ ⎡ ⎪ f 4 ⎢⎣ ⎩

ψxx − ∣ψ ∣2 ψ + V0 sn2(x , m ) ψ = 0

∣Φ∣2

1 2

i Φt + b10 Φxx + b20

19

18

+ g1′(t ) ∫ ⎡ f 2⎢ ⎣ a1 b10

⎤ (x − x 0 )⎥ ⎦

dx

⎫ ⎪ + g2′(t )⎬Φ = 0 ⎪ ⎭

5-3

26

2

3

0

0

(x − x0 )]

(x − x0 )] Bx(x, t )

i Φt + b10 Φxx + b20 ∣Φ∣2 Φ + b10 [cos2(x − x0 ) + i sin(x − x0 )] Φ = 0

a1 b10 a1 b10

0

0

17

f[

f′[

xΦ=0

0

4 a1 b10

2 a1 b20 c22

a2 [c6 g4′(t ) − g4″(t )]

i Φt + b10 Φxx + b20 ∣Φ∣2 Φ + [Veven(x ) + i Vodd(x )]Φ = 0

Φxx + b20 ∣Φ∣2 Φ +

16

a1 b20 c22 e−c6 t a2 c5

i Φt + b10 Φxx + b20 ∣Φ∣2 Φ ⎡ ⎛ + ⎢Bt (x , t ) + b10 Bx2(x , t ) − i b10 ⎜⎜ ⎢ ⎝ ⎣

i Φt +

15

14

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

5.1 Fundamental NLSE to Fundamental NLSE with Different Constant Coefficients If ψ (x , t ) is a solution of the fundamental NLSE, (2.1),

i ψt + a1 ψxx + a2 ∣ψ ∣2 ψ = 0, then

Φ( x , t ) =

⎞ a2 ⎛ a1 x , t⎟ ψ⎜ a22 ⎝ a11 ⎠

(5.1)

is a solution of

i Φt + a11 Φxx + a22 ∣Φ∣2 Φ = 0,

(5.2)

with arbitrary real constants a1, a2, a11 and a22. Example 1. sech(x) bright soliton Given

ψ (x , t ) = A0

2 a1 a2

⎡

2

sech[A0 (x − x0)] e i ⎣a1 A0

(t − t0) + ϕ0⎤⎦

is a solution of (2.1), then

Φ(x , t ) = A0

⎡ 2 a1 sech ⎢A0 a22 ⎣

⎤ ⎡ 2 a1 ⎤ (x − x0)⎥e i ⎣a1 A0 (t − t0) + ϕ0⎦ a11 ⎦

(5.3)

is a solution of (5.2), where a1 a22 > 0, a1 a11 > 0, A0, x0, t0, and ϕ0 are arbitrary real constants. Example 2. tanh(x) dark soliton Given

ψ (x , t ) = A0

−2 a1 a2

⎡

2

tanh[A0 (x − x0)] e−i ⎣2 a1 A0

(t − t0) + ϕ0⎤⎦

is a solution of (2.1), then

Φ(x , t ) = A0

⎡ −2 a1 tanh ⎢A0 a22 ⎣

⎤ 2 a1 ⎡ ⎤ (x − x0)⎥ e−i ⎣2 a1 A0 (t − t0) + ϕ0⎦ a11 ⎦

is a solution of (5.2), where a1 a22 < 0, a1 a11 > 0, A0, x0, t0, and ϕ0 are arbitrary real constants.

5-4

(5.4)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

5.2 Defocusing (Focusing) NLSE to Focusing (Defocusing) NLSE If ψ (x , t ) is a solution of the fundamental NLSE, (2.1),

i ψt + a1 ψxx + a2 ∣ψ ∣2 ψ = 0, then

Φ( x , t ) = ψ ( i x , − t )

(5.5)

i Φt + a1 Φxx − a2 ∣Φ∣2 Φ = 0,

(5.6)

is a solution of

with arbitrary real constants a1 and a2. Here ψ (x , t ) should be an even function in x. Example 1. (Figure 5.1) Given

ψ (x , t ) = A0

2 a1 a2

⎡

2

sech[A0 (x − x0)] e i ⎣a1 A0

(t − t0) + ϕ0⎤⎦

is a solution of (2.1), then

Φ(x , t ) = A0

2 2 a1 ⎡ ⎤ sech[i A0 (x − x0)] e i ⎣−a1 A0 (t − t0) + ϕ0⎦ a2

is a solution of (5.6), where a1 a2 > 0, A0, x0, t0, and ϕ0 are arbitrary real constants. Example 2. (Figure 5.2) Given

ψ (x , t ) =

1 a2

⎡ ⎤ 4 + i 8 (t − t 0 ) − 1⎥ e i [t − t0+ ϕ0] ⎢ 2 2 2 ⎣ 1 + 4 (t − t 0 ) + a 1 (x − x 0 ) ⎦

Figure 5.1. Plot of solution (5.7) with a1 = a2 = 1, A0 = 1, and x0 = t0 = ϕ0 = 0 .

5-5

(5.7)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Figure 5.2. Plot of solution (5.8) with a1 = a2 = 1 and x0 = t0 = ϕ0 = 0 . Animation available online at https:// iopscience.iop.org/book/978-0-7503-2428-1.

is a solution of (2.1), then

Φ( x , t ) =

⎡ ⎤ ⎢ ⎥ 1 ⎢ 4 − i 8 (t − t0 ) − 1⎥ e i [−(t − t0) + ϕ0] a2 ⎢ 1 + 4 (t − t )2 − 2 (x − x )2 ⎥ 0 0 ⎢⎣ ⎥⎦ a1

(5.8)

is a solution of (5.6), where a2 > 0, x0, t0, and ϕ0 are arbitrary real constants.

5.3 Galilean Transformation (Movable Solutions) If ψ (x , t ) is a solution of one of the three equations, fundamental NLSE, (2.1), NLSE with power law nonlinearity, (3.1), and NLSE with dual power law nonlinearity, (3.19), i ψt + a1 ψxx + a2 ∣ψ ∣2 ψ = 0, i ψt + a1 ψxx + a2 ∣ψ ∣n ψ = 0, i ψt + a1 ψxx + a2 ∣ψ ∣n ψ + a3 ∣ψ ∣m ψ = 0, then

Φ( x , t ) = ψ ( x − v t , t ) e

2 ⎡ ⎤ i ⎢ v (x − x 0 ) − v (t − t 0 )⎥ ⎣ 2 a1 ⎦ 4 a1

(5.9)

is a movable solution of the same equation

i Φt + a1 Φxx + a2 ∣Φ∣2 Φ = 0,

(5.10)

i Φt + a1 Φxx + a2 ∣Φ∣n Φ = 0,

(5.11)

i Φt + a1 Φxx + a2 ∣Φ∣n Φ + a3 ∣Φ∣m Φ = 0,

(5.12)

respectively, with real constants x0, t0, v, a1, a2, a3, n, and m.

5-6

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Example 1. sech(x,t) moving bright soliton (Figure 5.3) Given 2 a1 a2

ψ (x , t ) = A0

⎡

2

sech{A0 (x − x0)} e i ⎣a1 A0

(t − t0) + ϕ0⎤⎦

is a static solution of (2.1), then

Φ(x , t ) = A0

2 a1 sech{A0 [x − (x0 + v t )]} a2

(5.13)

⎡ ⎤ 4a 2 A 2 − v 2 i ⎢ v (x − x 0 ) + 1 0 (t − t0) + ϕ0⎥ 2 a1 4 a1 ⎣ ⎦ e

is a movable solution of (5.10), where a1 a2 > 0, A0, x0, t0, v, and ϕ0 are arbitrary real constants. Example 2. tanh(x,t) moving dark soliton (Figure 5.4) Given

ψ (x , t ) = A0

−2 a1 a2

⎡

2

tanh{A0 (x − x0)} e−i ⎣2 a1 A0

(t − t0) + ϕ0⎤⎦

is a static solution of (2.1), then

Figure 5.3. Moving bright soliton (5.13) with a1 = 1, a2 = 1/2 , A0 = 1, v = 1/2 , and x0 = t0 = ϕ0 = 0 . Animation available online at https://iopscience.iop.org/book/978-0-7503-2428-1.

Figure 5.4. Moving dark soliton (5.14) with a1 = 1/2 , a2 = −1, A0 = 1, v = 1/2 , and x0 = t0 = ϕ0 = 0 . Animation available online at https://iopscience.iop.org/book/978-0-7503-2428-1.

5-7

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Φ(x , t ) = A0

×

−2 a1 tanh{A0 [x − (x0 + v t )]} a2

⎡ ⎤ 8 a 2 A 2+v 2 − i ⎢ − v (x − x 0 ) + 1 0 (t − t0) + ϕ0⎥ 2 a1 4 a1 ⎣ ⎦ e

(5.14)

is a movable solution of (5.10), where a1 a2 < 0, A0, x0, t0, v and ϕ0 are arbitrary real constants. Example 3. Localization in x and t moving Peregrine soliton (Figure 5.5) Given ⎡ ⎤ 1 4 + i 8 (t − t 0 ) ψ (x , t ) = a ⎢ − 1⎥ e i [t − t0+ ϕ0] 2 2 2 (x − x 0 ) 2 ⎣ 1 + 4 (t − t 0 ) + ⎦ a1 is a static solution of (2.1), then

⎡ ⎤ ⎢ ⎥ 1 ⎢ 4 + i 8 (t − t0 ) − 1⎥ a2 ⎢ 1 + 4 (t − t )2 + 2 (x − x − v t )2 ⎥ 0 0 ⎢⎣ ⎥⎦ a1

Φ( x , t ) =

×e

2 ⎡ ⎤ i ⎢ v (x − x 0 ) − v (t − t 0 )⎥ ⎣ 2 a1 ⎦ 4 a1

(5.15)

e i [t − t0+ ϕ0]

is a moving solution of (5.10), where a2 > 0, x0, t0, v, and ϕ0 are arbitrary real constants. Example 4. sech(x,t) moving bright soliton Given

ψ (x , t ) =

{

2 A02 a1 (n + 2) a2 n 2

sech2[A0

1 n

}

(x − x0)]

⎡ 4 a1 A 2 ⎤ i ⎢ 2 0 (t − t0) + ϕ0⎥ ⎦ e ⎣ n

Figure 5.5. Moving Peregrine soliton (5.15) with a1 = a2 = 1, A0 = 1, v = 2, and x0 = t0 = ϕ0 = 0 . Animation available online at https://iopscience.iop.org/book/978-0-7503-2428-1.

5-8

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

is a static solution of (3.1), then 1

⎛ 2 A 2 a1 (n + 2) ⎞n 0 2{A [x − (x + v t )]}⎟ Φ( x , t ) = ⎜ sech 0 0 a2 n2 ⎝ ⎠ ×

(5.16)

⎡ 4 a1 A 2 ⎤ 2 ⎡ ⎤ 0 (t − t0) + ϕ0⎥ i ⎢ v (x − x 0 ) − v (t − t 0 ) ⎥ i ⎢ 2 n ⎣ ⎦ ⎣ ⎦ 2 a 4 a 1 1 e e

is a movable solution of (3.19), where a1 a2 (n + 2) > 0, A0, x0, t0, v, and ϕ0 are arbitrary real constants. Example 5. sech(x,t) moving flat-top soliton (Figure 5.6) Given

⎛ ψ (x , t ) = ⎜ ⎜ a2 ⎝

⎞n ⎟ e i [A1 (t − t0) + ϕ0] ⎟ ⎠ 1

A0 A1 (n + 2) ⎡ ⎤ A A0 + 2 cosh⎢n a 1 (x − x0)⎥ ⎣ ⎦ 1

{

}

is a static solution of (3.19), then 1

⎡ ⎤n ⎢ ⎥ ⎢ ⎥ A0 A1 (n + 2) Φ(x , t ) = ⎢ ⎛ ⎧ ⎫⎞ ⎥ ⎢ a ⎜A + 2 cosh⎨n A1 [x − (x + v t )]⎬⎟ ⎥ 0 ⎢⎣ 2 ⎝ 0 a1 ⎩ ⎭⎠ ⎥⎦ ×e

2 ⎡ ⎤ i ⎢ v (x − x 0 ) − v (t − t 0 ) ⎥ ⎣ 2 a1 ⎦ 4 a1

(5.17)

e i [A1 (t − t0) + ϕ0]

is a movable solution of (5.12), where A0 =

4 a 22 (n + 1) A1 δ (n + 2)2

δ = a3 +

,

a 22 (n + 1) , A1 (n + 2)2

m = 2 n, a1 A1 > 0,

Figure 5.6. Moving flat-top soliton (5.17) with a1 = A1 = 2 , a2 = 1, a3 = −0.06944444444 , n = 4, v = −1/3, and x0 = t0 = ϕ0 = 0 . Animation available online at https://iopscience.iop.org/book/978-0-7503-2428-1.

5-9

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

A1 δ (n + 1) > 0, x0, t0, A1, v, and ϕ0 are arbitrary real constants.

5.4 Function Coefficients If ψ (x , t ) is a solution of the fundamental NLSE, (2.1),

i ψt + a1 ψxx + a2 ∣ψ ∣2 ψ = 0, then

Φ(x , t ) = A(x , t ) e i B(x, t ) ψ [X (x , t ), T (x , t )]

(5.18)

i Φt + b1(x , t ) Φxx + b2(x , t ) ∣Φ∣2 Φ + [b3r(x , t ) + i b3i (x , t )] Φ = 0,

(5.19)

is a solution of

where

T (x , t ) = g1(t ), A(x , t ) =

B(x , t ) = g2(t ) −

g3(t )

,

Xx(x , t )

∫ 2 b1X(xt(,xt,)tX) xdx(x, t ) ,

b1(x , t ) =

b 2 (x , t ) =

b 3 r (x , t ) =

(5.20)

a1 g1′(t ) Xx2(x , t ) a2 g1′(t ) A2 (x , t )

(5.21)

(5.22)

,

(5.23)

,

(5.24)

∫ {g1″(t) Xt(x, t) Xx(x, t) − g1′(t) [Xtt(x, t) Xx(x, t) + Xt(x, t) Xxt(x, t)]} dx 2 a1 g1′2(t )

(5.25)

2 t) a1 g1′(t ) [2 Xx (x , t ) Xxxx (x , t ) − 3 Xxx (x , t )] + g2′(t ) + + , 4 4 a1 g1′(t ) 4 Xx ( x , t )

Xt2(x ,

b3i (x , t ) =

g ′(t ) Xxt(x , t ) − 3 , Xx(x , t ) g3(t )

g1(t ), g2(t ), g3(t ), and X (x , t ) are arbitrary real functions.

5.4.1 Constant Dispersion and Complex Potential If ψ (x , t ) is a solution of the fundamental NLSE, (2.1), 5-10

(5.26)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

i ψt + a1 ψxx + a2 ∣ψ ∣2 ψ = 0, then

Φ(x , t ) = A(x , t ) e i B(x, t ) ψ [X (x , t ), T (x , t )]

(5.27)

i Φt + b10 Φxx + b2(x , t ) ∣Φ∣2 Φ + [b3r(x , t ) + i b3i (x , t )] Φ = 0,

(5.28)

X (x , t ) = g4(t ) + g5(t ) x ,

(5.29)

is a solution of

where

b10 a1

T (x , t ) = g1(t ) = c0 +

g3(t )

A(x , t ) =

B(x , t ) = g2(t ) −

b3i (x , t ) =

+

,

(5.31)

4 b10 g5(t ) a2 b10 g53(t ) a1 g32(t ) g5′(t ) g5(t )

g4′2(t ) 4

(5.30)

⎡⎣2 g ′(t ) + x g ′(t )⎤⎦ x 4 5

b 2 (x , t ) =

b3r(x , t ) = g2′(t ) +

g5(t )

∫ g52(t ) dt,

b10 g52(t )

+

−

(5.33)

,

g3′(t ) g3(t )

,

2 g4′(t ) g5′(t ) − g5(t ) g4″(t )

2 g5′2(t ) − g5(t ) g5″(t ) 4 b10 g52(t )

(5.32)

,

2 b10 g52(t )

(5.34)

x (5.35)

x2,

g2(t ), g3(t ), g4(t ) and g5(t ) are arbitrary real functions and b10, a1, a2, and c0 are arbitrary real constants.

5.4.2 Constant Dispersion and Real Quadratic Potential If ψ (x , t ) is a solution of the fundamental NLSE, (2.1),

i ψt + a1 ψxx + a2 ∣ψ ∣2 ψ = 0,

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Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

then

Φ(x , t ) = c2

g5(t )

⎡ c2 i ⎢c1− 4 ⎢ 4 b10 e ⎣

∫ g52(t ) d t −

⎡ × ψ ⎢c3 + g5(t ) x + c4 ⎣

∫

⎤ 2 c 4 g52(t ) + g5′(t ) x ⎥ x ⎥ 4 b10 g5(t ) ⎦

g52(t )

b d t , c0 + 10 a1

∫

g52(t )

(5.36)

⎤ dt ⎥ ⎦

is a solution of

i Φt + b10 Φxx +

a2 b10 g5(t ) a1 c22

∣Φ∣2 Φ −

g5(t ) g5″(t ) − 2 g5′2(t ) 4 b10 g52(t )

x 2 Φ = 0,

(5.37)

where

g2(t ) = c1 −

c42 4 b10

∫ g52(t ) dt,

(5.38)

g3(t ) = c2 g5(t ), g4(t ) = c3 + c4

(5.39)

∫ g52(t ) dt,

(5.40)

c1, c2, c3, and c4 are arbitrary real constants, g5(t ) is an arbitrary real function of t. Example 1. sech(x) bright soliton Given

ψ (x , t ) = A0

2 a1 a2

⎡

2

sech[A0 (x − x0)] e i ⎣a1 A0

(t − t0) + ϕ0⎤⎦

is a solution of (2.1), then

Φ( x , t ) =

2 A02 a1 c22 g5(t ) sech A0 [c3 − x0 + g5(t ) x + c4 a2

{

∫ g52(t ) dt}

(5.41)

× e i ϕ(x, t ) is a solution of (5.37), where ϕ(x, t ) = A02 ⎡⎣a1 (c 0 − t0 ) + b10 ∫ g52(t ) dt ⎤⎦ −

c 42 4 b10

5-12

∫ g52(t ) dt −

⎡ ⎤ 2 c g 2(t ) + x g5′(t )⎥ x ⎣⎢ 4 5 ⎦ 4 b10 g5(t )

+ c1 + ϕ0 ,

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

a1 a2 > 0, A0, x0, t0, and ϕ0 are arbitrary real constants.

Case I: g5(t ) = α + β sin(γ t ) (Figure 5.7)

Φ(x , t ) =

⎛ ⎧ 2 A02 a1 c22 [α + β sin(γ t )] i ϕ(x, t ) e sech⎜ A0 ⎨ c3 − x0 + [α + β sin(γ t )] x a2 ⎝ ⎩ c4 [2 γ (2 α 2 + β 2) t − 8 α β cos(γ t ) − β 2 sin(2 γ t )] ⎫⎞ ⎬⎟ + 4γ ⎭⎠

(5.42)

is a solution of

i Φt + b10 Φxx +

a2 b10 [α + β sin(γ t )] ∣Φ∣2 Φ a1 c22

β γ 2 [3 β + β cos(2 γ t ) + 2 α sin(γ t )] 2 x Φ = 0, + 8 b10 [α + β sin(γ t )]2

(5.43)

where

ϕ(x , t ) = − −

β γ cos(γ t ) x 2 + 2 c4 [α + β sin(γ t )]2 x 4 b10 [α + β sin(γ t )] c42 [2 γ (2 α 2 + β 2 ) t − 8 α β cos(γ t ) − β 2 sin(2 γ t )] 16 b10 γ

A02 [4 γ a1 (c0 − t0) + b10 [2 γ (2 α 2 + β 2 ) t 4γ −8 α β cos(γ t ) − β 2 sin(2 γ t )]] + c1 + ϕ0 , +

α , β , and γ are arbitrary real constants.

Figure 5.7. Plot of solution (5.42). (a) 3D plot, (b) contour plot, with a1 = a2 = b10 = A0 = α = γ = c0 = c1 =c2 = c3 = c4 = 1, β = 6/10 , and x0 = t0 = ϕ0 = 0 . Animation available online at https://iopscience.iop.org/ book/978-0-7503-2428-1.

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Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Case II: g5(t ) = e γ t (Figure 5.8)

⎡ ⎛ c e2 γ t ⎞⎤ 2 a1 γ t e sech ⎢A0 ⎜ 4 + e γ t x − x0 + c3⎟⎥ e i ϕ(x, t ) (5.44) a2 ⎠⎦ ⎣ ⎝ 2γ

Φ(x , t ) = c2 A0 is a solution of

i Φt + b10 Φxx +

a2 b10 e γ t γ2 x 2 Φ = 0, ∣Φ∣2 Φ + 2 4 b10 a1 c2

(5.45)

where 1

ϕ(x , t ) = 8 b

10

γ

{−c4 e γ t (c4 e γ t + 4 γ x) + 2 γ [4 b10 (c1 + ϕ0) − γ x2 ]

+ 4 A02 b10 ⎡⎣ b10 e 2 γ t + 2 a1 γ (c0 − t0 ⎤⎦ + ϕ0 ,

)}

γ is an arbitrary real constant.

Example 2. tanh(x) dark soliton Given

ψ (x , t ) = A0

−2 a1 a2

⎡

2

tanh[A0 (x − x0)] e−i ⎣2 a1 A0

(t − t0) + ϕ0⎤⎦

is a solution of (2.1), then

Φ(x , t ) =

⎧ ⎡ −2 A02 a1 c22 g5(t ) tanh ⎨A0 ⎣⎢c3 − x0 + g5(t ) x + c4 ⎩ a2

∫ g52(t ) dt⎤⎦⎥}(5.46)

× e−i ϕ(x, t ) is a solution of (5.37), where

Figure 5.8. Plot of solution (5.44). (a) 3D plot, (b) contour plot, with a1 = a2 = b10 = A0 = c0 = c1 = c2 =c3 = c4 = 1, γ = 1/10 , and x0 = t0 = ϕ0 = 0 . Animation available online at https://iopscience.iop.org/book/ 978-0-7503-2428-1.

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Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

ϕ(x , t ) = 2 A02 ⎡⎣a1 (c0 − t0) + b10 ∫ g52(t ) dt ⎤⎦ +

c 42 4 b10

∫ g52(t ) dt +

⎡ 2 ′ ⎤ ⎢⎣2 c 4 g5 (t ) + x g5 (t )⎥⎦ x 4 b10 g5(t )

− c1 + ϕ0 ,

a1 a2 < 0, A0, x0, t0, and ϕ0 are arbitrary real constants.

Case I: g5(t ) = α + β sin(γ t ) (Figure 5.9)

Φ(x , t ) =

−2 A02 a1 c22 [α + β sin(γ t )] −i ϕ(x, t ) e a2 ⎡ ⎛ × tanh ⎢A0 ⎜ c3 − x0 + [α + β sin(γ t )] x ⎣ ⎝ +

(5.47)

c4 [2 γ (2 α 2 + β 2 ) t − 8 α β cos(γ t ) − β 2 sin(2 γ t )] ⎞⎤ ⎟⎥ 4γ ⎠⎦

is a solution of (5.43), where

ϕ(x , t ) =

β γ cos(γ t ) x 2 + 2 c 4 [α + β sin(γ t )] 2 x 4 b10 [α + β sin(γ t )]

+

A02 2γ

+

c 42 [2 γ (2 α 2 + β 2) t − 8 α β cos(γ t ) − β 2 sin(2 γ t )] 16 b10 γ

[4 γ a1 (c0 − t0) + b10 [2 γ (2 α 2 + β 2 ) t

−8 α β cos(γ t ) − β 2 sin(2 γ t )]] − c1 + ϕ0 , a1 a2 < 0, α , β , and γ are arbitrary real constants.

Figure 5.9. Plot of solution (5.47). (a) 3D plot, (b) contour plot, with a1 = b10 = A0 = c0 = c1 = c2 = c3 =c4 = α = β = γ = 1, a2 = −1, and x0 = t0 = ϕ0 = 0 . Animation available online at https://iopscience.iop.org/ book/978-0-7503-2428-1.

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Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Case II: g5(t ) = e γ t (Figure 5.10)

⎡ ⎛ c e2 γ t ⎞⎤ −2 a1 γ t e tanh ⎢A0 ⎜ 4 + e γ t x − x0 + c3⎟⎥ e i ϕ(x, t ) (5.48) a2 ⎠⎦ ⎣ ⎝ 2γ

Φ(x , t ) = c2 A0

is a solution of (5.45), where −1

ϕ(x , t ) = 8 b

10

γ

{c

4

e γ t (c4 e γ t + 4 γ x ) + 2 γ [4 b10 (ϕ0 − c1) + γ x 2 ]

+ 8 A02 b10 ⎡⎣b10 e 2 γ t + 2 a1 γ (c0 − t0)⎤⎦ + ϕ0 ,

}

a1 a2 < 0, γ is an arbitrary real constant.

Example 3. Two Bright Solitons Given

ψ (x , t ) =

1 a2

[ψ1(x , t ) + ψ2(x , t )]

is a solution of (2.1), then

Φ(x , t ) = c 2

g5(t ) [ψ1(x , t ) + ψ2(x , t )] a2

⎤ ⎡ 2 c 4 g52(t ) x + c42 g5(t ) ∫ g52(t ) d t + g5′(t )x 2 ⎥ ⎢ i ⎢c1− ⎥ 4 b10 g5(t ) ⎦, e ⎣

(5.49)

is a solution of (5.37), where ψ1(x , t ) = ψ2(x , t ) =

⎡ ⎤ ⎡ ⎤ M12 ⎣γ1−1(x, t ) + γ2*(x, t )⎦ − M22 ⎣γ2−1(x, t ) + γ2*(x, t )⎦ ⎡ ⎤⎡ ⎤ ⎡ ⎤⎡ ⎤ M12 M21 ⎣γ1*(x, t ) + γ2−1(x, t )⎦ ⎣γ1−1(x, t ) + γ2*(x, t )⎦ − M11 M22 ⎣γ1−1(x, t ) + γ1*(x, t )⎦ ⎣γ2−1(x, t ) + γ2*(x, t )⎦ ⎡ ⎤ ⎡ ⎤ −M11 ⎣γ1−1(x, t ) + γ1*(x, t )⎦ + M21 ⎣γ1*(x, t ) + γ2−1(x, t )⎦ ⎡ ⎤⎡ ⎤ ⎡ ⎤⎡ ⎤ M12 M21 ⎣γ1*(x, t ) + γ2−1(x, t )⎦ ⎣γ1−1(x, t ) + γ2*(x, t )⎦ − M11 M22 ⎣γ1−1(x, t ) + γ1*(x, t )⎦ ⎣γ2−1(x, t ) + γ2*(x, t )⎦

,

,

Figure 5.10. Plot of solution (5.48). (a) 3D plot, (b) contour plot, with a1 = b10 = A0 = c0 = c1 = c2 = c3 = c4 = 1, a2 = − 1, γ = 1/10 , and x0 = t0 = ϕ0 = 0 . Animation available online at https://iopscience.iop.org/ book/978-0-7503-2428-1.

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Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

M11 = 1/(λ1 + λ1*), M12 = 1/(λ1 + λ 2*), M21 = 1/(λ2 + λ1*), M22 = 1/(λ2 + λ 2*), λ1 = α1 + i ν1, λ2 = α2 + i ν2 ,

γ1(x , t ) =

⎧ λ2 ⎫ λ1 ⎡⎣c3 + g5(t) x + c 4 ∫ g52(t) dt⎤⎦ b i ⎨ 21 ⎡⎣c0− t0+ a101 ∫ g52(t ) dt⎤⎦ + ϕ01⎬ + − x01 λ1 2 a1 ⎭ e ⎩ ,

γ2(x , t ) =

⎧ λ2 ⎫ λ2 ⎡⎣c3 + g5(t) x + c4 ∫ g52(t) dt⎤⎦ b i ⎨ 22 ⎡⎣c0− t0+ a101 ∫ g52(t ) dt⎤⎦ + ϕ02⎬ + − x02 λ 2 2 a1 ⎩ ⎭ e .

Case I: g5(t ) = α + β sin(γ t ) (Figure 5.11)

α + β sin(γ t ) [ψ1(x , t ) + ψ2(x , t )] e i ϕ(x, t ) a2

Φ(x , t ) = c2

(5.50)

is a solution of (5.43), where

ϕ(x , t ) = − −

β γ cos(γ t ) x 2 + 2 c 4 [α + β sin(γ t )] 2 x 4 b10 [α + β sin(γ t )] c 42 [2 γ (2 α 2 + β 2) t − 8 α β cos(γ t ) − β 2 sin(2 γ t )] 16 b10 γ

γ1(x , t ) =

⎧ λ2 ⎫ b i ⎨ 21 ⎡⎣c0− t0+ 4 γ10a p(t )⎤⎦ + ϕ01⎬ + 1 ⎭ e ⎩

γ2(x , t ) =

⎧ λ2 ⎫ b i ⎨ 22 ⎡⎣c0− t0+ 4 γ10a p(t )⎤⎦ + ϕ02⎬ + 1 ⎩ ⎭ e

λ1 2 a1

λ2 2 a1

+ c1,

⎡c + (α + β sin(γ t )) x − c 4 p(t )⎤ − x λ ⎣ 3 ⎦ 01 1 4γ ⎡c + (α + β sin(γ t )) x − c 4 p(t )⎤ − x λ ⎣ 3 ⎦ 02 2 4γ

,

Figure 5.11. Plot of solution (5.50). (a) 3D plot, (b) contour plot, with a1 = A0 = 2 , a2 = b10 = c2 = c4 = α =γ = 1, β = 4/10 , c0 = c1 = c3 = 0 , α1 = 1, α2 = 2 , ν1 = 0 , ν2 = 1/2 , and x01 = x02 = ϕ01 = ϕ02 = 0 .

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Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

p(t ) = 8 α β cos(γ t ) + β 2 sin(2 γ t ) − 2 γ (2 α 2 + β 2 ) t , 2 A02 a1 c22 [α + β sin(γ t )]/a2 < 0, α , β , and γ are arbitrary real constants.

Case II: g5(t ) = e γ t (Figure 5.12) t

e2 [ψ (x , t ) + ψ2(x , t )] e−i a2 1

Φ( x , t ) =

1 (2 e t + 4 e t2 x + x 2) 8

(5.51)

is a solution of (5.43), where ⎧ λ2 ⎛ ⎫ b e2 γ t ⎞ i ⎨ 21 ⎜c0− t0+ 102 a γ ⎟ + ϕ01⎬ + 1 ⎠ ⎩ ⎝ ⎭

λ1 2 a1

⎡ c4 e 2 γ t ⎤ γt ⎢⎣c3+ e x + 2 γ ⎥⎦ − x01 λ1 ,

⎧ λ2 ⎛ ⎫ b e2 γ t ⎞ i ⎨ 22 ⎜c0− t0+ 102 a γ ⎟ + ϕ02⎬ + 1 ⎝ ⎠ ⎩ ⎭ e

λ2 2 a1

⎡ c4 e 2 γ t ⎤ γt ⎢⎣c3+ e x + 2 γ ⎥⎦ − x02 λ 2 ,

γ1(x , t ) = e γ2(x , t ) =

γ is an arbitrary real constant.

5.4.3 Constant Dispersion and Real Linear Potential If ψ (x , t ) is a solution of the fundamental NLSE, (2.1),

i ψt + a1 ψxx + a2 ∣ψ ∣2 ψ = 0, then ⎡

Φ(x , t ) =

c2

2

⎢ 7 ′ c 22 c5 ⎡ c5 x b10 c52 t ⎤ i ⎢⎣c1− 4 b10 c52 ∫ g4 (t ) d t − ⎥ ψ⎢ + g4(t ), c 0 + e c7 a1 c 72 ⎦ ⎣ c7

⎤ c 7 g4′(t ) x ⎥ 2 b10 c5 ⎥ ⎦

(5.52)

Figure 5.12. Plot of solution (5.51). (a) 3D plot, (b) contour plot, with a1 = A0 = 2 , a2 = b10 = c2 = c4 = 1, γ = 1/2 , c0 = c1 = c3 = 0 , α1 = 1, α2 = 2 , ν1 = 0 , ν2 = 1/2 , and x01 = x02 = ϕ01 = ϕ02 = 0 .

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Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

is a solution of

i Φt + b10 Φxx +

c7 g4″(t ) a2 b10 c5 x Φ = 0, ∣Φ∣2 Φ − 2 2 b10 c5 a1 c2 c7

(5.53)

where

g2(t ) = c1 −

1

∫ ⎡⎣c72 g4′2(t ) + 2 c6 c7 t g4′2(t ) + c62 t 2 g4′2(t )⎤⎦ dt,

4 b10 c52

g5(t ) =

c5 , c6 t + c7

(5.54) (5.55)

g4(t ) is an arbitrary real function of t, c1, c2, c5, c6, and c7 are arbitrary real constants. In (5.52) and (5.53), c6 is taken to be zero to obtain a solution with a norm proportional to that of ψ (x , t ).

Example 1. sech(x) bright soliton Given

ψ (x , t ) = A0

2 a1 a2

⎡

2

sech[A0 (x − x0)] e i ⎣a1 A0

(t − t0) + ϕ0⎤⎦

is a solution of (2.1), then

Φ(x , t ) = c2 A0

×

⎧ ⎡c ⎤⎫ 2 a1 c5 sech ⎨A0 ⎢ 5 x − x0 + g4(t )⎥⎬ ⎦⎭ a2 c7 ⎩ ⎣ c7 ⎪

⎪

⎧ ⎡ ⎤ ⎪ b c2 c72 i ⎨A02 a1 ⎢c0+ 10 52 (t − t0)⎥ − 2 ⎪ 4 a c b ⎣ ⎦ 1 7 10 c5 e ⎩

∫

⎫ ⎪ c7 g4′(t ) g4′ (t ) d t − x + c1+ ϕ0⎬ 2 b10 c5 ⎪ ⎭

(5.56)

2

is a solution of (5.53), where a1 a2 c5 c7 > 0, A0, x0, t0, and ϕ0 are arbitrary real constants. Case I: g4(t ) = α t 2 (Figure 5.13)

Φ(x , t ) = c2 A0

⎧ ⎡c ⎤⎫ 2 a1 c5 sech ⎨A0 ⎢ 5 x − x0 + α t 2⎥⎬ ⎦⎭ a2 c7 ⎩ ⎣ c7 ⎪

⎪

⎫ ⎧ ⎡ ⎤ c 2 α 2 t3 c α t b c2 ⎬ x + c1+ ϕ0⎪ − 7 i⎨ A02 a1 ⎢c0+ 10 52 (t − t0)⎥ − 7 ⎪ 2 b c 10 5 a1 c7 ⎣ ⎦ 3 b10 c5 ⎭ e ⎩ ⎪

×

⎪

5-19

(5.57)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Figure 5.13. Plot of (5.57) with a1 = a2 = A0 = α = c5 = c7 = 1, c2 = b10 = 2 , and c0 = c1 = x0 = t0 = ϕ0 = 0 . Animation available online at https://iopscience.iop.org/book/978-0-7503-2428-1.

is a solution of

i Φt + b10 Φxx +

a2 b10 c5 c α x Φ = 0. ∣Φ∣2 Φ − 7 b10 c5 a1 c22 c7

(5.58)

Example 2. tanh(x) dark soliton Given

ψ (x , t ) = A0

−2 a1 a2

⎡

2

tanh[A0 (x − x0)] e−i ⎣2 a1 A0

(t − t0) + ϕ0⎤⎦

is a solution of (2.1)], then

Φ(x , t ) = c2 A0

×

⎧ ⎡c ⎤⎫ −2 a1 c5 tanh ⎨A0 ⎢ 5 x − x0 + g4(t )⎥⎬ ⎦⎭ a2 c7 ⎩ ⎣ c7 ⎪

⎪

⎧ ⎡ ⎤ ⎪ b c2 c72 −i ⎨2 A02 a1 ⎢c0+ 10 52 (t − t0)⎥ + 2 ⎪ 4 a c b ⎣ ⎦ 1 7 10 c5 e ⎩

∫

⎫ ⎪ c7 g4′(t ) g4′ (t ) d t + x − c1+ ϕ0⎬ 2 b10 c5 ⎪ ⎭

(5.59)

2

is a solution of (5.53), where a1 a2 c5 c7 < 0, A0, x0, t0, and ϕ0 are arbitrary real constants. Case I: g4(t ) = α t 2 (Figure 5.14)

⎧ ⎡c ⎤⎫ −2 a1 c5 tanh ⎨A0 ⎢ 5 x − x0 + α t 2⎥⎬ ⎦⎭ a2 c7 ⎩ ⎣ c7 ⎪

Φ(x , t ) = c2 A0

⎪

⎫ ⎧ ⎡ ⎤ c 2 α 2 t3 c α t b c2 2 A02 a1 ⎢c0+ 10 52 (t − t0)⎥ + 7 x − c1+ ϕ0⎬ −i ⎨ + 7 ⎪ ⎪ 2 b c 10 5 a1 c7 ⎣ ⎦ 3 b10 c5 ⎭ e ⎩ ⎪

×

⎪

is a solution of (5.58).

5-20

(5.60)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Figure 5.14. Plot of (5.60) with a1 = A0 = α = c5 = c7 = 1, a2 = −1, c2 = b10 = 2 , and c0 = c1 = x0 = t0 = ϕ0 = 0 . Animation available online at https://iopscience.iop.org/book/978-0-7503-2428-1.

Example 3. Generalized First-Order Breather Given

ψ (x , t ) =

A0 a2

(1 −

8 λr A0

)

⎡

2

p(x , t ) e i ⎣A0

(t − t0) + ϕ0⎤⎦

is a solution of (2.1), then

Φ(x , t ) = A0

×

c22 c5 ⎛ ⎜1 − a2 c7 ⎝

⎞ 8 λr p(x , t )⎟ A0 ⎠

⎡ ⎛ ⎞ b c2t c72 i ⎢A02 ⎜c0+ 10 52 − t0⎟ − 2 ⎢ a c b 4 ⎝ ⎠ 1 7 10 c5 e ⎣

∫

(5.61)

⎤ c7 g4′(t ) x 2 g4′ (t ) d t − + c1+ ϕ0⎥ ⎥ 2 b10 c5 ⎦

is a solution of (5.53), where

p(x , t ) =

( A02 + Γ2) cos[q1(x, t )] + i ( A02 − Γ2) sin[q1(x, t )] + 2 A0 {Γr cosh[q2(x, t )] − i Γi sinh[q2(x, t )]} , 2 A0 Γr cos[q1(x, t )] + ( Γ2 + A02) cosh[q2(x, t )]

q1(x , t ) = δi +

⎡ c5 x + g4(t ) − x0 Δi − 2 c0 + 2 ⎢ c7 a 1 ⎣

(

b10 c52 t a1 c72

⎤ − t0 (Δi λi + Δr λr )⎥ , ⎦

)

q2(x , t ) = δr +

⎡ c5 x + g4(t ) − x0 2 ⎢ c7 a Δr − 2 c 0 + 1 ⎣

(

b10 c52 t a1 c72

)

(

− t0 Δr λi + 2 c 0 +

⎡ ⎤ Δr = Re⎣ 2 (λr − i λi )2 − A02 ⎦, ⎡ ⎤ Δi = Im⎣ 2 (λr − i λi )2 − A02 ⎦, Γr = Δr + 2 λr , Γi = Δi − 2 λi , Γ = Γ r2 + Γ i2 , a1 > 0,

5-21

b10 c52 t a1 c72

⎤ − t0 Δi λ r ⎥ , ⎦

)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

a2 > 0, A0, λr , λi , x0, t0, and ϕ0 are arbitrary real constants.

Case I: g4(t ) = α t 2 (Figure 5.15)

Φ(x , t ) = A0

×

c22 c5 ⎛ ⎜1 − a2 c7 ⎝

⎞ 8 λr p(x , t )⎟ A0 ⎠

(5.62)

⎡ ⎤ ⎛ ⎞ c α (3 c x + c α t 2) t b c2t 5 7 + c1+ ϕ0⎥ i ⎢A02 ⎜c0+ 10 52 − t0⎟ − 7 2 ⎢⎣ ⎥⎦ a c 3 b c ⎝ ⎠ 1 10 7 5 e

is a solution of (5.58), where

p(x , t ) =

2 2 2 2 (A0 + Γ ) cos[q1(x, t )] + i (A0 − Γ ) sin[q1(x, t )] + 2 A0 {Γr cosh[q2(x, t )] − i Γi sinh[q2(x, t )]} , 2 A0 Γr cos[q1(x, t )] + (Γ2 + A02) cosh[q2(x, t )]

q1(x , t ) = δi +

q2(x , t ) = δr +

⎡ c5 x + α t 2 − x 0 Δi − 2 c0 + 2 ⎢ c7 a 1 ⎣

(

⎡ c5 x + α t 2 − x 0 2 ⎢ c7 a Δr − 2 c0 + 1 ⎣

(

b10 c52 t a1 c 72

b10 c52 t a1 c72

⎤ − t0 (Δi λi + Δr λr )⎥ , ⎦

)

)

(

− t0 Δ r λ i + 2 c 0 +

b10 c52 t a1 c 72

⎤ − t0 Δi λ r ⎥ , ⎦

)

⎡ ⎤ Δr = Re⎣ 2 (λr − i λi )2 − A02 ⎦ , ⎡ ⎤ Δi = Im⎣ 2 (λr − i λi )2 − A02 ⎦, Γr = Δr + 2 λr , Γi = Δi − 2 λi , Γ = Γ r2 + Γ i2 , a1 > 0, a2 > 0, A0, λr , λi , x0, t0, and ϕ0 are arbitrary real constants.

Figure 5.15. Plot of (5.62) with a1 = a2 = A0 = α = c2 = c5 = c7 = b10 = 1, λr = −1/ 2 , λi = 1/1000 , and c0 = c1 = δr = δi = x0 = t0 = ϕ0 = 0 .

5-22

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Case II. g4(t ) = −α cos(β t ) (Figure 5.16)

c22 c5 [1 − p(x , t )] a2 c7

Φ(x , t ) = A0 ×e

i

{A02

(c 0 +

b10 c52 t a1 c72

(5.63)

c α β sin(β t ) x c72 α 2 β [2 β t − sin(2 β t )] − t 0) − 7 − + c1+ ϕ0} 2 b10 c5 16 b c 2 10

5

is a solution of (5.58), where p (x , t ) = 2 2 λ r

q1(x , t ) = δi +

q2(x, t ) = δr +

(A02 + Γ 2) cos[q1(x, t )] + i (A02 − Γ 2) sin[q1(x, t )] + 2 A0 {Γr cosh[q2(x, t )] − i Γi sinh[q2(x, t )]} 2 A02 Γr cos[q1(x, t )] + A0 (Γ 2 + A02 ) cosh[q2(x, t )]

⎡ c5c x − α cos(β t ) − x0 2⎢ 7 Δi − 2 c0 + a1 ⎣

(

⎡ c5 x − α cos(β t ) − x 0 ⎛ c 2 ⎢ 7 Δr − 2 ⎜c 0 + a ⎝ 1 ⎢⎣

b10 c52 t a1 c 72

b10 c52 t a1 c72

,

⎤ − t0 (Δi λi + Δr λr )⎥ , ⎦

)

⎞ ⎛ − t0⎟ Δr λi + 2 ⎜c 0 + ⎠ ⎝

b10 c52 t a1 c 72

⎤ ⎞ − t 0 ⎟ Δi λ r ⎥ , ⎠ ⎥⎦

Δr = Re[ 2 (λr − i λi )2 − A 02 ], Δi = Im[ 2 (λr − i λi )2 − A 02 ], Γr = Δr + 2 λr , Γi = Δi − 2 λi , Γ = Γ r2 + Γ i2 , a1 > 0, a2 > 0, A0 , λr , λi , x0, t0 , and ϕ0 are arbitrary real constants.

Figure 5.16. Plot of (5.63) with a1 = b10 = 1/2 , a 2 = 2 , A0 = 3/2 , α = 18/4 , c2 = c5 = c7 = 1, λr = −3/2 λi = 1/100 , and c0 = c1 = δr = δi = x0 = t0 = ϕ0 = 0 . (Used in the front cover page).

5-23

2,

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

5.4.4 Constant Nonlinearity and Complex Potential If ψ (x , t ) is a solution of the fundamental NLSE, (2.1),

i ψt + a1 ψxx + a2 ∣ψ ∣2 ψ = 0, then

Φ(x , t ) = A(x , t ) e i B(x, t ) ψ [X (x , t ), T (x , t )]

(5.64)

i Φt + b1(x , t ) Φxx + b20 ∣Φ∣2 Φ + [b3r(x , t ) + i b3i (x , t )] Φ = 0,

(5.65)

X (x , t ) = g4(t ) + g5(t ) x ,

(5.66)

is a solution of

where

b20 a2

T (x , t ) = g1(t ) = c0 +

A(x , t ) =

B(x , t ) = g2(t ) −

+

g5(t )

a1 b20 g32(t ) a2 g53(t ) g5′(t ) g5(t )

−

(5.67)

dt ,

,

(5.68)

4 a1 b20 g32(t )

b3i (x , t ) =

+

g3(t )

g5(t )

a2 ⎡⎣2 g4′(t ) + x g5′(t )⎤⎦ x g52(t )

b1(x , t ) =

b3r (x , t ) = g2′(t ) +

∫

g32(t )

(5.69)

,

(5.70)

,

g3′(t ) g3(t )

,

(5.71)

a2 g4′2(t ) g5(t ) 4 a1 b20 g32(t )

⎡ ⎤ a2 g5(t ) ⎣2 g5(t ) g3′(t ) g4′(t ) − g3(t ) g4′(t ) g5′(t ) − g3(t ) g4″(t ) g5(t )⎦ 2 a1 b20 g33(t ) ⎡ ⎤ a2 g5(t ) ⎣2 g3′(t ) g5(t ) g5′(t ) − g3(t ) g5′2(t ) − g3(t ) g5(t ) g5″(t )⎦ 4 a1 b20 g33(t )

g4(t ) and g5(t ) are arbitrary real functions, b20 and c0 are arbitrary real constants.

5-24

x 2,

x (5.72)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

5.4.5 Constant Nonlinearity and Real Quadratic Potential If ψ (x , t ) is a solution of the fundamental NLSE, (2.1),

i ψt + a1 ψxx + a2 ∣ψ ∣2 ψ = 0, then

Φ(x , t ) = c2

g5(t )

⎧ ⎡ a2 ⎣⎢2 c 4 g5(t ) x + g5′(t ) x 2 + c42 ⎪ i ⎨c1− 4 a1 b 20 c22 ⎪ e ⎩

⎡ × ψ ⎢c3 + g5(t ) x + c4 ⎣

⎫

∫ g5(t ) dt⎤⎦⎥ ⎪

∫ g5(t ) dt, c0 +

⎬ ⎪ ⎭

b20 c22 a2

∫

⎤ g5(t ) dt ⎥ ⎦

(5.73)

is a solution of

⎡ ⎤ a2 ⎣g5′2(t ) − g5(t ) g5″(t )⎦ a1 b20 c22 2 i Φt + Φxx + b20 ∣Φ∣ Φ + x 2 Φ = 0, a2 g5(t ) 4 a1 b20 c22 g5(t )

(5.74)

where g5(t ) is an arbitrary real function of t, c1, c2, c3, and c4 are arbitrary real constants. 5.4.6 Constant Nonlinearity and Real Linear Potential If ψ (x , t ) is a solution of the fundamental NLSE, (2.1),

i ψt + a1 ψxx + a2 ∣ψ ∣2 ψ = 0, then ⎧ ⎡ ⎤⎫ 2 a2 ⎢c52 c6 e c6 t x 2 + 2 c5 g4′(t ) x + g4′ (t ) e−c6 t dt⎥ ⎪ ⎪ ⎪ ⎣ ⎦⎪ ⎬ i ⎨c1− 2 a b c c 4 ⎪ ⎪ 1 20 2 5 ⎪ ⎪ ⎭ e c6 t e ⎩

∫

Φ(x , t ) = c2

c5 ⎡ b c 2 c (e c6 t − e c6) ⎤ ⎥ × ψ ⎢c5 e c6 t x + g4(t ), c0 + 20 2 5 a2 c6 ⎣ ⎦

(5.75)

is a solution of

⎡ ⎤ a2 ⎣c6 g4′(t ) − g4″(t )⎦ a1 b20 c22 e−c6 t i Φt + x Φ = 0, (5.76) Φxx + b20 ∣Φ∣2 Φ + 2 a2 c5 2 a1 b20 c2

5-25

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

where

g2(t ) = c1 −

a2 4 a1 b20 c22 c5

∫ g4′(t )2 e−c t dt, 6

g5(t ) = c5 e c6 t ,

(5.77) (5.78)

g4(t ) is an arbitrary real function of t, c1, c2, c5, and c6 are arbitrary real constants.

5.5 Solution-Dependent Transformation If ψ (x , t ) is a solution of the fundamental NLSE, (2.1),

i ψt + a1 ψxx + a2 ∣ψ ∣2 ψ = 0, then

Φ(x , t ) = A(x , t ) e i B(x, t ) ψ [X (x , t ), T (x , t )]

(5.79)

i Φt + b1(x , t ) Φxx + b2(x , t ) ∣Φ∣2 Φ + [b3r(x , t ) + i b3i (x , t )] Φ = 0,

(5.80)

is a solution of

where

T (x , t ) = g1(t ), b1(x , t ) =

b 2 (x , t ) =

b3r(x , t ) =

a1 g1′(t ) Xx2(x , t ) a2 g1′(t ) A2 (x , t )

(5.81) ,

(5.82)

,

(5.83)

a1 g1′(t )[A(x , t ) Bx2(x , t ) − Axx (x , t )] A(x , t ) Xx2(x , t )

+ Bt(x , t )

⎡ ψ (x , t ) ⎤ a1 g1′(t )[2 Ax (x , t ) Xx(x , t ) + A(x , t ) Xxx(x , t )] Re ⎢ x ⎥ (5.84) 2 ⎣ ψ (x , t ) ⎦ A(x , t )Xx (x , t ) ⎡ 2 a1 g1′(t )Bx(x , t ) ⎤ ⎡ ψ (x , t ) ⎤ +⎢ + Xt(x , t )⎥ Im ⎢ x ⎥, Xx(x , t ) ⎣ ⎦ ⎣ ψ (x , t ) ⎦

−

5-26

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

b3 i ( x , t ) = −

a1 g1′(t )[2 Ax (x , t ) Bx(x , t ) + A(x , t ) Bxx(x , t )] Xx2(x ,

t ) A(x , t ) ⎡ 2 a1 g1′(t ) Bx(x , t ) ⎤ ⎡ ψ (x , t ) ⎤ −⎢ + Xt(x , t )⎥Re ⎢ x ⎥ ⎣ Xx(x , t ) ⎦ ⎣ ψ (x , t ) ⎦ −

a1 g1′(t ) [2 Ax (x , t ) Xx(x , t ) + A(x , t ) Xxx(x , t )] Xx2(x ,

t ) A(x , t )

−

At (x , t ) A(x , t ) (5.85)

⎡ ψ (x , t ) ⎤ Im ⎢ x ⎥, ⎣ ψ (x , t ) ⎦

g1(t ), X (x , t ), and B (x , t ) are arbitrary real functions.

5.5.1 Special Case I: Stationary Solution, Constant Dispersion and Nonlinearity Coefficients 2

If ψ (x , t ) = A0 f (x ) ei A0 (2.1),

(t − t0 )

is a stationary solution of the fundamental NLSE,

i ψt + a1 ψxx + a2 ∣ψ ∣2 ψ = 0, then

Φ( x , t ) =

=A0

⎤ a2 i B(x, t ) ⎡ a1 (x − x0), (t − t0)⎥ e ψ⎢ b20 ⎣ b10 ⎦ ⎤ a2 ⎡ a1 2 (x − x0)⎥ e i [B(x, t ) + A0 (t − t0)] f⎢ b20 ⎣ b10 ⎦

(5.86)

(5.87)

is a solution of

i Φt + b10 Φxx + b20 ∣Φ∣2 Φ ⎡ ⎢ ⎢ + ⎢ Bt(x , t ) + b10 Bx2(x , t ) ⎢ ⎢ ⎣

(5.88)

⎛ 4a ⎞⎤ ⎡ a1 ⎤ 1 ⎜ ⎟⎥ (x − x0)⎥ Bx(x , t ) f ′⎢ ⎦ ⎜ b10 ⎣ b10 ⎟⎥ − i b10 ⎜ + Bxx(x , t )⎟⎥ Φ = 0, ⎡ a1 ⎤ ⎜ ⎟⎥ ( x − x0 ) ⎥ f⎢ ⎜ ⎟⎥ ⎣ b10 ⎦ ⎝ ⎠⎦ where B (x , t ) is an arbitrary real function, b10, b20, A0 and t0 are arbitrary real constants.

5-27

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

5.5.2 Special Case II: PT-Symmetric Potential 2

If ψ (x , t ) = A0 f (x ) ei A0 (2.1),

(t − t0 )

is a stationary solution of the fundamental NLSE,

i ψt + a2 ∣ψ ∣2 ψ = 0, then

⎤ a2 i B(x) ⎡ a1 (x − x0), (t − t0)⎥ e ψ⎢ b20 ⎣ b10 ⎦

Φ( x , t ) =

=A0

⎤ a2 ⎡ a1 2 (x − x0)⎥ e i [B(x) + A0 (t − t0)] f⎢ b20 ⎣ b10 ⎦

(5.89)

(5.90)

is a solution of

i Φt + b10 Φxx + b20 ∣Φ∣2 Φ + [Veven(x ) + i Vodd(x )]Φ = 0,

(5.91)

Veven = b10 B′2(x ),

(5.92)

where

and

Vodd

⎧ ⎪ ⎪ = −b10 ⎨ ⎪ ⎪ ⎩

⎫ ⎤ 4 a1 ⎡ a1 ⎪ (x − x0)⎥ B′(x ) f ′⎢ ⎪ b10 ⎣ b10 ⎦ + B″(x )⎬ , ⎡ a1 ⎤ ⎪ ( x − x0 ) ⎥ f⎢ ⎪ ⎣ b10 ⎦ ⎭

(5.93)

form a PT-symmetric potential for some choices of B (x ). Case I: B (x , t ) = sin(x − x0 ) 2 Given ψ (x , t ) = A0 ei [a2 A0 (t − t0 ) + ϕ0 ] is a stationary solution of the fundamental NLSE, (2.1),

i ψt + a1 ψxx + a2 ∣ψ ∣2 ψ = 0, then

Φ( x , t ) =

⎤ a2 i sin(x − x0) ⎡ a1 (x − x0), (t − t0)⎥ e ψ⎢ b20 ⎣ b10 ⎦

(5.94)

is a solution of

i Φt + b10 Φxx + b20 ∣Φ∣2 Φ + b10 [cos2 (x − x0 ) + i sin(x − x0 )] Φ = 0,

5-28

(5.95)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

where B (x ) is an arbitrary real function, b10, b20, A0, x0, and t0 are real constants. 5.5.3 Special Case III: Stationary Solution, Constant Dispersion and Nonlinearity Coefficients, and Real Potential 2

If ψ (x , t ) = A0 f (x ) ei [a2 A0

(t − t0 ) + ϕ0 ]

is a stationary solution of

i ψt + a1 ψxx + a2 ∣ψ ∣2 ψ = 0, then

Φ( x , t ) =

= A0

⎤ a2 i B(x, t ) ⎡ a1 (x − x0), (t − t0)⎥ e ψ⎢ b20 ⎣ b10 ⎦ ⎤ a2 ⎡ a1 2 (x − x0)⎥ e i [B(x, t ) + A0 (t − t0)] f⎢ b20 ⎣ b10 ⎦

(5.96)

(5.97)

is a solution of i Φt + b10 Φxx + b 20 ∣Φ∣2 Φ ⎧ ⎪ ⎪ b10 g12(t ) +⎨ + g1′(t ) ⎤ ⎪ 4 ⎡ a1 ⎪ f ⎢⎣ b (x − x0 )⎥⎦ ⎩ 10

∫

⎫ ⎪ ⎪ dx + g2′(t )⎬ Φ = 0, ⎡ a1 ⎤ ⎪ f 2⎢ (x − x0 )⎥ ⎪ ⎣ b10 ⎦ ⎭

(5.98)

where

B(x , t ) = g1(t )

∫

dx ⎡

⎤ a1 ( x − x0 ) ⎥ f ⎣ b10 ⎦ 2⎢

g1(t ) and g2(t ) are arbitrary real functions of t, b10, b20, A0, x0, and t0 are real constants.

5-29

+ g2(t ),

(5.99)

a2 a22

ψ

(

a1 a11

5-30

−2 a1 a22

2. Φ(x , t ) = A 0

2

⎤ (x − x0 )⎥⎦

⎤ ⎤ ⎡ 2 (x − x0 )⎥⎦ ei ⎣a1 A0 (t − t0) + ϕ0⎦

a1 a11

a1 a11

2.

Φ(x , t ) =

1 a2

1. Φ(x , t ) = A0

# Example ⎡

2

sech[i A0 (x − x0 )] ei ⎣−a1 A0

(t − t0 ) + ϕ0⎤⎦

⎡ ⎤ 4 − i 8 (t − t0 ) ⎢ ⎥ ei [−(t − t0) + ϕ0] − 1 2 2 2 ⎣ 1 + 4 (t − t0) − a1 (x − x0) ⎦

2 a1 a2

Equation: i Φt + a1 Φxx − a2 ∣Φ∣2 Φ = 0

Transformation: Φ(x , t ) = ψ (i x , − t )

Defocusing (Focusing) NLSE to Focusing (Defocusing) NLSE

(t − t0 ) + ϕ0⎤⎦

⎡ tanh ⎣⎢A0

⎡ sech ⎣⎢A0

× e−i ⎣2 a1 A0

⎡

2 a1 a22

1. Φ(x , t ) = A0

# Example

)

Name

Peregrine soliton, two solitons

(5.8)

(5.7)

–

a1 a2 > 0 , A0, x0, t0, and ϕ0 are arbitrary real constants a2 > 0, x0, t0, and ϕ0 are arbitrary real constants

Eq. #

Name

(5.4)

(5.3)

Eq. #

Conditions

a1 a22 < 0, a1 a11 > 0, A0, x0, t0, and ϕ0 are arbitrary real constants dark soliton

a1 a22 > 0, a1 a11 > 0, A0, x0, t0, and ϕ0 are arbitrary real constants bright soliton

Conditions

x , t , ψ is a solution of the fundamental NLSE (2.1)

Equation: i Φt + a11 Φxx + a22 ∣Φ∣2 Φ = 0

Transformation: Φ (x , t ) =

Fundamental NLSE to Fundamental NLSE with Different Constant Coefficients

Note: For lengthy conditions, the reader is referred to the solutions in sections 5.1–5.5.

5.6 Summary of Sections 5.1–5.5

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

5-31

Φ(x , t ) =

sech{A0 [x − (x0 + v t )]}

×e

1 a2

tanh{A0 [x − (x0 + v t )]}

2 ⎡ ⎤ i ⎢⎣ 2va (x − x0 ) − 4va (t − t0 )⎥⎦ 1 1

ei [t − t0 + ϕ0]

⎡ ⎤ 4 + i 8 (t − t0 ) ⎢ − 1⎥ 2 + 2 (x − x − v t )2 + − t t 1 4 ( ) 0 0 ⎣ ⎦ a1

⎡ ⎤ 8 a 2 A 2 + v2 (t − t0 ) + ϕ0⎥ −i ⎢− 2 va (x − x0 ) + 1 4 a0 1 1 ⎣ ⎦

−2 a1 a2

1.

Φ(x , t ) =

# Example

)

⎤ ⎡ 4 a1 A 2 i ⎢ 2 0 (t − t0 ) + ϕ0⎥ n ⎣ ⎦ e

sech2{A0 [x − (x0 + v t )]}

2 ⎡ ⎤ i v (x − x0 ) − 4va (t − t0 )⎦⎥ 1 e ⎢⎣ 2a1

2 A02 a1 (n + 2) a2 n2

×

(

Equation: i Φt + a1 Φxx + a2 ∣Φ∣n Φ = 0

3.

×e

2. Φ(x , t ) = A0

×

2 a1 a2

1 n

2 ⎡ ⎤ i ⎢⎣ 2 va (x − x0 ) − 4va (t − t0 )⎥⎦ 1 1

⎡ ⎤ 4a 2 A 2 − v 2 i ⎢ 2 va (x − x0 ) + 1 4 a0 (t − t0 ) + ϕ0⎥ 1 1 ⎣ ⎦ e

1. Φ(x , t ) = A0

# Example

Equation: i Φt + a1 Φxx + a2 ∣Φ∣2 Φ = 0

Transformation: Φ(x , t ) = ψ (x − v t , t ) e

Galilean Transformation (Movable Solutions)

Name

moving Peregrine soliton

a1 a2 (n + 2) > 0 , A0, x0, t0, v, and ϕ0 are arbitrary real constants moving bright soliton

Conditions

a2 > 0, x0, t0, v, and ϕ0 are arbitrary real constants

moving dark soliton

moving bright soliton

a1 a2 > 0 , A0, x0, t0, v, and ϕ0 are arbitrary real constants

a1 a2 < 0 , A0, x0, t0, v and ϕ0 are arbitrary real constants

Name

Conditions

(5.16)

Eq. #

(5.15)

(5.14)

(5.13)

Eq. #

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

{

×e

2 ⎡ ⎤ i ⎢⎣ 2va (x − x0 ) − 4va (t − t0 )⎥⎦ 1 1

1 n

4 a 22 (n + 1) A1 δ (n + 2)2

5-32

Equation 1: i Φt + b10 Φxx +

Transformation:

a1 c22

a2 b10 g5(t )

∣Φ∣2 Φ −

a 22 (n + 1) , A1 (n + 2)2

2

⎤

∫ g52(t )dt − 2c4g45b10(t)g+5(gt5′)(t)x x⎦⎥

4 b10 g52(t )

g5(t ) g5″(t ) − 2 g5 ′2(t )

x2 Φ = 0

∫

⎡ × ψ ⎢⎣c3 + g5(t )x + c4 g52(t )dt , c0 +

10

⎡ c2 i ⎢c1− 4b4

Φ(x , t ) = c2 g5(t ) e ⎣

Constant Dispersion and Real Quadratic Potential

Equation: i Φt + b10 Φxx + b2(x , t ) ∣Φ∣2 Φ + [b3r (x , t ) + i b3i (x , t )] Φ = 0

Constant Dispersion and Complex Potential

, δ = a3 +

m = 2 n , a1 A1 > 0 ,

b10 a1

∫ g52(t )dt⎤⎥⎦

A1 δ (n + 1) > 0 , x0, t0, A1, v, and ϕ0 are arbitrary real constants

A0 =

Conditions

Equation: i Φt + b1(x , t ) Φxx + b2(x , t ) ∣Φ∣2 Φ + [b3r (x , t ) + i b3i (x , t )] Φ = 0

Transformation: Φ(x , t ) = A(x , t ) ei B(x, t ) ψ [X (x , t ), T (x , t )]

ei [A1 (t − t0) + ϕ0]

⎤ ⎥ ⎞⎥ ⎟⎥ ⎠⎦

}

⎡ ⎢ A0 A1 (n + 2) Φ(x , t ) = ⎢ ⎛ A ⎢⎣ a2 ⎜⎝A0 + 2 cosh n a11 [x − (x0 + v t )]

Function Coefficients

1.

# Example

Equation: i Φt + a1 Φxx + a2 ∣Φ∣n Φ + a3 ∣Φ∣m Φ = 0

(Continued )

moving flat-top soliton

Name

(5.17)

Eq. #

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Φ(x , t ) =

Φ(x , t ) =

{

2 A02 a1 c22 g5(t ) a2

e i ϕ (x , t )

e−i ϕ(x, t )

}

5-33

c4 [2 γ (2 α 2 + β 2 ) t − 8 α β cos(γ t ) − β 2 sin(2 γ t )] }) 4γ

e i ϕ (x , t )

+

×

× sech(A0 {c3 − x0 + [α + β sin(γ t )] x

− c1 + ϕ0

∫ g52(t ) dt

∫ g52(t ) dt

A02 4γ

[4 γ a1 (c0 − t0 ) + b10 [2 γ (2 α 2 + β 2 ) t

c42 [2 γ (2 α 2 + β 2 ) t − 8 α β cos(γ t ) − β 2 sin(2 γ t )] 16 b10 γ

− 8 α β cos(γ t ) − β 2 sin(2 γ t )]] + c1 + ϕ0

+

−

x2 Φ = 0

β γ cos(γ t ) x 2 + 2 c4 [α + β sin(γ t )]2 x 4 b10 [α + β sin(γ t )]

β γ 2 [3 β + β cos(2 γ t ) + 2 α sin(γ t )] 8 b10 [α + β sin(γ t )]2

See text.

ϕ (x , t ) = −

2 A02 a1 c22 [α + β sin(γ t )] a2

4 b10 g5(t )

[2 c4 g52(t ) + x g5′(t )] x

c42 4 b10

4 b10

c42

a1 a2 < 0 , A0, x0, t0, and ϕ0 are arbitrary real constants

+

1.

Φ(x , t ) =

+ c1 + ϕ0

ϕ(x , t ) = 2 A02 [a1 (c0 − t0 ) + b10 ∫ g52(t ) dt ] +

Conditions

∣Φ∣2 Φ +

4 b10 g5(t )

⎡ 2 ′ ⎤ ⎣⎢2 c4 g5 (t ) + x g5 (t )⎦⎥ x

a1 a2 > 0 , A0, x0, t0, and ϕ0 are arbitrary real constants

−

ϕ(x , t ) = A02 ⎡⎣a1 (c0 − t0 ) + b10 ∫ g52(t ) dt ⎤⎦ −

Conditions

# Example

a1 c22

a2 b10 [α + β sin(γ t )]

2 c4 g52(t ) x + c42 g5(t ) ∫ g52(t ) d t + g5′(t )x 2 ] 4 b10 g5(t )

[ψ1(x , t ) + ψ2(x , t )]

Equation 2: i Φt + b10 Φxx +

× ei [c1−

g5(t ) a2

× tanh{A0 [c3 − x0 + g5(t ) x + c4 ∫ g52(t ) dt ]}

−2 A02 a1 c22 g5(t ) a2

× sech A0 [c3 − x0 + g5(t ) x + c4 ∫ g52(t ) dt ]

3. Φ(x , t ) = c 2

2.

1.

# Example

bright soliton

Name

two bright solitons

dark soliton

bright soliton

Name

(5.42)

Eq. #

(5.49)

(5.46)

(5.41)

Eq. #

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Φ(x , t ) =

c4 [2 γ (2

5-34

t

e2 a2

sin(2 γ t )]

Transformation: Φ(x , t ) =

)]

1 8

c22 c5 c7

ψ[

c5 x c7

γ2 4 b10

x2 Φ = 0

+ g4(t ), c0 +

t

a1 c72

b10 c52 t

(2 et + 4 e 2 x + x 2 )

]e

i [c1−

[4 γ a1 (c0 − t0 ) + b10 [2 γ (2 α 2 + β 2 ) t

{− c4 e γ t (c4 e γ t + 4 γ x ) + 2 γ [4 b10 (c1 + ϕ0 )

− γ x 2 ] + 4 A02 b10 [b10 e 2 γ t + 2 a1 γ (c0 − t0 )]} + ϕ0 ,

1 8 b10 γ

b10 [b10 e 2 γ t + 2 a1 γ (c0 − t0 )]} + ϕ0 ,

c 72 4 b10 c52

2

∫ g4′ (t ) dt −

See text.

c 7 g4′ (t ) x ] 2 b10 c5

γ is an arbitrary real constant

A02

{c4 e γ t (c4 e γ t + 4 γ x ) + 2 γ [4 b10 (ϕ0 − c1)

+ γ x2] + 8

−1 8 b10 γ

γ is an arbitrary real constant

ϕ (x , t ) =

A02 2γ

[2 γ (2 α 2 + β 2 ) t − 8 α β cos(γ t ) − β 2 sin(2 γ t )] 16 b10 γ

− 8 α β cos(γ t ) − β 2 sin(2 γ t )]] − c1 + ϕ0

+

+

c42

β γ cos(γ t ) x 2 + 2 c4 [α + β sin(γ t )]2 x 4 b10 [α + β sin(γ t )]

Conditions

See text.

ϕ (x , t ) =

+ e γ t x − x0 + c3)] ϕ(x , t ) =

+ e γ t x − x0 + c3)]

c4 e 2 γ t 2γ

[ψ1(x , t ) + ψ2(x , t )] e−i

e γ t tanh[A0 (

c4 e 2 γ t 2γ

∣Φ∣2 Φ +

Constant Dispersion and Real Linear Potential

Φ(x , t ) =

a2 b10 e γ t a1 c22

e γ t sech[A0 (

−2 a1 a2

× e i ϕ (x , t )

2. Φ(x , t ) = c2 A0

3.

2 a1 a2

× e i ϕ (x , t )

1. Φ(x , t ) = c2 A0

# Example

t ) − β2

[ψ1(x , t ) + ψ2(x , t )] ei ϕ(x, t )

t − 8 α β cos(γ 4γ

Equation 3: i Φt + b10 Φxx +

α + β sin(γ t ) a2

× e−i ϕ(x, t )

+

α 2 + β2)

× tanh[A0 (c3 − x0 + [α + β sin(γ t )] x

−2 A02 a1 c22 [α + β sin(γ t )] a2

3. Φ(x , t ) = c2

2.

(Continued )

(5.47)

(5.48)

(5.44)

Eq. #

two bright solitons (5.51)

dark soliton

bright soliton

Name

two bright solitons (5.50)

dark soliton

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Φ(x , t ) = c2 A0

5-35

×e

Φ(x , t ) = A0

×e

3.

×e

Φ(x , t ) = A0

×e

2 a1 c5 a2 c7

7

8 λr A0

a2 b10 c5 a1 c22 c7

(t − t0 )] −

c αt 10 c5

− b7

3 b10 c52

c 72 α 2 t3

c7 α b10 c5

xΦ=0

c 7 g4′ (t ) x + c1+ ϕ0 ] 2 b10 c5

x + c1+ ϕ0}

c αt 10 c5

+ b7 x − c1+ ϕ0}

b10 c52 t c α (3 c5 x + c 7 α t 2 ) t − t0 ) − 7 + c1+ ϕ0 ] 3 b10 c52 a1 c 72

p(x , t ))

(t − t0 )] +

8 λr A0

b10 c52 a1 c 72

7

c

3 b10 c52

c 72 α 2 t3

7

c

∣Φ∣2 Φ −

2

∫ g4′ (t )d t −

p(x , t ))

7

c

tanh{A0 [ c5 x − x0 + α t 2 ]}

(1 −

i [A02 (c0 +

xΦ=0

sech{A0 [ c5 x − x0 + α t 2 ]}

−i {2 A02 a1 [c0 +

c22 c5 a2 c7

2 b10 c5

c7 g4″(t )

2 b10 c5

c7 g4′(t )

(t − t0 )] −

x + c1 + ϕ0 ,

a1 c72

b10 c52 4 b10 c52

c72

∫ g4′2(t ) dt

See text.

–

–

See text.

+ 2 b10 c5

c7 g4′(t )

(t − t0 )] +

x − c1 + ϕ0

a1 c72

b10 c52 4 b10 c52

c72

∫ g4′2(t ) dt

a1 a2 c5 c7 > 0 , A0, x0, t0, and ϕ0 are arbitrary real constants

−

ϕ(x , t ) = A02 a1 [c0 +

tanh{A0 [ c5 x − x0 + g4(t )]} e−i ϕ(x, t ) ϕ(x , t ) = 2 A02 a1 [c0 +

b10 c52 a1 c 72

−2 a1 c5 a2 c7

∣Φ∣2 Φ −

c

b10c52t c2 − t0 ) − 7 2 4 b10 c5 a1 c 72

i {A02 a1 [c0 +

Φ(x , t ) = c2 A0

2. Φ(x , t ) = c2 A0

1.

a2 b10 c5 a1 c22 c7

sech{A0 [ c5 x − x0 + g4(t )]} ei ϕ(x, t )

(1 −

i [A02 (c0 +

c22 c5 a2 c7

−2 a1 c5 a2 c7

2 a1 c5 a2 c7

Equation 2: i Φt + b10 Φxx +

3.

2. Φ(x , t ) = c2 A0

1.

Equation 1: i Φt + b10 Φxx +

generalized firstorder breather

dark soliton

bright soliton

generalized firstorder breather

dark soliton

bright soliton

(5.62)

(5.60)

(5.57)

(5.61)

(5.59)

(5.56)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

a1 b20 c22 a2 g5(t )

Φxx + b20 ∣Φ∣2 Φ +

5-36

4 a1 b20 c22 g5(t )

i {c1−

ψ [c5e c6 tx + g4(t ), c0 +

}

x2 Φ = 0

a 2[c52c6e c6t x 2+ 2c5g4′ (t )x + ∫ g4′ 2(t )e−c6t dt ] 4a1b 20c 22c5

∫

2

a1 b10

a2 b20

e i B (x ) ψ [

a1 b10

a1 b10 a1 b10

(x − x0 )]

f[

a1 b10

(x − x0 )]

a2 b20

f[

a1 b10

∫ g5(t )dt ]

2

(x − x0 )] ei [B(x ) + A0

+ Bxx(x , t ))] Φ = 0

a2 b20

(x − x0 )] Bx(x, t )

(x − x0 ), (t − t0 )] = A0

f[

f′[

Equation: i Φt + b10 Φxx + b20 ∣Φ∣2 Φ + [Veven(x ) + i Vodd(x )]Φ = 0

Transformation 1: Φ(x , t ) =

Special Case 2: PT-symmetric Potential

4 a1 b10

(x − x0 ), (t − t0 )] = A0

Φ + [Bt (x , t ) + b10 Bx2(x , t ) − i b10 (

(t − t0 )]

e i B (x , t ) ψ [

ei [B(x, t ) + A0

a2 b20

Equation: i Φt + b10 Φxx + b20 ∣Φ∣2

Transformation: Φ(x , t ) =

a2

b20c 22

(t − t0 )]

b 20 c 22 c5 (e c6t − e c6 ) ] a 2 c6

ψ [c3 + g5(t )x + c4 g5(t )dt , c0 +

}

Special Case 1: Stationary Solution, Constant Dispersion and Nonlinearity Coefficients

Transformation: Φ(x , t ) = c 2 Solution-Dependent Transformation

c5e c6 t e

∫ g5(t )dt ]

a2 [g5′2(t ) − g5(t ) g5 ″ (t )]

4a1b20c 22

a2[2c4g5(t )x + g5′(t )x 2 + c42

Constant Nonlinearity and Real Linear Potential

Equation: i Φt +

Transformation: Φ(x , t ) = c2 g5(t ) e

i{c1−

Constant Nonlinearity and Real Quadratic Potential

Equation: i Φt + b1(x , t ) Φxx + b20 ∣Φ∣2 Φ + [b3r (x , t ) + i b3i (x , t )] Φ = 0

Transformation: Φ(x , t ) = A(x , t ) ei B(x, t ) ψ [X (x , t ), T (x , t )]

Constant Nonlinearity and Complex Potential

(Continued )

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

a2 b20

ei sin(x − x0) ψ [

a1 b10

(x − x0 ), (t − t0 )]

ei f 4[

(t − t0 )]

e i B (x , t ) ψ [

[B (x, t ) + A02

a2 b20

Equation: i Φt + b10 Φxx + b20 ∣Φ∣2 Φ + {

Transformation: Φ(x , t ) =

a1 b10

(x − x0 )]

+ g1′(t ) ∫ f 2[

a1 b10

a2 b20

a1 b10

(x − x0 )]

+ g2′(t )}Φ = 0

f[

(x − x0 )]

dx

(x − x0 ), (t − t0 )] = A0

b10 g12(t )

a1 b10

Special Case 3: Stationary Solution, Constant Dispersion and Nonlinearity Coefficients, and Real Potential

Equation: i Φt + b10 Φxx + b20 ∣Φ∣2 Φ + b10 [cos2(x − x0 ) + i sin(x − x0 )] Φ = 0

Transformation 2: Φ(x , t ) =

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

5-37

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

5.7 Other Equations: NLSE with Periodic Potentials 5.7.1 General Case: sn2(x , m ) Potential Equation:

i ψt +

1 ψ − ∣ψ ∣2 ψ + V0 sn2(x , m) ψ = 0, 2 xx

(5.100)

where ψ = ψ (x , t ) is the complex function profile, x and t are its two independent variables, V0 is a real constant.

Solutions: Solution 1. sn(x,m) solitary wave (SW)

ψ (x , t ) =

V0 + m sn(x − x0, m) e

⎡ (1 + m) (t − t0) ⎤ −i ⎢ + ϕ0⎥ 2 ⎣ ⎦,

(5.101)

where V0 + m ⩾ 0, x0, t0, and ϕ0 are arbitrary real constants. • Reference: [1], we corrected the constant prefactor and the exponential term. Solution 2. cn(x,m) SW

ψ (x , t ) =

−(V0 + m) cn(x − x0, m) e

⎡ ⎤ i ⎢⎣ V0+ m − 1 (t − t0) + ϕ0⎥⎦ 2 ,

(

)

(5.102)

where V0 + m ⩽ 0, x0, t0, and ϕ0 are arbitrary real constants. • Reference: [1], we corrected the constant prefactor and the exponential term. Solution 3. dn(x,m) SW

ψ (x , t ) =

⎡⎛ V ⎞ ⎤ ⎛ V ⎞ i 1 + 0 − m (t − t0) + ϕ0⎥ ⎦, −⎜1 + 0 ⎟ dn(x − x0, m) e ⎣⎢⎝ m 2 ⎠ ⎝ m⎠ ⎜

⎟

(5.103)

where V 1 + m0 ⩽ 0, x0, t0, and ϕ0 are arbitrary real constants. • Reference: [1], we corrected the constant prefactor and the exponential term. 5-38

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

5.7.2 Specific Case: sin2(x ) Potential Equation:

i ψt +

1 ψ − ∣ψ ∣2 ψ + V0 sin2(x ) ψ = 0, 2 xx

(5.104)

where V0 is a real constant. Solutions: Solution 1. sin(x)

ψ (x , t ) =

( t−2t +ϕ ),

(5.105)

⎡ ⎤ −i ⎢⎣ V0− 1 (t − t0) + ϕ0⎦⎥ 2 ,

(5.106)

V0 sin(x − x0) e

−i

0

0

where V0 > 0, x0, t0, and ϕ0 are arbitrary real constants. • Reference: [1]. Solution 2. cos(x)

ψ (x , t ) =

−V0 cos(x − x0) e

where V0 < 0, x0, t0, and ϕ0 are arbitrary real constants. • Reference: [1].

5-39

(

)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

5.8 Summary of Section 5.7

Equation

i ψt +

1 2

ψxx − ∣ψ ∣2 ψ + V0 sn2(x , m ) ψ = 0

# Solution

Conditions

1. ψ (x , t ) =

× e−i [

V0 + m sn(x − x0, m )

(1 + m ) (t − t0 ) + ϕ0 ] 2

2. ψ (x , t ) = − (V0 + m ) cn(x − x0, m ) 1 × ei [(V0+ m − 2 ) (t − t0) + ϕ0]

Name

Eq. #

V0 + m ⩾ 0 , x0, t0, and ϕ0 are solitary (5.101) arbitrary real constants wave V0 + m ⩽ 0 ,

solitary (5.102) wave

x0, t0, and ϕ0 are arbitrary real constants V0 ) dn(x m V0 m i [(1 + − ) ( t − t ) + ϕ0 ] , 0 m 2 e

3. ψ (x , t ) =

×

− (1 +

− x0 , m )

solitary wave

(5.103)

x0, t0, and ϕ0 are arbitrary real constants Equation

i ψt +

1 2

ψxx − ∣ψ ∣2 ψ + V0 sin2(x ) ψ = 0

# Solution 1. ψ (x , t ) =

V0 sin(x − x0 ) e−i

(t − t ) [ 2 0 + ϕ0 ]

Conditions

Name

Eq. #

V0 > 0,

–

(5.105)

–

(5.106)

x0, t0, and ϕ0 are arbitrary real constants 2. ψ (x , t ) =

1 − V0 cos(x − x0 ) e−i [(V0− 2 ) (t − t0) + ϕ0] V0 < 0,

x0, t0, and ϕ0 are arbitrary real constants

Reference [1] Bronski J C, Carr L D, Deconinck B and Kutz J N 2001 Bose–Einstein condensates in standing waves: The cubic nonlinear Schrödinger equation with a periodic potential Phys. Rev. Lett. 86 1402–5

5-40

IOP Publishing

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations Usama Al Khawaja and Laila Al Sakkaf

Chapter 6 Nonlinear Schrödinger Equation in (N + 1)-Dimensions

A Glance at Chapter 6

doi:10.1088/978-0-7503-2428-1ch6

6-1

ª IOP Publishing Ltd 2020

6-2

k k

k k

k k

k=1

∑ αk Φx x

N

k=1

∑ αk Φx x

N

k=1

∑ αk Φx x

+ a2 ∣Φ∣n Φ + a3 ∣Φ∣m Φ = 0

+ a2 ∣Φ∣n Φ = 0

+ a2 ∣Φ∣2 Φ = 0

(

(

7

8

i Φt + a1 Φrr +

i Φt + a1 Φrr +

2 r

1 r

)

)

Φ + a2 r ∣Φ∣2 Φ = 0

)

Φθθ + a2 r N − 1 ∣Φ∣2 Φ = 0 , for N = 1: Φ = Φ(r )

Φr + a2 r 2 ∣Φ∣2 Φ = 0

Φr −

1 r2

)

Φr + b2(r, t ) ∣Φ∣2 Φ + [b3r (r, t ) + i b3i (r, t )]Φ = 0

1 4r 2

Φr +

N−1 r

(

6

i Φt + a1 Φrr +

N−1 r

i Φt + b1(r, t ) Φrr +

(

i Φt + b1 Φx1x1 + b2 Φx2x2 + b3 Φx1x2 + b4 ∣Φ∣2 Φ = 0

i Φt +

i Φt +

i Φt +

5

4

3

2

1

N

Equation

A Statistical View of Chapter 6

1

1

1

0

2

8

4

14

Solutions Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

10

a1 b10 g1′ 3/2(t )

Total

2

⎡ − λ Z (r ) + a1 ⎣Z ″(r ) +

+

⎡ ′ ′ ″ ″ ⎤ ⎣⎢g2 (t ) g1 (t ) − g1 (t ) g2 (t )⎦⎥ r

1 r

α2 r2

⎬Φ=0 ⎪ ⎭

⎤ Z (r )⎦ + a2 Z 3(r ) = 0

16 b10 g1′ 2(t )

Z′(r ) −

+

∣Φ∣2 Φ

⎡ ⎤ 2⎫ ′ ″2 ⎣⎢3 g1 (t ) − 2 g1 (t ) g1‴(t )⎦⎥ r ⎪

4 a1 g1′(t )

g1′(t )

c02 r1 − N

a2

+ g4′(t )

Φr +

g2′ 2(t )

b10 (N − 1) r

⎧ ⎪ b (N − 1) (N − 3) + ⎨ 10 4 r 2 + ⎪ ⎩

i Φt + b10 Φrr +

10

9

34

1

2

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

6-3

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

6.1 (N + 1)-Dimensional NLSE with Cubic Nonlinearity If ψ (x , t ; a1, a2 ) is a solution to the fundamental NLSE, (2.1),

i ψt + a1 ψxx + a2 ∣ψ ∣2 ψ = 0, then

Φ(x1, x2 , … , xN , t ; α1, α2 , … , αN , a2 ) = ψ ⎡⎣ c1 (x1 − x01) + c2 (x2 − x02 ) + ⋯ + cN (xN − x0N ), t ; c12

α1 +

c12

α2 + ⋯ +

cN2

(6.1)

αN , a2⎤⎦

is a solution of N

i Φt +

∑ αk Φx x

k k

+ a2 ∣Φ∣2 Φ = 0,

(6.2)

k=1

with the replacements N

x→

∑ ck (xk − x0k ), k=1

t → t,

N

a1 →

∑ ck2 αk , k=1

a2 → a2 , where αk , a2, ck, and x0k are arbitrary real constants.

Example 1. sech(x1, x2 ) 2D bright soliton (Figure 6.1) Given

ψ (x , t ) = A0

2 2a1 ⎡ ⎤ sech[A0 (x − x0)] e i ⎣a1 A0 (t − t0) + ϕ0⎦ a2

is a solution of (2.1), then

Figure 6.1. 2D bright x01 = x02 = t0 = ϕ0 = 0 .

soliton

(6.3)

at

t

=

6-4

0,

with

α1 = α2 = A0 = c1 = c2 = a2 = 1

and

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Φ(x1, x2 , t ) = A0

(

2 c12 α1 + c22 α2 a2

)

sech{A0 [c1 (x1 − x01) + c2 (x2 − x02 )]} (6.3)

⎡ 2 2 ⎤ 2 × e i ⎣A0 ( c1 α1+ c2 α2) (t − t0) + ϕ0⎦

is a solution of

i Φt + α1 Φx1x1 + α2 Φx2x2 + a2 ∣Φ∣2 Φ = 0,

(6.4)

where a2 (c12 α1 + c22 α2 ) > 0, A0, t0, and ϕ0 are arbitrary real constants. Example 2. sech(x1, x2 , x3) 3D bright soliton Given 2 2 a1 ⎡ ⎤ sech[A0 (x − x0)] e i ⎣a1 A0 (t − t0) + ϕ0⎦ a2

ψ (x , t ) = A0 is a solution of (2.1), then

(

2 c12 α1 + c22 α2 + c32 α3

Φ(x1, x2 , x3, t ) = A0

)

a2 × sech{A0 [c1 (x1 − x01) + c2 (x2 − x02 )

(6.5)

⎡ 2 2 ⎤ 2 2 + c3 (x3 − x03)]} e i ⎣A0 ( c1 α1+ c2 α2+ c3 α3) (t − t0) + ϕ0⎦

is a solution of

i Φt + α1 Φx1x1 + α2 Φx2x2 + α3 Φx3x3 + a2 ∣Φ∣2 Φ = 0,

(6.6)

where a2 (c12 α1 + c22 α2 + c32 α3) > 0, A0, t0, and ϕ0 are arbitrary real constants. Example 3. tanh(x1, x2 ) 2D dark soliton (Figure 6.2) Given

ψ (x , t ) = A0

2 −2 a1 ⎡ ⎤ tanh[A0 (x − x0)] e−i ⎣2 a1 A0 (t − t0) + ϕ0⎦ a2

is a solution of (2.1), then

Φ(x1, x2 , t ) = A0

(

−2 c12 α1 + c22 α2

)

a2 × tanh{A0 [c1 (x1 − x01) + c2 (x2 − x02 )]} ⎡ ⎤ 2 2 2 × e−i ⎣2 A0 ( c1 α1+ c2 α2) (t − t0) + ϕ0⎦

6-5

(6.7)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Figure 6.2. 2D dark soliton (6.7) at t = 0, with α1 = α2 = A0 = c1 = c2 = 1, a2 = −1 and x01 = x02 = t0 = ϕ0 = 0 .

is a solution of (6.4), where a2 (c12 α1 + c22 α2 ) < 0, A0, t0, and ϕ0 are arbitrary real constants. Example 4. tanh(x1, x2 , x3) 3D dark soliton Given 2 −2 a1 ⎡ ⎤ tanh[A0 (x − x0)] e−i ⎣2 a1 A0 (t − t0) + ϕ0⎦ ψ (x , t ) = A0 a2 is a solution of (2.1), then

Φ(x1, x2 , x3, t ) = A0

(

−2 c12 α1 + c22 α2 + c32 α3

)

a2 × tanh{A0 [c1 (x1 − x01) + c2 (x2 − x02 )

(6.8)

⎡ ⎤ 2 2 2 2 + c3 (x3 − x03)]} e−i ⎣2 A0 ( c1 α1+ c2 α2+ c3 α3) (t − t0) + ϕ0⎦

is a solution of (6.6), where a2 (c12 α1 + c22 α2 + c32 α3) < 0, A0, t0, and ϕ0 are arbitrary real constants. Example 5. Periodicity in t and Localization in x1 and x2 2D Kuznetsov–Ma breather (Figure 6.3) Given ⎧ ⎫ ⎪ 1 ⎪ −p2 cos[ω (t − t0)] − 2 i p ν sin[ω (t − t0)] ⎨ ψ (x , t ) = − 1⎬ a2 ⎪ 2 cos[ω (t − t )] − 2 ν cosh ⎡ p (x − x )⎤ ⎪ 0 0 ⎥ ⎢⎣ 2 a1 ⎦ ⎩ ⎭

× e i [t − t0+ ϕ0] is a solution of (2.1), then

Φ(x1, x2 , t ) =

⎞ −1 ⎛ p2 cos[ω (t − t0)] + 2 i p ν sin[ω (t − t0)] + 1⎟ ⎜ 2 cos[ω (t − t0)] − q(x1, x2 ) a2 ⎝ ⎠ × e i [(t − t0) + ϕ0]

6-6

(6.9)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Figure 6.3. 2D Kuznetsov–Ma breather (6.9). (a) at t = −1, (b) at t = 0, (c) at t = 1, (d) at t = 2, (e) at t = 3, and (f) at t = 4. Used parameters: α1 = α2 = c1 = c2 = 3, a2 = 2 , x01 = x02 = t0 = ϕ0 = 0 , and ν = 1.5. Animation available online at https://iopscience.iop.org/book/978-0-7503-2428-1.

is a solution of (6.4), where ⎧ ⎫ p q(x1, x2 ) = 2 ν cosh ⎨ [c1 (x1 − x01) + c2 (x2 − x02 )]⎬, 2 2 ⎩ 2 (c1 α1 + c2 α2 ) ⎭ p = 2 ν2 − 1 , ω = p ν, ν > 1, a2 > 0, (c12 α1 + c22 α2 ) > 0, t0 and ϕ0 are arbitrary real constants.

Example 6. Periodicity in t and Localization in x1x2 , and x3 3D Kuznetsov–Ma breather Given

⎧ ⎫ ⎪ ⎪ 2 −p cos[ω (t − t0)] − 2 i p ν sin[ω (t − t0)] 1 ⎨ ψ (x , t ) = − 1⎬ a2 ⎪ 2 cos[ω (t − t )] − 2 ν cosh ⎡ p (x − x )⎤ ⎪ 0 0 ⎥ ⎢⎣ 2 a1 ⎦ ⎩ ⎭ × e i [t − t0+ ϕ0] is a solution of (2.1), then

Φ(x1, x2 , x3, t ) =

⎞ −1 ⎛ p2 cos[ω (t − t0)] + 2 i p ν sin[ω (t − t0)] + 1⎟ ⎜ 2 cos[ω (t − t0)] − q(x1, x2 , x3) a2 ⎝ ⎠ ×

e i [(t − t0) + ϕ0]

6-7

(6.10)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

is a solution of (6.6), where ⎧ q(x1, x2, x3) = 2 ν cosh ⎨ ⎩

p 2 (c12 α1 + c22 α2 + c32 α3)

⎫ × [c1 (x1 − x01) + c2 (x2 − x02 ) + c3 (x3 − x03)]⎬ ⎭

,

p = 2 ν2 − 1 , ω = p ν, ν > 1, a2 > 0, (c12 α1 + c22 α2 + c32 α3) > 0, t0 and ϕ0 are arbitrary real constants.

Example 7. Periodicity in x1 and x2 and Localization in t 2D Akhmediev breather (Figure 6.4) Given ⎧ ⎫ ⎪ 1 ⎪ κ 2 cosh[δ (t − t0)] + 2 i κ ν sinh[δ (t − t0)] ⎨ ψ (x , t ) = − 1⎬ e i [t − t0+ ϕ0] a2 ⎪ 2 cosh[δ (t − t )] − 2 ν cos ⎡ κ (x − x )⎤ ⎪ 0 0 ⎦ ⎥ ⎣⎢ 2 a1 ⎩ ⎭ is a solution of (2.1), then

Φ(x1, x2 , t ) =

⎞ 1 ⎛ κ 2 cosh[δ (t − t0)] + 2 i κ ν sinh[δ (t − t0)] − 1⎟ ⎜ 2 cosh[δ (t − t0)] − q(x1, x2 ) a2 ⎝ ⎠

(6.11)

× e i [(t − t0) + ϕ0]

Figure 6.4. 2D Akhmediev breather (6.11). (a) at t = −4 , (b) at t = −2 , (c) at t = 0, (d) at t = 2, and (e) at t = 4. Used parameters: α1 = α2 = c1 = c2 = 3, a2 = 2 , x01 = x02 = t0 = ϕ0 = 0 , and ν = 0.5. Animation available online at https://iopscience.iop.org/book/978-0-7503-2428-1.

6-8

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

is a solution of (6.4), where ⎧ ⎫ κ q(x1, x2 ) = 2ν cos ⎨ [c1 (x1 − x01) + c2 (x2 − x02 )]⎬, 2 2 ⎩ 2 (c1 α1 + c2 α2 ) ⎭ κ = 2 1 − ν2 , δ = κ ν, ν < 1, a2 > 0, (c12 α1 + c22 α2 ) > 0, t0 and ϕ0 are arbitrary real constants.

Example 8. Periodicity in x1, x2 , and x3 and Localization in t 3D Akhmediev breather. Given

ψ (x , t ) =

⎧ ⎫ ⎪ ⎪ 2 κ cosh[δ (t − t0)] + 2 i κ ν sinh[δ (t − t0)] 1 ⎨ − 1⎬ e i [t − t0+ ϕ0] a2 ⎪ 2 cosh[δ (t − t )] − 2 ν cos ⎡ κ (x − x )⎤ ⎪ 0 0 ⎦ ⎥ ⎣⎢ 2 a1 ⎩ ⎭

is a solution of (2.1), then

⎞ 1 ⎛ κ 2 cosh[δ (t − t0)] + 2 i κ ν sinh[δ (t − t0)] − 1⎟ ⎜ 2 cosh[δ (t − t0)] − q(x1, x2 , x3) a2 ⎝ ⎠ (6.12)

Φ(x1, x2 , x3, t ) =

× e i [(t − t0) + ϕ0] is a solution of (6.6), where q(x1, x2, x3) ⎧ ⎫, κ = 2 ν cos ⎨ [c1 (x1 − x01) + c2 (x2 − x02 ) + c3 (x3 − x03)]⎬ 2 2 2 ⎩ 2 (c1 α1 + c2 α2 + c3 α3) ⎭ κ = 2 1 − ν2 , δ = κ ν, ν < 1, a2 > 0, (c12 α1 + c22 α2 + c32 α3) > 0, t0 and ϕ0 are arbitrary real constants.

Example 9. Localization in t , x1, and x2 2D Peregrine soliton (Figure 6.5) Given

ψ (x , t ) =

⎡ ⎤ 1 ⎢ 4 + i 8 (t − t0 ) − 1⎥ e i [t − t0+ ϕ0] 2 ⎢ ⎥ 2 2 a2 1 + 4 (t − t0) + (x − x0) ⎣ ⎦ a1

6-9

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Figure 6.5. 2D Peregrine soliton (6.13). (a) at t = −2 , (b) at t = −1, (c) at t = 0, (d) at t = 1, and (e) at t = 2. Used parameters: α1 = α2 = c1 = c2 = a2 = 2 , and x01 = x02 = t0 = ϕ0 = 0 . Animation available online at https://iopscience.iop.org/book/978-0-7503-2428-1.

is a solution of (2.1), then ⎧ 1 ⎪ ⎨ Φ(x1, x2 , t ) = a 2 ⎪ 1 + 4 (t − t0 ) 2 + ⎩

⎫ ⎪ − 1⎬ ⎪(6.13) [c1 (x1 − x01) + c 2 (x2 − x02)]2 ⎭

4 + i 8 (t − t0 ) 2 c12 α1 + c 22 α 2

× e i [(t − t0 ) + ϕ0] ,

is a solution of (6.4), where a2 > 0, t0 and ϕ0 are arbitrary real constants. Example 10. Localization in t , x1, x2 , and x3 3D Peregrine soliton Given

ψ (x , t ) =

⎡ ⎤ 1 ⎢ 4 + i 8 (t − t0 ) − 1⎥ e i [t − t0+ ϕ0] ⎥ a2 ⎢ 1 + 4 (t − t0)2 + 2 (x − x0)2 ⎣ ⎦ a1

is a solution of (2.1), then

Φ(x1, x2 , x3, t ) =

⎫ 1 ⎧ 4 + i 8 (t − t0 ) ⎨ ⎬ e i [(t − t0) + ϕ0] − 1 a2 ⎩ 1 + 4 (t − t0)2 + q(x1, x2 , x3) ⎭

is a solution of (6.6), where 2 q(x1, x2, x3) = 2 2

c1 α1 + c2 α2 + c32 α3

(6.14)

[c1 (x1 − x01) + c2 (x2 − x02 ) + c3 (x3 − x03)]2 ,

6-10

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

a2 > 0, t0 and ϕ0 are arbitrary real constants.

6.2 (N + 1)-Dimensional NLSE with Power Law Nonlinearity If ψ (x , t ; a1, a2 ) is a solution to the NLSE with power law nonlinearity, (3.1),

i ψt + a1 ψxx + a2 ∣ψ ∣n ψ = 0, then (6.1) is a solution of N

i Φt +

∑ αk Φx x

k k

+ a2 ∣Φ∣n Φ = 0,

(6.15)

k=1

with the replacements N

∑ ck (xk − x0k ),

x→

k=1

t → t,

N

a1 →

∑ ck2 αk , k=1

a2 → a2 , where αk , a2, and n are real constants.

Example 1. sech(x1, x2 ) 2D bright soliton. Given 1

⎡

⎧ 2 A 2 a1 (n + 2) ⎫ n i ⎢ 4 a1 2A0 0 2 ψ (x , t ) = ⎨ sech [A0 (x − x0)]⎬ e ⎣ n a2 n2 ⎩ ⎭ is a solution of (3.1), then ⎛ 2 A 2 (n + 2) c 2 α + c 2 α 0 1 1 2 2 Φ(x1, x2 , t ) = ⎜ 2 ⎜ a2 n ⎝

(

2

⎤ (t − t0) + ϕ0⎥ ⎦

) 1

⎞n ⎟ × sech2{A0 [ c1 (x1 − x01) + c2 (x2 − x02 )]} ⎟ ⎠ ⎡ 4 A 2 c 2 α +c 2 α ⎤ 1 2 0 1 2 ⎢ ⎥ − + ϕ t t ( ) 0 0⎥ i⎢ n2 ⎣ ⎦ e

(

×

(6.16)

)

is a solution of

i Φt + α1 Φx1x1 + α2 Φx2x2 + a2 ∣Φ∣n Φ = 0, where a2 (n + 2) (c12 α1 + c22 α2 ) > 0, A0, t0, and ϕ0 are arbitrary real constants. 6-11

(6.17)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Example 2. sech(x1, x2 , x3) 3D bright soliton. Given 1

⎡

⎧ 2 A 2 a1 (n + 2) ⎫ n i ⎢ 4 a1 A0 0 2[A (x − x )]⎬ e ⎣ n 2 ψ (x , t ) = ⎨ sech 0 0 a2 n2 ⎩ ⎭

2

⎤ (t − t0) + ϕ0⎥ ⎦

is a solution of (3.1), then

⎛ 2 A 2 (n + 2) c 2 α + c 2 α + c 2 α 0 1 1 2 2 3 3 Φ(x1, x2 , x3, t ) = ⎜ 2 ⎜ a2 n ⎝

(

)

× sech2{A0 [c1 (x1 − x01) + c2 (x2 − x02 ) 1 ⎞n

+ c3 (x3 − x03)]}⎟⎟ ⎠

(6.18)

⎡ 4 A 2 c 2 α +c 2 α +c 2 α ⎤ 1 2 3 0 1 2 3 ⎢ (t − t0) + ϕ0⎥ i⎢ 2 ⎥ n ⎦ e ⎣

(

)

is a solution of

i Φt + α1 Φx1x1 + α2 Φx2x2 + α3 Φx3x3 + a2 ∣Φ∣n Φ = 0,

(6.19)

where a2 (n + 2) (c12 α1 + c22 α2 + c32 α3) > 0, A0, t0, and ϕ0 are arbitrary real constants.

6.3 (N + 1)-Dimensional NLSE with Dual Power Law Nonlinearity If ψ (x , t ; a1, a2, a3) is a solution of the NLSE with dual power law nonlinearity, (3.19),

i ψt + a1 ψxx + a2 ∣ψ ∣n ψ + a3 ∣ψ ∣m ψ = 0, then (6.1) is a solution of N

i Φt +

∑ αk Φx x

k k

+ a2 ∣Φ∣n Φ + a3 ∣Φ∣m Φ = 0,

k=1

with the replacements N

x→

∑ ck (xk − x0k ), k=1

t → t,

N

a1 →

∑ ck2 αk , k=1

a2 → a2 , a3 → a3, where αk , a2, a3, n, and m are real constants. 6-12

(6.20)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Example 1. sech(x1, x2 ) 2D flat-top soliton (Figure 6.6) Given 1

⎛ ⎞n ⎜ ⎟ A0 A1 (n + 2) ⎜ ⎟ e i [A1 (t − t0) + ϕ0] ψ (x , t ) = ⎜ ⎧ ⎡ ⎤ ⎟ A ⎜ a2 ⎨A0 + 2 cosh ⎢⎣n a 1 (x − x0)⎥⎦ ⎟ 1 ⎝ ⎩ ⎠

}

is a solution of (3.19), then ⎡ ⎢ Φ(x1, x2, t ) = ⎢ ⎢ ⎢⎣ a2 A0 + 2 a2 cosh

1

{n

⎤n ⎥ A0 A1 (n + 2) ⎥ A1 ⎥ (6.21) [ c ( x − x ) + c ( x − x )] 1 1 01 2 2 02 2 2 ⎥⎦ c 1 α1 + c 2 α 2

}

× e i [A1 (t − t0) + ϕ0]

is a solution of

i Φt + α1 Φx1x1 + α2 Φx2x2 + a2 ∣Φ∣n Φ + a3 ∣Φ∣m Φ = 0,

(6.22)

where A0 =

2a2 n+2

n+1 A1 δ

,

a 22 (n + 1) , A1(n + 2)2 2 2 A1 (c1 α1 + c2 α2 )

δ = a3 +

> 0, A1 δ (n + 1) > 0, m = 2n , t0, A1, and ϕ0 are arbitrary real constants.

Figure 6.6. Plot of solution (6.21) at t = 0. (a) 2D flat-top soliton with a3 = −0.0694444 , (b) 2D bright soliton with a3 = 0.0305556 . Values of the other parameters are: A1 = 2 , n = 4, α1 = α2 = 5, c1 = c2 = a2 = 1 and x01 = x02 = t0 = ϕ0 = 0 .

6-13

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Example 2. sech(x1, x2 , x3) 3D flat-top soliton. Given 1

⎛ ⎞n ⎜ ⎟ A0 A1 (n + 2) ⎜ ⎟ e i [A1 (t − t0) + ϕ0] ψ (x , t ) = ⎜ ⎧ ⎡ ⎤ ⎟ A ⎜ a2 ⎨A0 + 2 cosh ⎢⎣n a 1 (x − x0)⎥⎦ ⎟ 1 ⎝ ⎩ ⎠

}

is a solution of (3.19), then 1

⎡ ⎤n A0 A1 (n + 2) Φ(x1, x2 , x3, t ) = ⎢ ⎥ e i [A1 (t − t0) + ϕ0] ⎣ a2 A0 + q(x1, x2 , x3) ⎦

(6.23)

is a solution of

i Φt + α1 Φx1x1 + α2 Φx2x2 + α3 Φx3x3 + a2 ∣Φ∣n Φ + a3 ∣Φ∣m Φ = 0,

(6.24)

where

q(x1, x2 , x3)= 2a2 cosh

A0 =

{n

2a2 n+2

A1 (c12 α1 + c 22 α 2 + c32 α3)

n+1 A1 δ

,

a 22

(n + 1) , A1 (n + 2)2 A1 (c12 α1 + c22 α2 +

δ = a3 +

}

[c1 (x1 − x01) + c2 (x2 − x02 ) + c3 (x3 − x03)] ,

c32 α3) > 0,

A1 δ (n + 1) > 0, m = 2n , t0, A1, and ϕ0 are arbitrary real constants.

Example 3. tanh(x1, x2 ) 2D dark soliton (Figure 6.7) Given 1

⎡

⎛ 2a A 2 (n + 2) ⎞ n i⎢ 4 a1 2A0 ψ (x , t ) = ⎜ 1 0 2 {1 − tanh[A0 (x − x0)]}⎟ e ⎣ n a2 n ⎝ ⎠

2

⎤ (t − t0) + ϕ0⎥ ⎦

is a solution of (3.19), then 1

⎡ 2A 2 (n + 2) c 2 α + c 2 α ⎤n 1 0 1 2 2 ⎢ (1 − tanh{A0 [c1(x1 − x 01) + c 2(x2 − x 02 )]})⎥ Φ(x1, x2, t ) = ⎢ ⎥ a2 n 2 ⎣ ⎦ (6.25)

(

⎤ ⎡ 4 A 2 c 2 α +c 2 α 0 1 1 2 2 ⎥ ⎢ (t − t0) + ϕ0⎥ i⎢ n2 ⎦ e ⎣

(

×

)

)

6-14

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Figure 6.7. 2D dark soliton (6.25) at t = 0, with α1 = α2 = c1 = c2 = a2 = 1, a3 = −1, x01 = x02 = t0 = ϕ0 = 0 , and n = 2.5.

is a solution of (6.22), where A0 =

a2 n 2 (n + 2)

−(n + 1) , a3 (c12 α1 + c22 α2 ) α1 + c22 α2 ) > 0, α1 + c22 α2 ) < 0,

a2 (n + 2) (c12 a3 (n + 1) (c12 m = 2 n, t0 and ϕ0 are arbitrary real constants.

Example 4. tanh(x1, x2 , x3) 3D dark soliton Given 1

⎡

⎛ 2a A 2 (n + 2) ⎞ n i⎢ 4 a1 2A0 ψ (x , t ) = ⎜ 1 0 2 {1 − tanh[A0 (x − x0)]}⎟ e ⎣ n a2 n ⎠ ⎝

2

⎤ (t − t0) + ϕ0⎥ ⎦

is a solution of (3.19), then

⎡ 2A 2 (n + 2) c 2 α + c 2 α + c 2 α 0 1 1 2 2 3 3 Φ(x1, x2 , x3, t ) = ⎢ 2 ⎢ a2 n ⎣ × (1 − tanh{A0 [c1 (x1 − x01) + c2 (x2 − x02 )

(

)

1 ⎤n

⎥ + c3 (x3 − x03)]}) ⎥ ⎦

⎡ 4 A 2 c 2 α +c 2 α +c 2 α ⎤ 1 2 3 0 1 2 3 ⎢ (t − t0) + ϕ0⎥ i⎢ 2 ⎥ n ⎦ e⎣

is a solution of (6.24), where A0 =

a2 n 2 (n + 2)

−(n + 1) , a3 (c12 α1 + c22 α2 + c32 α3) α1 + c22 α2 + c32 α3) α1 + c22 α2 + c32 α3)

a2 (n + 2) (c12 > 0, 2 a3 (n + 1) (c1 < 0, m = 2n , t0 and ϕ0 are arbitrary real constants.

6-15

(

)

(6.26)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

6.4 Galilean Transformation in (N + 1)-Dimensions (Movable Solutions) If ψ (x , t ; a1) is a solution of one of the three equations, fundamental NLSE, (2.1), NLSE with power law nonlinearity, (3.1), and NLSE with dual power law nonlinearity, (3.19),

i ψt + a1 ψxx + a2 ∣ψ ∣2 ψ = 0, i ψt + a1 ψxx + a2 ∣ψ ∣n ψ = 0, i ψt + a1 ψxx + a2 ∣ψ ∣n ψ + a3 ∣ψ ∣m ψ = 0, then

Φ(x1, x2 , … , xN , t ; α1, α2 , … , αN )=

{

ψ c1 [x1 − (x01 + v1 t )] + c2 [x2 − (x02 + v2 t )] + ⋯ + cN [xN − (x0N + vN t )], t ; c12 α1 + c12 α2 + ⋯ + cN2 αN ×

}

⎡ v ⎤ ⎛ v2 v2 v2 ⎞ v v i ⎢ 1 (x1− x01) + 2 (x2 − x02) +⋯+ N (xN − x0N ) − ⎜ 1 + 2 +⋯+ N ⎟ (t − t0)⎥ 2 α2 2 αN 4 α1 4 α 2 4 αN ⎠ ⎢⎣ 2 α1 ⎥⎦ ⎝ e

is a movable solution of the N

i Φt +

∑ αk Φx x

k k

+ a2 ∣Φ∣2 Φ = 0,

k=1 N

i Φt +

∑ αk Φx x

k k

+ a2 ∣Φ∣n Φ = 0,

k=1 N

i Φt +

∑ αk Φx x

k k

+ a2 ∣Φ∣n Φ + a3 ∣Φ∣m Φ = 0,

k=1

respectively, with the replacements N

x→

∑ ck [xk − (x0k + vk t )], k=1

t → t,

N

a1 →

∑ ck2 αk , k=1

a2 → a2 , a3 → a3,

N

i

v2

∑ 2vαkk (xk − x0k ) − 4αkk (t − t0 )

, ψ → ψ e k=1 where αk , ck, x0k , vk, a2, a3, n, m, and t0 are real constants.

6-16

(6.27)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Example 1. sech(x1, x2 , t ) moving 2D bright soliton (Figure 6.8) Given

ψ (x , t ) = A0

2 2 a1 ⎡ ⎤ sech[A0 (x − x0)] e i ⎣a1 A0 (t − t0) + ϕ0⎦ a2

is a static solution of (2.1), then

(

2 c12 α1 + c22 α2

Φ(x1, x2 , t ) = A0

)

a2 × sech(A0 {c1 [x1 − (x01 + v1 t )] + c2 [x2 − (x02 + v2 t )]}) ⎫ ⎧ v (x − x ) v (x − x ) ⎡ v 2 α +v 2 α ⎤ 1 1 01 i⎨ + 2 2 02 + ⎢A02 c12 α1+ c22 α 2 − 1 2 2 1 ⎥ (t − t0) + ϕ0⎬ ⎪ ⎪ 2 2 α α 4 α α 1 2 1 2 ⎣ ⎦ ⎭ e ⎩ ⎪

×

(6.28)

(

)

⎪

is a movable solution of (6.4), where a2 (c12 α1 + c22 α2 ) > 0, A0, t0, and ϕ0 are arbitrary real constants. Example 2. sech(x1, x2 , x3, t ) moving 3D bright soliton Given

ψ (x , t ) = A0

2 2a1 ⎡ ⎤ sech[A0 (x − x0)] e i ⎣a1 A0 (t − t0) + ϕ0⎦ a2

is a static solution of (2.1), then Φ(x1, x2, x3, t ) = A0

(

2 c12 α1 + c 22 α2 + c32 α3 a2

)

sech(A0 {c1 [x1 − (x 01 + v1 t )] (6.29)

+ c 2 [x2 − (x 02 + v2 t )] + c 3 [x3 − (x 03 + v3 t )]}) e i ϕ(x1, x2, x3, t )

is a movable solution of (6.6), where

Figure 6.8. Moving 2D bright soliton (6.28) at t = 0, with a2 = α1 = α2 = A0 = c1 = c2 = 1, v1 = v2 = 1/2 , and x01 = x02 = t0 = ϕ0 = 0 . The arrow shows the direction of motion. Animation available online at https:// iopscience.iop.org/book/978-0-7503-2428-1.

6-17

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

ϕ(x1, x2 , x3, t ) =

v1 (x1 − x01) v (x − x02 ) v (x − x03) + 2 2 + 3 3 2α1 2α2 2α3 2 ⎡ v α α + v22 α1 α3 + v32 α1 α2 ⎤ ⎥ + ⎢A02 c12 α1 + c22 α2 + c32 α3 − 1 2 3 4 α1 α2 α3 ⎣ ⎦ × (t − t0) + ϕ0 ,

(

)

a2 (c12 α1 + c22 α2 + c32 α3) > 0, A0, t0, and ϕ0 are arbitrary real constants.

Example 3. tanh(x1, x2 , t ) moving 2D dark soliton (Figure 6.9) Given

ψ (x , t ) = A0

2 −2a1 ⎡ ⎤ tanh[A0 (x − x0)] e−i ⎣2a1 A0 (t − t0) + ϕ0⎦ a2

is a static solution of (2.1), then

Φ(x1, x2 , t ) = A0

(

−2 c12 α1 + c22 α2

)

a2 × tanh(A0 {c1 [x1 − (x01 + v1 t )] + c2 [x2 − (x02 + v2 t )]}) ⎫ ⎧ v (x − x ) v (x − x ) ⎡ v 2 α +v 2 α ⎤ ⎬ ⎨ 1 1 01 + 2 2 02 − ⎢2 A02 c12 α1+ c22 α 2 + 1 2 2 1 ⎥ (t − t0) + ϕ0⎪ i⎪ 2α1 2α 2 4 α1 α 2 ⎦ ⎣ ⎭ ⎩ e ⎪

×

(6.30)

(

)

⎪

is a movable solution of (6.4), where a2 (c12 α1 + c22 α2 ) < 0, A0, t0, and ϕ0 are arbitrary real constants.

Figure 6.9. Moving 2D dark soliton (6.30) at t = 0, with α1 = α2 = A0 = c1 = c2 = 1, a2 = −1, v1 = v2 = 1/2 , and x01 = x02 = t0 = ϕ0 = 0 . The arrow shows the direction of motion. Animation available online at https:// iopscience.iop.org/book/978-0-7503-2428-1.

6-18

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Example 4. tanh(x1, x2 , x3, t ) moving 3D dark soliton Given

ψ (x , t ) = A0

2 −2 a1 ⎡ ⎤ tanh[A0 (x − x0)] e−i ⎣2 a1 A0 (t − t0) + ϕ0⎦ a2

is a static solution of (2.1), then

Φ(x1, x2 , x3, t ) = A0

(

− 2 c12 α1 + c22 α2 + c32 α3

) tanh(A {c [x − (x 0

a2

1

1

01

+ v1 t )] (6.31)

+ c2 [x2 − (x02 + v2 t )] + c3 [x3 − (x03 + v3 t )]}) e i ϕ(x1, x2, x3, t )

is a movable solution of (6.6), where

ϕ(x1, x2 , x3, t ) = ϕ0 +

v1 (x1 − x01) v (x − x02 ) v (x − x03) + 2 2 + 3 3 2α1 2α2 2α3

⎡ v 2 α2 α3 + v22 α1 α3 + v32 α1 α2 ⎤ ⎥ − ⎢2A02 (c12 α1 + c22 α2 + c32 α3) + 1 4 α1 α2 α3 ⎣ ⎦ × (t − t0), a2 (c12 α1 + c22 α2 + c32 α3) < 0, A0, t0, and ϕ0 are arbitrary real constants.

Example 5. sech(x1, x2 , t ) moving 2D bright soliton Given 1

⎡

⎧ 2 A 2 a1 (n + 2) ⎫ n i ⎢ 4 a1 2A0 0 2 ψ (x , t ) = ⎨ sech [A0 (x − x0)]⎬ e ⎣ n a2 n2 ⎩ ⎭ is a static solution of (3.1), then ⎛ 2A 2 (n + 2) c 2 α + c 2 α 0 1 1 2 2 Φ(x1, x2 , t ) = ⎜ 2 ⎜ a2 n ⎝

(

2

⎤ (t − t0) + ϕ0⎥ ⎦

) 1

× sech2(A0 {c1 [x1 − (x01 + v1 t )] + c2 [x2 − (x02

⎧ ⎫ ⎡ 4 A 2 c 2 α +c 2 α ⎤ ⎪ v (x − x ) v (x − x ) ⎪ 1 2 0 1 2 v 2 α +v 2 α i ⎨ 1 1 01 + 2 2 02 + ⎢ − 1 2 2 1 ⎥ (t − t0) + ϕ0⎬ 2 ⎢ 2 α1 2 α2 4 α1 α 2 ⎥ n ⎪ ⎪ ⎣ ⎦ ⎭ e ⎩

(

×

⎞n (6.32) + v2 t )]})⎟⎟ ⎠

6-19

)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

is a movable solution of (6.17), where a2 (n + 2) (c12 α1 + c22 α2 ) > 0, A0, t0, and ϕ0 are arbitrary real constants.

Example 6. sech(x1, x2 , x3, t ) moving 3D bright soliton Given ⎡

1

⎧ 2 A 2 a1 (n + 2) ⎫ n i ⎢ 4 a1 A0 0 2[A (x − x )]⎬ e ⎣ n 2 ψ (x , t ) = ⎨ sech 0 0 a2 n2 ⎩ ⎭

2

⎤ (t − t0) + ϕ0⎥ ⎦

is a static solution of (3.1), then 1

⎛ 2 A 2 (n + 2) c 2 α + c 2 α + c 2 α ⎞n 0 1 1 2 2 3 3 ⎜ 2 Φ(x1, x2, x3, t ) = sech [q(x1, x2, x3, t )]⎟ (6.33) ⎜ ⎟ a2 n2 ⎝ ⎠

(

)

× e i ϕ(x1, x2, x3, t )

is a movable solution of (6.19),where

ϕ(x1, x2 , x3, t ) = ϕ0 +

v1 (x1 − x01) 2 α1

+

v 2 (x2 − x02) 2 α2

+

v3 (x3 − x03) 2 α3

⎡ 4 A02 ( c12 α1 + c 22 α2 + c32 α3) +⎢ n2 ⎣ v2 α α +v2 α α +v2 α α ⎤ − 1 2 3 4 α2 α1 α3 3 1 2 ⎦⎥ × (t − t0), 1 2 3 q(x1, x2, x3, t ) = A0 {c1 [x1 − (x01 + v1 t )] , + c2 [x2 − (x02 + v2 t )] + c3 [x3 − (x03 + v3 t )]} a2 (n + 2) (c12 α1 + c22 α2 + c32 α3) > 0, A0, t0, and ϕ0 are arbitrary real constants.

Example 7. sech(x1, x2 , t ) moving 2D flat-top soliton (Figure 6.10) Given 1

⎛ ⎞n ⎜ ⎟ A0 A1 (n + 2) ⎜ ⎟ e i [A1 (t − t0) + ϕ0] ψ (x , t ) = ⎜ ⎧ ⎡ ⎤ ⎟ A1 ⎜ a2 ⎨A0 + 2 cosh ⎢⎣n a (x − x0)⎥⎦ ⎟ 1 ⎝ ⎩ ⎠

}

6-20

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Figure 6.10. Moving 2D flat-top soliton (6.34) at t = 0, with a2 = 1, A1 = 2 , c1 = c2 = α1 = α2 = 1, a3 = −0.06944444444 , n = 4, v1 = v2 = 1/2 , and x01 = x02 = t0 = ϕ0 = 0 . The arrow shows the direction of motion. Animation available online at https://iopscience.iop.org/book/978-0-7503-2428-1.

is a static solution of (3.19), then 1

⎡ A0 A1 (n + 2) ⎤ n Φ(x1, x2 , t ) = ⎢ ⎥ ⎣ a2 A0 + q(x1, x2 , t ) ⎦ ×

(6.34)

⎡ v (x − x ) v (x − x ) ⎛ ⎤ v 2 α +v 2 α ⎞ i ⎢ 1 1 01 + 2 2 02 + ⎜A1− 1 2 2 1 ⎟ (t − t0) + ϕ0⎥ α α 2 2 4 α α ⎥⎦ 1 2 1 2 ⎝ ⎠ e ⎢⎣

is a movable solution of (6.22), where ⎛ q(x1, x2, t ) = 2 a2 cosh ⎜n ⎝

A0 =

2 a2 n+2

n+1 A1 δ

A1 c12 α1 + c 22 α 2

⎞ {c1 [x1 − (x 01 + v1 t )] + c 2 [x2 − (x 02 + v2 t )]}⎟ , ⎠

,

a 22

(n + 1) , A1 (n + 2)2 2 2 A1 (c1 α1 + c2 α2 )

δ = a3 +

> 0, A1 δ (n + 1) > 0, m = 2 n, t0, A1, and ϕ0 are arbitrary real constants.

Example 8. sech(x1, x2 , x3, t ) moving 3D flat-top soliton Given 1

⎛ ⎞n ⎜ ⎟ A0 A1 (n + 2) ⎜ ⎟ e i [A1 (t − t0) + ϕ0] ψ (x , t ) = ⎜ ⎧ ⎡ ⎤ ⎟ A ⎜ a2 ⎨A0 + 2 cosh ⎢n a 1 (x − x0)⎥ ⎟ ⎣ ⎦ ⎠ 1 ⎝ ⎩

}

is a static solution of (3.19), then 1

⎡ ⎤n A0 A1 (n + 2) Φ(x1, x2 , x3, t ) = ⎢ ⎥ e i ϕ(x1, x2, x3, t ) ⎣ a2 A0 + q(x1, x2 , x3) ⎦

6-21

(6.35)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

is a movable solution of (6.24), where

ϕ(x1, x2 , x3, t ) =

v1 (x1 − x01) 2 α1

(

+ A1 −

+

v 2 (x2 − x02) 2 α2

+

v3 (x3 − x03) 2 α3

v12 α 2 α3 + v 22 α1 α3 + v 32 α1 α 2 4 α1 α 2 α3

⎛ q(x1, x2 , x3, t ) = 2 a2 cosh ⎜n ⎝

) (t − t ) + ϕ ,

A1 2 2 2 (c1 α1 + c 2 α 2 + c3 α3)

0

0

{c1 [x1 − (x01 + v1 t )]

⎞ + c2 [x2 − (x02 + v2 t )] + c3 [x3 − (x03 + v3 t )]} ⎟ , ⎠

A0 =

2 a2 n+2

n+1 A1 δ

,

a 22

(n + 1) , A1 (n + 2)2 A1 (c12 α1 + c22 α2 +

δ = a3 +

c32 α3) > 0,

A1 δ (n + 1) > 0, m = 2 n, t0, A1, and ϕ0 are arbitrary real constants.

6.5 NLSE in (2 + 1)-Dimensions with Φx x Term 1 2

If ψ (x , t ) is a solution to the fundamental NLSE, (2.1),

i ψt + a1 ψxx + a2 ∣ψ ∣2 ψ = 0, then

Φ(x1, x2 , t ) =

a2 ψ [X (x1, x2 ), t ], b4

(6.36)

is a solution of

i Φt + b Φx1x1 + b2 Φx2x2 + b3 Φx1x2 + b4 ∣Φ∣2 Φ = 0,

(6.37)

where

X (x1, x2 ) = c0 −

1 b3 c 1 − 2 b1

(

)

4 a1 b1 − 4 b1 b2 c12 + b32 c12 x1 + c1 x2 ,

a2 b4 > 0, 4 a1 b1 (1 − b2 c12 ) > b32 c12 , b1, b2, b3, b4, c0 and c1 are arbitrary real constants.

6-22

(6.38)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Example 1. sech(x1, x2 ) bright soliton Given

ψ (x , t ) = A0

2 2 a1 ⎡ ⎤ sech[A0 (x − x0)] e i ⎣a1 A0 (t − t0) + ϕ0⎦ a2

is a solution of (2.1), then ⎧ ⎡ b 3 c1 − 4 a1 b1 − 4 b1 b 2 c12 + b 32 c12 (x1 − x 01) ⎪ ⎢ 2a1 Φ(x1, x2, t ) = A0 sech⎨A0 ⎢c 0 − b4 2b1 ⎪ ⎢ ⎩ ⎣ (6.39) ⎤⎫ ⎪ ⎥ 2 ⎡ ⎤ + c1 (x2 − x 02 ) ⎥⎬ e i ⎣a1 A0 (t − t0) + ϕ0⎦ ⎥⎦⎪ ⎭

(

)

is a solution of (6.37), where a1 b4 > 0, 2 2 2 (4 a1 b1 − 4 b1 b2 c1 + b3 c1 ) > 0, A0, t0, and ϕ0 are arbitrary real constants. Example 2. tanh(x1, x2 ) dark soliton Given

ψ (x , t ) = A0

2 −2 a1 ⎡ ⎤ tanh[A0 (x − x0)] e−i ⎣2 a1 A0 (t − t0) + ϕ0⎦ a2

is a solution of (2.1), then ⎧ ⎡ 4 a1 b1 − 4 b1 b2 c12 + b 32 c12 − b3 c1 (x1 − x01) ⎪ ⎢ − 2a1 Φ(x1, x2 , t ) = A0 tanh ⎨A0⎢c0 + b4 2 b1 ⎪ ⎢ ⎩ ⎣ (6.40) ⎤⎫ ⎥⎪ 2 ⎡ ⎤ + c1 (x2 − x02) ⎥⎬ e−i ⎣2 a1 A0 (t − t0 ) + ϕ0⎦ ⎪ ⎥⎦ ⎭

(

)

is a solution of (6.37), where a1 b4 < 0, (4 a1 b1 − 4 b1 b2 c12 + b32 c12 ) > 0, A0, t0, and ϕ0 are arbitrary real constants.

6-23

(N + 1)-Dimensional NLSE with Cubic Nonlinearity

6-24

4.

3.

2.

1.

#

N

k=1

2

)

⎡

2

(

)

2 2 2 (c1 α1+c2 α2+c3 α3) (t−t0 )+ϕ0⎤⎦

−2 c12 α1 + c 22 α 2 a2

× e i ⎣A0

2

(

)

−2 c12 α1 + c 22 α 2 + c32 α3 a2

2 2 (c1 α1+c2 α2 ) (t−t0 )+ϕ0⎤⎦

2

× e−i ⎣2 A0

⎡

2 2 2 (c1 α1+c2 α2+c3 α3) (t−t0 )+ϕ0⎤⎦

× tanh{A0 [c1 (x1 − x01) + c2 (x2 − x02 ) + c3 (x3 − x03)]}

Φ(x1, x2, x3, t ) = A0

× e−i ⎣2 A0

⎡

× tanh{A0 [c1 (x1 − x01) + c2 (x2 − x02 )]}

Φ(x1, x2, t ) = A0

(

2 c12 α1 + c 22 α 2 + c32 α3 a2

2 2 (c1 α1+c2 α2 ) (t−t0 )+ϕ0⎤⎦

× sech{A0 [c1 (x1 − x01) + c2 (x2 − x02 ) + c3 (x3 − x03)]}

Φ(x1, x2, x3, t ) = A0

× e i ⎣A0

⎡

× sech{A0 [c1 (x1 − x01) + c2 (x2 − x02 )]}

)

+ a2 ∣Φ∣2 Φ = 0

2 c12 α1 + c 22 α 2 a2

(

k k

∑ αk Φx x

Φ(x1, x2, t ) = A0

Example

Equation: i Φt +

Name

3D dark a2 (c12 α1 + c22 α2 + c32 α3) < 0 , soliton A0, t0, and ϕ0 are arbitrary real constants

2D dark a2 (c12 α1 + c22 α2 ) < 0 , A0, t0, and ϕ0 are arbitrary real constants soliton

3D bright a2 (c12 α1 + c22 α2 + c32 α3) > 0 , A0, t0, and ϕ0 are arbitrary real constants soliton

2D bright a2 (c12 α1 + c22 α2 ) > 0 , A0, t0, and ϕ0 are arbitrary real constants soliton

Conditions

Transformation: Φ(x1, x2, … , xN , t ; α1, α2, … , αN ) = Φ[c1 (x1 − x01) + c2 (x2 − x02 ) + ⋯ + cN (xN − x0N ), t ; c12 α1 + c12 α2 + ⋯ + cN2 αN , a2 ]

6.6 Summary of Sections 6.1–6.5

(6.8)

(6.7)

(6.5)

(6.3)

Eq. #

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

8.

7.

6.

5.

−1 ⎛ p2 cos[ω (t − t0)] + 2 i p ν sin[ω (t − t0)] ⎜ a2 ⎝ 2 cos[ω (t − t0)] − q(x1, x2, x3)

) ⎞ + 1⎟ ⎠

⎫ ⎪ [c1 (x1 − x01) + c2 (x2 − x02 )]⎬ ⎪ ⎭

⎞ + 1⎟ e i [(t−t0)+ϕ0 ], ⎠

1 ⎛ κ 2 cosh[δ (t − t0)] + 2 i κ ν sinh[δ (t − t0)] ⎜ a2 ⎝ 2 cosh[δ (t − t0)] − q(x1, x2 )

⎞ − 1⎟ ⎠

6-25

1 ⎛ κ 2 cosh[δ (t − t0)] + 2 i κ ν sinh[δ (t − t0)] ⎜ a2 ⎝ 2 cosh[δ (t − t0)] − q(x1, x2, x3)

⎞ − 1⎟ ⎠

× e i [(t−t0)+ϕ0 ], q(x1, x2, x3) = 2 ν cos ⎧ ⎫ ⎪ ⎪ κ ⎨ [c1 (x1 − x01) + c2 (x2 − x02 ) + c3 (x3 − x03)]⎬ ⎪ ⎪ 2 α +c2 α +c2 α 2 c ⎩ ( 1 1 2 2 3 3) ⎭

Φ(x1, x2, x3, t ) =

× e i [(t−t0)+ϕ0 ], ⎧ ⎫ ⎪ ⎪ κ q(x1, x2 ) = 2 ν cos ⎨ [c1 (x1 − x01) + c2 (x2 − x02 )]⎬ ⎪ ⎪ 2 2 ⎩ 2 (c1 α1 + c2 α2) ⎭

Φ(x1, x2, t ) =

× e i [(t−t0)+ϕ0 ], q(x1, x2, x3) = 2 ν cosh ⎧ ⎫ ⎪ ⎪ p ⎨ [c1 (x1 − x01) + c2 (x2 − x02 ) + c3 (x3 − x03)]⎬ ⎪ ⎪ 2 2 2 ⎩ 2 (c1 α1 + c2 α2 + c3 α3) ⎭

Φ(x1, x2, x3, t ) =

(

2 c12 α1 + c 22 α 2

p

−1 ⎛ p2 cos[ω (t − t0)] + 2 i p ν sin[ω (t − t0)] ⎜ a2 ⎝ 2 cos[ω (t − t0)] − q(x1, x2 )

⎧ ⎪ q(x1, x2 ) = 2 ν cosh ⎨ ⎪ ⎩

Φ(x1, x2, t ) =

ν 2 − 1 , ω = p ν , ν > 1, a2 > 0,

ν 2 − 1 , ω = p ν , ν > 1, a2 > 0,

1 − ν 2 , δ = κ ν , ν < 1, a2 > 0

1 − ν 2 , δ = κ ν , ν < 1, a2 > 0,

(c12 α1 + c22 α2 + c32 α3) > 0, t0 and ϕ0 are arbitrary real constants

κ=2

(c12 α1 + c22 α2 ) > 0, t0 and ϕ0 are arbitrary real constants

κ=2

(c12 α1 + c22 α2 + c32 α3) > 0, t0 and ϕ0 are arbitrary real constants

p=2

(c12 α1 + c22 α2 ) > 0, t0 and ϕ0 are arbitrary real constants

p=2

3D Akhmediev (6.12) breather

2D Akhmediev (6.11) breather

3D Kuznetsov– (6.10) Ma breather

2D Kuznetsov– (6.9) Ma breather

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

q(x1, x2, x3) =

2 c12 α1 + c 22 α 2 + c32 α3

t0 and ϕ0 are arbitrary real constants

a2 > 0,

t0 and ϕ0 are arbitrary real constants

a2 > 0,

(N + 1)-Dimensional NLSE with Power Law Nonlinearity

[c1 (x1 − x01) + c2 (x2 − x02 ) + c3 (x3 − x03)]2

⎞ − 1⎟ e i [(t−t0)+ϕ0 ], ⎠

[c1 (x1 − x01) + c2 (x2 − x02 )]2

⎞ − 1⎟ e i [(t−t0)+ϕ0 ], ⎠

4 + i 8 (t − t 0 ) 1 ⎛ ⎜ a2 ⎝ 1 + 4 (t − t0)2 + q(x1, x2, x3)

2 c12 α1 + c 22 α 2

4 + i 8 (t − t 0 ) 1 ⎛ ⎜ a2 ⎝ 1 + 4 (t − t0)2 + q(x1, x2 )

Φ(x1, x2, x3, t ) =

q(x1, x2 ) =

Φ(x1, x2, t ) =

3D Peregrine soliton

2D Peregrine soliton

6-26

2.

1.

#

N

k=1

k k

∑ αk Φx x

+ a2 ∣Φ∣n Φ = 0

⎡

2

2

2

⎤

1

×e

⎡ 4 A 2 (c 2 α1+ c22 α2 + c32 α3) ⎤ i⎢ 0 1 (t−t0 )+ϕ0⎥ n2 ⎣ ⎦

⎞n × sech2{A0 [c1 (x1 − x01) + c2 (x2 − x02 ) + c3 (x3 − x03)]}⎟ ⎠

⎛ 2 A 2 (n + 2) (c12 α1 + c22 α2 + c32 α3) Φ(x1, x2, x3, t ) = ⎜ 0 a 2 n2 ⎝

⎞ n i ⎢ 4 A0 (c1 α21+ c2 α2) (t−t0)+ϕ0⎥ n ⎦ + c2 (x2 − x02 )]}⎟ e ⎣ ⎠

1

⎛ 2 A 2 (n + 2) (c12 α1 + c22 α2) Φ(x1, x2, t ) = ⎜ 0 sech2{A0 [ c1 (x1 − x01) a 2 n2 ⎝

Example

Equation: i Φt +

c22

Name

N = 3, 3D bright a2 (n + 2) (c12 α1 + c22 α2 soliton + c32 α3) > 0 , A0, t0, and ϕ0 are arbitrary real constants

2D bright a2 (n + 2) α1 + α2 ) > 0 , N = 2 , A0, t0, and ϕ0 are arbitrary real constants soliton

(c12

Conditions

Transformation: Φ(x1, x2, … , xN , t ; α1, α2, … , αN ) = ψ [c1 (x1 − x01) + c2 (x2 − x02 ) + ⋯ + cN (xN − x0N ), t ; c12 α1 + c12 α2 + ⋯ + cN2 αN , a2 ]

10.

9.

(Continued )

(6.18)

(6.16)

Eq. #

(6.14)

(6.13)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

N

k k

6-27

3.

2.

1

×e⎣

⎡ 4 A 2 (c 2 α1+ c 2 α2 ) ⎤ i⎢ 0 1 2 2 (t−t0 )+ϕ0⎥ n ⎦

⎤n (1 − tanh{A0 [c1 (x1 − x01) + c2 (x2 − x02 )]})⎥ ⎦

⎡ 2 A 2 (n + 2) (c12 α1 + c22 α2) Φ(x1, x2, t ) = ⎢ 0 a 2 n2 ⎣

1

⎡ A A (n + 2) ⎤ n Φ(x1, x2, x3, t ) = ⎢⎣ a A 0+ q1(x , x , x ) ⎦⎥ e i [A1 (t−t0)+ϕ0 ], 2 0 1 2 3 ⎧ A1 q(x1, x2, x3) = 2a2 cosh ⎨n (c12 α1 + c 22 α 2 + c32 α3) ⎩ ⎫ [c1 (x1 − x01) + c2 (x2 − x02 ) + c3 (x3 − x03)]⎬ ⎭

⎡ A A (n + 2) ⎤ n Φ(x1, x2, t ) = ⎢⎣ a A0 +1 q(x , x ) ⎦⎥ e i [A1 (t−t0)+ϕ0 ], 2 0 1 2 ⎧ ⎫ A1 q(x1, x2 ) = 2a2 cosh ⎨n [c1 (x1 − x01) + c2 (x2 − x02 )]⎬ 2 2 c1 α1 + c 2 α 2 ⎩ ⎭

1.

1

+ a2 ∣Φ∣n Φ + a3 ∣Φ∣m Φ = 0

Example

k=1

∑ αk Φx x

#

Equation: i Φt +

2 a2 n+2

a 2 (n + 1) n+1 , δ = a3 + 2 , A1 δ A1(n + 2)2 c22 α2 ) > 0 , A1 δ (n + 1) >

0,

n+1 A1 δ

a2 n 2 (n + 2)

−(n + 1) a3 (c12 α1 + c 22 α 2 )

,

a2 (n + 2) (c12 α1 + c22 α2 ) > 0 , N = 2 , a3 (n + 1) (c12 α1 + c22 α2 ) < 0 , m = 2 n , t0 and ϕ0 are arbitrary real constants

A0 =

A1 (c12 α1 + c22 α2 + c32 α3) > 0 , A1 δ (n + 1) > 0 , m = 2 n , t0, A1, and ϕ0 are arbitrary real constants

A1(n + 2)2

,

, N = 3,

a 22 (n + 1)

2 a2 n+2

δ = a3 +

A0 =

m = 2 n, N = 2, t0, A1, and ϕ0 are arbitrary real constants

A1 (c12 α1 +

A0 =

Conditions

2D dark soliton (6.25)

(6.23)

(6.21)

2D flat-top soliton

3D flat-top soliton

Eq. #

Name

Transformation: Φ(x1, x2, … , xN , t ; α1, α2, … , αN ) = ψ [c1 (x1 − x01) + c2 (x2 − x02 ) + ⋯ + cN (xN − x0N ), t ; c12 α1 + c12 α2 + ⋯ + cN2 αN , a2 ]

(N + 1)-Dimensional NLSE with Dual Power Law Nonlinearity

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Transformation:

⎡

2

2

2

2

⎤

6-28

1.

#

N

k=1

k k

×e

×

⎧ v (x − x ) v (x − x ) ⎡ ⎫ v 2 α 2 + v 2 α1 ⎤ i ⎨ 1 21α 01 + 2 22α 02 +⎢A02 (c12 α1+c22 α2 )− 1 4 α α2 ⎥ (t−t0 )+ϕ0⎬ 1 2 ⎦ 1 2 ⎣ ⎩ ⎭ e

c22

α1 + α2 ) > 0, N = 2 , A0, t0, and ϕ0 are arbitrary real constants

a2 (c12

Conditions

⎡ v ⎤ ⎛ v2 v2 v2 ⎞ v v i ⎢ 2 α1 (x1−x01)+ 2 α2 (x2−x02 )+⋯+ 2 αN (xN −x0N )−⎜ 4 1α + 4 α2 +⋯+ 4 αN ⎟ (t−t0 )⎥ N⎠ N 2 2 ⎝ 1 ⎣ 1 ⎦

× sech(A0 {c1 [x1 − (x01 + v1 t )] + c2 [x2 − (x02 + v2 t )]})

,

a2 (n + 2) (c12 α1 + c22 α2 + c32 α3) > 0, a3 (n + 1) (c12 α1 + c22 α2 + c32 α3) < 0, m = 2 n , N = 3, t0 and ϕ0 are arbitrary real constants

+ ⋯ + cN [xN − (x0N + vN t )], t ; c12 α1 + c12 α2 + ⋯ + cN2 αN , a2}

+ a2 ∣Φ∣2 Φ = 0

2 (c12 α1 + c 22 α 2 ) a2

∑ αk Φx x

Φ(x1, x2, t ) = A0

Example

Equation: i Φt +

a2 n 2 (n + 2)

−(n + 1) a3 (c12 α1 + c 22 α 2 + c32 α3)

A0 =

Galilean Transformation in (N + 1)-Dimensions (Movable Solutions)

⎤ n i ⎢ 4 A0 (c1 α1+ c22 α2+ c3 α3) (t−t0)+ϕ0⎥ n ⎦ + c3 (x3 − x03)]})⎥ e ⎣ ⎦

1

(1 − tanh{A0 [c1 (x1 − x01) + c2 (x2 − x02 )

⎡ 2 A 2 (n + 2) (c12 α1 + c22 α2 + c32 α3) Φ(x1, x2, x3, t ) = ⎢ 0 a 2 n2 ⎣

Φ(x1, x2, … , xN , t ; α1, α2, … , αN , a2 ) = Φ{c1 [x1 − (x01 + v1 t )] + c2 [(x2 − x02 + v2 t )]

4.

(Continued )

Eq. # moving 2D (6.28) bright soliton

Name

3D dark soliton (6.26)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

6-29

×

+

v2 (x2 − x02 ) 2 α2

+

∑ αk Φx x

k k

+ a2 ∣Φ∣n Φ = 0

× (t − t0 ) + ϕ0 ,

v12 α 2 α3 + v 22 α1 α3 + v32 α1 α 2 ⎤ ⎥ 4 α1 α 2 α3 ⎦

v3 (x3 − x03) 2 α3

⎡ − ⎢2 A02 (c12 α1 + c22 α2 + c32 α3) + ⎣

v1 (x1 − x01) 2 α1

N = 3,

a2 (c12 α1 + c22 α2 + c32 α3) > 0,

A0, t0, and ϕ0 are arbitrary real constants

N = 3,

a2 (c12 α1 + c22 α2 + c32 α3) < 0,

a2 (c12 α1 + c22 α2 ) < 0, N = 2 , A0, t0, and ϕ0 are arbitrary real constants

(t − t0 ) A0, t0, and ϕ0 are arbitrary real constants

+ c2 [x2 − (x02 + v2 t )] + c3 [x3 − (x03 + v3 t )]}) e i ϕ(x1, x2, x3, t ),

× tanh(A0 {c1 [x1 − (x01 + v1 t )]

−2 (c12 α1 + c 22 α 2 + c32 α3) a2

⎧ v (x − x ) v (x − x ) ⎡ ⎫ v 2 α 2 + v 2 α1 ⎤ i ⎨ 1 21α 01 + 2 22α 02 −⎢2 A02 (c12 α1+c22 α2 )+ 1 4 α α2 ⎥ (t−t0 )+ϕ0⎬ 1 2 ⎦ 1 2 ⎣ ⎩ ⎭ e

ϕ(x1, x2, x3, t ) =

k=1

+

v3 (x3 − x03) 2 α3 2 v α 2 α3 + v 22 α1 α3 + v32 α1 α 2 ⎤ c32 α3) − 1 ⎥ 4 α1 α 2 α3 ⎦

v2 (x2 − x02 ) 2 α2

× tanh(A0 {c1 [x1 − (x01 + v1 t )] + c2 [x2 − (x02 + v2 t )]})

−2 (c12 α1 + c 22 α 2 ) a2

Φ(x1, x2, x3, t ) = A0

N

+

⎡ + ⎢A02 (c12 α1 + c22 α2 + ⎣

v1 (x1 − x01) 2 α1

+ c2 [x2 − (x02 + v2 t )] + c3 [x3 − (x03 + v3 t )]}) e i ϕ(x1, x2, x3, t ),

ϕ(x1, x2, x3, t ) = ϕ0 +

Φ(x1, x2, t ) = A0

2 (c12 α1 + c 22 α 2 + c32 α3) a2

× sech(A0 {c1 [x1 − (x01 + v1 t )]

Φ(x1, x2, x3, t ) = A0

Equation: i Φt +

4.

3.

2.

moving 3D (6.31) dark soliton

moving 2D (6.30) dark soliton

moving 3D (6.29) bright soliton

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

6-30

#

1

⎪

⎪

1

Example

k=1

k k

∑ αk Φx x

N

v1 (x1 − x01) 2 α1

+

v2 (x2 − x02 ) 2 α2

v3 (x3 − x03) 2 α3

v12 α 2 α3 + v 22 α1 α3 + v32 α1 α 2 ⎤ ⎥ 4 α1 α 2 α3 ⎦

+

+ a2 ∣Φ∣n Φ + a3 ∣Φ∣m Φ = 0

⎡ 4 A 2 (c 2 α1 + c22 α2 + c32 α3) +⎢ 0 1 − n2 ⎣

ϕ(x1, x2, x3, t ) = ϕ0 +

× e i ϕ(x1, x2, x3, t ), q(x1, x2, x3, t ) = A0 {c1 [x1 − (x01 + v1 t )] + c2 [x2 − (x02 + v2 t )] + c3 [x3 − (x03 + v3 t )]},

⎛ 2 A 2 (n + 2) (c12 α1 + c22 α2 + c32 α3) ⎞n Φ(x1, x2, x3, t ) = ⎜ 0 sech2[q(x1, x2, x3, t )]⎟ 2 a2 n ⎝ ⎠

×

⎪

⎪

⎧ ⎫ ⎡ 4 A 2 (c 2 α1+ c 2 α2) v 2 α2 + v 2 α1 ⎤ v (x − x ) v (x − x ) i ⎨ 1 21α 01 + 2 22α 02 +⎢ 0 1 2 2 − 1 4 α α2 ⎥ (t−t0 )+ϕ0⎬ 1 2 ⎥ 1 2 n ⎢⎣ ⎦ ⎩ ⎭ e

(t − t0 )

Conditions

A0, t0, and ϕ0 are arbitrary real constants

N = 3,

a2 (n + 2) (c12 α1 + c22 α2 + c32 α3) > 0,

N = 2, A0, t0, and ϕ0 are arbitrary real constants

a2 (c12 α1 + c22 α2 ) > 0,

⎛ 2 A 2 (n + 2) (c12 α1 + c22 α2) Φ(x1, x2, t ) = ⎜ 0 a 2 n2 ⎝

× sech2(A0 {c1 [x1 − (x01 + v1 t )] + c2 [x2 − (x02 + v2 t )]})) n

Conditions

Example

Equation: i Φt +

2.

1.

#

(Continued )

Eq. #

Name

Eq. #

moving 3D (6.33) bright soliton

moving 2D (6.32) bright soliton

Name

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

6-31

1

1

⎛ + ⎜A1 − ⎝

+

+

{c1 [x1 − (x01 + v1 t )]

Φ[X (x1, x2 ), t ],

X (x1, x2 ) = c0 −

Equation: i Φt + b Φx1x1 + b2 Φx2x2 + b3 Φx1x2 + b4 ∣Φ∣2 Φ = 0

a2 b4

2 a2 n+2

1 2b

(b

3 c1

−

n+1 A1 δ

, δ = a3 + A1(n + 2)2

a 22 (n + 1)

,

2 a2 n+2 n+1 A1 δ

, δ = a3 +

A1(n + 2)2

a 22 (n + 1)

,

4 a1 b − 4 b b2 c12 + b32 c12 x1 + c1 x2

)

t0, A1, and ϕ0 are arbitrary real constants

A1 δ (n + 1) > 0, m = 2 n , N = 3,

A1 (c12 α1 + c22 α2 + c32 α3) > 0 ,

A0 =

A1 δ (n + 1) > 0, m = 2 n , t0, A1, and ϕ0 are arbitrary real constants

A1 (c12 α1 + c22 α2 ) > 0 , N = 2 ,

A0 =

NLSE in (2 + 1)-Dimensions with Φx1x2 Term

⎞ + c2 [x2 − (x02 + v2 t )] + c3 [x3 − (x03 + v3 t )]}⎟⎟ ⎠

A1 (c12 α1 + c 22 α 2 + c32 α3)

v2 (x2 − x02 ) 2α 2

v3 (x3 − x03) 2α 3 v12 α 2 α3 + v 22 α1 α3 + v32 α1 α 2 ⎞ ⎟ (t − t0 ) 4 α1 α 2 α3 ⎠

v1 (x1 − x01) 2α1

⎛ q(x1, x2, x3, t ) = 2 a2 cosh ⎜n ⎝

ϕ(x1, x2, x3, t ) = + ϕ0 ,

{c1 [x1 − (x01 + v1 t )]

⎡ A A (n + 2) ⎤ n Φ(x1, x2, x3, t ) = ⎢⎣ a A 0+ q1(x , x , x ) ⎥⎦ e i ϕ(x1, x2, x3, t ), 2 0 1 2 3

⎞ + c2 [x2 − (x02 + v2 t )]}⎟⎟ ⎠

A1 c12 α1 + c 22 α 2

⎡ v (x − x ) v (x − x ) ⎛ ⎤ v 2 α 2 + v 2 α1 ⎞ i ⎢ 1 21α 01 + 2 22α 02 +⎜A1− 1 4 α α2 ⎟ (t−t0 )+ϕ0⎥ 1 2 ⎠ 1 2 ⎝ ⎣ ⎦, e

A0 A1 (n + 2) ⎤ n ⎥ 2 A0 + q(x1, x2, t ) ⎦

⎛ q(x1, x2, t ) = 2a2 cosh ⎜n ⎝

×

⎡ Φ(x1, x2, t ) = ⎣⎢ a

Transformation: Φ(x1, x2, t ) =

2.

1.

moving 3D flat-top soliton

moving 2D flat-top soliton

(6.35)

(6.34)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

2.

1.

#

2 a1 b4

⎡ ⎢c − ⎢ 0 ⎣

⎡

2

(b3 c1 −

−2a1 b4

⎡ ⎢c + ⎢ 0 ⎣

⎡

(

)

)

2

(t−t0 )+ϕ0⎤⎦

2 b1

4a1 b1 − 4 b1 b2 c12 + b32 c12 − b3 c1 (x1 − x01)

2 b1

4 a1 b1 − 4 b1 b2 c12 + b32 c12 (x1 − x01)

(t−t0 )+ϕ0⎤⎦

+ c1 (x2 − x02 )]} e i ⎣2a1 A0

⎧ ⎪ × tanh ⎨A0 ⎪ ⎩

Φ(x1, x2, t ) = A0

+ c1 (x2 − x02 )]} e i ⎣a1 A0

⎧ ⎪ × sech ⎨A0 ⎪ ⎩

Φ(x1, x2, t ) = A0

Example

(Continued )

A0, t0, and ϕ0 are arbitrary real constants

dark soliton

bright soliton

a1 b4 > 0 , (4 a1 b1 − 4 b1 b2 c12 + b32 c12 ) > 0,

a1 b4 < 0 , (4 a1 b1 − 4 b1 b2 c12 + b32 c12 ) > 0,

Name

Conditions

(6.40)

(6.39)

Eq. #

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

6-32

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

6.7 (N + 1)-Dimensional Isotropic NLSE with Cubic Nonlinearity in Polar Coordinate System If ψ (x , t ) is a solution of the fundamental NLSE in 1D, (2.1),

i ψt + a1 ψxx + a2 ∣ψ ∣2 ψ = 0, then

Φ(r , t ) = A(r , t ) e i B(r, t ) ψ [R(r , t ), T (r , t )]

(6.41)

is a solution of ⎛ N−1 ⎞ i Φt + b1(r , t )⎜Φrr + Φr⎟ + b2(r , t ) ∣Φ∣2 Φ + [b3r (r , t ) + i b3i (r , t )]Φ = 0, (6.42) ⎝ ⎠ r

where

T (r , t ) = g1(t ), A(r , t ) =

B(r , t ) = g3(t ) −

g2(t ) r

∫

b1(r , t ) =

b 2 (r , t ) =

b3i (r , t ) =

b 3 r (r , t ) =

(6.43)

1−N 2

,

(6.44)

Rt(r , t ) Rr(r , t ) dr , 2a1 g1′(t )

(6.45)

Rr(r , t )

a1 g1′(t ) R r2(r , t ) a2 g1′(t ) A2 (r , t )

,

(6.46)

,

(6.47)

g ′(t ) Rrt(r , t ) − 2 , Rr(r , t ) g2(t )

1 2g1″(t ) 4a1 g1′2(t )

{

− 2g1′(t )

∫ Rt(r, t ) Rr(r, t ) dr

(6.48)

(6.49)

∫ [Rtt(r, t ) Rr(r, t ) + Rt(r, t ) Rrt(r, t )] dr + g1′(t ) Rt2(r, t )

a12 g1′3(t ) ⎡ 2 ⎢ (N − 1) (N − 3) R r (r , t ) r 2 R r4(r , t ) ⎣ ⎤ − 3r 2 R rr2 (r , t ) + 2r 2 Rr(r , t ) Rrrr(r , t ) ⎦⎥ , +

}

g1(t ), g2(t ), and R(r, t ) are arbitrary real functions.

6-33

(6.50)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

6.7.1 Angular Dependence If ψ (x , t ) is a solution to the fundamental NLSE, (2.1), then

Φ(r , θ , t ) = r (1 − N )/2 e i

1 2

−(N − 1)(N − 3) θ

ψ (r , t )

(6.51)

is a solution to the NLSE

⎛ ⎞ 1 N−1 i Φt + a1 ⎜Φrr + Φr + 2 Φθθ ⎟ + a2 r N − 1 ∣Φ∣2 Φ = 0. ⎝ ⎠ r r

(6.52)

This is obtained from the previous section with the special choices: g1 = t , g2 = 1, g3 = 0, R = r . Remarks: 1. The angular term Φθθ /r 2 vanishes in 1D and 3D. 2. The prefactor r (1 − N )/2 in Φ diverges at r = 0 as r−1/2 and r−1 for 2D and 3D, respectively. This divergence may be removed with certain solutions of the fundamental NLSE, ψ , such as the tanh(r ) solution. Example 1. General Case Given

ψ (x , t ) = A0

−2 a1 2 tanh[A0 (x − x0)] e−i [2 a1 A0 (t − t0) + ϕ0] a2

is a solution of (2.1), then Φ(r , θ , t ) = r (1 − N )/2 e i

1 2

−(N − 1)(N − 3) θ

⎡ ⎢A0 ⎣

⎤ − 2a1 −i (2a1 A02 t + ϕ0) e tanh(A0 r )⎥ (6.53) a2 ⎦

is a solution of (6.52), where a1 a2 < 0, A0 and ϕ0 are arbitrary real constants. Example 2. 2D vortex soliton (Figure 6.11)

Φ(r , θ , t ) = r −1/2 e i

a1 θ

⎡ ⎢A0 ⎣

⎤ −2a1 −i (2a1 A02 t + ϕ0) e tanh(A0 r )⎥ a2 ⎦

(6.54)

is a solution of

⎛ 1 1 ⎞ i Φt + a1 ⎜Φrr + Φr − 2 ⎟ + a2 r ∣Φ∣2 Φ = 0, ⎝ 4r ⎠ r where a1 a2 < 0, A0 and ϕ0 are arbitrary real constants. 6-34

(6.55)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Figure 6.11. Vortex soliton (6.54) at θ = t = 0 , with a1 = A0 = 3/2 , a2 = −1/2 , and ϕ0 = 0 .

Figure 6.12. Plot of solution (6.56) at t = 0 with a1 = A0 = 3/2 , a2 = −1/2 , and ϕ0 = 0 .

Example 3. 3D (Figure 6.12)

⎡ Φ(r , t ) = r −1 ⎢A0 ⎣

⎤ −2a1 −i (2a1 A02 t + ϕ0) e tanh(A0 r )⎥ a2 ⎦

(6.56)

is a solution to

⎛ 2 ⎞ i Φt + a1 ⎜Φrr + Φr⎟ + a2 r 2 ∣Φ∣2 Φ = 0, ⎝ r ⎠

(6.57)

where a1 a2 < 0, A0 and ϕ0 are arbitrary real constants. 6.7.2 Constant Dispersion and Real Potential If ψ (x , t ) is a solution of the fundamental NLSE, (2.1),

i ψt + a1 ψxx + a2 ∣ψ ∣2 ψ = 0, then ⎧ [4 ⎪ i ⎨g4(t ) − 1−N ⎪ r 2 e ⎩

Φ(r , t ) = c0 g1′1/4(t ) ⎡ × ψ ⎢g2(t ) + ⎢⎣

a1 g1′(t ) b10

⎫ b10 g1′(t ) g2′(t ) + a1 g1″(t ) r ] r ⎪ ⎬ ⎪ 8 b10 a1 g1′(t ) ⎭

⎤ r , g1(t )⎥ ⎥⎦

6-35

(6.58)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

is a solution of i Φt + b10 Φrr +

a2 g1′(t ) b10 (N − 1) Φr + 2 1 − N ∣Φ∣2 Φ r c0 r

⎧ ⎡ ′ ⎤ ⎪ b10 (N − 1) (N − 3) g2′2(t ) ⎣g2 (t ) g1″(t ) − g1′(t ) g2″(t )⎦ r ′ ⎨ + + + g4 (t ) + (6.59) 4r 2 4a1 g1′(t ) 2 a1 b10 g1′3/2(t ) ⎪ ⎩ ⎡ ″2 ⎤ 2⎫ ⎣3g1 (t ) − 2g1′(t ) g1‴(t )⎦ r ⎪ ⎬ Φ = 0, + 16b10 g1′2(t ) ⎪ ⎭

where

T (r , t ) = g1(t ), R(r , t ) = g2(t ) +

a1 g1′(t )

A(r , t ) = g3(t ) r

B(r , t ) = g4(t ) −

(6.60)

b10 1−N 2 ,

⎡ ⎣4 b10 g1′(t ) g2′(t ) + 8b10

(6.62) ⎤ a1 g1″(t ) r⎦ r

a1 g1′(t )

a2 g1′(t ) A2 (r , t )

,

(6.63)

(6.64)

b1(r , t ) = b10 ,

b 2 (r , t ) =

(6.61)

r,

,

(6.65)

⎡ ′ ⎤ g2′2(t ) ⎣g2 (t ) g1″(t ) − g1′(t ) g2″(t )⎦ r b10 (N − 1) (N − 3) b 3 r (r , t ) = + + g4′(t ) + 4 r2 4 a1 g1′(t ) 2 a1 b10 g1′3/2(t ) (6.66) ⎡ ″2 ⎤ 2 ‴ ′ 3 g ( t ) − 2 g ( t ) g ( t ) r 1 1 ⎣ 1 ⎦ + , 2 16 b10 g1′ (t )

g3(t ) = c0 g1′1/4(t ), g1(t ), g2(t ), and g4(t ) are arbitrary real functions, c0 and b10 are arbitrary real constants.

6-36

(6.67)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Example 1. sech(r, t ) bright soliton Given

ψ (x , t ) = A0

2 2 a1 ⎡ ⎤ sech[A0 (x − x0)] e i ⎣a1 A0 (t − t0) + ϕ0⎦ a2

is a solution of (2.1), then

Φ(r , t ) = c0 A0

×

⎧ ⎡ ⎪ 1−N 2a1 1/4 g1′ (t ) r 2 sech ⎨A0 ⎢g2(t ) + ⎪ a2 ⎩ ⎢⎣

⎪ a1 g1′(t ) ⎤⎫ r⎥⎬ ⎪ b10 ⎥⎦⎭

⎫ ⎪ r− r 2 + ϕ0⎬ ⎪ 8 b10 g1′(t ) a1 b10 g1′(t ) ⎭

⎧ ⎪ i ⎨A02 a1 [g1(t ) − t0 ] + g4(t ) − ⎪ 2 e ⎩

g2′(t )

(6.68)

g1″(t )

is a solution of (6.59), where a1 a2 > 0, a1 b10 > 0, A0, t0, and ϕ0 are arbitrary real constants. Example 2. tanh(r, t ) dark soliton Given

ψ (x , t ) = A0

2 −2a1 ⎡ ⎤ tanh[A0 (x − x0)] e−i ⎣2 a1 A0 (t − t0) + ϕ0⎦ a2

is a solution of (2.1), then

Φ(r , t ) = c0 A0

×

⎧ ⎡ ⎪ 1−N −2a1 1/4 g1′ (t ) r 2 tanh ⎨A0 ⎢g2(t ) + ⎪ a2 ⎩ ⎢⎣

⎧ ⎪ −i ⎨2 A02 a1 [g1(t ) − t0 ] − g4(t ) + ⎪ 2 e ⎩

⎪ a1 g1′(t ) ⎤⎫ r⎥⎬ ⎪ b10 ⎥⎦⎭

⎫ ⎪ r+ r 2 + ϕ0⎬ ⎪ 8 b10 g1′(t ) a1 b10 g1′(t ) ⎭ g2′(t )

is a solution of (6.59), where a1 a2 < 0, a1 b10 > 0, A0, t0, and ϕ0 are arbitrary real constants.

6-37

g1″(t )

(6.69)

N−1 r

(

6-38

1

(

1. Φ(r, θ , t ) = r−1/2 e i

# Example: 2D

a1 θ

1 r

⎡ ⎢⎣A0

Equation (2): i Φt + a1 Φrr +

−2a1 a2

Φr −

1

2

)+a 2

e−i (2a1 A0

4r 2

t+ϕ0

⎦

) tanh(A0 r )⎤⎥

r ∣Φ∣2 Φ = 0

A0 and ϕ0 are arbitrary real constants

a1 a2 < 0 ,

Conditions

A0 and ϕ0 are arbitrary real constants

Eq. #

(6.53)

Eq. #

dark(vortex) soliton (6.54)

Name

dark soliton

1

a1 a2 < 0 ,

)

Φθθ + a2 r N −1 ∣Φ∣2 Φ = 0

1. Φ(r, θ , t ) = r (1−N )/2 e i 2 −(N −1)(N −3) θ ⎡ ⎤ −2a1 −i (2a1 A 2 t+ϕ ) 0 0 tanh(A r ) × ⎢A0 e 0 ⎥⎦ a2 ⎣

r2

Name

Φr +

t)

Conditions

N−1 r

−(N −1)(N −3) θ ψ (r ,

Angular Dependence

# Example: General Case

Equation (1): i Φt + a1 Φrr +

1 2

)

Φr + b2(r, t ) ∣Φ∣2 Φ + [b3r (r, t ) + i b3i (r, t )]Φ = 0

Transformation: Φ(r, θ , t ) = r (1−N )/2 e i

Equation: i Φt + b1(r, t ) Φrr +

(

Transformation: Φ(r, t ) = A(r, t ) e i B (r, t ) ψ [R(r, t ), T (r, t )]

(N + 1)-Dimensional NLSE with Cubic Nonlinearity in a Polar Coordinate System

6.8 Summary of Section 6.7

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

(

6-39

1.

b10 (N − 1) r

Φr +

a2

g1′(t ) 1 2 c 0 r −N

×

2 a1 a2

g1′1/4(t ) r

1−N 2

⎧ ⎪ i ⎨A02 a1 [g1(t )−t0 ]+g4(t )− ⎪ 2 e ⎩

Φ(r, t ) = c0 A0

a1 b10 g1′(t )

g2′(t )

r−

⎧ ⎪ sech ⎨A0 ⎪ ⎩ g1″(t )

a1 g1′(t ) b10

⎫ ⎪ r 2+ϕ0⎬ ⎪ ⎭

⎡ ⎢g2(t ) + ⎢⎣

a1 g1′(t )

⎬ ⎪ ⎪ ⎭

⎡ ψ ⎢g2(t ) + ⎣

⎤⎫ ⎪ r ⎥⎬ ⎪ ⎥⎦⎭

a1 g1′(t ) b10

⎤ r, g1(t )⎥ ⎦

A0, t0, and ϕ0 are arbitrary real constants

a1 a2 > 0 , a1 b10 > 0,

Conditions

⎡g ′ (t ) g ″(t ) − g ′(t ) g ″(t )⎤ r ⎣ 2 1 1 2 ⎦ 2 a1 b10 g1′ 3/2(t )

8 b10

⎤ ⎫ b10 g1′(t ) g2′(t ) + a1 g1″(t ) r ⎥ r ⎪ ⎥⎦

8 b10 g1′(t )

A0 and ϕ0 are arbitrary real constants Constant Dispersion and Real Potential

∣Φ∣2

⎡ ⎧ ⎢4 ⎪ ⎢⎣ i ⎨g4(t )− ⎪ ⎪ e ⎩

⎦

) tanh(A0 r )⎤⎥

⎧ b (N − 1) (N − 3) g ′ 2 (t ) Φ + ⎨ 10 + 2 + g4′(t ) + 2 4 a1 g1′(t ) 4r ⎩ ⎡3 g ″ 2(t ) − 2 g ′(t ) g (t )⎤ r 2 ⎫ ⎣ 1 1 1‴ ⎦ ⎬Φ=0 + 16 b10 g1′ 2(t ) ⎭

# Example

Equation:

i Φt + b10 Φrr +

1−N 2

t+ϕ0

Transformation: Φ(r, t ) = c0 g1′1/4(t ) r

2

e−i (2a1 A0

a1 a2 < 0 ,

−2a1 a2

1. Φ(r, t ) = r−1 ⎡A ⎢⎣ 0

)

Φr + a2 r 2 ∣Φ∣2 Φ = 0

Conditions

2 r

# Example: 3D

Equation (3): i Φt + a1 Φrr +

bright soliton

Name

dark soliton

Name

(6.68)

Eq.#

(6.56)

Eq.#

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

×

−2 a1 a2

g1′1/4(t ) r

1−N 2

⎡ ⎢c + ⎢ 0 ⎣

(

8 b10 g1′(t )

g1″(t )

⎫ ⎪ r 2+ϕ0⎬ ⎪ ⎭

)

a1 g1′(t ) b10

⎤⎫ ⎪ r ⎥⎬ ⎪ ⎥⎦⎭

2 b1

4 a1 b1 − 4 b1 b2 c12 + b32 c12 − b3 c1 (x1 − x01)

r+

⎤⎫ ⎪ ⎡ 2 ⎤ + c1 (x2 − x02 )⎥⎬ e i ⎣2 a1 A0 (t−t0)+ϕ0⎦ ⎥⎪ ⎦⎭

⎧ ⎪ × tanh ⎨A0 ⎪ ⎩

−2 a1 b4

a1 b10 g1′(t )

g2′(t )

⎧ ⎡ ⎪ tanh ⎨A0 ⎢g2(t ) + ⎪ ⎩ ⎢⎣

⎧ ⎪ −i ⎨2 A02 a1 [g1(t )−t0 ]−g4(t )+ ⎪ 2 e ⎩

Φ(r, t ) = c0 A0

2. Φ(x , x , t ) = A 1 2 0

2.

(Continued )

dark soliton

A0, t0, and ϕ0 are arbitrary real constants

a1 b4 < 0 , (4 a1 b1 − 4 b1 b2 c12 + b32 c12 ) > 0, dark soliton

A0, t0, and ϕ0 are arbitrary real constants

a1 a2 < 0 , a1 b10 > 0,

(6.40)

(6.69)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

6-40

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

6.9 Power Series Solutions to (2 + 1)-Dimensional NLSE with Cubic Nonlinearity in a Polar Coordinate System The function

Φ ( r , θ , t ) = Z ( r ) e i (λ t + α θ )

(6.70)

with ⎛ 25 [ − a2 (a 0) 3 − a1 b0 + a1 a 0 α 2 + a 0 λ ] ⎞ ⎛5 b ⎞ Z (r ) = a 0 + ⎜ 0 ⎟ r + ⎜ ⎟ r2 ⎝ 4 ⎠ 32 a1 ⎝ ⎠ ⎛ 15 [a2 (a 0) 2 (a 0 − 3 b0) − 3 a1 a 0 α 2 + a1 b0 (2 + α 2) + (b0 − a 0) λ ] ⎞ +⎜ ⎟ r3 64 a1 ⎝ ⎠ 5 a12 [a 0 α 2 (68 + 7 α 2) − 3 b0 (12 + 13 α 2)] + (6.71) 512 a12 + a1 − a2 a 0 ⎡⎣ − 33 a 0 b0 + 42 (b0) 2 + 2 (a 0) 2 (9 + 14 α 2)⎤⎦

(

{

+ ⎡⎣ − 11 b0 + 2 a 0 (9 + 7 α 2)⎤⎦ λ

}

+ 7 a 0⎡⎣ 3a 22 (a 0)4 − 4 a2 (a 0) 2 λ + λ2⎤⎦ r 4 + O 5(r )

)

is a stationary solution of

⎡ ⎤ 1 1 i Φt + a1 ⎢Φrr + Φr + 2 Φθθ ⎥ + a2 ∣Φ∣2 Φ = 0, ⎣ ⎦ r r

(6.72)

where λ , α , a0, and b0 are arbitrary real constants. The solution is obtained using an Iterative Power Series (IPS) method [1, 2], which is briefly described as follows: The function Z (r ) obeys the ordinary differential equation

⎡ ⎤ 1 α2 −λ Z (r ) + a1 ⎢Z ″(r ) + Z ′(r ) − 2 Z (r )⎥ + a2 Z 3(r ) = 0. ⎣ ⎦ r r

(6.73)

A convergent power series solution is obtained by the following algorithm: 1. Set initial values a0 and b0. 2. Expand Z (r ) in power series around the arbitrary real r0: Z (r ) = a 0 + b0 n (r − r0 ) + ∑ j =max2 cj (r − r0 ) j . 3. Substitute in (6.71) to obtain the recursion relation for cn in terms of a0 and b0. 4. Calculate Z(Δ) and Z′(Δ), where Δ = (r − r0 )/I , and I is an integer larger than 1. 5. Assign: a 0 = Z(Δ) and b0 = Z′(Δ). 6. Obtain cn in terms of a0 and b0. 7. Repeat steps 2–6 I times. 8. At the Ith step, a0 will correspond to the power series of Z.

6-41

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Figure 6.13. Stationary power series solutions of (6.73) with a different number of nodes. (a) nodeless solution with a 0 = 1.1986644033, (b) single-node solution with a 0 = 1.4605146251, root at r = 1.46 , (c) double-node solution with a 0 = 1.6326670699 , roots at r = 1.17, 4.46, and (d) triple-node solution with a 0 = 1.7642003085, roots at r = 1.02, 3.75, 7.64 . IPS parameters used are: n max = 2 , I = 1700, and Δ = 0.01.

Solution (6.71) is obtained with I = 4, n max = 2, and for arbitrary a1, a2, λ , α , a0, and b0. 6.9.1 Family of Infinite Number of Localized Solutions Using the IPS method, described above, we tune the parameter a0 to obtain a family of infinite number of solutions differing by the number of nodes. In Figure 6.13, we present some plots showing the nodeless, single-node, double-node, and triple-node solutions of (6.73). Other parameters are fixed to b0 = 0, a1 = 1, a2 = 2, λ = 1, and α = 0.

References [1] Al Khawaja U and Al-Mdallal Q M 2018 Convergent power series of and solutions to nonlinear differential equations Int. J. Differ. Equ. 2018 1–10 [2] Al Sakkaf L Y, Al-Mdallal Q M and Al Khawaja U 2018 A Numerical algorithm for solving higher-order nonlinear BVPs with an application on fluid flow over a shrinking permeable infinite long cylinder Complexity 2018 1–11

6-42

IOP Publishing

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations Usama Al Khawaja and Laila Al Sakkaf

Chapter 7 Coupled Nonlinear Schrödinger Equations

A Glance at Chapter 7

doi:10.1088/978-0-7503-2428-1ch7

7-1

ª IOP Publishing Ltd 2020

7-2

9

8

7

6

5

4

N

k=1

∑ b1 j ∣σj∣2 ∣ψ1∣2 ψ1 = 0, j = 1, 2, … , N

+

c2 ∣ψ2∣2 ) ψ2 = 0

− g2 ∣Φ2

∣2 ) Φ2 = 0

i Φ2t + Φ2 xx + 2 (a21 ∣Φ1∣2 + a22 ∣Φ2∣2 ) Φ2 + 2 (b21 Φ1 Φ*2 + b22 Φ2 Φ1*) Φ2 = 0

i Φ1t + Φ1xx + 2 (a11 ∣Φ1∣2 + a12 ∣Φ2∣2 ) Φ1 + 2 (b11 Φ1 Φ*2 + b12 Φ2 Φ1*) Φ1 = 0,

i Φ2t + Φ2 xx +

(g1 ∣Φ1∣2

i Φ1t + Φ1xx + (g1 ∣Φ1∣2 − g2 ∣Φ2∣2 ) Φ1 = 0,

i ψ2t + a2 ψ2 xx +

(c1 ∣ψ1∣2

i ψ1t + a1 ψ1xx + (b1 ∣ψ1∣2 + b2 ∣ψ2∣2 ) ψ1 = 0,

i ψ3t + a3 ψ3xx + (d1 ∣ψ1∣2 + d3 ∣ψ3∣2 ) ψ3 = 0

i ψ1t + a1 ψ1xx + (b1 ∣ψ1∣2 + b3 ∣ψ3∣2 ) ψ1 = 0,

i Φ2t + Φ2 xx + (g1 ∣Φ1∣2 − g2 ∣Φ2∣2 ) Φ2 − g0 (g1 + g2 ) Φ2 + 2 g0 g1 Φ1 = 0

i Φ1t + Φ1xx + (g1 ∣Φ1∣2 − g2 ∣Φ2∣2 ) Φ1 + g0 (g1 + g2 ) Φ1 − 2 g0 g2 Φ2 = 0,

i ψ1t + b0 j ψ1xx +

0

1

0

2

0

0

0

i ψ1t + b0 ψ1xx − (c1 + c2 ) ∣ψ1∣2 ψ1 = 0

3

ψ2 = 0

13

0

+

c2 ∣ψ2∣2 )

i ψ1t + b0 ψ1xx + (c1 + c2∣σ∣2 ) ∣ψ1∣2 ψ1 = 0

i ψ2t + c0 ψ2 xx +

(c1 ∣ψ1∣2

i ψ1t + b0 ψ1xx + (b1 ∣ψ1∣2 + b2 ∣ψ2∣2 ) ψ1 = 0,

Solutions

2

1

Equation

A Statistical View of Chapter 7

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

7-3

Total

18

17

16

15

14

13

12

11

10

N

k k

k k

N

k=1

k=1

N

∑ b1 j ∣σj + 1∣2 ∣ψ1∣2 ψ1 = 0

+

N

k k

− 2 (c1 + c2 ) ∣ψ1∣2 ψ1 = 0

+ (c1 + c2∣σ∣2 ) ∣ψ1∣2 ψ1 = 0

∑ b10k ψ1x x

k=1

∑ c0k ψ1x x

N

k=1

∑ c0k ψ1x x

18

N i Φ2t + ∑k = 1 Φ2 xkxk − 2(a + b ) (∣Φ1∣2 + ∣Φ2∣2 ) Φ2 + 2 ((a + i b ) Φ1 Φ*2 + (a − i b ) Φ2 Φ1*) Φ2 = 0

i Φ1t + ∑k = 1 Φ1xkxk − 2(a + b ) (∣Φ1∣2 + ∣Φ2∣2 ) Φ1 + 2 ((a + i b ) Φ1 Φ*2 + (a − i b ) Φ2 Φ1*) Φ1 = 0,

N

i Φ2t + ∑k = 1 Φ2 xkxk + 2 (a21 ∣Φ1∣2 + a22 ∣Φ2∣2 ) Φ2 + 2 (b21 Φ1 Φ*2 + b22 Φ2 Φ1*) Φ2 = 0

N

i Φ1t + ∑k = 1 Φ1xkxk + 2 (a11 ∣Φ1∣2 + a12 ∣Φ2∣2 ) Φ1 + 2 (b11 Φ1 Φ*2 + b12 Φ2 Φ1*) Φ1 = 0,

N

i Φ2t + ∑k = 1 Φ2 xkxk + (g1 ∣Φ1∣2 − g2 ∣Φ2∣2 ) Φ2 = 0

N

i Φ1t + ∑k = 1 Φ1xkxk + (g1 ∣Φ1∣2 − g2 ∣Φ2∣2 ) Φ1 = 0,

N

i Φ2t + ∑k = 1 Φ2 xkxk + (g1 ∣Φ1∣2 − g2 ∣Φ2∣2 ) Φ2 − g0 (g1 + g2 ) Φ2 + 2 g0 g1 Φ1 = 0

N

i Φ1t + ∑k = 1 Φ1xkxk + (g1 ∣Φ1∣2 − g2 ∣Φ2∣2 ) Φ1 + g0 (g1 + g2 ) Φ1 − 2 g0 g2 Φ2 = 0,

i ψ1t +

i ψ1t −

i ψ1t +

i ψ2t + ∑k = 1 c0k ψ2 xkxk + (c1 ∣ψ1∣2 + c2 ∣ψ2∣2 ) ψ2 = 0

N

i ψ1t + ∑k = 1 b0k ψ1xkxk + (b1 ∣ψ1∣2 + b2 ∣ψ2∣2 ) ψ1 = 0,

N

i Φ2t + Φ2 xx − 2(a + b ) (∣Φ1∣2 + ∣Φ2∣2 ) Φ2 + 2 ((a + i b ) Φ1 Φ*2 + (a − i b ) Φ2 Φ1*) Φ2 = 0

i Φ1t + Φ1xx − 2(a + b ) (∣Φ1∣2 + ∣Φ2∣2 ) Φ1 + 2 ((a + i b ) Φ1 Φ*2 + (a − i b ) Φ2 Φ1*) Φ1 = 0,

17

0

0

0

0

0

0

0

1

0

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

7.1 Fundamental Coupled NLSE Manakov System Equation:

i ψ1t + b0 ψ1xx + (b1 ∣ψ1∣2 + b2 ∣ψ2∣2 ) ψ1 = 0, i ψ2t + c0 ψ2xx + (c1 ∣ψ1∣2 + c2 ∣ψ2∣2 ) ψ2 = 0,

(7.1)

where ψj = ψj (x , t ) is the complex function profile, x and t are its two independent variables, b0, c0, b1, c1, b2, and c2 are real constants. Solutions: Solution 1. Constant Amplitude I continuous wave (CW), t- and x-independent phase

ψ1(x , t ) = A0 e iϕ1, ψ2(x , t ) = B0 e iϕ2,

(7.2)

where b1 = − c2 = −

b2 B02 A02 c1 A02 B02

, ,

A0, B0, ϕ1, and ϕ2 are arbitrary real constants. Solution 2. Constant Amplitude II CW, t-dependent phase

ψ1(x , t ) = A0 e i [A1 (t − t0) + ϕ1], ψ2(x , t ) = B0 e i [B1 (t − t0) + ϕ2],

(7.3)

where A1 = A02 b1 + B02 b2 , B1 = A02 c1 + B02 c2 , A0, B0, t0, ϕ1, and ϕ2 are arbitrary real constants. Solution 3. Constant Amplitude III CW, x-dependent phase

ψ1(x , t ) = A0 e i [A1 (x − x0) + ϕ1], ψ2(x , t ) = B0 e i [B1 (x − x0) + ϕ2], where A1 = ±

A02 b1 + B02 b2 b0

,

B1 = ±

A02 c1 + B02 c2 c0

,

7-4

(7.4)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

b0 (A02 b1 + B02 b2 ) > 0, c0 (A02 c1 + B02 c2 ) > 0, A0, B0, x0, ϕ1, and ϕ2 are arbitrary real constants.

Solution 4. Constant Amplitude IV CW, t- and x-dependent phase

ψ1(x , t ) = A0 e i [A1 (t − t0) + A2 (x − x0) + ϕ1], ψ2(x , t ) = B0 e i [B1 (t − t0) + B2 (x − x0) + ϕ2],

(7.5)

where A1 = −A22 b0 + A02 b1 + B02 b2 , B1 = −B22 c0 + A02 c1 + B02 c2 , A0, B0, A2, B2, x0, t0, ϕ1, and ϕ2 are arbitrary real constants. Solution 5. Rational Solution decaying wave (DW)

ψ1(x , t ) = ψ2(x , t ) =

A0 e i ϕ1(x, t ), 2 [A1 + t − t0 ] B0 e i ϕ2(x, t ), 2 [B1 + t − t0 ]

(7.6)

where ϕ1(x , t ) =

(b0 A2 + x − x0 )2 4 b0 (A1 + t − t0 )

+ ϕ2(x , t ) =

b2 B02

+

b1 A02 2

ln[2(A1 + t − t0 )]

,

ln[2 (B1 + t − t0 )] + ϕ01

2 (c 0 B 2 + x − x 0 ) 2 4 c0 (B1 + t − t0 )

+

c1 A02 2

ln[2(A1 + t − t0 )]

, c B2 + 2 2 0 ln[2 (B1 + t − t0 )] + ϕ02 A0, A1, A2, B0, B1, B2, x0, t0, ϕ01, and ϕ02 are arbitrary real constants. Solution 6. tanh(x , t )-sech(x , t ) dark-bright soliton (Figure 7.1)

ψ1(x , t ) = (A0 tanh{A1 [x − x0 − v (t − t0 )]} + i A2 ) e i [b1 (t − t0) + ϕ1], 2

ψ2(x , t ) = B0 sech{A1 [x − x0 − v (t − t0)]} e i [(A1 + b2) (t − t0) + θ(x, t ) + ϕ2], where

⎡ A0 = cos ⎢tan−1 ⎣ A1 =

b1 (1 − b22 ) 2 (1 + b1 b2 )

( )⎤⎥⎦, −v 2 A1

−

v2 4

,

7-5

(7.7)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Figure 7.1. Dark-bright soliton (7.7) at t = 0. Blue is ψ1 and red is ψ2 with b2 = 1/2 , v = 1/2 , and x0 = t0 = ϕ1 = ϕ2 = 0 . Animation available online at https://iopscience.iop.org/book/978-0-7503-2428-1.

⎡ A2 = sin ⎢tan−1 ⎣

( )⎤⎦⎥, −v 2 A1

2 A12 (b1 + b2 )

B0 =

θ (x , t ) = b0 = −1, c0 = 1, c1 = 1/b1, c2 = b2 , b1 b2 = 1,

1 − b22 v [x − 2

,

x0 −

v 2

(t − t0 )],

b1 (1 − b22 ) v2 > 4, 2 (1 + b1 b2 ) 2 A12 (b1 + b2 ) (1

− b22 ) > 0, x0, t0, ϕ1, ϕ2, and v are arbitrary real constants.

• Reference: [2]. Solution 7. (Figure 7.2) ψ1(x , t ) = A0 1 − 4 ⎡⎣α12(x , t ) + α 22(x , t ) − α1(x , t ) + i α2(x , t )⎤⎦ e −δ(x, t )

( × { ⎡⎣2 α (x , t ) + 2 α (x , t ) − 2 α (x , t ) + 1⎤⎦ e , + (β + β ) e } }) × e 2 1

2 3

2 2

2 4

2 δ(x, t ) −1

1

−δ(x, t )

i [A1 x + (2 A02 − A12 ) t ]

(7.8)

ψ2(x , t ) = − 4 A0 {β3 [α1(x , t ) − 1] − β4 α2(x , t ) + i [β3 α2(x , t ) + β4 α1(x , t ) − β4]} ×

( {2 ⎡⎣α (x, t ) + α (x, t )⎤⎦ − 2 α (x, t ) + 1} e 2 1

2 2

2

2

× e i [A1 x+ (3 A0 − A1 )] +

1

δ(x, t ) 2 ,

7-6

−δ(x, t )

(

)

+ β32 + β 42 e 2 δ(x, t )

−1

)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Figure 7.2. Plot of solution (7.8) at t = 0. Blue is ψ1 and red is ψ2 with A0 = A1 = β1 = β2 = β3 = β4 = 1. Animation available online at https://iopscience.iop.org/book/978-0-7503-2428-1.

where 2 A (x − 2 A t ) δ (x , t ) = 0 3 1 , α1(x , t ) = β1 A0 + A0 (x − 2 A1 t ), α2(x , t ) = β2 A0 − 2 A02 t , b0 = c0 = 1, b1 = b2 = c1 = c2 = 2, A0, A1, β1, β2 , β3, and β4 are arbitrary real constants. • Reference: [3]. Solution 8. (Figure 7.3) ⎛ −6 ψ1(x, t ) = A0 ⎜ ⎜ ⎝

3 A0 δ (x, t ) − 36

12 A02 δ 2(x, t ) + 8

3 ⎤⎦ − 3

3 A0 δ (x, t ) + 144 A04 t 2 + 5

⎞ 2 2 3 )⎟⎟ × e i [A1 x + (16 A0 − A1 ) t ], ⎠

− (1 + i ⎛ −6 ψ2(x, t ) = A0 ⎜ ⎜ ⎝

3 A02 t + i ⎡⎣ 36 A02 t + 6 A0 δ (x, t ) + 5

3 A0 δ (x, t ) + 36

− (1 − i

3 A02 t + i ⎡⎣ 36 A02 t − 6 A0 δ (x, t ) − 5

12 A02 δ 2(x, t ) + 8

3 A0 δ (x, t ) + 144 A04 t 2 + 5

⎞ 2 2 3 )⎟⎟ × e i [A2 x + (16 A0 − A2 ) t ], ⎠

where δ (x , t ) = x + 6 A3 t , b0 = c0 = 1, b1 = b2 = c1 = c2 = 2, A0 = A2 + 3 A3, A1 = A2 − 2 A0, A2 and A3 are arbitrary real constants. • Reference: [3].

7-7

3 ⎤⎦ − 3

(7.9)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Figure 7.3. Plot of solution (7.9). Blue is ψ1 and red is ψ2 with A2 = A3 = 1. (a) at t = −1, (b) at t = 0, and (c) at t = 1.

Figure 7.4. Plot of solution (7.10). Blue is ψ1 and red is ψ2 with A2 = A3 = 1. (a) at t = −1, (b) at t = 0, and (c) at t = 1.

Solution 9. (Figure 7.4)

⎡ α1(x , t ) + i β1(x , t ) ⎤ ⎡ 2 2 ⎤ − (1 + i 3 )⎥ e i ⎣A1 x + ( 16 A0 − A1 ) t⎦, ψ1(x , t ) = A0 ⎢ γ (x , t ) ⎣ ⎦ ⎡ α2(x , t ) + i β2(x , t ) ⎤ ⎡ 2 2 ⎤ ψ2(x , t ) = A0 ⎢ − (1 − i 3 )⎥ e i ⎣A2 x + ( 16 A0 − A2 ) t⎦, γ (x , t ) ⎣ ⎦ where α1(x , t ) = − 864

3 A06 t 3 − 144

−216 A04 t 2 − 12

3 t 3 − 144

−216 A04 t 2 − 12

3 A04 δ 2(x, t ) t

3 A03 δ3(x , t ) − 144 A03 δ (x , t ) t

− 18 A02 δ 2(x , t ) − 12 α2(x , t ) = 864 A06

3 A05 δ (x , t ) t 2 − 72 3 A02 t + 3, 3 A05 δ (x , t ) t 2 + 72

3 A04 δ 2(x, t ) t

3 A03 δ3(x , t ) + 144 A03 δ (x , t ) t

− 18 A02 δ 2(x , t ) + 12

3 A02 t + 3,

β1(x , t ) = 864 A06 t 3 + 144 A05 δ (x , t ) t 2 + 72 A04 δ 2(x, t ) t + 312 +12 A03 δ3(x , t ) + 96

3 A03 δ (x , t ) t + 18

+ 108 A02 t + 12 A0 δ (x , t ) + β2(x , t ) =

(7.10)

864 A06 t 3 − 144 A05 δ (x , t ) t 2 −12 A03 δ3(x , t ) + 96 3 A03 + 108 A02 t − 12 A0 δ (x , t ) − 7-8

3 A04 t 2

3 A02 δ 2(x, t )

3, + 72 A04 δ 2(x, t ) t − 312 δ (x , t ) t − 18 3,

3 A04 t 2

3 A02 δ 2(x, t )

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

γ (x , t ) = 1728 A08 t 4 + 288 A06 δ 2(x , t ) t 2 + 384 + 12 A04 δ 4(x , t ) + 432 A04 t 2 + 16

3 A05 δ (x , t ) t 2

3 A03 δ3(x , t ) + 24 A02 δ 2(x , t )

+ 4 3 A0 δ (x , t ) + 1, A0 = A2 + 3 A3, A1 = A2 − 2 A0, A2 = −6A3, δ (x , t ) = x + 6 A3 t , b0 = c0 = 1, b1 = b2 = c1 = c2 = 2, A3 is an arbitrary real constant.

• Reference: [3]. Solution 10. sech2(x ) (Figure 7.5)

ψ1(x , t ) = {A0 sech2[A1 (x − x0)] + A3} e−i [ω1 (t − t0) + ϕ1], ψ2(x , t ) = B0 sech2[A1 (x − x0)] e−i [ω2 (t − t0) + ϕ2], where A3 =

(7.11)

−2 A0 , 3 2 2 A1 ,

ω1 = ω2 = −2 A12 ,

b1 = b2 =

−9 A12 2 A02 9 A12 2 B02

,

,

b0 = c0 = 1, b1 = c1, b2 = c2 , A0 ≠ 0, B0 ≠ 0, A1, x0, t0, ϕ1, and ϕ2 are arbitrary real constants.

• Reference: [4], taken from the nonlocal case.

Figure 7.5. Plot of solution (7.11) at t = 0. Blue is ψ1 and red is ψ2 with A0 = 3/2 , B0 = A1 = 1, b1 = b2 = 2 , and x0 = t0 = ϕ1 = ϕ2 = 0 .

7-9

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Figure 7.6. Solitary wave (7.12) at t = 0. Blue is ψ1 and red is ψ2 with A0 = 3, B0 = 2 , A1 = B1 = 1, b2 = c2 = 2 , m = 1/2 , and x0 = t0 = ϕ1 = ϕ2 = 0 .

Solution 11. cd(x , m )-nd(x , m ) solitary wave (SW) (Figure 7.6)

ψ1(x , t ) = A0

m cd[A1 (x − x0), m ] e−i [ω1 (t − t0) + ϕ1],

ψ2(x , t ) = B0

1 − m nd[A1 (x − x0), m ] e−i [ω2 (t − t0) + ϕ2],

(7.12)

where ω1 = −(1 − m ) A12 − b1 A02 , ω2 = −(2 − m ) A12 − c1 A02 , b0 = c0 = 1, b1 = c1 =

−2 A12 + b2 B02 −2

A02 + c2 B02

A12

A02

,

,

0 < m ⩽ 1, A0 ≠ 0, B0, x0, t0, ϕ1, and ϕ2 are arbitrary real constants.

• Reference: [4], taken from the nonlocal case. Solution 12. Weierstrass Elliptic Function I: ℘(z, g2 , g3) SW

ψ1(x , t ) = Φ1(z ) e i ϕ1(z, t ), ψ2(x , t ) = Φ2(z ) e i ϕ2(z, t ),

(7.13)

where

Φ1(z ) = F ℘ 2(z , g2, g3) + A0 ℘(z , g2, g3) + A1 , Φ2(z ) = G ℘ 2(z , g2, g3) + B0 ℘(z , g2, g3) + B1 ,

(7.14)

which satisfy

⎡ ⎤ − A22 b0 + ⎣ b1 Φ12(z ) + b2 Φ22(z ) − γ ⎦ Φ14(z ) + b0 Φ13(z ) Φ″1 (z ) = 0, ⎡ ⎤ − B22 c0 + ⎣b2 Φ12(z ) + b1−1 Φ22(z ) − β ⎦ Φ24(z ) + c0 Φ32(z ) Φ″2(z ) = 0,

7-10

(7.15)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

with ϕ1(z, t ) =

b0 α 2

ϕ2(z, t ) =

c0 α 2

⎡ z − b0 A2 ⎢⎣∫ ⎡ z − c0 B2 ⎢⎣∫

dz ⎤ ⎦ Φ12(z ) ⎥

+ ϕ01(t ),

dz ⎤ ⎦ Φ 22(z ) ⎥

+ ϕ02(t ),

c1 = b2 , c2 = b1−1, z = x − c t,

( )t + ϕ (t ) = (β + )t + ϕ

ϕ01(t ) = γ +

b0 α 2 4

ϕ02

c0 α 2 4

011, 022 ,

γ = b1 A1 + b2 B1 − 3 β = b2 A1 + b1−1 B1 − A1 =

4 A02 − F 2 g2 , 4F

A2 = − B0 =

A b0 F0 , B 3 c0 G0 ,

3 F2 4

(

A02 F2

−

g2 3

) (4

A03 F3

−

g2 A0 F

− g3 ,

)

−

g2 3

) (4

B03 G3

−

g2 B0 G

− g3 ,

−6 b0 − b1 A0 , b2 3 G2 4

(

(B02 − B1 G2

G)

B2 = − g2 =

4

G=

−b1 F , b2

B02 G2

)

,

F ≠ 0, b b b1 = 0c 2 , 0

b2 = 1, b0, c0, A0, B1, g3, α, c, ϕ011, and ϕ022 are arbitrary real constants.

• Reference: [4]. Solution 13. Weierstrass Elliptic Function II: ℘(z, g2 , g3) SW

ψ1(x , t ) = Φ1(z ) e i ϕ1(z, t ), ψ2(x , t ) = Φ2(z ) e i ϕ2(z, t ),

(7.16)

where

Φ1(z ) = A0 ℘(z , g2, g3) + A1 , Φ2(z ) = B0 ℘(z , g2, g3) + B1 , which satisfy (7.15), with ⎡ b α ϕ1(z, t ) = 02 z − b0 A2 ⎢⎣∫

dz ⎤ ⎦ Φ12(z ) ⎥

+ ϕ01(t ),

7-11

(7.17)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

ϕ2(z, t ) =

c0 α 2

⎡ z − c0 B2 ⎢⎣∫

dz ⎤ ⎦ Φ 22(z ) ⎥

+ ϕ02(t ),

c1 = b2 , c2 = b1−1, z = x − c t,

( )t + ϕ (t ) = (β + )t + ϕ

ϕ01(t ) = γ +

b0 α 2 4

011,

c0 α 2 022 , 4 b2 (B1 A0 − A1 B0 ) − 3 b0 A1 , γ= A0 b2 (A1 B0 − B1 A0 ) − 3 c0 B1 , β= B0 2 (b0 c0 b1 − b0 ) , A0 = b1 (1 − b22 )

ϕ02

A2 = − B0 =

A02 4

(

−A1 g2 A0

+ g3 +

4 A13 A03

),

2 (b0 b2 − c0 b1) , 1 − b22

B2 = −

B02 4

(

−B1 g2 B0

+ g3 +

4 B13 B03

),

b0, b1, b2, c0, A1, B1, g2, g3, α, c, ϕ011, and ϕ022 are arbitrary real constants. • Reference: [4].

7-12

A02

b2 B02

, c2 = − B02

c1 A02

,

7-13

4. ψ1(x , t ) = A0 ei [A1 (t − t0) + A2 (x − x0) + ϕ1], ψ2(x , t ) = B0 ei [B1 (t − t0) + B2 (x − x0) + ϕ2]

A1 = − A22 b0 + A02 b1 + B02 b2 , B1 = − B22 c0 + A02 c1 + B02 c2 , A0, B0, A2, B2, x0, t0, ϕ1, and ϕ2 are arbitrary real constants

b0 (A02 b1 + B02 b2 ) > 0, c0 (A02 c1 + B02 c2 ) > 0 , A0, B0, x0, ϕ1, and ϕ2 are arbitrary real constants

, B1 = ±

,

A02 c1 + B02 c2 c0

A02 b1 + B02 b2 b0

3. ψ1(x , t ) = A0 ei [A1 (x − x0) + ϕ1], ψ2(x , t ) = B0 ei [B1 (x − x0) + ϕ2]

A1 = ±

A1 = A02 b1 + B02 b2 , B1 = A02 c1 + B02 c2 , A0, B0, t0, ϕ1, and ϕ2 are arbitrary real constants

A0, B0, ϕ1, and ϕ2 are arbitrary real constants

b1 = −

Conditions

2. ψ1(x , t ) = A0 ei [A1 (t − t0) + ϕ1], ψ2(x , t ) = B0 ei [B1 (t − t0) + ϕ2]

1. ψ1(x , t ) = A0 ei ϕ1, ψ2(x , t ) = B0 ei ϕ2

# Solution

(7.2)

Eq. #

(7.5) continuous wave, t- and x-dependent phase

continuous wave, (7.4) x-dependent phase

continuous wave, (7.3) t-dependent phase

continuous wave, t- and x-independent phase

Name

Equation (1): i ψ1t + b0 ψ1xx + (b1 ∣ψ1∣2 + b2 ∣ψ2∣2 ) ψ1 = 0 , i ψ2t + c0 ψ2 xx + (c1 ∣ψ1∣2 + c2 ∣ψ2∣2 ) ψ2 = 0

Note: For lengthy conditions, the reader is referred to the solutions in section 7.1.

7.2 Summary of Section 7.1

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

e i ϕ 2 (x , t )

B0 2 [B1 + t − t0 ]

ψ2(x , t ) =

7-14

+ ϕ02 ,

+ t − t0 )] +

+

2

ln[2(A1

ln[2 (B1 + t − t0 )]

c1 A02 2

c2 B02

+

ln[2(A1

ln[2 (B1 + t − t0 )]

b1 A02 2

b2 B02 2

[c0 B2 + (x − x0 )]2 4 c0 [B1 + t − t0 ]

+ ϕ01,

+ t − t0 )] +

[b0 A2 + (x − x0 )]2 4 b0 [A1 + t − t0 ]

− t0 )],

1 − b 22

2 A12 (b1 + b2 )

b1 (1 − b 22 ) 2 (1 + b1 b2 )

,

−

v2 4

−1 i [A x + (3 A 2 − A 2 )] + δ(x, t ) 0 1 2 e 1

α22(x ,

× e−δ(x, t ) + (β32 + β42 ) e 2 δ(x, t ))

α12(x ,

,

2 A (x − 2 A t )

0, c0 = 1, c1 = 1/b1, c2 = b2 , b1 b2 = 1, x0, t0, ϕ1, ϕ2 , and v are arbitrary real constants

θ (x , t ) =

v v [x − x 0 − 2 ( t 2 2 b (1 − b ) v2 b0 = − 1, 2 1(1 + b b2 ) > 4 , 1 2 2 A12 (b1 + b2 ) (1 − b22 ) >

1

−v

A2 = sin[tan−1( 2 A )], B0 =

1

−v

A0 = cos[tan−1( 2 A )], A1 =

A0, A1, A2, B0, B1, B2, x0, t0, ϕ01 and ϕ02 are arbitrary real constants

ϕ2(x , t ) =

ϕ1(x , t ) =

δ (x , t ) = 0 3 1 , α1(x , t ) = β1 A0 + A0 (x − 2 A1 t ), × e−δ(x, t ) {[2 t) + 2 t ) − 2 α1(x , t ) + 1] e−δ(x, t ) 2 2 α2(x , t ) = β2 A0 − 2 A02 t , b0 = c0 = 1, 2 2 2 δ ( x , t ) − 1 i [ A x + (2 A − A ) t ] 0 1 + ( β3 + β 4 ) e } )e 1 , b1 = b2 = c1 = c2 = 2 , A0, A1, β1, β2 , β3, and β4 are arbitrary real ψ2(x , t ) = − 4 A0 {β3 [α1(x , t ) − 1] − β4 α2(x , t ) + i [β3 α2(x , t ) + β4 α1(x , t ) − β4 ]} ({2 [α12(x , t ) + α22(x , t )] − 2 α1(x , t ) + 1} constants

7. ψ1(x , t ) = (1 − 4 [α12(x , t ) + α22(x , t ) − α1(x , t ) + i α2(x , t )]

×

2 ei [(A1 + b2) (t − t0) + θ (x, t ) + ϕ2]

ψ2(x , t ) = B0 sech{A1 [(x − x0 ) − v (t − t0 )]}

6. ψ1(x , t ) = (A0 tanh{A1 [(x − x0 ) − v (t − t0 )]} + i A2 ) ei [b1 (t − t0) + ϕ1],

ei ϕ1(x, t ),

A0 2 [A1 + t − t0 ]

5. ψ1(x , t ) =

(Continued )

–

dark-bright soliton

decaying wave

(7.8)

(7.7)

(7.6)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

(

−6

(

−6

2

3 A02 2 δ (x , t ) + 8 2

3 A0 δ (x, t ) + 144 A04 t 2 + 5

t + i [36 A02 t + 6 A0 δ (x, t ) + 5

2

7-15

11. ψ1(x , t ) = A0 ψ2(x , t ) = B0

m cd[A1 (x − x0 ), m ] e−i [ω1 (t − t0) + ϕ1], 1 − m nd[A1 (x − x0 ), m ] e−i [ω2 (t − t0) + ϕ2]

ψ2(x , t ) = B0 sech2[A1 (x − x0 )] e−i [ω2 (t − t0) + ϕ2]

10. ψ1(x , t ) = {A0 sech2[A1 (x − x0 )] + A3} e−i [ω1 (t − t0) + ϕ1],

9. ψ (x , t ) = A0 ⎡ α1(x, t ) + i β1(x, t ) − (1 + i 3 )⎤ ei [A1 x + (16 A02 − A12 ) t ], 1 ⎣ ⎦ γ (x , t ) 2 2 ⎡ α2(x, t ) + i β2(x, t ) ⎤ ψ2(x , t ) = A0 ⎣ − (1 − i 3 )⎦ ei [A2 x + (16 A0 − A2 ) t ] γ (x , t )

3 )) ei [A2 x + (16 A0 − A2 ) t ]

2

3 A0 δ (x, t ) + 144 A04 t 2 + 5

3]−3

3]−3

3 A02 t + i [36 A02 t − 6 A0 δ (x, t ) − 5

12 A02 δ 2(x, t ) + 8

3 A0 δ (x, t ) + 36

3 )) ei [A1 x + (16 A0 − A1 ) t ],

12

A02

3 A0 δ (x, t ) − 36

− (1 − i

ψ2(x , t ) = A0

− (1 + i

8. ψ (x , t ) = A 0 1

2 A02

2 B02

9 A12

,

ω1 = 2 A12 , ω2 = − 2 A12 ,

, b2 =

−2 A0 , 3 −9 A12

A02

, 0 < m ⩽ 1,

A02

−2 A12 + b2 B02

,

A0, B0, x0, t0, ϕ1, and ϕ2 are arbitrary real constants

c1 =

−2 A12 + c2 B02

b0 = c0 = 1, b1 =

ω1 = − (1 − m ) A12 − b1 A02 , ω2 = − (2 − m ) A12 − c1 A02 ,

b0 = c0 = 1, b1 = c1, b2 = c2 , A0 ≠ 0, B0 ≠ 0, A1, B1, x0, t0, ϕ1, and ϕ2 are arbitrary real constants

b1 =

A3 =

See text.

δ (x , t ) = x + 6 A3 t , b0 = c0 = 1, b1 = b2 = c1 = c2 = 2 , A0 = A2 + 3 A3, A1 = A2 − 2 A0 , A2 and A3 are arbitrary real constants

(7.12)

(7.11)

–

solitary wave

(7.10)

(7.9)

–

–

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

7-16

A0 ℘(z , g2, g3) + A1 ,

B0 ℘(z , g2, g3) + B1

Φ2 (z ) =

G ℘2(z , g2, g3) + B0 ℘(z , g2, g3) + B1

Φ2 (z ) =

13. Φ1(z ) =

F ℘2(z , g2, g3) + A0 ℘(z , g2, g3) + A1 ,

12. Φ1(z ) = c0 α 2 4

A02 F2

4

−b1 F , b2

,

−

g2 3

g2 3

F ≠ 0 , b1 =

G)

(B02 − B1 G2

B02 G2

(

−

−

−

g2 B0 G

)t+ϕ

,

,

4

B02

(

−B1 g2 B0

1 − b 22

+ g3 +

+ g3 +

B03

4 B13

A03

4 A13

,

t + ϕ011,

),

),

b0, b1, b2, c0, c, A1, B1, g2, g3, α, ϕ011, and ϕ022 are arbitrary real constants

B2 = −

B0 =

A02

b1 (1 − b 22 )

−A g ( A1 2 4 0 2 (b0 b2 − c0 b1)

A2 = −

A0 =

β=

γ=

c0

α2

b0 α 2 ) 4

022 4 b2 (B1 A0 − A1 B0 ) − 3 b0 A1 , A0 b2 (A1 B0 − B1 A0 ) − 3 c0 B1 , B0 2 (b0 c0 b1 − b0 )

ϕ02(t ) = (β +

ϕ01(t ) = (γ +

)

− g3 ,

)

− g3 ,

4 A02 − F 2 g2 , 4F

g2 A0 F

b2 = 1,

B03 G3

b0 b2 , c0

) (4

) (4

A1 =

b0, c0, c, A0, B1, g3, α, ϕ011, and ϕ022 are arbitrary real constants

G=

g2 =

(

3 G2 4

B2 = −

B0 =

3 F2 4

−6 b0 − b1 A0 , b2

A2 = −

β = b2 A1 + b1−1 B1 − 3 A03 F3

A0 , F B0 c0 G ,

022 ,

t + ϕ011,

)t +ϕ

b0 α 2 ) 4

γ = b1 A1 + b2 B1 − 3 b0

ϕ02(t ) = β +

(

ϕ01(t ) = (γ +

Weierstrass elliptic function II

Weierstrass elliptic function I

Equation (2): − A22 b0 + [b1 Φ12(z ) + b2 Φ22(z ) − γ ] Φ14(z ) + b0 Φ13(z ) Φ1″(z ) = 0 , − B22 c0 + [b2 Φ12(z ) + b1−1 Φ22(z ) − β ] Φ24(z ) + c0 Φ32(z ) Φ″2(z ) = 0

(Continued )

(7.17)

(7.14)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

7.3 Symmetry Reductions 7.3.1 Symmetry Reduction I From Manakov System to Fundamental NLSE The CNLSE, (7.1),

i ψ1t + b0 ψ1xx + (b1 ∣ψ1∣2 + b2 ∣ψ2∣2 ) ψ1 = 0, i ψ2t + b0 ψ2 xx + (c1 ∣ψ1∣2 + c2 ∣ψ2∣2 ) ψ2 = 0, transforms to the scalar NLSE

i ψ1t + b0 ψ1xx + (c1 + c2∣σ∣2 ) ∣ψ1∣2 ψ1 = 0,

(7.18)

with the replacements ψ2(x , t ) = σ ψ1(x , t ), b1 = c1 + (c2 − b2 ) ∣σ∣2 , where σ is an arbitrary complex constant. Conclusion: If ψ1(x , t ) is a solution of the fundamental NLSE

i ψ1t + a1 ψ1xx + a2 ∣ψ1∣2 ψ1 = 0, then

(ψ1, ψ2 ) = (ψ1, σ ψ1)

(7.19)

is a solution of the CNLSE

i ψ1t + b0 ψ1xx + (b1 ∣ψ1∣2 + b2 ∣ψ2∣2 ) ψ1 = 0, i ψ2t + b0 ψ2 xx + (c1 ∣ψ1∣2 + c2 ∣ψ2∣2 ) ψ2 = 0, with a1 = b0, a2 = c1 + c2∣σ∣2 , b1 = c1 + (c2 − b2 ) ∣σ∣2 . 7.3.2 Symmetry Reduction II From Manakov System to Fundamental NLSE The CNLSE, (7.1),

i ψ1t + b0 ψ1xx + (b1 ∣ψ1∣2 + b2 ∣ψ2∣2 ) ψ1 = 0, i ψ2t − b0 ψ2 xx + (c1 ∣ψ1∣2 + c2 ∣ψ2∣2 ) ψ2 = 0, transforms to the scalar NLSE

i ψ1t + b0 ψ1xx − (c1 + c2 ) ∣ψ1∣2 ψ1 = 0, with the replacements 7-17

(7.20)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

ψ2(x , t ) = ei ϕ ψ1*(x , t ), b1 = −(c1 + c2 + b2 ),

where ϕ is an arbitrary real constant. Conclusion: If ψ1(x , t ) is a solution of the fundamental NLSE

i ψ1t + b0 ψ1xx + a2 ∣ψ1∣2 ψ1 = 0, then

(

)

(ψ1, ψ2 ) = ψ1, e i ϕ ψ1*

(7.21)

is a solution of the CNLSE

i ψ1t + b0 ψ1xx + (b1 ∣ψ1∣2 + b2 ∣ψ2∣2 ) ψ1 = 0, i ψ2t − b0 ψ2 xx + (c1 ∣ψ1∣2 + c2 ∣ψ2∣2 ) ψ2 = 0, with a2 = −(c1 + c2 ). 7.3.3 Symmetry Reduction III From Vector NLSE to Fundamental NLSE The vector CNLSE

i ψj t + b0 j ψj xx

⎛N ⎞ + ⎜⎜ ∑ b1k ∣ψk∣2 ⎟⎟ ψj = 0, ⎝ k=1 ⎠

j = 1, 2, … , N ,

(7.22)

transforms to the scalar NLSE N

i ψ1t + b0 j ψ1xx +

∑ b1 j ∣σj∣2 ∣ψ1∣2 ψ1 = 0, k=1

with the replacement: ψj (x , t ) = σj ψ1(x , t ), where σj are arbitrary complex constants, σ1 = 1. Conclusion: If ψ1(x , t ) is a solution of the fundamental NLSE

i ψ1t + a1 ψ1xx + a2 ∣ψ1∣2 ψ1 = 0, then

(ψ1, ψ2, ψ3, …) = (ψ1, σ2 ψ1, σ3 ψ1, …)

7-18

(7.23)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

is a solution of the vector CNLSE

⎛N ⎞ i ψj t + b0 j ψj xx + ⎜⎜ ∑ b1k ∣ψk∣2 ⎟⎟ ψj = 0, ⎝ k=1 ⎠

j = 1, 2, … , N ,

with b0 j = a1, N

a2 =

∑b1 j ∣σj∣2. j=1

7.3.4 Symmetry Reduction IV From Three Coupled NLSEs to Manakov System The three CNLSEs,

i ψ1t + a1 ψ1xx + (b1 ∣ψ1∣2 + b2 ∣ψ2∣2 + b3 ∣ψ3∣2 ) ψ1 = 0, i ψ2t + a2 ψ2 xx + (c1 ∣ψ1∣2 + c2 ∣ψ2∣2 + c3 ∣ψ3∣2 ) ψ2 = 0, 2

2

(7.24)

2

i ψ3t + a3 ψ3xx + (d1 ∣ψ1∣ + d2 ∣ψ2∣ + d3 ∣ψ3∣ ) ψ3 = 0, transform to the CNLSE

i ψ1t + a1 ψ1xx + (b1 ∣ψ1∣2 + b3 ∣ψ3∣2 ) ψ1 = 0,

(7.25)

i ψ3t + a3 ψ3xx + (d1 ∣ψ1∣2 + d3 ∣ψ3∣2 ) ψ3 = 0, with the replacements ψ2(x , t ) = σ ψ1(x , t ), c1 = b1 + (b2 − c2 ) ∣σ∣2 , a1 = a2 , c3 = b3,

where a1,2,3, b1,2,3, c1,2,3, and d1,2,3 are real constants, σ is an arbitrary complex constant.

Example 1. tanh(x , t )-tanh(x , t )-sech(x , t ) dark-dark-bright soliton (Figure 7.7)

ψ1(x , t ) = (A0 tanh{A1 [x − x0 − v (t − t0 )]} + i A2 ) e i [b1 (t−t0)+ϕ1], ψ2(x , t ) = σ2 ψ1(x , t ), ψ3(x , t ) = B0 sech{A1 [x − x0 − v (t − t0)]} e

7-19

i [(A12 +b 2 ) (t −t0 )+θ (x, t )+ϕ2]

(7.26) ,

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Figure 7.7. Dark-dark-bright soliton (7.26) at t = 0. Blue is ψ1, red is ψ2 , and green is ψ3 with b2 = 1/2 , v = 1/2 , σ2r = σ2i = 1, and x0 = t0 = ϕ1 = ϕ2 = 0 . Animation available online at https://iopscience.iop.org/book/9780-7503-2428-1.

are a solution of

i ψ1t + a1 ψ1xx + (b11 ∣ψ1∣2 + b12 ∣ψ2∣2 + b13 ∣ψ3∣2 ) ψ1 = 0, i ψ2t + a2 ψ2 xx + (c11 ∣ψ1∣2 + c12 ∣ψ2∣2 + c13 ∣ψ3∣2 ) ψ2 = 0, 2

2

2

i ψ3t + a3 ψ3xx + (d11 ∣ψ1∣ + d12 ∣ψ2∣ + d13 ∣ψ3∣ ) ψ3 = 0, where

⎡ A0 = cos ⎢tan−1 ⎣ A1 =

b1 (1 − b22 ) 2 (1 + b1 b2 )

⎡ A2 = sin ⎢tan−1 ⎣ B0 =

( )⎤⎦⎥, −v 2 A1

−

v2 4

,

( )⎤⎦⎥, −v 2 A1

2 A12 (b1 + b2 ) 1 − b22 v [x − 2

, v

θ (x , t ) = x0 − 2 (t − t0 )], b0 = −1, c0 = 1, c1 = 1/b1, c2 = b2 , b1 b2 = 1, a1 = a2 = b0 , a3 = c0, b11 = b1 − b12 ∣σ2∣2 , b13 = b2 , d11 = c1 − d12 ∣σ2∣2 , d13 = c2 , c13 = b13, c11 = b11 + (b12 − c12 ) ∣σ2∣2 , σ2 = σ2r + i σ2i , b1 (1 − b22 ) 2 (1 + b1 b2 )

>

v2 , 4

(b1 + b2 ) (1 − b22 ) > 0, x0, t0, ϕ1, ϕ2, σ2r , σ2i , and v are arbitrary real constants. 7-20

(7.27)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Figure 7.8. Dark-bright-bright soliton (7.28) at t = 0. Blue is ψ1, red is ψ2 , and green is ψ3 with b2 = 1/2 , v = 1/2 , σ2r = σ2i = 1, and x0 = t0 = ϕ1 = ϕ2 = 0 . Animation available online at https://iopscience.iop.org/ book/978-0-7503-2428-1.

Example 2. tanh(x , t )-sech(x , t )-sech(x , t ) dark-bright-bright soliton (Figure 7.8)

ψ1(x , t ) = (A0 tanh{A1 [x − x0 − v (t − t0 )]} + i A2 ) e i [b1 (t−t0)+ϕ1], ψ2(x , t ) = σ2 ψ3(x , t ), ψ3(x , t ) = B0 sech{A1 [x − x0 − v (t − t0)]} e are a solution of (7.27), where ⎡ −v ⎤ A0 = cos ⎢tan−1 2 A ⎥⎦, ⎣ 1

( )

A1 =

b1 (1 − b22 ) 2 (1 + b1 b2 )

⎡ A2 = sin ⎢tan−1 ⎣ B0 =

−

,

( )⎤⎦⎥, −v 2 A1

2 A12 (b1 + b2 ) 1 − b22 v [(x 2

v2 4

, v

θ (x , t ) = − x0 ) − 2 (t − t0 )], b0 = −1, c0 = 1, c1 = 1/b1, c2 = b2 , b1 b2 = 1, a1 = a2 = b0 , a3 = c0, b11 = b1 − b12 ∣σ2∣2 , b13 = b2 , d11 = c1 − d12 ∣σ2∣2 , d13 = c2 , c13 = b13, c11 = b11 + (b12 − c12 ) ∣σ2∣2 , σ = σ2r + i σ2i , b1 (1 − b22 ) 2 (1 + b1 b2 )

>

v2 , 4

7-21

i [(A12 +b 2 ) (t −t0 )+θ (x, t )+ϕ2]

(7.28) ,

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

(b1 + b2 ) (1 − b22 ) > 0, x0, t0, ϕ1, ϕ2, σ2r , σ2i , and v are arbitrary real constants.

7.3.5 Symmetry Reduction V From Vector NLSE to Manakov System The vector CNLSE

⎛N ⎞ i ψj t + a1 j ψj xx + ⎜⎜ ∑ bj k ∣ψk∣2 ⎟⎟ ψj = 0, ⎝ k=1 ⎠

j = 1, 2, … , N ,

(7.29)

transforms to the CNLSE

i ψ1t + a1 ψ1xx + (b1 ∣ψ1∣2 + b2 ∣ψ2∣2 ) ψ1 = 0, i ψ2t + a2 ψ2 xx + (c1 ∣ψ1∣2 + c2 ∣ψ2∣2 ) ψ2 = 0, with the replacements: ⎧ σk ψ1(x , t ), ψk (x , t ) = ⎨ ⎩ σk ψm + 1(x , t ),

(7.30)

k = 1, 2, … , m, σ1 = 1, k = m + 1, … , N , σm + 1 = 1,

m

b1 =

∑ bj k ∣σk∣2,

j = 1, 2, … , m,

k=1 N

b2 =

∑

bj k ∣σk∣2 , j = 1, 2, … , m,

k=m+1 m

c1 =

∑ bj k ∣σk∣2,

j = m, m + 1, … , N ,

k=1 N

c2 =

∑

bj k ∣σk∣2 , j = m, m + 1, … , N ,

k=m+1

⎧ a1, a1 j = ⎨ ⎩ a2,

j = 1, 2, … , m, j = m + 1, m + 2, … , N ,

where a1 and b1k are real constants, σk is an arbitrary complex constant.

7.4 Scaling Transformations 7.4.1 Linear and Nonlinear Coupling 7.4.1.1 General Case If (ψ1, ψ2 ) is a solution of

i ψ1t + ψ1xx + (b1 ∣ψ1∣2 + b2 ∣ψ2∣2 ) ψ1 = 0, i ψ2t + ψ2 xx + (b1 ∣ψ1∣2 + b2 ∣ψ2∣2 ) ψ2 = 0, then 7-22

(7.31)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Φ1(x , t ) =

b1 ψ (x , t ) e i g0 (g1−g2) t + g1 − g2 1

g2 b2 ψ (x , t ) e−i g0 (g1−g2) t , g1 (g2 − g1) 2

Φ2 (x , t ) =

b1 ψ (x , t ) e i g0 (g1−g2) t + g1 − g2 1

g1 b2 ψ (x , t ) e−i g0 (g1−g2) t g2 (g2 − g1) 2

(7.32)

is a solution of

i Φ1t + Φ1xx + (g1 ∣Φ1∣2 − g2 ∣Φ2∣2 ) Φ1 + g0 (g1 + g2 ) Φ1 − 2 g0 g2 Φ2 = 0, i Φ2t + Φ2 xx + (g1 ∣Φ1∣2 − g2 ∣Φ2∣2 ) Φ2 − g0 (g1 + g2 ) Φ2 + 2 g0 g1 Φ1 = 0,

(7.33)

where b1 (g1 − g2 ) > 0, b2 g1 g2 (g2 − g1) > 0, b1, b2, g0, g1, and g2 are real constants. 7.4.1.2 Specific Case I: Manakov System to Another Manakov System If (ψ1, ψ2 ) is a solution of

i ψ1t + ψ1xx + (b1 ∣ψ1∣2 + b2 ∣ψ2∣2 ) ψ1 = 0, i ψ2t + ψ2 xx + (b1 ∣ψ1∣2 + b2 ∣ψ2∣2 ) ψ2 = 0,

(7.34)

then

Φ1(x , t ) =

b1 ψ1(x , t ) + g1 − g2

g2 b2 ψ2(x , t ), g1 (g2 − g1)

Φ2 (x , t ) =

b1 ψ (x , t ) + g1 − g2 1

g1 b2 ψ (x , t ) g2 (g2 − g1) 2

(7.35)

is a solution of

i Φ1t + Φ1xx + (g1 ∣Φ1∣2 − g2 ∣Φ2∣2 ) Φ1 = 0, i Φ2t + Φ2 xx + (g1 ∣Φ1∣2 − g2 ∣Φ2∣2 ) Φ2 = 0,

(7.36)

where b1 (g1 − g2 ) > 0, b2 g1 g2 (g2 − g1) > 0, b1, b2, g1 and g2 are real constants. 7.4.1.3 Specific Case II: Manakov System to the Same Manakov System Superposition Principle for a Nonlinear System If (ψ1, ψ2 ) is a solution of

i ψ1t + ψ1xx + (b1 ∣ψ1∣2 + b2 ∣ψ2∣2 ) ψ1 = 0, i ψ2t + ψ2 xx + (b1 ∣ψ1∣2 + b2 ∣ψ2∣2 ) ψ2 = 0,

7-23

(7.37)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

then

Φ1(x , t ) =

⎡ ⎤ b2 b1 ψ2(x , t )⎥ , ⎢ψ1(x , t ) − ⎦ b1 b1 + b2 ⎣

Φ2 ( x , t ) =

b1 [ψ (x , t ) + ψ2(x , t )] b1 + b2 1

(7.38)

is also a solution of (7.37), where b1 + b2 ≠ 0, b1 (b1 + b2 ) > 0. Example 1. sech2(x ) (Figure 7.9) Given ψ1(x , t ) = {A0 sech2[A1 (x − x0 )] + A3} e−i [ω1 (t−t0)+ϕ1], ψ2(x , t ) = B0 sech2[A1 (x − x0 )] e−i [ω2 (t−t0)+ϕ2 ] is a solution of (7.1), then

Φ1(x , t ) =

Φ2 (x , t ) =

⎛ b1 ⎜ {A0 sech2[A1 (x − x0)] + A3} e−i [ω1 (t−t0)+ϕ1] b1 + b2 ⎝ ⎞ b − 2 B0 sech2[A1 (x − x0)] e−i [ω2 (t−t0)+ϕ2]⎟ , ⎠ b1

(7.39)

b1 ( {A0 sech2[A1 (x − x0)] + A3} e−i [ω1 (t−t0)+ϕ1] b1 + b2 + B0 sech2[A1 (x − x0)] e−i [ω2 (t−t0)+ϕ2])

is a solution of (7.37), where −2 A A3 = 3 0 ,

Figure 7.9. Plot of solution (7.39). (a) t = 0, (b) x = 0, with A0 = A1 = B1 = 1, B0 = 3/2 , and x0 = t0 = ϕ1 = ϕ2 = 0. Blue is ϕ1 and red is ϕ2.

7-24

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

ω1 = 2 A12 , ω2 = −2 A12 , −9 A12

b1 =

2 A02 9 A12

b2 =

2 B02

,

,

b0 = c0 = 1, b1 = c1, b2 = c2 , b1 + b2 ≠ 0, b1 (b1 + b2 ) > 0, A0 ≠ 0, B0 ≠ 0, A1, B1, x0, t0, ϕ1, and ϕ2 are arbitrary real constants.

7.4.2 Complex Coupling 7.4.2.1 General Case If (ψ1, ψ2 ) is a solution of

i ψ1t + ψ1xx + q1 (q2 ∣ψ1∣2 + q3 ∣ψ2∣2 ) ψ1 = 0, i ψ2t + ψ2 xx + q1 (q2 ∣ψ1∣2 + q3 ∣ψ2∣2 ) ψ2 = 0,

(7.40)

then

Φ1(x , t ) = c1 ψ1(x , t ) + c2 ψ2(x , t ), Φ2(x , t ) = c3 ψ1(x , t ) + c4 ψ2(x , t )

(7.41)

is a solution of i Φ1t + Φ1xx + 2 (a11 ∣Φ1∣2 + a12 ∣Φ2∣2 ) Φ1 + 2 (b11 Φ1 Φ*2 + b12 Φ2 Φ1*) Φ1 = 0, i Φ2t + Φ2 xx + 2 (a21 ∣Φ1∣2 + a22 ∣Φ2∣2 ) Φ2 + 2 (b21 Φ1 Φ*2 + b22 Φ2 Φ1*) Φ2 = 0,

where q1 =

2 (c1 c4 − c2 c3) (c2* c3* − c1* c4*) c1* c2 c3 c4* − c1 c2* c3* c4

,

q2 = (a − i b ) c1* c3 − (a + i b ) c1 c3*, q3 = a (c2 c4* − c2* c4 ) + i b (c2* c4 + c2 c4*), a12 =

b12 c1* c2* (c2 c3 − c1 c4 ) + b11 c1 c2 (c1* c4* − c2* c3*) c1 c2* c3* c4 − c1* c2 c3 c4* b11 c3* c4* (c2 c3 − c1 c4 ) + b12 c3 c4 (c1* c4* − c2* c3*)

,

, a11 = c1 c2* c3* c4 − c1* c2 c3 c4* , a22 = a12 b12 = a − i b, b21 = a + i b, b11 = b21, b22 = b12 , c1 c2* c3* c4 should not be pure real or pure imaginary, 7-25

(7.42)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

c2 c3 − c1 c4 ≠ 0, cj, j = 1, 2, 3, 4, are complex constants, a and b are real constants.

7.4.2.2 Specific Case If (ψ1, ψ2 ) is a solution of

i ψ1t + ψ1xx − 4 (b ∣ψ1∣2 + a ∣ψ2∣2 ) ψ1 = 0, i ψ2t + ψ2 xx − 4 (b ∣ψ1∣2 + a ∣ψ2∣2 ) ψ2 = 0,

(7.43)

then

Φ1(x , t ) = ψ1(x , t ) + ψ2(x , t ), Φ2(x , t ) = ψ1(x , t ) + i ψ2(x , t )

(7.44)

is a solution of

i Φ1t + Φ1xx − 2(a + b) (∣Φ1∣2 + ∣Φ2∣2 ) Φ1 ⎡ ⎤ + 2 ⎣(a + i b) Φ1 Φ*2 + (a − i b) Φ2 Φ*1 ⎦ Φ1 = 0, i Φ2t + Φ2 xx − 2(a + b) (∣Φ1∣2 + ∣Φ2∣2 ) Φ2 ⎡ ⎤ + 2 ⎣(a + i b) Φ1 Φ*2 + (a − i b) Φ2 Φ1*⎦ Φ2 = 0,

(7.45)

where a and b are real constants. 7.4.3 Function Coefficients 7.4.3.1 General Case If (ψ1, ψ2 ) is a solution of

i ψ1t + a11 ψ1xx + (a12 ∣ψ1∣2 + a13 ∣ψ2∣2 ) ψ1 = 0, i ψ2t + a21 ψ2 xx + (a22 ∣ψ1∣2 + a23 ∣ψ2∣2 ) ψ2 = 0,

(7.46)

then

Φ1(x , t ) = A(x , t ) e i B1(x,t ) ψ1[X (x , t ), T (x , t )], Φ2(x , t ) = A(x , t ) e i B2(x,t ) ψ2[X (x , t ), T (x , t )]

(7.47)

is a solution of

i Φ1t + b11(x , t )Φ1xx + [b12(x , t )∣Φ1 ∣2 + b13(x , t )∣Φ2 ∣2 ]Φ1 + [b14r(x , t ) + ib14i (x , t )]Φ1 = 0, i Φ2t + b21(x , t )Φ2xx + [b22(x , t )∣Φ1 ∣2 + b23(x , t )∣Φ2 ∣2 ]Φ2 + [b24r(x , t ) + ib24i (x , t )]Φ2 = 0,

7-26

(7.48)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

where

T (x , t ) = g1(t ), A(x , t ) =

B1(x , t ) = g2(t ) −

b12(x , t ) =

B2(x , t ) = g2(t ) −

,

a11 g1′(t ) Xx2(x , t )

(7.50)

a12 g1′(t ) A2 (x , t )

(7.51)

,

(7.52)

,

(7.53)

a13 g1′(t ) Xx(x , t ) g32(t )

,

(7.54)

∫ 2 b21(xX,t(tx), Xtx)(x, t ) dx,

b21(x , t ) =

b22(x , t ) =

b23(x , t ) =

Xx(x , t )

∫ 2 b11(xX,t(tx), Xtx)(x, t ) dx,

b11(x , t ) =

b13(x , t ) =

g3(t )

(7.49)

a21 g1′(t ) Xx2(x , t ) a22 g1′(t ) A2 (x , t )

,

(7.56)

,

(7.57)

a23 g1′(t ) Xx(x , t ) g32(t )

(7.55)

,

(7.58)

⎛ 1 ⎜2 g ″(t ) Xt(x , t ) Xx(x , t ) dx 4 a11 g1′2(t ) ⎝ 1 ⎧ [Xtt(x , t ) Xx(x , t ) + Xt(x , t ) Xxt(x , t )] dx + g1′(t )⎨ −2 ⎩

∫

b14 r(x , t ) = g2′(t ) +

∫

+

Xt2(x ,

(7.59)

2 ⎫⎞ a11 g1′2(t ) ⎡ 2 ⎤ t) + 4 ⎣ −3 Xxx(x , t ) + 2 Xx(x , t ) Xxxx(x , t )⎦⎬⎟⎟ , X x (x , t ) ⎭⎠ ⎪

⎪

7-27

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

b14i (x , t ) =

g ′(t ) Xxt(x , t ) − 3 , Xx(x , t ) g3(t )

(7.60)

⎛ 1 ⎜ 2 g ″(t ) Xt(x , t ) Xx(x , t ) dx 2 4 a21 g1′ (t ) ⎝ 1 ⎧ [Xtt(x , t ) Xx(x , t ) + Xt(x , t ) Xxt(x , t )] dx + g1′(t )⎨ −2 ⎩

∫

b24 r(x , t ) = g2′(t ) +

∫

+ Xt2(x , t ) +

(7.61)

2 ⎫⎞ a21 g1′2(t ) ⎡ 2 ⎤⎬⎟ , 3 ( , ) 2 ( , ) ( , ) − X x t + X x t X x t x xxx xx ⎣ ⎦ ⎟ Xx4(x , t ) ⎭⎠ ⎪

⎪

b24i (x , t ) =

g ′(t ) Xxt(x , t ) − 3 , Xx(x , t ) g3(t )

(7.62)

a11, a12, a13, a21, a22, and a23 are arbitrary real constants. 7.4.3.2 Specific Case: Constant Dispersion and Real Quadratic Potential If (ψ1, ψ2 ) is a solution of

i ψ1t + a11 ψ1xx + (a12 ∣ψ1∣2 + a13 ∣ψ2∣2 ) ψ1 = 0,

(7.63)

i ψ2t + a21 ψ2 xx + (a22 ∣ψ1∣2 + a23 ∣ψ2∣2 ) ψ2 = 0, then

Φ1(x , t ) = e Φ2 (x , t ) =

γ (t ) 2

γ (t ) e 2

e−

i x 2 γ ′ (t ) a11

i x 2 γ ′ (t ) e− a21

⎡ 1 ψ1⎢e γ (t ) x , ⎣ 4

∫ e2 γ(t) dt⎥⎦,

⎤

⎡ 1 ψ2⎢e γ (t ) x , ⎣ 4

⎤ e 2 γ (t ) dt ⎥ ⎦

∫

(7.64)

is a solution of

i Φ1t + a11 Φ1xx + e γ (t ) (a12 ∣Φ1∣2 + a13 ∣Φ2∣2 ) Φ1 +

4 [γ′2 (t ) − γ ″(t )] x 2 Φ1 = 0, a11

i Φ2t + a21 Φ2 xx + e γ (t ) (a22 ∣Φ1∣2 + a23 ∣Φ2∣2 ) Φ2 +

4 [γ′2 (t ) − γ ″(t )] x 2 Φ2 = 0, a21

where γ (t ) is an arbitrary real function.

7-28

(7.65)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Remark: This case is obtained from the general case with the specifications: X (x , t ) = c (t ) x , g1(t ) = ∫ c 2(t ) dt , g2(t ) = 0, g3(t ) = c(t ), c (t ) = e γ (t ) .

7-29

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

7.5 Summary of Sections 7.3–7.4

Symmetry Reductions Symmetry Reduction I: From Manakov system to fundamental NLSE Transformation: ψ2(x , t ) = σ ψ1(x , t ), b1 = c1 + (c2 − b2 ) ∣σ∣2 , (ψ1, ψ2 ) is a solution of the fundamental CNLSE Equation: i ψ1t + b0 ψ1xx + (c1 + c2∣σ∣2 ) ∣ψ1∣2 ψ1 = 0 Symmetry Reduction II: From Manakov system to fundamental NLSE Transformation: ψ2(x , t ) = ei ϕ ψ1*(x , t ), b1 = − (c1 + c2 + b2 ) Equation: i ψ1t + b0 ψ1xx − (c1 + c2 ) ∣ψ1∣2 ψ1 = 0 Symmetry Reduction III: From vector NLSE to fundamental NLSE Transformation: ψj (x , t ) = σj ψ1(x , t ) N

Equation: i ψ1t + b0 j ψ1xx +

∑ b1 j ∣σj∣2 ∣ψ1∣2 ψ1 = 0 k=1

Symmetry Reduction IV: From three coupled NLSEs to Manakov system Transformation: ψ2(x , t ) = σ ψ1(x , t ), c1 = b1 + (b2 − c2 ) ∣σ∣2 , a1 = a2, c3 = b3 Equation: i ψ1t + a1 ψ1xx + (b1 ∣ψ1∣2 + b3 ∣ψ3∣2 ) ψ1 = 0,i ψ3t + a3 ψ3xx + (d1 ∣ψ1∣2 + d3 ∣ψ3∣2 ) ψ3 = 0 Symmetry Reduction V: From vector NLSE to Manakov system Transformation: ⎧ σk ψ1(x , t ), ψk (x , t ) = ⎨ ⎩ σk ψm + 1(x , t ),

k = 1, 2, … , m , σ1 = 1, k = m + 1, … , N , σm + 1 = 1,

m

b1 =

∑ bj k ∣σk∣2 ,

j = 1, 2, … , m ,

k=1 N

b2 =

∑

bj k ∣σk∣2 , j = 1, 2, … , m ,

k=m+1 m

c1 =

∑ bj k ∣σk∣2 ,

j = m , m + 1, … , N ,

k=1 N

c2 =

∑

bj k ∣σk∣2 , j = m , m + 1, … , N ,

k=m+1

⎧ a1, a1 j = ⎨ ⎩ a2,

j = 1, 2, … , m , j = m + 1, m + 2, … , N

Equation: i ψ1t + a1 ψ1xx + (b1 ∣ψ1∣2 + b2 ∣ψ2∣2 ) ψ1 = 0, i ψ2t + a2 ψ2 xx + (c1 ∣ψ1∣2 + c2 ∣ψ2∣2 ) ψ2 = 0

7-30

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Scaling Transformations Linear and Nonlinear Coupling General Case Transformation: Φ1(x , t ) =

Φ2 (x , t ) =

b1 g1 − g2

b1 g1 − g2

ψ1(x , t ) ei g0 (g1− g2) t +

ψ1(x , t ) ei g0 (g1− g2) t +

g1 b2 g2 (g2 − g1)

g2 b2 g1 (g2 − g1)

ψ2(x , t ) e−i g0 (g1− g2) t ,

ψ2(x , t ) e−i g0 (g1− g2) t

Equation: i Φ1t + Φ1xx + (g1 ∣Φ1∣2 − g2 ∣Φ2∣2 ) Φ1 + g0 (g1 + g2 ) Φ1 − 2 g0 g2 Φ2 = 0, i Φ2t + Φ2 xx + (g1 ∣Φ1∣2 − g2 ∣Φ2∣2 ) Φ2 − g0 (g1 + g2 ) Φ2 + 2 g0 g1 Φ1 = 0 Specific Case I: Manakov System to Another Manakov System Transformation: Φ1(x , t ) =

Φ2 (x , t ) =

b1 g1 − g2

b1 g1 − g2

ψ1(x , t ) +

g2 b2 g1 (g2 − g1)

ψ1(x , t ) +

g1 b2 g2 (g2 − g1)

ψ2(x , t ),

ψ2(x , t )

Equation: i Φ1t + Φ1xx + (g1 ∣Φ1∣2 − g2 ∣Φ2∣2 ) Φ1 = 0, i Φ2t + Φ2 xx + (g1 ∣Φ1∣2 − g2 ∣Φ2∣2 ) Φ2 = 0 Specific Case II: Manakov System to the Same Manakov System Superposition Principle for a Nonlinear System Transformation: Φ1(x , t ) =

b1 b1 + b2

⎡ψ (x , t ) − ⎣ 1

b2 b1

ψ2(x , t )⎤⎦ , Φ2(x , t ) =

b1 b1 + b2

× [ψ1(x , t ) + ψ2(x , t )] Complex Coupling General Case Transformation: Φ1(x , t ) = c1 ψ1(x , t ) + c2 ψ2(x , t ), Φ2(x , t ) = c3 ψ1(x , t ) + c4 ψ2(x , t ) Equation: i Φ1t + Φ1xx + 2 (a11 ∣Φ1∣2 + a12 ∣Φ2∣2 ) Φ1 + 2 (b11 Φ1 Φ*2 + b12 Φ2 Φ1*) Φ1 = 0, i Φ2t + Φ2 xx + 2 (a21 ∣Φ1∣2 + a22 ∣Φ2∣2 ) Φ2 + 2 (b21 Φ1 Φ*2 + b22 Φ2 Φ1*) Φ2 = 0 Specific Case Transformation: Φ1(x , t ) = ψ1(x , t ) + ψ2(x , t ), Φ2(x , t ) = ψ1(x , t ) + i ψ2(x , t ) Equation: i Φ1t + Φ1xx − 2(a + b ) (∣Φ1∣2 + ∣Φ2∣2 ) Φ1 + 2 [(a + i b ) Φ1 Φ*2 + (a − i b ) Φ2 Φ1*] Φ1 = 0,

i Φ2t + Φ2 xx − 2(a + b ) (∣Φ1∣2 + ∣Φ2∣2 ) Φ2 + 2 [(a + i b ) Φ1 Φ*2 + (a − i b ) Φ2 Φ1*] Φ2 = 0 Function Coefficients General Case Transformation: Φ1(x , t ) = A(x , t ) ei B1(x, t ) ψ1[X (x , t ), T (x , t )], Φ2(x , t ) = A(x , t ) ei B2(x, t ) ψ2[X (x , t ), T (x , t )] (Continued)

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Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

(Continued )

Symmetry Reductions Equation: i Φ1t + b11(x , t ) Φ1xx + [b12(x , t ) ∣Φ1∣2 + b13(x , t ) ∣Φ2∣2 ] Φ1 + [b14 r (x , t ) + i b14i (x , t )]

Φ1 = 0, i Φ2t + b21(x , t )Φ2 xx + [b22(x , t ) ∣Φ1∣2 + b23(x , t ) ∣Φ2∣2 ] Φ2 + [b24 r (x , t ) + i b24i (x , t )i ] Φ2 = 0, Specific Case: Constant Dispersion and Real Quadratic Potential Transformation: Φ1(x , t ) = e

Φ2 (x , t ) = e

γ (t ) 2

e−

i x 2 γ ′ (t ) a21

γ (t ) 2

e−

i x 2 γ ′ (t ) a11

ψ2⎡⎣e γ (t ) x ,

1 4

ψ1⎡⎣e γ (t ) x ,

1 4

∫ e 2 γ (t ) dt ⎤⎦,

∫ e 2 γ (t ) dt ⎤⎦

Equation: i Φ1t + a11 Φ1xx + e γ (t ) [a12 ∣Φ1∣2 + a13 ∣Φ2∣2 ] Φ1 +

i Φ2t + a21 Φ2 xx + e γ (t ) [a22 ∣Φ1∣2 + a23 ∣Φ2∣2 ] Φ2 +

4 a21

4 a11

[γ ′2 (t ) − γ ″(t )] x 2 Φ1 = 0,

[γ ′2 (t ) − γ ″(t )] x 2 Φ2 = 0

7.6 (N + 1)-Dimensional Coupled NLSE (N + 1)-Dimensional Manakov System 7.6.1 Reduction to 1D Manakov System If (ψ1, ψ2 ) is a solution of the Manakov system

i ψ1t + a1 ψ1xx + (b1 ∣ψ1∣2 + b2 ∣ψ2∣2 ) ψ1 = 0, i ψ2t + a2 ψ2 xx + (c1 ∣ψ1∣2 +

c2 ∣ψ2∣2 )

ψ2 = 0,

ψ1 = ψ1(x , t ; a1, b1, b2 ), ψ2 = ψ2(x , t ; a2 , c1, c2 ),

(7.66)

then N ⎡N ⎤ Φ1(x1, x2 , … , xN , t ) = ψ1⎢ ∑ dk (xk − xk0), t ; ∑ d k2 b0k , b1, b2⎥ , ⎢⎣ k = 1 ⎥⎦ k=1 N ⎡N ⎤ ⎢ Φ2(x1, x2 , … , xN , t ) = ψ2 ∑ dk (xk − xk0), t ; ∑ d k2 c0k , c1, c2⎥ ⎢⎣ k = 1 ⎥⎦ k=1

(7.67)

is a solution of N

i Φ1t +

∑ b0k Φ1x x

k k

(

)

+ b1 ∣Φ1∣2 + b2 ∣Φ2∣2 Φ1 = 0,

k=1 N

i Φ 2t +

(7.68)

∑ c0k Φ2 x x

k k

(

)

+ c1 ∣Φ1∣2 + c2 ∣Φ2∣2 Φ2 = 0,

k=1

7-32

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

where Φj = Φj (x1, x2, … , xN , t ) is the complex function profile, j = 1, 2, b0 k , c0 k , dk, b1, c1, b2, and c2 are real constants. Example 1. tanh(x1, x2 )-sech(x1, x2 ) 2D dark-bright soliton (Figure 7.10) Given ψ1(x , t ) = (A0 tanh{A1 [x − x0 − v (t − t0 )]} + i A2 ) ei [b1 (t − t0 ) + ϕ1], 2

ψ2(x , t ) = B0 sech{A1 [x − x0 − v (t − t0 )]} ei [(A1 + b2 ) (t − t0 ) + θ (x, t ) + ϕ2 ]

is a solution of (7.1), then

Φ1(x1, x2 , t ) = (A0 tanh{A1 [d1 (x1 − x01) + d2 (x2 − x02 ) − v (t − t0)]} + i A2 ) × e i [b1 (t − t0) +Φ1], Φ2(x1, x2 , t ) = B0 sech{A1 [d1 (x1 − x01) + d2 (x2 − x02 ) − v (t − t0)]}

(7.69)

2

× e i [(A1 + b2) (t − t0) + θ(x1, x2, t ) + ϕ2] is a solution of

i Φ1t + b01 Φ1x1x1 + b02 Φ1x2x2 + (b1 ∣Φ1∣2 + b2 ∣Φ2∣2 ) Φ1 = 0, i Φ2t + c01 Φ2 x1x1 + c02 Φ2 x2x2 + (c1 ∣Φ1∣2 + c2 ∣Φ2∣2 ) Φ2 = 0, where

⎡ A0 = cos ⎢tan−1 ⎣ A1 =

b1 (1 − b22 ) 2 (1 + b1 b2 )

⎡ A2 = sin ⎢tan−1 ⎣ B0 =

(7.70)

( )⎤⎥⎦, −v 2 A1

−

v2 4

,

( )⎤⎦⎥, −v 2 A1

2 A12 (b1 + b2 )

1 − b22 v θ (x1, x2, t ) = 2 ⎡⎣d1

, (x1 − x01) + d2 (x2 − x02 ) −

v 2

(t − t0 )⎤⎦,

Figure 7.10. 2D dark-bright soliton (7.69) at t = − 10 with v = b2 = 1/2 , d1 = d 2 = 1, b02 = c02 = 1/4 , and x01 = x02 = t0 = ϕ1 = ϕ2 = 0 . Yellow is Φ1 and green is Φ2. Animation available online at https://iopscience. iop.org/book/978-0-7503-2428-1.

7-33

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

b01 = c01 =

−1 − d 22 b02 d12 1 − d 22 c02 d12

,

,

c1 = 1/b1, c2 = b2 , b1 b2 = 1, b1 (1 − b22 ) 2 (1 + b1 b2 )

>

v2 , 4

(b1 + b2 ) (1 − b22 ) > 0, d1 ≠ 0, x01, x02, t0, ϕ1, ϕ2, d2, and v are arbitrary real constants.

7.7 Symmetry Reductions of (N + 1)-Dimensional CNLSE to Scalar NLSE 7.7.1 Symmetry Reduction I From (N + 1)-Dimensional Manakov System to (N + 1)-Dimensional Fundamental NLSE The (N + 1)-dimensional CNLSE, (7.68), N

i ψ1t +

∑ b0k ψ1x x

k k

+ (b1 ∣ψ1∣2 + b2 ∣ψ2∣2 ) ψ1 = 0,

∑ c0k ψ2 x x

+ (c1 ∣ψ1∣2 + c2 ∣ψ2∣2 ) ψ2 = 0,

k=1 N

i ψ2 t +

k k

k=1

transforms to the scalar (N + 1)-dimensional NLSE N

i ψ1t +

∑ c0k ψ1x x

k k

+ (c1 + c2∣σ∣2 ) ∣ψ1∣2 ψ1 = 0,

(7.71)

k=1

with the replacements: ψ2(x1, x2, … , xN , t ) = σ ψ1(x1, x2, … , xN , t ), b1 = c1 + (c2 − b2 ) ∣σ∣2 , b0k = c0k , where σ is an arbitrary complex constant. Conclusion: If ψ1(x1, x2, … , xN , t ) is a solution of the (N + 1)-dimensional NLSE N

i ψ1t +

∑ αk ψ1x x

k k

+ a2 ∣ψ1∣2 ψ1 = 0,

(7.72)

k=1

then

(ψ1, ψ2 ) = (ψ1, σ ψ1)

7-34

(7.73)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

is a solution of the (N + 1)-dimensional CNLSE N

∑ b0k ψ1x x

i ψ1t +

k k

+ (b1 ∣ψ1∣2 + b2 ∣ψ2∣2 ) ψ1 = 0,

∑ c0k ψ2 x x

+ (c1 ∣ψ1∣2 + c2 ∣ψ2∣2 ) ψ2 = 0,

k=1 N

i ψ2 t +

k k

k=1

with αk = b0k = c0k , a2 = c1 + c2∣σ∣2 , b1 = c1 + (c2 − b2 ) ∣σ∣2 . 7.7.2 Symmetry Reduction II From (N + 1)-Dimensional Manakov System to (N + 1)-Dimensional Fundamental NLSE The (N + 1)-dimensional CNLSE, (7.68), N

∑ b0k ψ1x x

i ψ1t +

k k

+ (b1 ∣ψ1∣2 + b2 ∣ψ2∣2 ) ψ1 = 0,

∑ c0k ψ2 x x

+ (c1 ∣ψ1∣2 + c2 ∣ψ2∣2 ) ψ2 = 0,

k=1 N

i ψ2 t +

k k

k=1

transforms to the scalar (N + 1)-dimensional NLSE N

i ψ1t −

∑ c0k ψ1x x

k k

− 2 (c1 + c2 ) ∣ψ1∣2 ψ1 = 0,

(7.74)

k=1

with the replacements: ψ2(x1, x2, … , xN , t ) = ei ϕ ψ1*(x1, x2, … , xN , t ), b1 = −(c1 + c2 + b2 ), b0k = −c0k , where ϕ is an arbitrary real constant. Conclusion: If ψ1(x1, x2, … , xN , t ) is a solution of the (N + 1)-dimensional NLSE N

i ψ1t +

∑ αk ψ1x x

k k

+ a2 ∣ψ1∣2 ψ1 = 0,

(7.75)

k=1

then

(ψ1, ψ2 ) = (ψ1, e i ϕ ψ1*) is a solution of the (N + 1)-dimensional CNLSE

7-35

(7.76)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

N

∑ b0k ψ1x x

i ψ1t +

k k

+ (b1 ∣ψ1∣2 + b2 ∣ψ2∣2 ) ψ1 = 0,

∑ c0k ψ2 x x

+ (c1 ∣ψ1∣2 + c2 ∣ψ2∣2 ) ψ2 = 0,

k=1 N

i ψ2 t +

k k

k=1

with αk = b0k = −c0k , a2 = −2 (c1 + c2 ). 7.7.3 Symmetry Reduction III From (N + 1)-Dimensional Vector NLSE to (N + 1)-Dimensional Fundamental NLSE The generalized (N + 1)-dimensional CNLSE

⎛M ⎞ 2 ⎟ ψ = 0, ⎜ b ψ + b ∣ ψ ∣ ∑ j 0k j xkxk ⎜∑ 1k k ⎟ j ⎝ k=1 ⎠ k=1 N

i ψj t +

j = 1, 2, … , M ,

(7.77)

transforms to the scalar (N + 1)-dimensional NLSE N

i ψ1t +

N

∑ b10k ψ1x x

k k

k=1

+

∑ b1 j ∣σj +1∣2 ∣ψ1∣2 ψ1 = 0,

(7.78)

k=1

with the replacement:

ψj (x1, x2 , … , xN , t ) = σj ψ1(x1, x2 , … , xN , t ), where σj are arbitrary complex constants, σ1 = 1. Conclusion: If ψ1(x1, x2, … , xN , t ) is a solution of the (N + 1)-dimensional NLSE N

i ψ1t +

∑ αk ψ1x x

k k

+ a2 ∣ψ1∣2 ψ1 = 0,

k=1

then

(ψ1, ψ2, ψ3, …) = (ψ1, σ2 ψ1, σ3 ψ1, …) is a solution of the generalized (N + 1)-dimensional CNLSE N ⎛M ⎞ i ψj t + ∑ b0k j ψj x x + ⎜⎜ ∑ b1k ∣ψk∣2 ⎟⎟ ψj = 0, j = 1, 2, … , M , k k ⎝ k=1 ⎠ k=1

7-36

(7.79)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

with αk = b10k , N

a2 =

∑b1 j ∣σj + 1∣2. j=1

Notes: The (N + 1)-dimensional three coupled NLSEs reduce to (N + 1)-dimensional Manakov system in a similar manner to the above described symmetry reductions, (7.7.1–7.7.3). The (N + 1)-dimensional vector NLSE reduces to (N + 1)-dimensional Manakov system in a similar manner to the above described symmetry reductions, (7.7.1–7.7.3).

7.8 (N + 1)-Dimensional Scaling Transformations 7.8.1 Linear and Nonlinear Coupling 7.8.1.1 General Case If (ψ1, ψ2 ) is a solution of N

i ψ1t +

∑ ψ1x x

k k

+ (b1 ∣ψ1∣2 + b2 ∣ψ2∣2 ) ψ1 = 0,

k=1 N

i ψ2 t +

(7.80)

∑ ψ2 x x

k k

+

(b1 ∣ψ1∣2

+

b2 ∣ψ2∣2 )

ψ2 = 0,

k=1

then

b1 ψ (x1, x2 , … , xN , t ) e i g0 (g1− g2) t g1 − g2 1

Φ1(x1, x2 , … , xN , t ) =

g2 b2 ψ (x1, x2 , … , xN , t ) e−i g0 (g1− g2) t , g1 (g2 − g1) 2

+

(7.81)

b1 ψ (x1, x2 , … , xN , t ) e i g0 (g1− g2) t g1 − g2 1

Φ2(x1, x2 , … , xN , t ) = +

g1 b2 ψ (x1, x2 , … , xN , t ) e−i g0 (g1− g2) t g2 (g2 − g1) 2

is a solution of N

i Φ1t +

∑ Φ1x x

k k

+ (g1 ∣Φ1∣2 − g2 ∣Φ2∣2 ) Φ1 + g0 (g1 + g2 ) Φ1 − 2 g0 g2 Φ2 = 0,

k =1 N

i Φ 2t +

(7.82)

∑ Φ2 x x

k k

+

(g1 ∣Φ1∣2

− g2 ∣Φ2

∣2 )

Φ2 − g0 (g1 + g2 ) Φ2 + 2 g0 g1 Φ1 = 0,

k =1

7-37

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

where b1 (g1 − g2 ) > 0, b2 g1 g2 (g2 − g1) > 0, b1, b2, g0, g1, and g2 are real constants. 7.8.1.2 Specific Case I: (N + 1)-Dimensional Manakov System to Another (N + 1)-Dimensional Manakov System If (ψ1, ψ2 ) is a solution of N

i ψ1t +

∑ ψ1x x

k k

+ (b1 ∣ψ1∣2 + b2 ∣ψ2∣2 ) ψ1 = 0,

k=1 N

i ψ2 t +

∑ ψ2 x x

k k

+ (b1 ∣ψ1∣2 + b2 ∣ψ2∣2 ) ψ2 = 0,

k=1

then

b1 ψ1(x1, x2 , … , xN , t ) g1 − g2

Φ1(x1, x2 , … , xN , t ) =

g2 b2 ψ2(x1, x2 , … , xN , t ), g1 (g2 − g1)

+

(7.83)

b1 ψ (x1, x2 , … , xN , t ) g1 − g2 1

Φ2(x1, x2 , … , xN , t ) = +

g1 b2 ψ (x1, x2 , … , xN , t ) g2 (g2 − g1) 2

is a solution of N

i Φ1t +

∑ Φ1x x

k k

+ (g1 ∣Φ1∣2 − g2 ∣Φ2∣2 ) Φ1 = 0,

k=1 N

i Φ 2t +

(7.84)

∑ Φ2 x x

k k

+

(g1 ∣Φ1∣2

− g2 ∣Φ2

∣2 )

Φ2 = 0,

k=1

where b1 (g1 − g2 ) > 0, b2 g1 g2 (g2 − g1) > 0, b1, b2, g1, and g2 are real constants. 7.8.1.3 Specific Case II: (N + 1)-Dimensional Manakov System to the Same (N + 1)-Dimensional Manakov System If (ψ1, ψ2 ) is a solution of N

i ψ1t +

∑ ψ1x x

k k

+ (b1 ∣ψ1∣2 + b2 ∣ψ2∣2 ) ψ1 = 0,

k=1

7-38

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

N

i ψ2 t +

∑ ψ2 x x

k k

+ (b1 ∣ψ1∣2 + b2 ∣ψ2∣2 ) ψ2 = 0,

k=1

then ⎡ ⎤ b2 ψ 2(x1, x2, … , xN , t )⎥ , ⎢ψ1(x1, x2, … , xN , t ) − ⎣ ⎦ b1

Φ1(x1, x2, … , xN , t ) =

b1 b1 + b 2

Φ2(x1, x2, … , xN , t ) =

b1 [ψ1(x1, x2, … , xN , t ) + ψ 2(x1, x2, … , xN , t )] b1 + b 2

(7.85)

is also a solution of (7.34), where b1 + b2 ≠ 0. 7.8.2 Complex Coupling 7.8.2.1 General Case If (ψ1, ψ2 ) is a solution of N

i ψ1t +

+ q1 {q2 ∣ψ1∣2 + q3 ∣ψ2∣2 } ψ1 = 0,

∑ ψ1x x

k k

k=1 N

i ψ2 t +

(7.86)

∑ ψ2 x x

k k

+ q1 {q2 ∣ψ1∣2 + q3 ∣ψ2∣2 } ψ2 = 0,

k=1

then

Φ1(x1, x2 , … , xN , t ) = c1 ψ1(x1, x2 , … , xN , t ) + c2 ψ2(x1, x2 , … , xN , t ), Φ2(x1, x2 , … , xN , t ) = c3 ψ1(x1, x2 , … , xN , t ) + c4 ψ2(x1, x2 , … , xN , t )

(7.87)

is a solution of N

i Φ1t +

∑ Φ1x x

k k

+ 2 (a11 ∣Φ1∣2 + a12 ∣Φ2∣2 ) Φ1

k=1

+ 2 (b11 Φ1 Φ*2 + b12 Φ2 Φ*1 ) Φ1 = 0,

(7.88)

N

i Φ 2t +

∑ Φ2 x x

k k

+2

(a21 ∣Φ1∣2

+ a22 ∣Φ2

k=1

+ 2 (b21 Φ1 Φ*2 + b22 Φ2 Φ*1 ) Φ2 = 0, where q1 =

2 (c1 c4 − c2 c3) (c2* c3* − c1* c4*) c1* c2 c3 c4* − c1 c2* c3* c4

,

q2 = (a − i b ) c1* c3 − (a + i b ) c1 c3*, q3 = a (c2 c4* − c2* c4 ) + i b (c2* c4 + c2 c4*), a12 = a11 =

b12 c1* c2* (c2 c3 − c1 c4 ) + b11 c1 c2 (c1* c4* − c2* c3*) c1 c2* c3* c4 − c1* c2 c3 c4* b11 c3* c4* (c2 c3 − c1 c4 ) + b12 c3 c4 (c1* c4* − c2* c3*) c1 c2* c3* c4 − c1* c2 c3 c4*

7-39

, ,

∣2 )

Φ2

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

a22 = a12 , b12 = a − i b, b21 = a + i b, b11 = b21, b22 = b12 , c1 c2* c3* c4 should not be pure real or pure imaginary, c2 c3 − c1 c4 ≠ 0, cj, j = 1, 2, 3, 4 are complex constants, a and b are real constants.

7.8.2.2 Specific Case If (ψ1, ψ2 ) is a solution of N

∑ ψ1x x

i ψ1t +

k k

− 4 (b ∣ψ1∣2 + a ∣ψ2∣2 ) ψ1 = 0,

k=1 N

(7.89)

∑ ψ2 x x

i ψ2 t +

k k

− 4 (b ∣ψ1∣2 + a ∣ψ2∣2 ) ψ2 = 0,

k=1

then

Φ1(x1, x2 , … , xN , t ) = ψ1(x1, x2 , … , xN , t ) + ψ2(x1, x2 , … , xN , t ), Φ2(x1, x2 , … , xN , t ) = ψ1(x1, x2 , … , xN , t ) + i ψ2(x1, x2 , … , xN , t )

(7.90)

is a solution of N

i Φ1t +

∑ Φ1x x

k k

− 2(a + b) (∣Φ1∣2 + ∣Φ2∣2 ) Φ1

k=1

+ 2 [(a + i b) Φ1 Φ*2 + (a − i b) Φ2 Φ*1 ] Φ1 = 0, N

i Φ 2t +

∑ Φ2 x x

k k

− 2(a + b)

(∣Φ1∣2

+ ∣Φ2

∣2 )

(7.91)

Φ2

k=1

+ 2 [(a + i b) Φ1 Φ*2 + (a − i b) Φ2 Φ*1 ] Φ2 = 0, where a and b are real constants.

7.9 Summary of Sections 7.7–7.8

(N + 1)-Dimensional Symmetry Reductions (N +1)-Dimensional Symmetry Reduction I: From (N + 1)-dimensional Manakov system to (N + 1)-dimensional fundamental NLSE Transformation: ψ2(x1, x2, … , xN , t ) = σ ψ1(x1, x2, … , xN , t ), b1 = c1 + (c2 − b2 ) ∣σ∣2 , b0k = c0k , (ψ1, ψ2 ) is a solution of the (N + 1)-dimensional CNLSE

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Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

N

∑ c0k ψ1x x

Equation: i ψ1t +

k k

+ (c1 + c2∣σ∣2 ) ∣ψ1∣2 ψ1 = 0

k=1

(N +1)-Dimensional Symmetry Reduction II: From (N + 1)-dimensional Manakov system to (N + 1)-dimensional fundamental NLSE Transformation: ψ2(x1, x2, … , xN , t ) = ei ϕ ψ1*(x1, x2, … , xN , t ), b1 = − (c1 + c2 + b2 ), b0k = − c0k N

∑ c0k ψ1x x

Equation: i ψ1t −

k k

− 2 (c1 + c2 ) ∣ψ1∣2 ψ1 = 0

k=1

(N +1)-Dimensional Symmetry Reduction III: From (N + 1)-dimensional vector NLSE to (N + 1)dimensional fundamental NLSE Transformation: ψj (x1, x2, … , xN , t ) = σj ψ1(x1, x2, … , xN , t ) N

N

∑ b10k ψ1x x

Equation: i ψ1t +

k k

+

k=1

∑ b1 j ∣σj + 1∣2 ∣ψ1∣2 ψ1 = 0 k=1

(N +1)-Dimensional Scaling Transformations Linear and Nonlinear Coupling General Case b1 g1 − g2

Φ1(x1, x2, … , xN , t ) = Transformation:

g2 b2 g1 (g2 − g1)

+ b1 g1 − g2

Φ2(x1, x2, … , xN , t ) = +

ψ1(x1, x2, … , xN , t ) ei g0 (g1− g2) t ψ2(x1, x2, … , xN , t ) e−i g0 (g1− g2) t ,

ψ1(x1, x2, … , xN , t ) ei g0 (g1− g2) t

g1 b2 g2 (g2 − g1)

ψ2(x1, x2, … , xN , t ) e−i g0 (g1− g2) t

N

Equation: i Φ1t + N

i Φ2 t +

∑ Φ1x x

k k

+ (g1 ∣Φ1∣2 − g2 ∣Φ2∣2 ) Φ1 + g0 (g1 + g2 ) Φ1 − 2 g0 g2 Φ2 = 0,

k=1

∑ Φ2 x x

k k

+ (g1 ∣Φ1∣2 − g2 ∣Φ2∣2 ) Φ2 − g0 (g1 + g2 ) Φ2 + 2 g0 g1 Φ1 = 0

k=1

Specific Case I: (N +1)-Dimensional Manakov System to Another (N + 1)-Dimensional Manakov System b1 g1 − g2

Φ1(x1, x2, … , xN , t ) = Transformation:

ψ1(x1, x2, … , xN , t )

g2 b2 g1 (g2 − g1)

+

b1 g1 − g2

Φ2(x1, x2, … , xN , t ) = +

ψ2(x1, x2, … , xN , t ),

ψ1(x1, x2, … , xN , t )

g1 b2 g2 (g2 − g1)

ψ2(x1, x2, … , xN , t ) (Continued)

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Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

(Continued )

(N + 1)-Dimensional Symmetry Reductions N

∑ Φ1x x

Equation: i Φ1t +

k k

N

∑ Φ2 x x

i Φ2 t +

+ (g1 ∣Φ1∣2 − g2 ∣Φ2∣2 ) Φ1 = 0,

k=1

+ (g1 ∣Φ1∣2 − g2 ∣Φ2∣2 ) Φ2 = 0

k k

k=1

Specific Case II: (N + 1)-Dimensional Manakov System to the Same (N + 1)-Dimensional Manakov System Superposition Principle to a Nonlinear System Transformation: Φ1(x1, x2, … , xN , t ) =

Φ2(x1, x2, … , xN , t ) =

b1 b1 + b2

b1 b1 + b2

⎡ ψ (x , x , … , x , t ) + N ⎣ 1 1 2

b2 b1

ψ2(x1, x2, … , xN , t )⎤⎦ ,

[ψ1(x1, x2, … , xN , t ) + ψ2(x1, x2, … , xN , t )]

Complex Coupling General Case Transformation: Φ1(x1, x2, … , xN , t ) = c1 ψ1(x1, x2, … , xN , t ) + c2 ψ2(x1, x2, … , xN , t ), Φ2(x1, x2, … , xN , t ) = c3 ψ1(x1, x2, … , xN , t ) + c4 ψ2(x1, x2, … , xN , t ) N

∑ Φ1x x

Equation: i Φ1t +

k k

+ 2 (a11 ∣Φ1∣2 + a12 ∣Φ2∣2 ) Φ1 + 2 (b11 Φ1 Φ*2 + b12 Φ2 Φ1*) Φ1 = 0,

k=1 N

i Φ2 t +

∑ Φ2 x x

k k

+ 2 (a21 ∣Φ1∣2 + a22 ∣Φ2∣2 ) Φ2 + 2 (b21 Φ1 Φ*2 + b22 Φ2 Φ1*) Φ2 = 0

k=1

Specific Case Transformation: Φ1(x1, x2, … , xN , t ) = ψ1(x1, x2, … , xN , t ) + ψ2(x1, x2, … , xN , t ), Φ2(x1, x2, … , xN , t ) = ψ1(x1, x2, … , xN , t ) + i ψ2(x1, x2, … , xN , t ) Equation: N

i Φ1t + N

i Φ2 t +

∑ Φ1x x

k k

− 2(a + b ) (∣Φ1∣2 + ∣Φ2∣2 ) Φ1 + 2 ((a + i b ) Φ1 Φ*2 + (a − i b ) Φ2 Φ1*) Φ1 = 0,

k=1

∑ Φ2 x x

k k

− 2(a + b ) (∣Φ1∣2 + ∣Φ2∣2 ) Φ2 + 2 ((a + i b ) Φ1 Φ*2 + (a − i b ) Φ2 Φ1*) Φ2 = 0

k=1

References [1] Gordon J P 1983 Interaction forces among solitons in optical fibers Opt. Lett. 8 596–8 [2] Buryak A V, Kivshar Y S and Parker D F 1996 Coupling between dark and bright solitons Phys. Lett. A 215 57–62

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Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

[3] Guo B L and Ling L M 2011 Rogue wave, breathers and bright-dark-rogue solutions for the coupled Schrödinger equations Chin. Phys. Lett. 28 110202 [4] Khare A and Saxena A 2015 Periodic and hyperbolic soliton solutions of a number of nonlocal nonlinear equations J. Math. Phys. 56 032104–27 [5] Porubov A V and Parker D F 1999 Some general periodic solutions to coupled nonlinear Schrödinger equations Wave Motion 29 97–109

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IOP Publishing

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations Usama Al Khawaja and Laila Al Sakkaf

Chapter 8 Discrete Nonlinear Schrödinger Equation

A Glance at Chapter 8

doi:10.1088/978-0-7503-2428-1ch8

8-1

ª IOP Publishing Ltd 2020

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

A Statistical View of Chapter 8 Equation

Solutions a2 ∣ ψn

∣2

ψn

35

1

i ψnt + ψn + 1 + ψn − 1 − 2 ψn +

2

i ψnt + ψn + 1 + ψn − 1 − 2 ψn + a2 F [∣ψn∣2 ] ψn = 0

6

3

i ψnt + ψn + 1 + ψn − 1 − 2 ψn + a2 ∣ψn∣2 ψn = 0

6

4

i ψnt + ψn + 1 + ψn − 1 − 2 ψn + a2 (ψn + 1 + ψn − 1) ∣ψn∣2 = 0

5

i ψnt + a1 (ψn + 1 + ψn − 1 − 2 ψn ) + a2 ∣ψn∣2 ψn + (a3 ∣ψn∣2 + a 4 ∣ψn∣4 )(ψn + 1 + ψn − 1) = 0

5

6

i ψnt + a1 (ψn + 1 + ψn − 1 − 2 ψn ) + f [ψn − 1, ψn, ψn + 1] = 0

7

7

i ψ1nt + ψ1n + 1 + ψ1n − 1 − 2 ψ1n + (μ1 ∣ψ1n∣2 + μ2 ∣ψ2 n∣2 ) (ψ1n + 1 + ψ1n − 1 +

1 + μ ∣ ψn ∣2

i ψ2 nt + [ψ2 n + 1 + ψ2 n − 1 − (2 + + ψ2 n − 1 + (

8

ν 2 − 2 μ2 ) μ2

−

ν2 ) μ2

ν1 − 2 μ1 μ1

ψ1n ) = 0,

14

ψ2 n ] + (μ1 ∣ψ1n∣2 + μ2 ∣ψ2 n∣2 ) [ψ2 n + 1

i ψ1nt + ψ1n + 1 + ψ1n − 1 − 2 ψ1n + (μ1 ∣ψ1n∣2 + μ2 ∣ψ2 n∣2 ) (ψ1n + 1 + ψ1n − 1) = 0, 2 μ2 μ1

18

ψ2 n + (μ1 ∣ψ1n∣2 + μ2 ∣ψ2 n∣2 ) (ψ2 n + 1 + ψ2 n − 1) = 0

i ψ1nt + ψ1n + 1 + ψ1n − 1 − 2 ψ1n + i ψ2 nt + ψ2 n + 1 + ψ2 n − 1 − 2 ψ2 n +

Total

μ12

12

ψ2 n ] = 0

i ψ2 nt + ψ2 n + 1 + ψ2 n − 1 −

9

ν1 μ2

=0

ν1 (μ1 ∣ ψ1n ∣2 +μ2 ∣ ψ2 n ∣2 ) ψ1n μ1 (1 + μ1 ∣ ψ1n ∣2 +μ2 ∣ ψ2 n ∣2 ) ⎡ ⎢ν 2 − ⎢⎣

ν1 μ22 μ12

3

= 0,

⎤ + ν2 (μ1 ∣ ψ1n ∣2 +μ2 ∣ ψ2 n ∣2 )⎥ ψ2 n ⎥⎦

μ2 (1 + μ1 ∣ ψ1n ∣2 +μ2 ∣ ψ2 n ∣2 )

=0

9

106

8.1 Discrete NLSE with Saturable Nonlinearity Equation:

i ψnt + ψn + 1 + ψn − 1 − 2 ψn +

a2 ∣ψn∣2 ψn = 0, 1 + μ ∣ψn∣2

(8.1)

where ψn = ψ (n, t ) is the complex function profile, the integer site index, n, and t are its two independent variables, a2 and μ are real constants. Solutions: 8.1.1 Nonstaggered Solutions Solution 1. Constant Amplitude discrete continuous wave (CW), t- and n-dependent phase 8-2

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

ψ (n , t ) = A0 e i[A1 (n − n 0) − A2 (t − t0) + ϕ0],

(8.2)

where A2 = 4 sin2(A1/2) −

a2 A02 1 + μ A02

,

A0, A1, t0, n0, and ϕ0 are arbitrary real constants. • Reference: [2]. Solution 2. sec(n)

ψ (n , t ) = A0 sec[A1 (n − n 0)] e−i [A2 (t − t0) + ϕ0], where A0 = A2 =

(8.3)

sin(A1) , −μ 2 μ − a2 , μ

a2 = 2 μ cos(A1), μ < 0, A1, t0, n0, and ϕ0 are arbitrary real constants.

• Reference: [3]. Solution 3. tan(n)

ψ (n , t ) = A0 tan[A1 (n − n 0)] e−i [A2 (t − t0) + ϕ0], where A0 =

(8.4)

tan(A1) , −μ

A2 = 2 − 2 sec 2(A1), a2 = 2 μ sec 2(A1), μ < 0, A1, t0, n0, and ϕ0 are arbitrary real constants.

• Reference: [3], we corrected the expression of A0. Solution 4. sech(n) discrete bright soliton (Figure 8.1)

ψ (n , t ) = A0 sech[A1 (n − n 0)] e−i [A2 (t − t0) + ϕ0], where A0 =

sinh(A1) , μ 2μ−a A2 = μ 2 , a sech(A ) μ= 2 2 1

> 0,

8-3

(8.5)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Figure 8.1. Discrete bright soliton (8.5) at t = 0. (a) Absolute value, (b) real part, with a2 = A1 = 1 and n 0 = t0 = ϕ0 = 0 . The lines are guides for the eye.

A1, t0, n0, and ϕ0 are arbitrary real constants. • Reference: [2]. Solution 5. csch(n)

ψ (n , t ) = A0 csch[A1 (n − n 0)] e−i [A2 (t − t0) + ϕ0], where A0 = A2 =

(8.6)

sinh(A1) , −μ 2 μ − a2 , μ

a2 = 2 μ cosh(A1), μ < 0, A1, t0, n0, and ϕ0 are arbitrary real constants.

• Reference: [3]. Solution 6. tanh(n) discrete dark soliton (Figure 8.2)

ψ (n , t ) = A0 tanh[A1 (n − n 0)] e−i [A2 (t − t0) + ϕ0], where A0 = A2 =

tanh(A1) , −μ 2 μ − a2 , μ

a2 = 2 μ sech2(A1), μ < 0, A1, t0, n0, and ϕ0 are arbitrary real constants.

• Reference: [3].

8-4

(8.7)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Figure 8.2. Discrete dark soliton (8.7) at t = 0. (a) Absolute value, (b) Real part, with A1 = 1, μ = −1, and n 0 = t0 = ϕ0 = 0 . The lines are guide for the eye.

Solution 7. coth(n)

ψ (n , t ) = A0 coth[A1 (n − n 0)] e−i [A2 (t − t0) + ϕ0], where A0 = A2 =

(8.8)

tanh(A1) , −μ 2 μ − a2 , μ

a2 = 2 μ sech2(A1), μ < 0, A1, t0, n0, and ϕ0 are arbitrary real constants.

• Reference: [3]. Solution 8. sn(n,m) discrete solitary wave (SW) (Figure 8.3)

ψ (n , t ) = A0 sn[A1 (n − n 0), m ] e−i [A2 (t − t0) + ϕ0], where A0 = A2 =

(8.9)

m sn(A1, m ) , −μ 2 μ − a2 , μ

a2 = 2 μ cn(A1, m ) dn(A1, m ), μ < 0, 0 ⩽ m ⩽ 1, A1, t0, n0, and ϕ0 are arbitrary real constants.

• Reference: [3], we corrected the expression of A0. Solution 9. cn(n,m) discrete SW (Figure 8.4)

ψ (n , t ) = A0 cn[A1 (n − n 0), m ] e−i [A2 (t − t0) + ϕ0],

8-5

(8.10)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Figure 8.3. Discrete solitary wave (8.9) at t = 0. (a) Absolute value, (b) real part, with A1 = 1/3, μ = −1/2 , m = 1/2 , and n 0 = t0 = ϕ0 = 0 . The lines are guides for the eye.

Figure 8.4. Discrete solitary wave (8.10) at t = 0. (a) Absolute value, (b) real part, with a2 = 1, A1 = 1/3, m = 1/2 , and n 0 = t0 = ϕ0 = 0 . The lines are guides for the eye.

where A0 =

m sn(A1, m ) , μ dn(A1, m ) 2μ−a A2 = μ 2 , a dn2(A , m ) μ = 22 cn(A ,1m ) > 1

0,

0 ⩽ m ⩽ 1, A1, t0, n0, and ϕ0 are arbitrary real constants.

• Reference: [2]. Solution 10. dn(n,m) discrete SW (Figure 8.5)

ψ (n , t ) = A0 dn[A1 (n − n 0), m ] e−i [A2 (t − t0) + ϕ0], where A0 =

sn(A1, m ) , μ cn(A1, m ) 2 μ − a2 A2 = μ , a cn2(A , m ) μ = 22 dn(A 1, m ) > 1

0,

8-6

(8.11)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Figure 8.5. Discrete solitary wave (8.11) at t = 0. (a) Absolute value, (b) real part, with a2 = 1, A1 = 1/3, m = 1/2 , and n 0 = t0 = ϕ0 = 0 . The lines are guides for the eye.

0 ⩽ m ⩽ 1, A1, t0, n0, and ϕ0 are arbitrary real constants.

• Reference: [2]. Solution 11. ns(n,m) discrete SW

ψ (n , t ) = A0 ns[A1 (n − n 0), m ] e−i [A2 (t − t0) + ϕ0], where A0 = A2 =

(8.12)

sn(A1, m ) , −μ 2 μ − a2 , μ

a2 = 2 μ dn(A1, m ) cn(A1, m ), μ < 0, 0 ⩽ m ⩽ 1, A1, t0, n0, and ϕ0 are arbitrary real constants.

• Reference: [3]. Solution 12. cs(n,m) discrete SW

ψ (n , t ) = A0 cs[A1 (n − n 0), m ] e−i [A2 (t − t0) + ϕ0], where A0 = A2 =

a2 =

sn(A1, m ) , −μ cn(A1, m ) 2 μ − a2 , μ 2 μ dn(A1, m ) , cn2(A1, m )

μ < 0, 0 ⩽ m ⩽ 1, A1, t0, n0, and ϕ0 are arbitrary real constants.

• Reference: [3].

8-7

(8.13)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Solution 13. ds(n,m) discrete SW

ψ (n , t ) = A0 ds[A1 (n − n 0), m ] e−i [A2 (t − t0) + ϕ0], where A0 = A2 =

a2 =

(8.14)

sn(A1, m ) , −μ dn(A1, m ) 2 μ − a2 , μ 2 μ cn(A1, m ) , dn2(A1, m )

μ < 0, 0 ⩽ m ⩽ 1, A1, t0, n0, and ϕ0 are arbitrary real constants.

• Reference: [3]. Solution 14. cd(n,m) discrete SW

ψ (n , t ) = A0 cd[A1 (n − n 0), m ] e−i [A2 (t − t0) + ϕ0], where A0 = A2 =

(8.15)

m sn(A1, m ) , −μ 2 μ − a2 , μ

a2 = 2 μ cn(A1, m ) dn(A1, m ), μ < 0, 0 ⩽ m ⩽ 1, A1, t0, n0, and ϕ0 are arbitrary real constants.

• Reference: [3], we corrected the expression of A0. Solution 15. dc(n,m) discrete SW

ψ (n , t ) = A0 dc[A1 (n − n 0), m ] e−i [A2 (t − t0) + ϕ0], where A0 = A2 =

sn(A1, m ) , −μ 2 μ − a2 , μ

a2 = 2 μ cn(A1, m ) dn(A1, m ), μ < 0, 0 ⩽ m ⩽ 1, A1, t0, n0, and ϕ0 are arbitrary real constants.

• Reference: [3].

8-8

(8.16)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Solution 16. dn(n,m) + cn(n,m) discrete SW

ψ (n , t ) =

where A0 =

{

A0 B dn[A1 (n − n 0), m ] + 0 2 2

×

e−i [A2 (t − t0) + ϕ0],

}

m cn[A1 (n − n 0), m ]

(8.17)

2 , cs(A1, m ) + ds(A1, m )

B0 = ±A0, A2 = 2 − a2 , 4 a2 = cn(A , m ) + dn(A , m ) , 1

1

μ = 1, 0 ⩽ m ⩽ 1, A1, t0, and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer.

• Reference: [1], taken from the nonlocal case. 8.1.2 Staggered Solutions If ψ (n, t ; a2 ) is a nonstaggered solution of (8.1), then

ψs(n , t , a2 ) = ( −1)n ψ *(n , t , −a2 ) e−4 i (t − t0)

(8.18)

is a staggered solution of the same equation, where ψ * is the complex conjugate of the nonstaggered solution. Solution 1. Constant Amplitude staggered discrete CW, t- and n-dependent phase

ψs(n , t ) = ( −1)n A0 e−i[A1 (n − n 0) − (A2 − 4) (t − t0) + ϕ0],

(8.19)

where A2 = 4 sin2(A1/2) +

a2 A02 1 + μ A02

,

A0, A1, t0, n0, and ϕ0 are arbitrary real constants. Solution 2. cos(n) I

ψs(n , t ) = A0 cos[π (n − n 0)] e−i [A2 (t − t0) + ϕ0], where A2 = 4 −

a2 A02 1 + μ A02

,

A0, t0, n0, and ϕ0 are arbitrary real constants. • Reference: [4].

8-9

(8.20)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Solution 3. cos(n) II

⎡π ⎤ ψs(n , t ) = A0 cos ⎢ (n − n 0)⎥ e ±i [A2 (t − t0) + ϕ0], ⎣2 ⎦

(8.21)

where A2 = −2 +

A02 a2 A02

μ + sec 2 [π (n − n 0 ) / 2]

,

A0, t0, n0, and ϕ0 are arbitrary real constants. • Reference: [4]. Solution 4. cos(n)-sin(n)

⎧ ⎡π ⎤⎫ ⎡π ⎤ ψs(n , t ) = A0 ⎨cos ⎢ (n − n 0)⎥ − sin ⎢ (n − n 0)⎥⎬ e−i [A2 (t − t0) + ϕ0], ⎣2 ⎦ ⎣2 ⎦⎭ ⎩

(8.22)

where A2 = 2 −

a2 A02 1 + μ A02

,

A0, t0, and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer. • Reference: [4]. Solution 5. sec(n)

ψs(n , t ) = ( −1)n A0 sec[A1 (n − n 0)] e i [A2 (t − t0) − 4 (t − t0) + ϕ0], where A0 = A2 =

(8.23)

sin(A1) , −μ 2 μ + a2 , μ

a2 = −2 μ cos(A1), μ < 0, A1, t0, n0, and ϕ0 are arbitrary real constants.

Solution 6. tan(n)

ψs(n , t ) = ( −1)n A0 tan[A1 (n − n 0)] e i [A2 (t − t0) − 4 (t − t0) + ϕ0], where A0 =

tan(A1) , −μ

A2 = 2 − 2 sec 2(A1), a2 = −2 μ sec 2(A1), μ < 0, A1, t0, n0, and ϕ0 are arbitrary real constants.

8-10

(8.24)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Figure 8.6. Staggered discrete bright soliton (8.25) at t = 0. (a) Absolute value, (b) real part, with a2 = −1, A1 = 1, and n 0 = t0 = ϕ0 = 0 . The lines are guides for the eye.

Solution 7. sech(n) staggered discrete bright soliton (Figure 8.6)

ψs(n , t ) = ( −1)n A0 sech[A1 (n − n 0)] e i [A2 (t − t0) − 4 (t − t0) + ϕ0],

(8.25)

where A0 =

sinh(A1) , μ 2 μ + a2 A2 = μ , a sech(A ) μ=− 2 2 1

> 0, A1, t0, n0, and ϕ0 are arbitrary real constants.

Solution 8. csch(n)

ψs(n , t ) = ( −1)n A0 csch[A1 (n − n 0)] e i [A2 (t − t0) − 4 (t − t0) + ϕ0], where A0 = A2 =

(8.26)

sinh(A1) , −μ 2 μ + a2 , μ

a2 = −2 μ cosh(A1), μ < 0, A1, t0, n0, and ϕ0 are arbitrary real constants.

Solution 9. tanh(n) staggered discrete dark soliton (Figure 8.7)

ψs(n , t ) = ( −1)n A0 tanh[A1 (n − n 0)] e i [A2 (t − t0) − 4 (t − t0) + ϕ0], where A0 = A2 =

tanh(A1) , −μ 2 μ + a2 , μ

a2 = −2 μ sech2(A1),

8-11

(8.27)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Figure 8.7. Staggered discrete dark soliton (8.27) at t = 0. (a) Absolute value, (b) real part, with A1 = 1, μ = −1, and n 0 = t0 = ϕ0 = 0 . The lines are guides for the eye.

μ < 0, A1, t0, n0, and ϕ0 are arbitrary real constants.

Solution 10. coth(n)

ψs(n , t ) = ( −1)n A0 coth[A1 (n − n 0)] e i [A2 (t − t0) − 4 (t − t0) + ϕ0], where A0 = A2 =

(8.28)

tanh(A1) , −μ 2 μ + a2 , μ

a2 = −2 μ sech2(A1), μ < 0, A1, t0, n0, and ϕ0 are arbitrary real constants.

Solution 11. sn(n,m) staggered discrete SW (Figure 8.8)

ψs(n , t ) = ( −1)n A0 sn[A1 (n − n 0), m ] e i [A2 (t − t0) − 4 (t − t0) + ϕ0], where A0 = A2 =

(8.29)

m sn(A1, m ) , −μ 2 μ + a2 , μ

a2 = −2 μ cn(A1, m ) dn(A1, m ), μ < 0, 0 ⩽ m ⩽ 1, A1, t0, n0, and ϕ0 are arbitrary real constants.

Solution 12. cn(n,m) staggered discrete SW (Figure 8.9)

ψs(n , t ) = ( −1)n A0 cn[A1 (n − n 0), m ] e i [A2 (t − t0) − 4 (t − t0) + ϕ0], where A0 =

m sn(A1, m ) , μ dn(A1, m )

8-12

(8.30)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Figure 8.8. Staggered discrete solitary wave (8.29) at t = 0. (a) Absolute value, (b) real part, with A1 = 1/3, μ = −1/2 , m = 1/2 , and n 0 = t0 = ϕ0 = 0 . The lines are guides for the eye.

Figure 8.9. Staggered discrete solitary wave (8.30) at t = 0. (a) Absolute value, (b) real part, with a2 = −1, A1 = 1/3, m = 1/2 , and n 0 = t0 = ϕ0 = 0 . The lines are guides for the eye. 2 μ + a2 , μ 2 a dn (A , m ) − 22 cn(A ,1m ) 1

A2 =

μ=

> 0,

0 ⩽ m ⩽ 1, A1, t0, n0, and ϕ0 are arbitrary real constants.

Solution 13. dn(n,m) staggered discrete SW (Figure 8.10)

ψs(n , t ) = ( −1)n A0 dn[A1 (n − n 0), m ] e i [A2 (t − t0) − 4 (t − t0) + ϕ0],

(8.31)

where A0 =

sn(A1, m ) , μ cn(A1, m ) 2μ+a A2 = μ 2 , a cn2(A , m ) μ = − 22 dn(A 1, m ) > 1

0,

0 < m ⩽ 1, A1, t0, n0, and ϕ0 are arbitrary real constants.

Solution 14. ns(n,m) staggered discrete SW

ψs(n , t ) = ( −1)n A0 ns[A1 (n − n 0), m ] e i [A2 (t − t0) − 4 (t − t0) + ϕ0],

8-13

(8.32)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Figure 8.10. Staggered discrete solitary wave (8.31) at t = 0. (a) Absolute value, (b) real part, with a2 = −1, A1 = 1/3, m = 1/2 , and n 0 = t0 = ϕ0 = 0 . The lines are guides for the eye.

where A0 = A2 =

sn(A1, m ) , −μ 2 μ + a2 , μ

a2 = −2 μ dn(A1, m ) cn(A1, m ), μ < 0, 0 ⩽ m ⩽ 1, A1, t0, n0, and ϕ0 are arbitrary real constants.

Solution 15. cs(n,m) staggered discrete SW

ψs(n , t ) = ( −1)n A0 cs[A1 (n − n 0), m ] e i [A2 (t − t0) − 4 (t − t0) + ϕ0],

(8.33)

where A0 =

sn(A1, m ) , −μ cn(A1, m ) 2μ+a A2 = μ 2 , 2 μ dn(A , m ) a2 = − cn2(A , 1m ) , 1

μ < 0, 0 ⩽ m ⩽ 1, A1, t0, n0, and ϕ0 are arbitrary real constants.

Solution 16. ds(n,m) staggered discrete SW

ψs(n , t ) = ( −1)n A0 ds[A1 (n − n 0), m ] e i [A2 (t − t0) − 4 (t − t0) + ϕ0], where A0 =

sn(A1, m ) , −μ dn(A1, m ) 2 μ + a2 A2 = μ , 2 μ cn(A , m ) a2 = − dn2(A ,1m ) , 1

μ < 0,

8-14

(8.34)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

0 ⩽ m ⩽ 1, A1, t0, n0, and ϕ0 are arbitrary real constants.

Solution 17. cd(n,m) staggered discrete SW

ψs(n , t ) = ( −1)n A0 cd[A1 (n − n 0), m ] e i [A2 (t − t0) − 4 (t − t0) + ϕ0], where A0 = A2 =

(8.35)

m sn(A1, m ) , −μ 2 μ + a2 , μ

a2 = −2 μ cn(A1, m ) dn(A1, m ), μ < 0, 0 ⩽ m ⩽ 1, A1, t0, n0, and ϕ0 are arbitrary real constants.

Solution 18. dc(n,m) staggered discrete SW

ψs(n , t ) = ( −1)n A0 dc[A1 (n − n 0), m ] e i [A2 (t − t0) − 4 (t − t0) + ϕ0], where A0 = A2 =

sn(A1, m ) , −μ 2 μ + a2 , μ

a2 = −2 μ cn(A1, m ) dn(A1, m ), μ < 0, 0 ⩽ m ⩽ 1, A1, t0, n0, and ϕ0 are arbitrary real constants.

8-15

(8.36)

8-16

5. ψ (n, t ) = A0 csch[A1 (n − n 0 )] e−i [A2 (t − t0 ) + ϕ0]

4. ψ (n, t ) = A0 sech[A1 (n − n 0 )] e−i [A2 (t − t0 ) + ϕ0]

3. ψ (n, t ) = A0 tan[A1 (n − n 0 )] e−i [A2 (t − t0 ) + ϕ0]

2. ψ (n, t ) = A0 sec[A1 (n − n 0 )] e−i [A2 (t − t0 ) + ϕ0]

sin(A1) , −μ

A2 =

2 μ − a2 , μ

A2 = 2 − 2 sec 2(A1),

sec 2(A1),

tan(A1) , −μ

> 0,

A2 =

2 μ − a2 , μ

sinh(A1) , −μ

A2 =

2 μ − a2 , μ

a2 = 2 μ cosh(A1), μ < 0 , A1, t0, and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

A0 =

A1, t0, and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

μ=

A0 =

sinh(A1) , μ a2 sech(A1) 2

a2 = 2 μ μ < 0, A1, t0, ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

A0 =

a2 = 2 μ cos(A1), μ < 0 , A1, t0, and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

A0 =

A2 = 4 sin2(A1/2) − 1 + μ A 2 , A0, A1, t0, and ϕ0 are 0 arbitrary real constants, n0 is an arbitrary real integer

=0

1. ψ (n, t ) = A0 ei[A1 (n − n0 ) − A2 (t − t0 ) + ϕ0]

1 + μ ∣ ψn ∣2

a2 ∣ ψn ∣2 ψn

Conditions a2 A02

Equation: i ψnt + ψn + 1 + ψn − 1 − 2 ψn +

# Solution

Nonstaggered Solutions

8.2 Summary of Section 8.1

–

discrete bright soliton

–

–

discrete constant wave, t- and n-dependent phase

Name

(8.6)

(8.5)

(8.4)

(8.3)

(8.2)

Eq. #

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

8-17

11. ψ (n, t ) = A0 ns[A1 (n − n 0 ), m ] e−i [A2 (t − t0 ) + ϕ0]

10. ψ (n, t ) = A0 dn[A1 (n − n 0 ), m ] e−i [A2 (t − t0 ) + ϕ0]

9. ψ (n, t ) = A0 cn[A1 (n − n 0 ), m ] e−i [A2 (t − t0 ) + ϕ0]

8. ψ (n, t ) = A0 sn[A1 (n − n 0 ), m ] e−i [A2 (t − t0 ) + ϕ0]

7. ψ (n, t ) = A0 coth[A1 (n − n 0 )] e−i [A2 (t − t0 ) + ϕ0]

6. ψ (n, t ) = A0 tanh[A1 (n − n 0 )] e−i [A2 (t − t0 ) + ϕ0]

tanh(A1) , −μ

A2 =

2 μ − a2 , μ

tanh(A1) , −μ

A2 =

2 μ − a2 , μ

m sn(A1, m ) , −μ

A2 =

2 μ − a2 , μ

2 μ − a2 , μ

0 , 0 ⩽ m ⩽ 1,

A2 =

2 μ − a2 , μ

0 ⩽ m ⩽ 1,

=

sn(A1, m ) , −μ

A2 =

2 μ − a2 , μ

a2 = 2 μ dn(A1, m ) cn(A1, m ), μ < 0, 0 ⩽ m ⩽ 1, A1, t0, and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

A0 =

A1, t0, and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

μ=

A0 =

sn(A1, m ) , A2 μ cn(A1, m ) a2 cn2(A1, m ) > 0, 2 dn(A1, m )

A1, t0, and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

μ=

m sn(A1, m ) , μ dn(A1, m ) a2 dn2(A1, m ) > 2 cn(A1, m )

A0 =

a2 = 2 μ cn(A1, m ) dn(A1, m ), μ < 0, 0 ⩽ m ⩽ 1, A1, t0, and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

A0 =

a2 = 2 μ sech2(A1), μ < 0, A1, t0, and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

A0 =

a2 = 2 μ sech2(A1), μ < 0, A1, t0, and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

A0 =

discrete solitary wave

discrete solitary wave

discrete solitary wave

discrete solitary wave

–

discrete dark soliton

(8.12)

(8.11)

(8.10)

(8.9)

(8.8)

(8.7)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

8-18

16. ψ (n, t ) =

A0 2

dn[A1 (n − n 0 ), m ] +

× e−i [A2 (t − t0 ) + ϕ0]

{

B0 2

}

m cn[A1 (n − n 0 ), m ]

15. ψ (n, t ) = A0 dc[A1 (n − n 0 ), m ] e−i [A2 (t − t0 ) + ϕ0]

14. ψ (n, t ) = A0 cd[A1 (n − n 0 ), m ] e−i [A2 (t − t0 ) + ϕ0]

13. ψ (n, t ) = A0 ds[A1 (n − n 0 ), m ] e−i [A2 (t − t0 ) + ϕ0]

12. ψ (n, t ) = A0 cs[A1 (n − n 0 ), m ] e−i [A2 (t − t0 ) + ϕ0]

(Continued ) sn(A1, m ) , −μ cn(A1, m ) 2 μ dn(A1, m ) ,μ cn2(A1, m ) 2 μ − a2 , μ

< 0, 0 ⩽ m ⩽ 1,

A2 =

sn(A1, m ) , −μ dn(A1, m ) 2 μ cn(A1, m ) ,μ dn2(A1, m ) 2 μ − a2 , μ

< 0 , 0 ⩽ m ⩽ 1,

A2 =

m sn(A1, m ) , −μ

A2 =

2 μ − a2 , μ

sn(A1, m ) , −μ

A2 =

2 μ − a2 , μ

1

μ = 1, 0 ⩽ m ⩽ 1, A1, t0, and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

1

2 , cs(A1, m ) + ds(A1, m )

B0 = ±A0 , A2 = 2 − a2 , 4 a2 = cn(A , m) + dn(A , m) ,

A0 =

a2 = 2 μ cn(A1, m ) dn(A1, m ), μ < 0 , 0 ⩽ m ⩽ 1, A1, t0, and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

A0 =

a2 = 2 μ cn(A1, m ) dn(A1, m ), μ < 0 , 0 ⩽ m ⩽ 1, A1, t0, and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

A0 =

A1, t0, and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

a2 =

A0 =

A1, t0, and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

a2 =

A0 =

discrete solitary wave

discrete solitary wave

discrete solitary wave

discrete solitary wave

discrete solitary wave

(8.17)

(8.16)

(8.15)

(8.14)

(8.13)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

}

8-19

6. ψs (n, t ) = (−1) n A0 tan[A1 (n − n 0 )] ei [A2 (t − t0 ) − 4 (t − t0 ) + ϕ0]

5. ψs (n, t ) = (−1) n A0 sec[A1 (n − n 0 )] ei [A2 (t − t0 ) − 4 (t − t0 ) + ϕ0]

1 + μ A02

,

a2 A02 1 + μ A02

,

A02 a2 A02 μ + sec 2 [π (n − n0 ) / 2]

,

,

sin(A1) , −μ

A2 =

2 μ + a2 , μ

tan(A1) , −μ

A2 = 2 − 2 sec 2(A1),

a2 = −2 μ sec 2(A1), μ < 0 , A1, t0, and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

A0 =

a2 = −2 μ cos(A1), μ < 0 , A1, t0, and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

A0 =

A0, t0, and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

1 + μ A02

a2 A02

A0, t0, and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

A2 = −2 +

A0, t0, and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

A2 = 4 −

4. ψs (n, t ) = A0 cos ⎡⎣ π (n − n 0 )⎤⎦ − sin ⎡⎣ π (n − n 0 )⎤⎦ e−i [A2 (t − t0 ) + ϕ0] A2 = 2 − 2 2

{

π 3. ψs (n, t ) = A0 cos ⎡⎣ 2 (n − n 0 )⎤⎦ e±i [A2 (t − t0 ) + ϕ0]

2. ψs (n, t ) = A0 cos[π (n − n 0 )] e−i [A2 (t − t0 ) + ϕ0]

A2 = 4 sin2(A1/2) +

1. ψs (n, t ) = (−1) n A0 e−i[A1 (n − n0 ) − (A2 − 4) (t − t0 ) + ϕ0]

a2 A02

A0, A1, t0, ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

Conditions

# Solution

Staggered Solutions: ψs (n, t, a2 ) = (−1) n ψ *(n, t, −a2 ) e−4 i (t − t0 )

–

–

–

–

–

staggered discrete constant wave, t- and n-dependent phase

Name

(8.24)

(8.23)

(8.22)

(8.21)

(8.20)

(8.19)

Eq. #

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

8-20

12. ψs (n, t ) = (−1) n A0 cn[A1 (n − n 0 ), m ] ei [A2 (t − t0 ) − 4 (t − t0 ) + ϕ0]

11. ψs (n, t ) = (−1) n A0 sn[A1 (n − n 0 ), m ] ei [A2 (t − t0 ) − 4 (t − t0 ) + ϕ0]

10. ψs (n, t ) = (−1) n A0 coth[A1 (n − n 0 )] ei [A2 (t − t0 ) − 4 (t − t0 ) + ϕ0]

9. ψs (n, t ) = (−1) n A0 tanh[A1 (n − n 0 )] ei [A2 (t − t0 ) − 4 (t − t0 ) + ϕ0]

8. ψs (n, t ) = (−1) n A0 csch[A1 (n − n 0 )] ei [A2 (t − t0 ) − 4 (t − t0 ) + ϕ0]

7. ψs (n, t ) = (−1) n A0 sech[A1 (n − n 0 )] ei [A2 (t − t0 ) − 4 (t − t0 ) + ϕ0]

(Continued ) 2 μ + a2 , μ

sinh(A1) , −μ

A2 =

2 μ + a2 , μ

tanh(A1) , −μ

A2 =

2 μ + a2 , μ

tanh(A1) , −μ

A2 =

2 μ + a2 , μ

m sn(A1, m ) , −μ

A2 =

2 μ + a2 , μ

2 cn(A1, m )

a2 dn2(A1, m )

m sn(A1, m ) , μ dn(A1, m )

2 μ + a2 , μ

> 0 , 0 ⩽ m ⩽ 1,

A2 =

A1, t0, and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

μ=−

A0 =

a2 = −2 μ cn(A1, m ) dn(A1, m ), μ < 0 , 0 ⩽ m ⩽ 1, A1, t0, and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

A0 =

a2 = −2 μ sech2(A1) , μ < 0, A1, t0, and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

A0 =

a2 = −2 μ sech2(A1) , μ < 0, A1, t0, and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

A0 =

a2 = −2 μ cosh(A1), μ < 0 , A1, t0, and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

A0 =

A1, t0, and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

> 0,

A2 =

a2 sech(A1) 2

sinh(A1) , μ

μ=−

A0 =

staggered discrete solitary wave

staggered discrete solitary wave

–

staggered discrete dark soliton

–

staggered discrete bright soliton

(8.30)

(8.29)

(8.28)

(8.27)

(8.26)

(8.25)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

8-21

18. ψs (n, t ) = (−1) n A0 dc[A1 (n − n 0 ), m ] ei [A2 (t − t0 ) − 4 (t − t0 ) + ϕ0]

17. ψs (n, t ) = (−1) n A0 cd[A1 (n − n 0 ), m ] ei [A2 (t − t0 ) − 4 (t − t0 ) + ϕ0]

16. ψs (n, t ) = (−1) n A0 ds[A1 (n − n 0 ), m ] ei [A2 (t − t0 ) − 4 (t − t0 ) + ϕ0]

15. ψs (n, t ) = (−1) n A0 cs[A1 (n − n 0 ), m ] ei [A2 (t − t0 ) − 4 (t − t0 ) + ϕ0]

14. ψs (n, t ) = (−1) n A0 ns[A1 (n − n 0 ), m ] ei [A2 (t − t0 ) − 4 (t − t0 ) + ϕ0]

13. ψs (n, t ) = (−1) n A0 dn[A1 (n − n 0 ), m ] ei [A2 (t − t0 ) − 4 (t − t0 ) + ϕ0]

2 μ + a2 , μ

> 0 , 0 < m ⩽ 1,

A2 =

sn(A1, m ) , −μ

A2 =

2 μ + a2 , μ

μ < 0, 0 ⩽ m ⩽ 1,

2 μ cn(A1, m ) , dn2(A1, m )

a2 = −

m sn(A1, m ) , −μ

A2 =

2 μ + a2 , μ

sn(A1, m ) , −μ

A2 =

2 μ + a2 , μ

a2 = −2 μ cn(A1, m ) dn(A1, m ), μ < 0 , 0 ⩽ m ⩽ 1, A1, t0, and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

A0 =

a2 = −2 μ cn(A1, m ) dn(A1, m ), μ < 0 , 0 ⩽ m ⩽ 1, A1, t0, and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

A0 =

A1, t0, and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

A2 =

sn(A1, m ) , −μ dn(A1, m )

A0 =

2 μ + a2 , μ

A1, t0, and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

μ < 0 , 0 ⩽ m ⩽ 1,

2 μ dn(A1, m ) , cn2(A1, m )

a2 = −

2 μ + a2 , μ

A2 =

sn(A1, m ) , −μ cn(A1, m )

A0 =

a2 = −2 μ dn(A1, m ) cn(A1, m ), μ < 0 , 0 ⩽ m ⩽ 1, A1, t0, and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

A0 =

A1, t0, and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

μ=

sn(A1, m ) , μ cn(A1, m )

a cn2(A , m ) − 22 dn(A 1, m) 1

A0 =

staggered discrete solitary wave

staggered discrete solitary wave

staggered discrete solitary wave

staggered discrete solitary wave

staggered discrete solitary wave

staggered discrete solitary wave

(8.36)

(8.35)

(8.34)

(8.33)

(8.32)

(8.31)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

8.3 Short-period Solutions with General, Kerr, and Saturable Nonlinearities Equations: Case I: DNLS with General Nonlinearity (GN)

i ψnt + ψn + 1 + ψn − 1 − 2 ψn + a2 F [∣ψn∣2 ] ψn = 0,

(8.37)

Case II: DNLS with Kerr Nonlinearity (KN)

i ψnt + ψn + 1 + ψn − 1 − 2 ψn + a2 ∣ψn∣2 ψn = 0,

(8.38)

Case III: DNLS with Saturable Nonlinearity (SN)

i ψnt + ψn + 1 + ψn − 1 − 2 ψn + a2

∣ψn∣2 ψn = 0, 1 + μ ∣ψn∣2

(8.39)

where a2 and μ are real constants, ψn = ψ (n, t ) is the complex function profile, the integer site index, n, and t are its two independent variables, F is a general real function. For other solutions of Case III, see section 8.1. General Solutions:

ψ (n , t ) = A0 (… , c0, c1, c2 , c3, …) e i [A2 (t − t0) + ϕ0],

(8.40)

where A0, ϕ0, t0, cj, j = 0, 1, 2, 3, … are arbitrary real constants. For specific values of A2, short-period solutions are obtained as summarized in table 8.1.

8.4 Ablowitz–Ladik Equation Equation:

i ψnt + ψn + 1 + ψn − 1 − 2 ψn + a2 (ψn + 1 + ψn − 1) ∣ψn∣2 = 0,

(8.41)

where a2 is a real constant. Solutions: Solution 1. Constant Amplitude discrete CW, t- and n-dependent phase

ψ (n , t ) = A0 e i[A1 (n − n 0) + (A2 − 2) (t − t0) + ϕ0], where A2 = 2 cos(A1) (1 + a2 A02 ), A0, A1, t0, n0, and ϕ0 are arbitrary real constants.

8-22

(8.42)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Table 8.1. Short-period solutions to general, Kerr, and saturable nonlinearities of the discrete NLSE (8.37, 8.38, 8.39), where F is a general real function and, A0, t0, and ϕ0 are arbitrary real constants.

Period Nonlinearity* Condition on A2 1

2

3

4

6

Solution

a2 F [A02 ] a2A02 a2 A02

GN KN SN

A2 = 0 − A2 = 0 −

GN KN SN

A2 = 4 − a2 F [A02 ] ψ (n , t ) = A0 (… , 1, − 1, …) ei [A2 (t − t0) + ϕ0] A2 = 4 − a2A02

GN KN SN

A2 = 3 − a2 F [A02 ] ψ (n , t ) = A0 (… , 1, 0, − 1, …) ei [A2 (t − t0) + ϕ0] A2 = 3 − a2A02

GN KN SN

i [A (t − t ) + ϕ ] A2 = 2 − a2 F [A02 ] ψ (n , t ) = A0 (… , 1, 1, − 1, − 1, …) e 2 0 0 A2 = 2 − a2A02 a A2 ψ (n , t ) = A0 (… , 1, 0, − 1, 0, …) ei [A2 (t − t0) + ϕ0] A =2− 2 0

GN KN SN

A2 = 1 − a2 F [A02 ] ψ (n , t ) = A0 (… , 1, 1, 0, − 1, − 1, 0, …) ei [A2 (t − t0) + ϕ0] A2 = 1 − a2A02

A2 = 0 −

A2 = 4 −

A2 = 3 −

2

1 + μ A02

a2 A02 1 + μ A02

a2 A02 1 + μ A02

1 + μ A02

A2 = 1 −

* GN: General Nonlinearity,

ψ (n , t ) = A0 (… , 1, 1, 1, 1, …) ei [A2 (t − t0) + ϕ0]

a2 A02 1 + μ A02

a2 F [∣ψ ∣2 ],

KN: Kerr Nonlinearity,

a2 ∣ψ ∣2,

SN: Saturable Nonlinearity,

∣ψ ∣2

a2 . 1 + μ ∣ ψ ∣2

Solution 2. sech(n) discrete bright soliton (Figure 8.11)

ψ (n , t ) = A0 sech[A1 (n − n 0)] e−i[(A2 + 2) (t − t0) + ϕ0],

(8.43)

where A2 = −2 cosh(A1), a2 =

sinh2(A1) , A02

A0 ≠ 0, A1, t0, n0, and ϕ0 are arbitrary real constants.

• Reference: [1], taken from the nonlocal case. Solution 3. tanh(n) discrete dark soliton (Figure 8.12)

ψ (n , t ) = A0 tanh[A1 (n − n 0)] e−i[(A2 + 2) (t − t0) + ϕ0],

8-23

(8.44)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Figure 8.11. Discrete bright soliton (8.43) at t = 0 with A0 = A1 = 1, and n 0 = t0 = ϕ0 = 0 . The lines are guides for the eye.

Figure 8.12. Discrete dark soliton (8.44) at t = 0 with A0 = A1 = 1, and n 0 = t0 = ϕ0 = 0 . The lines are guides for the eye.

where A2 = −2 sech2(A1), a2 =

−tanh2(A1) , A02

A0 ≠ 0, A1, t0, n0, and ϕ0 are arbitrary real constants.

• Reference: [1], taken from the nonlocal case. Solution 4. sn(n,m) discrete SW (Figure 8.13)

ψ (n , t ) = A0

m sn[A1 (n − n 0), m ] e−i[(A2 + 2) (t − t0) + ϕ0],

where A2 = −2 cn(A1, m ) dn(A1, m ), −1 , a2 = 2 2 A0 ns (A1, m )

0 ⩽ m ⩽ 1, A0 ≠ 0, A1, t0, n0, and ϕ0 are arbitrary real constants.

• Reference: [1], taken from the nonlocal case.

8-24

(8.45)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Figure 8.13. Discrete solitary wave (8.45) at t = 0 with A0 = 1, A1 = 1/3, m = 1/2 , and n 0 = t0 = ϕ0 = 0 . The lines are guides for the eye.

Figure 8.14. Discrete solitary wave (8.46) at t = 0 with A0 = 1, A1 = 1/3, m = 1/2 , and n 0 = t0 = ϕ0 = 0 . The lines are guides for the eye.

Solution 5. cn(n,m) discrete SW (Figure 8.14)

ψ (n , t ) = A0 where A2 = a2 =

m cn[A1 (n − n 0), m ] e−i[(A2 + 2) (t − t0) + ϕ0],

(8.46)

−2 cn(A1, m ) , dn2(A1, m ) 1 , A02 ds2(A1, m )

0 ⩽ m ⩽ 1, A0 ≠ 0, A1, t0, n0, and ϕ0 are arbitrary real constants.

• Reference: [1], taken from the nonlocal case. Solution 6. dn(n,m) discrete SW (Figure 8.15)

ψ (n , t ) = A0 dn[A1 (n − n 0), m ] e−i[(A2 + 2) (t − t0) + ϕ0], where A2 = a2 =

−2 dn(A1, m ) , cn2(A1, m ) 1 , A02 cs2(A1, m )

0 ⩽ m ⩽ 1,

8-25

(8.47)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Figure 8.15. Discrete solitary wave (8.47) at t = 0 with A0 = 1, A1 = 1/3, m = 1/2 , and n 0 = t0 = ϕ0 = 0 . The lines are guides for the eye.

A0 ≠ 0, A1, t0, n0, and ϕ0 are arbitrary real constants.

• Reference: [1], taken from the nonlocal case. Solution 7. dn(n,m) + cn(n,m) discrete SW

ψ (n , t ) =

{

}

A0 B m dn[A1 (n − n 0), m ] + 0 cn[A1 (n − n 0), m ] 2 2

(8.48)

× e−i[(A2 + 2) (t − t0) + ϕ0], where A2 = a2 =

−4 , cn(A1, m ) + dn(A1, m ) 4 , A02 [ds(A1, m ) + cs(A1, m )]2

B0 = ±A0, 0 ⩽ m ⩽ 1, A0 ≠ 0, A1, t0, n0, and ϕ0 are arbitrary real constants.

• Reference: [1], taken from the nonlocal case. Solution 8. cd(n,m) discrete SW

ψ (n , t ) = A0 where A2 = a2 =

m cd[A1 (n − n 0), m ] e−i[(A2 + 2) (t − t0) + ϕ0],

2 ns(A1, m ) [cs2(A1, m ) − ds2(A1, m )] [cs(2 A1, m ) + ds(2 A1, m )] , ds(A1, m ) cs(A1, m ) [cs(A1, m )ds(A1, m ) − 2 cs(A1, m ) ns(A1, m )] 3 2 cs(2 A1, m ) ds(A1, m ) ns(A1, m ) − cs (A1, m ) , A02 [cs(A1, m ) ds2(A1, m ) − 2 cs(2 A1, m ) ds(A1, m ) ns(A1, m )]

0 ⩽ m ⩽ 1, A0 ≠ 0, A1, t0, n0, and ϕ0 are arbitrary real constants.

• Reference: [1], taken from the nonlocal case.

8-26

(8.49)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Solution 9. Periodicity in n and Localization in t discrete Akhmediev breather (Figure 8.16)

⎧ ⎫ 2 + κ2 2 (t − t )] ⎪ ⎪ cos[α (n − n 0)] + i sinh[2 κ 0 ⎪ ⎪ 1 + κ2 ⎬ ψ (n , t ) = κ ⎨ ⎪ 2 + κ2 ⎪ cosh[2 κ 2 (t − t0)] − cos[α (n − n 0)] ⎪ ⎪ ⎩ 1 + κ2 ⎭ × e i [2 κ

2

(8.50)

(t − t0) + ϕ0 ],

where 1 α = cos−1 1 + κ 2 , t0, κ , n0, and ϕ0 are arbitrary real constants.

(

)

• Reference: [5]. Solution 10. Localization in n and t discrete Peregrine soliton (Figure 8.17)

Figure 8.16. Discrete Akhmediev breather (8.50). (a) t = 0, (b) n = 0, with κ = 1 and n 0 = t0 = ϕ0 = 0 . The lines are guides for the eye. Animation available online at https://iopscience.iop.org/book/978-0-7503-2428-1.

Figure 8.17. Discrete Peregrine soliton (8.51). (a) t = 0, (b) n = 0, with n 0 = t0 = ϕ0 = 0 . The lines are guides for the eye. Animation available online at https://iopscience.iop.org/book/978-0-7503-2428-1.

8-27

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

⎧ ⎫ 8[1 + 4i (t − t0)] − 1⎬e i [(t − t0) + ϕ0], ψ (n , t ) = ⎨ 2 2 ⎩ 1 + 4(n − n 0) + 32(t − t0) ⎭

(8.51)

where t0, n0, and ϕ0 are arbitrary real constants. • Reference: [5]. Solution 11. Periodicity in n and t (Figure 8.18)

⎧ ⎫ m dn[2 κ 2 (t − t0), sin2(θ )] cn[A0 (n − n 0), m2 ] ⎪ ⎪ ⎪ ⎪ + i A1 sin(θ ) sn[2 κ 2 (t − t0 ), sin2(θ )] ⎬ ψ (n , t ) = κ ⎨ 2 2 2 ⎪ A1 − m sin(θ ) cn[2 κ (t − t0 ), sin (θ )] cn[A0 (n − n 0 ), m ] ⎪ (8.52) ⎪ ⎪ ⎩ ⎭ × e i [2 κ

2

sin(θ ) (t − t0) + ϕ0 ],

where A1 = {(1 − m2 ) [1 − sin2(θ )]}1/4 , κ= θ=

m

1 − m2 sn2(A0 , m2 )

,

1 − sin2(θ ) cn(A0 , m2 ) 1 1 tan−1{ [ 2 2 m 1 − m2 1 + cn (A0 , m )

− m2 ]},

0 < m < 1, A0, t0, n0, and ϕ0 are arbitrary real constants.

• Reference: [5].

Figure 8.18. Plot of solution (8.52). (a) t = 0, (b) n = 0, with A0 = 1, m = 1/2 , and n 0 = t0 = ϕ0 = 0 . The lines are guides for the eye.

8-28

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Solution 12. Periodicity in t and Localization in n discrete Kuznetsov–Ma breather (Figure 8.19)

⎧ (1 + κ α ) cos[α (t − t )] + κ α α ⎫ 1 4 0 1 2 ⎪ ⎪ ⎪ cosh[α5 (n − n 0)] + i α3 sin[α 4 (t − t0)] ⎪ ⎬ ψ (n , t ) = ⎨ ⎪ −α1 cos[α 4 (t − t0)] − α1 α2 cosh[α5 (n − n 0)] ⎪ ⎪ ⎪ ⎩ ⎭ × e i [2 κ where α1 =

2

(8.53)

(t − t0) + ϕ0 ],

κ , (1 + κ 2 ) [cosh(α5 ) − 1] tanh2

( α25 )

α2 =

1+

α3 =

1 + 2 κ α1 ,

κ2

,

α4 = 4 (1 + κ 2 ) sinh

( α2 ) 5

κ2 1 + κ2

+ sinh2

( α2 ) , 5

α5 > 0, κ , t0, n0, and ϕ0 are arbitrary real constants.

• Reference: [5].

Figure 8.19. Discrete Kuznetsov–Ma breather (8.53). (a) t = 0, (b) n = 0, with κ = α5 = 3/2 , n 0 = t0 = ϕ0 = 0 . The lines are guides for the eye. Animation available online at https://iopscience.iop.org/book/978-0-75032428-1.

8-29

8-30

4. ψ (n , t ) = A0

m sn[A1 (n − n 0 ), m ] e−i [(A2 + 2) (t − t0) + ϕ0]

3. ψ (n , t ) = A0 tanh[A1 (n − n 0 )] e−i [(A2 + 2) (t − t0) + ϕ0]

2. ψ (n , t ) = A0 sech[A1 (n − n 0 )] e−i [(A2 + 2) (t − t0) + ϕ0]

A2 = 2 cos(A1) (1 + a2 A02 ),

1. ψ (n , t ) = A0 ei [A1 (n − n0) + (A2 − 2) (t − t0) + ϕ0]

Name

Eq. #

A02

, A0 ≠ 0,

A02

, A0 ≠ 0 ,

A0 ns (A1, m )

A1, t0, and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

A2 = − 2 cn(A1, m ) dn(A1, m ), −1 , 0 ⩽ m ⩽ 1, A0 ≠ 0, a2 = 2 2

A1, t0, and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

a2 =

−tanh2(A1)

A2 = − 2 sech2(A1),

A1, t0, and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

a2 =

sinh2(A1)

A2 = − 2 cosh(A1),

discrete solitary wave

discrete dark soliton

discrete bright soliton

(8.45)

(8.44)

(8.43)

discrete (8.42) continuous A0, A1, t0, and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer wave, t- and ndependent phase

Conditions

# Solution

Equation i ψnt + ψn + 1 + ψn − 1 − 2 ψn + a2 (ψn + 1 + ψn − 1) ∣ψn∣2 = 0

Note: For lengthy conditions, the reader is referred to the solutions in section 8.4.

8.5 Summary of Section 8.4

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

m cn[A1 (n − n 0 ), m ] e−i [(A2 + 2) (t − t0) + ϕ0]

A0 2

8-31

2

m

cn[A1 (n − n 0 ), m ]

m cd[A1 (n − n 0 ), m ] e−i [(A2 + 2) (t − t0) + ϕ0]

B0

}

{

8 [1 + 4 i (t − t0 )] 1 + 4 (n − n 0 )2 + 32 (t − t0 )2

}

− 1 ei [2 (t − t0) + ϕ0]

⎧ cos[α (n − n )] + i 2 + κ 2 sinh[2 κ 2 (t − t )] ⎫ 0 0 ⎪ ⎪ 2 1 + κ2 ⎬ ei [2 κ (t − t0) + ϕ0] ψ (n , t ) = κ ⎨ 2 ⎪ 2 + κ2 cosh[2 κ 2 (t − t0)] − cos[α (n − n0)] ⎪ ⎩ 1+κ ⎭

10. ψ (n , t ) =

9.

dn[A1 (n − n 0 ), m ] +

× e−i [(A2 + 2) (t − t0) + ϕ0]

{

8. ψ (n , t ) = A0

7. ψ (n , t ) =

6. ψ (n , t ) = A0 dn[A1 (n − n 0 ), m ] e−i [(A2 + 2) (t − t0) + ϕ0]

5. ψ (n , t ) = A0

−2 cn(A1, m ) , dn2(A1, m )

a2 =

1 , A02 ds2(A1, m )

−2 dn(A1, m ) , cn2(A1, m )

a2 =

1 , A02 cs2(A1, m )

−4 , cn(A1, m ) + dn(A1, m ) 4 , A02 [ds(A1, m ) + cs(A1, m )]2

)

t0 and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

(

discrete solitary wave

discrete solitary wave

discrete solitary wave

(8.50)

(8.49)

(8.48)

(8.47)

(8.46)

discrete Peregrine (8.51) soliton

discrete solitary wave 1 discrete α = cos−1 1 + κ 2 , Akhmediev t0, κ , and ϕ0 are arbitrary real constants, breather n0 is an arbitrary real integer

See text

B0 = ± A0 , 0 ⩽ m ⩽ 1, A0 ≠ 0, A1, t0, and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

a2 =

A2 =

0 ⩽ m ⩽ 1, A0 ≠ 0, A1, t0, and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

A2 =

0 ⩽ m ⩽ 1, A0 ≠ 0, A1, t0, and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

A2 =

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

12. ψ (n , t ) =

{

sin(θ ) (t − t0 ) + ϕ0 ]

(t − t0

), sin2(θ )] cn[A0 (n − n 0

2

(t − t0 ) + ϕ0 ]

}

), m2 ]

sin(θ ) sn[2 κ 2 (t − t0 ), sin2(θ )]

(1 + κ α1) cos[α4 (t − t0 )] + κ α1 α2 cosh[α5 (n − n 0 )] + i α3 sin[α4 (t − t0 )] −α1 cos[α4 (t − t0 )] − α1 α2 cosh[α5 (n − n 0 )]

× ei [2 κ

{

2

A1 − m sin(θ ) cn[2

κ2

m dn[2 κ 2 (t − t0 ), sin2(θ )] cn[A0 (n − n 0 ), m2 ] + i A1

× ei [2 κ

11. ψ (n , t ) = κ

(Continued )

sn2(A0 , m2 )

m

1 1 − m2 0

1/4

}

1 [ 1 + cn2(A , m2) − m2 ] ,

2

1+

8-32 κ2 1 + κ2 2

α5 2

α5 2

1 + 2 κ α1 ,

α5 > 0 , κ , t0, and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

×

α3 =

( ) + sinh ( ) ,

α5 2

( ),

κ2

tanh2

κ , (1 + κ 2 ) [cosh(α5) − 1]

α4 = 4 (1 + κ 2 ) sinh

α2 =

α1 =

A0, t0, and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

{

1 − sin2(θ ) cn(A0 , m2 )

m

2

1 − m2

θ = tan−1

1

} Aκ == {(1 − m ) [1 − sin, (0θ )]}< m ,< 1,

discrete Kuznetsov– Ma breather

–

(8.53)

(8.52)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

8.6 Cubic-quintic Discrete NLSE Equation:

i ψnt + a1 (ψn + 1 + ψn − 1 − 2 ψn ) + a2 ∣ψn∣2 ψn + (a3 ∣ψn∣2 + a 4 ∣ψn∣4 )(ψn + 1

(8.54)

+ ψn − 1) = 0, where ψn = ψ (n, t ) is the complex function profile, the integer site index, n, and t are its two independent variables, a1, a2, a3, and a4 are real constants. Solutions: Solution 1. Constant Amplitude discrete CW, t- and n-dependent phase

ψ (n , t ) = A0 e i [A1 (n − n 0) + A2 (t − t0) + ϕ0],

(8.55)

where A2 = −2 a1 + A02 a2 + 2 cos(A1) (a1 + A02 a3 + A04 a 4 ), A0, A1, t0, n0, and ϕ0 are arbitrary real constants. Solution 2. sech(n) I discrete bright soliton (Figure 8.20)

ψ (n , t ) = A0 sech[A1 (n − n 0)] e i [A2 (t − t0) + ϕ0],

(8.56)

where A0 = ±

a3 (a 22 − 4 a1 a 4 ) + a32 − 4 a1 a 4 (a 22 + 4 a1 a 4 )

⎛ −a3 + A1 = sech−1 ⎜ ⎝ A2 = −2 a1 −

2 a4

2 a1

a32 − 4 a1 a 4 a2

⎞ ⎟, ⎠

a2 (a3 + a32 − 4 a1 a 4 ) 2 a4

,

,

Figure 8.20. Discrete bright soliton (8.56) t = 0. (a) Absolute value, (b) real part, with n 0 = ϕ0 = 0 , a1 = a2 = a3 = 1, and a 4 = −1/10 . The lines are guides for the eye.

8-33

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

a32 − 4 a 4 a1 ⩾ 0, a1 > 0,

a3 (a 22 − 4 a1 a 4 ) + a32 − 4 a1 a 4 (a 22 + 4 a1 a 4 ) > 0, t0, n0, and ϕ0 are arbitrary real constants.

• Reference: [6]. Solution 3. sech(n) II staggered discrete bright soliton (Figure 8.21)

ψ (n , t ) = ( −1)n A0 sech[A1 (n − n 0)] e i [A2 (t − t0) + ϕ0],

(8.57)

where A0 = ±

a3 (a 22 − 4 a1 a 4 ) + a32 − 4 a1 a 4 (a 22 + 4 a1 a 4 )

⎛ a3 − A1 = sech−1 ⎜ ⎝ A2 = −2 a1 −

2 a4 a32 − 4 a1 a 4 a2

2 a1

,

⎞ ⎟, ⎠

a2 (a3 + a32 − 4 a1 a 4 ) 2 a4

,

a32 − 4 a 4 a1 ⩾ 0, a1 > 0,

a3 (a 22 − 4 a1 a 4 ) + a32 − 4 a1 a 4 (a 22 + 4 a1 a 4 ) > 0, t0, n0, and ϕ0 are arbitrary real constants.

• Reference: [6]. Solution 4. tanh(n) I discrete dark soliton (Figure 8.22)

ψ (n , t ) = A0 tanh[A1 (n − n 0)] e i [A2 (t − t0) + ϕ0],

(8.58)

where A0 = ±

a2 + a3 + a32 − 4 a1 a 4 −2 a 4

,

Figure 8.21. Staggered discrete bright soliton (8.57) t = 0. (a) Absolute value, (b) real part, with n 0 = ϕ0 = 0 , a1 = a3 = 1, a2 = −4/10 , and a 4 = −1/10 . The lines are guides for the eye.

8-34

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Figure 8.22. Discrete dark soliton (8.58) t = 0. (a) Absolute value, (b) real part, with n 0 = ϕ0 = 0 , a1 = −18/10 , a2 = −8/10 , a3 = 1, and a 4 = −18/10 . The lines are guides for the eye.

Figure 8.23. Staggered discrete dark soliton (8.59) at t = 0. (a) Absolute value, (b) real part, with n 0 = ϕ0 = 0 , a1 = 8/10 , a2 = a3 = 1, and a 4 = −25/100 . The lines are guides for the eye.

⎛ A1 = cosh−1 ⎜⎜ ⎝ A2 = −2 a1 − a32

a3 + a32 − 4 a1 a 4 −a2

⎞ ⎟⎟, ⎠

a2 (a3 − a32 − 4 a1 a 4 ) 2 a4

,

− 4 a 4 a1 ⩾ 0,

a 4 (a2 + a3 +

a32 − 4 a1 a 4 ) < 0,

a2 (a3 + a32 − 4 a1 a 4 ) < 0, t0, n0, and ϕ0 are arbitrary real constants.

• Reference: [6]. Solution 5. tanh(n) II staggered discrete dark soliton (Figure 8.23)

ψ (n , t ) = ( −1)n A0 tanh[A1 (n − n 0)] e i [A2 (t − t0) + ϕ0],

8-35

(8.59)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

where A0 = ±

a2 − a3 − a32 − 4 a1 a 4

⎛ A1 = cosh−1 ⎜⎜ ⎝ A2 = −2 a1 − a32

2 a4

,

a3 + a32 − 4 a1 a 4 a2

⎞ ⎟⎟, ⎠

a2 (a3 − a32 − 4 a1 a 4 ) 2 a4

,

− 4 a 4 a1 ⩾ 0,

(a2 − a3 −

a32 − 4 a1 a 4 ) > 0,

a2 (a3 + a32 − 4 a1 a 4 ) > 0, t0, n0, and ϕ0 are arbitrary real constants.

• Reference: [6].

8-36

3. ψ (n , t ) = ( − 1) n A0 sech[A1 (n − n 0 )] ei [A2 (t − t0) + ϕ0]

2. ψ (n , t ) = A0 sech[A1 (n − n 0 )] ei [A2 (t − t0) + ϕ0]

8-37 − 4 a 4 a1 ⩾ 0 , a1 > 0 ,

2 a4

a2 (a3 + a32 − 4 a1 a 4 )

,

,

2 a4

⎞ ⎟, ⎠

2 a1

− 4 a 4 a1 ⩾ 0 , a1 > 0 ,

2 a4

a2 (a3 + a32 − 4 a1 a 4 )

a2

a32 − 4 a1 a 4

,

,

a3 (a 22 − 4 a1 a 4 ) + a32 − 4 a1 a 4 (a 22 + 4 a1 a 4 ) > 0 , t0 and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

a32

A2 = − 2 a1 −

A1 =

⎛ a3 − ⎜ ⎝

a3 (a 22 − 4 a1 a 4 ) + a32 − 4 a1 a 4 (a 22 + 4 a1 a 4 )

sech−1

A0 = ±

a3 (a 22 − 4 a1 a 4 ) + a32 − 4 a1 a 4 (a 22 + 4 a1 a 4 ) > 0 , t0 and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

a32

A2 = − 2 a1 −

a2

⎞ ⎟, ⎠

2 a1

a32 − 4 a1 a 4

2 a4

a3 (a 22 − 4 a1 a 4 ) + a32 − 4 a1 a 4 (a 22 + 4 a1 a 4 )

⎛ −a3 + A1 = sech−1 ⎜ ⎝

A0 = ±

(8.56)

staggered discrete bright soliton (8.57)

discrete bright soliton

A2 = − 2 a1 + A02 a2 + 2 cos(A1) (a1 + A02 a3 + A04 a 4 ), discrete continuous wave, t- and (8.55) n-dependent phase A0, A1, t0, and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

Eq. #

1. ψ (n , t ) = A0 ei [A1 (n − n0) + A2 (t − t0) + ϕ0]

Name

Conditions

# Solution

Equation i ψnt + a1 (ψn + 1 + ψn − 1 − 2 ψn ) + a2 ∣ψn∣2 ψn + (a3 ∣ψn∣2 + a 4 ∣ψn∣4 )(ψn + 1 + ψn − 1) = 0

8.7 Summary of Section 8.6

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

5. ψ (n , t ) = ( − 1) n A0 tanh[A1 (n − n 0 )] ei [A2 (t − t0) + ϕ0]

4. ψ (n , t ) = A0 tanh[A1 (n − n 0 )] ei [A2 (t − t0) + ϕ0]

(Continued )

⎞ ⎟⎟, ⎠ ,

− 4 a 4 a1 ⩾ 0 , a 4 (a2 + a3 +

2 a4

a2 (a3 − a32 − 4 a1 a 4 )

−a2

a32 − 4 a1 a 4 ) < 0 ,

8-38 − 4 a 4 a1 ⩾ 0 , (a2 − a3 −

2 a4

a2 (a3 − a32 − 4 a1 a 4 )

a2

⎞ ⎟⎟, ⎠

a32 − 4 a1 a 4 ) > 0 ,

,

a2 (a3 + a32 − 4 a1 a 4 ) > 0 , t0 and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

a32

A2 = − 2 a1 −

,

a3 + a32 − 4 a1 a 4

2 a4

a2 − a3 − a32 − 4 a1 a 4

⎛ A1 = cosh−1 ⎜⎜ ⎝

A0 = ±

a2 (a3 + a32 − 4 a1 a 4 ) < 0 , t0 and ϕ0 are arbitrary real constants, n0 is an arbitrary real integer

a32

A2 = − 2 a1 −

,

a3 + a32 − 4 a1 a 4

−2 a 4

a2 + a3 + a32 − 4 a1 a 4

⎛ A1 = cosh−1 ⎜⎜ ⎝

A0 = ±

staggered discrete dark soliton

discrete dark soliton

(8.59)

(8.58)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

8.8 Generalized Discrete NLSE Equation:

i ψnt + a1 (ψn + 1 + ψn − 1 − 2 ψn ) + f [ψn − 1, ψn, ψn + 1] = 0,

(8.60)

where f [ψn−1, ψn, ψn+1] = α1 ∣ψn∣2 ψn + α2 ∣ψn∣2 (ψn+1 + ψn−1) + α3 ψn2 (ψn*+1 + ψn*−1) + α4 ψn (∣ψn+1∣2 + ∣ψn−1∣2 ) + α5 ψn (ψn*+1 ψn−1 + ψn*−1 ψn+1) + α6 ψn* (ψn2+1 + ψn2−1) + α7 ψn* ψn+1 ψn−1 + α8 (∣ψn+1∣2 ψn+1 + ∣ψn−1∣2 ψn−1) + α9 (ψn*−1 ψn2+1 + ψn*+1 ψn2−1) + α10 (∣ψn+1∣2 ψn−1 + ∣ψn−1∣2 ψn+1) + α11 (∣ψn−1 ψn∣ + ∣ψn ψn+1∣) ψn + α12 (ψn+1 ∣ψn+1 ψn∣ + ψn−1 ∣ψn ψn−1∣) + α13 (ψn+1 ∣ψn−1 ψn∣ + ψn−1 ∣ψn ψn+1∣) + α14 (ψn+1 ∣ψn−1 ψn+1∣ + ψn−1 ∣ψn−1 ψn+1∣), ψn = ψ (n, t ) is the complex function profile, the integer site index, n, and t are its two independent variables, a1 and α1, … , α14 are real constants.

Solutions: Solution 1. sech(n) discrete bright soliton (Figure 8.24)

ψ (n , t ) = A0 sech[β (n + γ )] e−i (A2 t + ϕ0),

(8.61)

where α1 = α8 = 0, A2 = 2 a1 [1 − cosh(β )], α4 = −α12 + α13 + α5 − α6 +

α7 2

+ cosh(β ) [α2 + α3 + α11 −

a1 sinh2(β ) ], A02

α10 = −α9 − α14 − (2 α13 + 2 α5 + α7 ) cosh(β ) − 2 (α2 + α3 + α11) cosh2(β ), +

2 a1 cosh2(β ) sinh2(β ) A02

Figure 8.24. Discrete bright soliton (8.61) at t = 0 with a1 = A0 = A1 = 2 , α2 = α3 = α5 = α7 = α9 = α11= α12 = α13 = α14 = β = γ = 1, and ϕ0 = 0 . The lines are guides for the eye.

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α7 = −2 α13 − 2 α5, α11 = −α2 − α3, a1, A0, A1, α2 , α3, α5, α6, α9, α12, α13, α14 , γ , and ϕ0 are arbitrary real constants.

• Reference: [7]. Solution 2. sech(n,t) moving discrete bright soliton (Figure 8.25)

ψ (n , t ) = A0 sech[β (n − v t + γ )] e i (A1 n − A2 t + ϕ0),

(8.62)

where a1 = 1, α1 = α8 = 0, 2 a sin(A1) sinh(β ) , v= 1 β A2 = 2 a1 [1 − cos(A1) cosh(β )], α2 = α3 +

sinh2(β ) , A02

α4 = α6 − (α10 − α9 ) cos(A1) sech(β ), 1 α6 = − 4 sec(A1) [2 α12 − α10 sech(β ) + α14 sech(β )+ α9 csc(A1) sech(β ) sin(3 A1)], α7 = −2 [α13 cos(A1) + α5 cos(2 A1)] − [(α10 + α14 ) cos(A1) + α9 cos(3 A1)] × sech(β ) + 2 cosh(β )[−α11 − 2 α3 cos(A1) +

(a1 − 1) cos(A1) sinh2(β ) A02

],

A0, A1, α3, α5, α9, α10, α11, α12 , α13, α14 , γ , and ϕ0 are arbitrary real constants. • Reference: [7].

Figure 8.25. Moving discrete bright soliton (8.62) with A0 = A1 = 2 , a1 = α3 = α5 = α9 = α10 = α11 = α12 = α13 = α14 = β = γ = 1, and ϕ0 = 0 . (a) at t = −1, (b) at t = 0, and (c) at t = 1. The lines are guides for the eye. Animation available online at https://iopscience.iop.org/book/978-0-7503-2428-1.

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Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Solution 3. tanh(n) discrete dark soliton (Figure 8.26)

ψ (n , t ) = A0 tanh[β (n + γ )] e−i (A2 t + ϕ0),

(8.63)

where α1 = α8 = 0, 2 A2 = 1 + coth2(β ) [A02 (α10 − α11 + α14 − α2 − α3 + α9 ) + a1], α4 =

−1 1 + cosh(2 β ) −1

α5 =

[α6 + (α10 + α14 + α6 + α9 ) cosh(2 β ) + 2 α12 cosh2(β )],

2 A02 [1 + tanh2(β )]

×{A02 (α10 + 2 α11 + 2 α13 + α14 + 2 α2 + 2 α3 + α7 + α9 ) + [A02 (2 α10 + 2 α14 + 2 α13 + α7 + 2 α9 ) + 2 a1] × tanh2(β ) − A02 (α10 + α14 + α9 ) tanh4(β )}, a1, A0, A1, γ , α2 , α3, α6 , α7, α9, α10, α11, α12, α13, α14 , and ϕ0 are arbitrary real constants.

• Reference: [7]. Solution 4. tanh(n,t) moving discrete dark soliton (Figure 8.27)

ψ (n , t ) = A0 tanh[β (n − v t + γ )] e i (A1 n − A2 t + ϕ0),

(8.64)

where α1 = α8 = α14 = γ = 0, v=

2 A02 (α2 − α3) coth(β ) sin(A1) , β

Figure 8.26. Discrete dark soliton (8.63) at t = 0 with a1 = 2 , A0 = α2 = α3 = α6 = α7 = α9 = α10 = α11= β = γ = 1, α12 = 7 , α13 = α14 = 2 , and ϕ0 = 0 . The lines are guides for the eye.

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Figure 8.27. Moving discrete dark soliton (8.64) with a1 = A0 = A1 = α2 = α5 = α9 = α10 = α11 = α12 = α13 = β = 1, and γ = ϕ0 = 0 . (a) at t = −0.5, (b) at t = 0, and (c) at t = 0.5. The lines are guides for the eye. Animation available online at https://iopscience.iop.org/book/978-0-7503-2428-1.

A2 =

2 1 + coth2(β )

(a1 + 2 A02 α10 cos(A1) − 2 A02 α9 cos(A1)

−A02 [α11 + 2 α2 cos(A1)] coth4(β ) + A02 [α11 + (α2 + α3) cos(A1)] csch2(β ) + coth2(β ) {a1 + A02 [α11 + (α2 + α3) cos(A1)] csch2(β )}), α3 =

1 A02

[−A02 α10 + A02 α14 − a1 coth2(β )

+A02 α4 =

1 2

α6 =

1 4 A02

{α6

4

α2 coth (β ) +

sech2(β )

A02

, 4

α9 csc(A1) sin(3 A1)] tanh (β )

+ [α6 − 2 (α10 − α9 ) cos(A1)] [1 + tanh2(β )]},

[a1 + 2 A02 α12 + a1 coth2(β ) − A02 α2 coth2(β )

+ A02 α3 coth2(β ) − A02 α2 coth4(β ) + A02 α3 coth4(β )] sec(A1), α7 =

1 A02 (1 + coth2(β ))

( −4 A02 α10 cos(A1) − 2 A02 α13 cos(A1) + 4 A02 α9 cos(A1) −a1 cos(A1) − 2 A02 α5 cos(2 A1) −{[A02 (2 α10 + 2 α13 + 3 α2 + α3 − 2 α9 ) + 2 a1] cos(A1) +2 A02 [α11 + α5 cos(2 A1)]} coth2(β ) +[2 A02 (α2 − α3) − a1] cos(A1) coth4(β ) +A02 (α2 − α3) cos(A1) coth6(β ) +2 A02 (α10 − α9 ) cos(A1) tanh2(β )), a1, A0, A1, α2 , α5, α9, α10, α11, α12, α13, γ , and ϕ0 are arbitrary real constants.

• Reference: [7].

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Solution 5. sin(n,t)

ψ (n , t ) = A0 sin[β (n − v t + γ )] e i (A1 n − A2 t + ϕ0),

(8.65)

where α1 = α8 = 0, v =

2 sin(β ) 2 {A0 β

α6 sin(2 A1) sin(β ) sin(2 β ) + sin(A1)

[a1 + A02 (α2 − α3 + 4 α8) sin2(β ) − 4 A02 α8 sin4(β ) + A02 α12 sin(β ) sin(2 β )]}, A2 = 2 a1 − A02 cos(A1) cos(β ) ⎡ 2 a1 ⎢⎣α2 + α3 + α8 + A02 + α12 cos(β ) − (α2 + α3) cos(2 β ) − α12 cos(3 β ) − α8 cos(4 β )] −A02 sin2(β ){α1 + 2 α4 + 2 α6 cos(2 A1) + α6 cos[2 (A1 − β )] + 2 α11 cos(β )} +2 α4 cos(2 β ) + α6 cos[2 (A1 + β )], α7 = − α1 + α5 + α6 − 2 (α5 − α6 ) cos2(A1) + (α6 − α5) cos(2 A1) − 2 α11 cos(β ) + 2 cos(A1) [α12 − α13 − 2 cos(β ) (α10 + α3 − α8 − α9 )] − 2α4 cos(2β ) + 2α6 cos(2β ), α14 = α10 − α2 + α3 − 3 α8 − 2 cos(β ) [α12 + 2 α6 cos(A1)] − α9 csc(A1) sin(3 A1) + 4 α8 sin2(β ), γ = β = 1, a1, A0, A1, α2 , α3, α4, α5, α6, α9, α10, α11, α12, α13, and ϕ0 are arbitrary real constants.

• Reference: [7]. Solution 6. dn(n,t,m) moving discrete SW (Figure 8.28)

ψ (n , t ) = A0 dn[β (n − v t + γ ), m ] e i (A1 n − A2 t + ϕ0), where a1 = 1, α1 = α8 = α11 = α12 = α13 = α14 = 0, v=

2 A02 β

(α2 − α3) sin(A1) cs(β, m ),

8-43

(8.66)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Figure 8.28. Moving discrete solitary wave (8.66) with a1 = A1 = α2 = α9 = α10 = β = 1, α3 = 2 , m = 1/2 , and γ = ϕ0 = 0 . (a) at t = −0.5, (b) at t = 0, and (c) at t = 0.5. The lines are guides for the eye.

sin(A1) q1 − q2 − q3 q4

A0 =

,

q1 = (α2 − α3) sin(A1) cs2(β, m ), q2 = α6 sin(2 A1) ds(β, m ) ns(β, m ), q3 = α9 sin(3 A1) − α10 sin(A1), q4 = cs2(2 β , m ) + ds(2 β , m ) ns(2 β , m ), A2 = A02 {−2 (α2 + α3) cos(A1) ds(β, m ) ns(β, m )

+ 2 [α4 + α6 cos(2 A1)] cs2(β, m ) − [2 α5 cos(2 A1) + α7 ] cs2(β, m ) + α4 = −α6 cos(2 A1) −

α5 = −

α7 cos(2 A1)

+

2 }, A02

[α9 cos(3 A1) + α10 cos(A1)] cs(2 β , m ) , cs(β , m )

1 2 cs(β , m ) cs(2 β , m ) cos(2 A1)

{

cos(A1) A02

− (α2 + α3) cos(A1) cs2(β, m )

+ [α4 + α6 cos(2 A1)] ds(β, m ) ns(β , m )

}

+ [α9 cos(3 A1) + α10 cos(A1)] [ds(2 β , m ) ns(2 β , m ) − cs2(2 β , m )], , α6 =

α7 =

[α sin(3 A1) − α10 sin(A1)] cs(2β , m ) − 9 , sin(2 A1) cs(β , m ) 1 (2 (α2 cs2(β , m ) cs(2 β, m ) [sec(2 A1) − 1]

+ α3 + α9 + α10 − 1) cs2(β, m ) cs(2 β, m )

− 2 α3 cos(A1) cs3(β, m ) sec(2 A1) + 2 (α9 − α10 ) cos(A1) cs(2 β, m ) ds(β, m ) ns(β, m ) sec(2 A1) + cs(β , m ) {− 2 [α9 + (α10 + 2 α9 ) cos(2 A1)] cs2(2 β, m ) sec(A1) + 2(α10 − α9 ) cos(A1) ds(2 β, m ) ns(2 β, m ) sec(2 A1)}), 0 ⩽ m ⩽ 1, A1, β , γ , α2 , α3, α9, α10, and ϕ0 are arbitrary real constants.

• Reference: [8].

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Solution 7. cn(n,t,m) moving discrete SW (Figure 8.29)

m cn[β (n − v t + γ ), m ] e i (A1 n − A2 t + ϕ0),

ψ (n , t ) = A0

(8.67)

where a1 = 1, α1 = α8 = α11 = α12 = α13 = α14 = 0, v=

2 A02 β

sin(A1) p1 − p2 − p3 p4

A0 = p1 p2 p3 p4

(α2 − α3) sin(A1) ds(β, m ),

,

= (α2 − α3) sin(A1) ds2(β, m ), = α6 sin(2 A1) cs(β, m ) ns(β, m ), = α9 sin(3 A1) − α10 sin(A1), = ds2(2 β, m ) + cs(2 β , m ) ns(2 β , m ),

A2 = A02 {−2 (α2 + α3) cos(A1) cs(β, m ) ns(β, m ) +2 [α4 + α6 cos(2 A1)] ds2(β, m ) −[2 α5 cos(2 A1) + α7 ] ds2(β, m ) + α4 = −α6 cos(2 A1) −

α5 = −

α7 2 cos(2 A1)

+

2

},

A02 [α9 cos(3 A1) + α10 cos(A1)] ds(2 β , m ) , ds(β , m )

1 2 ds(β , m ) ds(2 β , m ) cos(2 A1)

{

cos(A1) A02

− (α2 + α3) cos(A1) ds2(β, m )

+ [α4 + α6 cos(2 A1)] cs(β, m ) ns(β , m ) + [α9 cos(3 A1) + α10 cos(A1)] [cs(2 β , m ) ns(2 β , m )

}

− ds2(2 β, m )] ,

Figure 8.29. Moving discrete solitary wave (8.67) with a1 = A1 = α2 = α9 = α10 = β = 1, α3 = 2 , m = 1/2 , and γ = ϕ0 = 0 . (a) at t = −0.5, (b) at t = 0, and (c) at t = 0.5. The lines are guides for the eye.

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[α9 sin(3 A1) − α10 sin(A1)] ds(2β , m ) , sin(2 A1) ds(β , m ) 1 (2 (α2 ds2(β , m ) ds(2 β , m ) [sec(2 A1) − 1]

α6 = −

α7 =

+ α3 + α9 + α10 − 1) ds2(β, m ) ds(2 β, m )

− 2 α3 cos(A1) ds3(β, m ) sec(2 A1) + 2 (α9 − α10 ) cos(A1) ds(2 β, m ) cs(β, m ) ns(β, m ) sec(2 A1) + ds(β , m ) {−2 [α9 + (α10 + 2 α9 ) cos(2 A1)] ds2(2 β, m ) sec(A1) + 2(α10 − α9 ) cos(A1) cs(2 β , m ) ns(2 β, m ) sec(2 A1)}), 0 ⩽ m ⩽ 1, A1, β , γ , α2 , α3, α9, α10, and ϕ0 are arbitrary real constants.

• Reference: [8].

8-46

8-47

= = = = = = =

ψ (n , ψ (n , ψ (n , ψ (n , ψ (n , ψ (n , ψ (n ,

1. 2. 3. 4. 5. 6. 7.

t) t) t) t) t) t) t)

Solution

#

A0 A0 A0 A0 A0 A0 A0

Equation

sech[β (n + γ )] e−i (A2 t + ϕ0) sech[β (n − v t + γ )] ei (A1 n − A2 t + ϕ0) tanh[β (n + γ )] e−i (A2 t + ϕ0) tanh[β (n − v t + γ )] ei (A1 n − A2 t + ϕ0) sin[β (n − v t + γ )] ei (A1 n − A2 t + ϕ0) dn[β (n − v t + γ ), m ] ei (A1 n − A2 t + ϕ0) m cn[β (n − v t + γ ), m ] ei (A1 n − A2 t + ϕ0)

See See See See See See See

text text text text text text text

Conditions

discrete moving discrete moving – moving moving

Name

discrete solitary wave discrete solitary wave

bright soliton discrete bright soliton dark soliton discrete dark soliton

+ α10 (∣ψn + 1∣2 ψn − 1 + ∣ψn − 1∣2 ψn + 1) + α11 (∣ψn − 1 ψn∣ + ∣ψn ψn + 1∣) ψn + α12 (ψn + 1 ∣ψn + 1 ψn∣ + ψn − 1 ∣ψn ψn − 1∣) + α13 (ψn + 1 ∣ψn − 1 ψn∣ + ψn − 1 ∣ψn ψn + 1∣) + α14 (ψn + 1 ∣ψn − 1 ψn + 1∣ + ψn − 1 ∣ψn − 1 ψn + 1∣)

+ α8 (∣ψn + 1∣2 ψn + 1 + ∣ψn − 1∣2 ψn − 1) + α9 (ψn*− 1 ψn2+ 1 + ψn*+ 1 ψn2− 1)

+ α6 ψn* (ψn2+ 1 + ψn2− 1) + α7 ψn* ψn + 1 ψn − 1

+ α 4 ψn (∣ψn + 1∣2 + ∣ψn − 1∣2 ) + α5 ψn (ψn*+ 1 ψn − 1 + ψn*− 1 ψn + 1)

f [ψn − 1, ψn, ψn + 1] = α1 ∣ψn∣2 ψn + α2 ∣ψn∣2 (ψn + 1 + ψn − 1) + α3 ψn2 (ψn*+ 1 + ψn*− 1)

i ψnt + a1 (ψn + 1 + ψn − 1 − 2 ψn ) + f [ψn − 1, ψn, ψn + 1] = 0,

Note: For lengthy conditions, the reader is referred to the solutions in section 8.8.

8.9 Summary of Section 8.8

(8.61) (8.62) (8.63) (8.64) (8.65) (8.66) (8.67)

Eq. #

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

8.10 Coupled Salerno Equations Equation: i ψ1nt + ψ1n + 1 + ψ1n − 1 − 2 ψ1n ⎛ ⎞ ν1 − 2 μ1 ψ1n⎟ = 0, + (μ1 ∣ψ1n∣2 + μ2 ∣ψ2 n∣2 ) ⎜ψ1n + 1 + ψ1n − 1 + μ1 ⎝ ⎠ ⎡ ⎛ ⎞ ν1 μ2 ν i ψ2 nt + ⎢ψ2 n + 1 + ψ2 n − 1 − ⎜⎜2 + − 2 ⎟⎟ ψ2 n] 2 ⎢⎣ μ2 ⎠ μ1 ⎝ ⎤ ⎡ ⎛ ν2 − 2 μ2 ⎞ + (μ1 ∣ψ1n∣2 + μ2 ∣ψ2 n∣2 ) ⎢ψ2 n + 1 + ψ2 n − 1 + ⎜ ⎟ ψ2 n⎥ = 0, ⎥⎦ ⎢⎣ μ2 ⎠ ⎝

(8.68)

where ψj = ψj (n, t ) is the complex function profile, j = 1, 2, the integer site index, n, and t are its two independent variables, μ1, μ2 , ν1, and ν2 are real constants. Solutions: Solution 1. dn(n,m)-sn(n,m) discrete SW (Figure 8.30)

ψ1(n , t ) = A0 dn[A1 (n + n 0), m ] e−i (ω1 t + ϕ1), ψ2(n , t ) = B0 where ω1 = ω2 = μ1 = μ2 =

ν1 , μ1 ν1 μ2

(8.69)

,

μ12 −1 , A02 μ1 A02 B02

m sn[A1 (n + n 0), m ] e−i (ω2 t + ϕ2),

,

0 ⩽ m ⩽ 1,

Figure 8.30. Discrete solitary wave (8.69) at t = 0. Blue is ψ1 and red is ψ2 with A0 = B0 = A1 = 1, ν1 = ν2 = 2 , m = 1/2 , and n 0 = t0 = ϕ1 = ϕ2 = 0 . The lines are guides for the eye.

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A0, A1, B0, n0, ϕ1, and ϕ2 are arbitrary real constants. • Reference: [9]. Solution 2. cn(n,m)-sn(n,m) discrete SW (Figure 8.31)

where ω1 = ω2 = μ1 = μ2 =

ν1 , μ1 ν1 μ2

ψ1(n , t ) = A0

m cn[A1 (n + n 0), m ] e−i (ω1 t + ϕ1),

ψ2(n , t ) = B0

m sn[A1 (n + n 0), m ] e−i (ω2 t + ϕ2),

(8.70)

,

μ12 −1 , m A02 2 μ1 A0

,

B02

0 ⩽ m ⩽ 1, A0, A1, B0, n0, ϕ1, and ϕ2 are arbitrary real constants.

• Reference: [9]. Solution 3. sech(n)-tanh(n) discrete bright-dark soliton (Figure 8.32)

ψ1(n , t ) = A0 sech[A1 (n + n 0)] e−i (ω1 t + ϕ1), ψ2(n , t ) = B0 tanh[A1 (n + n 0)] e−i (ω2 t + ϕ2), where ω1 = ω2 = μ1 = μ2 =

ν1 , μ1 ν1 μ2

,

μ12 −1 , A02 μ1 A02 B02

(8.71)

,

Figure 8.31. Discrete solitary wave (8.70) at t = 0. Blue is ψ1 and red is ψ2 with A0 = B0 = A1 = 1, ν1 = ν2 = 2 , m = 1/2 , and n 0 = t0 = ϕ1 = ϕ2 = 0 . The lines are guides for the eye.

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Figure 8.32. Discrete bright-dark soliton (8.71) at t = 0. Blue is ψ1 and red is ψ2 with A0 = B0 = A1 = 1, ν1 = ν2 = 2 , and n 0 = t0 = ϕ1 = ϕ2 = 0 . The lines are guides for the eye.

A0, A1, B0, n0, ϕ1, and ϕ2 are arbitrary real constants. • Reference: [9]. Solution 4. dn2(n,m)-sn(n,m) dn(n,m) discrete SW

{

}

ψ1(n , t ) = A0 dn2[A1 (n + n 0), m ] + A2 e−i (ω1 t + ϕ1), ψ2(n , t ) = B0

(8.72)

m sn[A1 (n + n 0), m ] dn[A1 (n + n 0), m ] e−i (ω2 t + ϕ2),

where A0 = −2 A2 , ν ω1 = μ1 , 1

ω2 = μ1 = μ2 =

ν1 μ2

,

μ12 −4 , A02 μ1 A02 B02

,

0 ⩽ m ⩽ 1, A1, A2, B0, n0, ϕ1, and ϕ2 are arbitrary real constants.

• Reference: [9]. Solution 5. dn2(n,m)-sn(n,m) cn(n,m) discrete SW

ψ1(n , t ) = {A0 dn2[A1 (n + n 0), m ] + A2 } e−i (ω1 t + ϕ1), ψ2(n , t ) = B0 m sn[A1 (n + n 0), m ] cn[A1 (n + n 0), m ] e−i (ω2 t + ϕ2), where A0 = ω1 = ω2 = μ1 = μ2 =

−2 A2 , 2−m ν1 , μ1 ν1 μ2

,

μ12 −4 , m2 A02 2 μ1 A0 B02

,

8-50

(8.73)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

0 ⩽ m ⩽ 1, A1, A2, B0, n0, ϕ1, and ϕ2 are arbitrary real constants.

• Reference: [9]. Solution 6. sech2(n)-sech(n) tanh(n) (Figure 8.33)

ψ1(n , t ) = {A0 sech2[A1 (n + n 0)] + A2 } e−i (ω1 t + ϕ1), ψ2(n , t ) = B0 sech[A1 (n + n 0)] tanh[A1 (n + n 0)] e−i (ω2 t + ϕ2),

(8.74)

where A0 = −2 A2 , ν ω1 = μ1 , 1

ω2 = μ1 = μ2 =

ν1 μ2

,

μ12 −4 , A02 μ1 A02 B02

,

A1, A2, B0, n0, ϕ1, and ϕ2 are arbitrary real constants. • Reference: [9]. Solution 7. Rational Solution I

ψ1(n , t ) = ψ2(n , t ) = where ω1 = ω2 = μ1 = μ2 =

ν1 , μ1 ν1 μ2

1 + n2 B0 n 1 + n2

e−i (ω1 t + ϕ1), (8.75) e−i (ω2 t + ϕ2),

,

μ12 −1 , A02 μ1 A02 B02

A0

,

Figure 8.33. Plot of solution (8.74) at t = 0. Blue is ψ1 and red is ψ2 with A1 = A2 = 1, B0 = 3/2 , ν1 = ν2 = 2 , m = 1/2 , and n 0 = t0 = ϕ1 = ϕ2 = 0 . The lines are guides for the eye.

8-51

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

A0, B0, ϕ1, and ϕ2 are arbitrary real constants. • Reference: [9]. Solution 8. Rational Solution II

1 + n2 e−i (ω1 t + ϕ1), 1 + n2 + n 4 B0 n2 e−i (ω2 t + ϕ2), 1 + n2 + n 4

ψ1(n , t ) = A0 ψ2(n , t ) = where ω1 = ω2 = μ1 = μ2 =

ν1 , μ1 ν1 μ2

,

μ12 −1 , A02 μ1 A02 B02

(8.76)

,

A0, B0, ϕ1, and ϕ2 are arbitrary real constants. • Reference: [9]. Solution 9. Rational Solution III

ψ1(n , t ) = A0 ψ2(n , t ) = where ω1 = ω2 = μ1 =

ν1 , μ1 ν1 μ2 μ12 −1 , A02

2 + n2 −i (ω1 t + ϕ ) 1 , e 1 + n2 B0

1 + n2

(8.77)

e−i (ω2 t + ϕ2),

,

μ2 = 1, A0, B0, ϕ1, and ϕ2 are arbitrary real constants.

• Reference: [9]. Solution 10. cos(n)-sin (n)

ψ1(n , t ) = A0 cos[A1 (n + n 0)] e−i (ω1 t + ϕ1), ψ2(n , t ) = B0 sin[A1 (n + n 0)] e−i (ω2 t + ϕ2),

8-52

(8.78)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

where ω1 = ω2 = μ1 = μ2 =

ν1 , μ1 ν1 μ2

,

μ12 −1 , A02 μ1 A02 B02

,

A0, B0, n0, ϕ1, and ϕ2 are arbitrary real constants. • Reference: [9]. Solution 11. nd(n,m)-sd(n,m) discrete SW

ψ1(n , t ) = A0 nd[A1 (n + n 0), m ] e−i (ω1 t + ϕ1), ψ2(n , t ) = B0

m sd[A1 (n + n 0), m ] e−i (ω2 t + ϕ2),

(8.79)

where A0 = ω1 = ω2 =

μ2 B02 , ∣ μ1 ∣ ν1 , μ1 ν1 μ2 μ12

,

μ1 = −1, μ2 = 1, 0 ⩽ m ⩽ 1, A1, A2, B0, n0, ϕ1, and ϕ2 are arbitrary real constants.

• Reference: [9]. Solution 12. cosh(n)-sinh(n)

ψ1(n , t ) = A0 cosh[A1 (n + n 0)] e−i (ω1 t + ϕ1), ψ2(n , t ) = B0 sinh[A1 (n + n 0)] e−i (ω2 t + ϕ2), where A0 = ω1 = ω2 =

μ2 B02 , ∣ μ1 ∣ ν1 , μ1 ν1 μ2 μ12

,

μ1 = −1, μ2 = 1,

8-53

(8.80)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

A1, B0, n0, ϕ1, and ϕ2 are arbitrary real constants. • Reference: [9]. Solution 13. discrete SW

ψ1(n , t ) = {A0 nd2[A1 (n + n 0), m ] + A2 } e−i (ω1 t + ϕ1), B m sn[A1 (n + n 0), m ] −i (ω2 t + ϕ ) 2 , e ψ2(n , t ) = 0 2 dn [A1 (n + n 0), m ]

(8.81)

where A0 = −2 A2 , B0 = ω1 = ω2 = μ1 =

∣ μ1 ∣ A02 μ2 ν1 , μ1 ν1 μ2 μ12 −4 , A02

,

,

μ2 > 0 , 0 ⩽ m ⩽ 1, A1, A2, n0, ϕ1, and ϕ2 are arbitrary real constants.

• Reference: [9]. Solution 14.

ψ1(n , t ) = {A0 cosh2[A1 (n + n 0)] + A2 } e−i (ω1 t + ϕ1), ψ2(n , t ) = B0 sinh[A1 (n + n 0)] cosh[A1 (n + n 0)] e−i (ω2 t + ϕ2), where A0 = −2 A2 , B0 = ω1 = ω2 = μ1 =

∣ μ1 ∣ A02 μ2 ν1 , μ1 ν1 μ2 μ12 −4 , A02

,

,

μ2 > 0 , A1, A2, n0, ϕ1, and ϕ2 are arbitrary real constants.

• Reference: [9].

8-54

(8.82)

μ12

ν1 μ2

−

ν2 ) μ2

1

8-55

m sn[A1 (n + n 0 ), m ] e−i (ω2 t + ϕ2)

ψ2(n , t ) = B0

ψ2(n , t ) = B0

−1 , A02

μ2 =

ω2 =

,

B02

μ1 A02

μ12

ν1 μ2

ψ2 n ) = 0

, 0 ⩽ m ⩽ 1,

ν2 − 2 μ 2 ) μ2

,

B02

μ1 A02

μ12

ν1 μ2

μ2 =

ω2 =

−1 , m A02

ν1 , μ1

, 0 ⩽ m ⩽ 1,

ν1 , μ1

ω2 =

μ12

ν1 μ2

, μ1 =

−1 , A02

μ2 =

B02

μ1 A02

,

μ2 =

B02

μ1 A02

ω2 =

μ12

ν1 μ2

, , 0 ⩽ m ⩽ 1,

ν1 , μ1

A1, A2, B0, ϕ1, and ϕ2 are arbitrary real constants, n0 is an arbitrary real integer

−4 , A02

A0 = − 2 A2 , ω1 =

A0, A1, B0, ϕ1, and ϕ2 are arbitrary real constants, n0 is an arbitrary real integer

ω1 =

A0, A1, B0, ϕ1, and ϕ2 are arbitrary real constants, n0 is an arbitrary real integer

μ1 =

ω1 =

A0, A1, B0, ϕ1, and ϕ2 are arbitrary real constants, n0 is an arbitrary real integer

μ1 =

m sn[A1 (n + n 0 ), m ] dn[A1 (n + n 0 ), m ] e−i (ω2 t + ϕ2) μ1 =

4. ψ1(n , t ) = {A0 dn2[A1 (n + n 0 ), m ] + A2 } e−i (ω1 t + ϕ1),

ψ2(n , t ) = B0 tanh[A1 (n + n 0 )] e−i (ω2 t + ϕ2)

3. ψ1(n , t ) = A0 sech[A1 (n + n 0 )] e−i (ω1 t + ϕ1),

m cn[A1 (n + n 0 ), m ] e−i (ω1 t + ϕ1),

m sn[A1 (n + n 0 ), m ] e−i (ω2 t + ϕ2)

2. ψ1(n , t ) = A0

ψ2(n , t ) = B0

ω1 =

ν1 , μ1

Conditions

ψ2 n ] + (μ1 ∣ψ1n∣2 + μ2 ∣ψ2 n∣2 ) (ψ2 n + 1 + ψ2 n − 1 + (

1. ψ1(n , t ) = A0 dn[A1 (n + n 0 ), m ] e−i (ω1 t + ϕ1),

# Solution

i ψ2 nt + [ψ2 n + 1 + ψ2 n − 1 − (2 +

Equation ν −2μ i ψ1nt + (ψ1n + 1 + ψ1n − 1 − 2 ψ1n ) + (μ1 ∣ψ1n∣2 + μ2 ∣ψ2 n∣2 ) (ψ1n + 1 + ψ1n − 1 + 1 μ 1 ψ1n ) = 0,

8.11 Summary of Section 8.10

Eq. #

(8.71)

discrete solitary (8.72) wave

discrete brightdark soliton

discrete solitary (8.70) wave

discrete solitary (8.69) wave

Name

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

8-56

A0

2 + n2 −i (ω1 t + ϕ ) 1 , e 1 + n2 B0 e−i (ω2 t + ϕ2) 1 + n2

ψ2(n , t ) = B0 sin[A1 (n + n 0 )] e−i (ω2 t + ϕ2)

10. ψ1(n , t ) = A0 cos[A1 (n + n 0 )] e−i (ω1 t + ϕ1),

ψ2(n , t ) =

9. ψ (n , t ) = A 0 1

ψ2(n , t ) =

e−i (ω2 t + ϕ2)

e−i (ω1 t + ϕ1),

1 + n2 e−i (ω1 t + ϕ1), 1 + n2 + n 4 B0 n 2 e−i (ω2 t + ϕ2) 1 + n2 + n 4

1 + n2

1 + n2 B0 n

8. ψ (n , t ) = A 0 1

ψ2(n , t ) =

7. ψ1(n , t ) =

ψ2(n , t ) = B0 sech[A1 (n + n 0 )] tanh[A1 (n + n 0 )] e−i (ω2 t + ϕ2)

6. ψ1(n , t ) = {A0 sech2[A1 (n + n 0 )] + A2 } e−i (ω1 t + ϕ1),

ψ2(n , t ) = B0 m sn[A1 (n + n 0 ), m ] cn[A1 (n + n 0 ), m ] e−i (ω2 t + ϕ2)

5. ψ1(n , t ) = {A0 dn2[A1 (n + n 0 ), m ] + A2 } e−i (ω1 t + ϕ1),

(Continued )

−4 , m2 A02

−2 A2 , 2−m

μ2 =

ω1 = B02

μ12

ν1 μ2

,

, 0 ⩽ m ⩽ 1,

ω2 =

μ1 A02

ν1 , μ1

−4 , A02

μ2 =

,

ν1 , μ1

ω2 = μ12

ν1 μ2

,

ν1 , μ1

ω2 = μ12

ν1 μ2

, μ1 =

−1 , A02

μ2 = B02

μ1 A02

,

ν1 , μ1

ω2 = μ12

ν1 μ2

, μ1 =

−1 , A02

μ2 =

B02

μ1 A02

,

ν1 , μ1

ω2 =

μ12

ν1 μ2

, μ1 =

−1 , A02

μ2 = 1,

ν1 , μ1

ω2 =

μ12

ν1 μ2

, μ1 =

−1 , A02

μ2 =

B02

μ1 A02

,

A0, B0, ϕ1, and ϕ2 are arbitrary real constants, n0 is an arbitrary real integer

ω1 =

A0, B0, ϕ1, and ϕ2 are arbitrary real constants

ω1 =

A0, B0, ϕ1, and ϕ2 are arbitrary real constants

ω1 =

A0, B0, ϕ1, and ϕ2 are arbitrary real constants

ω1 =

A1, A2, B0, ϕ1, and ϕ2 are arbitrary real constants, n0 is an arbitrary real integer

μ1 =

μ1 A02 B02

A0 = − 2 A2 , ω1 =

A1, A2, B0, ϕ1, and ϕ2 are arbitrary real constants, n0 is an arbitrary real integer

μ1 =

A0 =

–

–

–

–

–

(8.78)

(8.77)

(8.76)

(8.75)

(8.74)

discrete solitary (8.73) wave

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

m sd[A1 (n + n 0 ), m ] e−i (ω2 t + ϕ2)

B0

m sn[A1 (n + n 0 ), m ] dn2[A1 (n + n 0 ), m ]

e−i (ω2 t + ϕ2)

8-57

ψ2(n , t ) = B0 sinh[A1 (n + n 0 )] cosh[A1 (n + n 0 )] e−i (ω2 t + ϕ2)

14. ψ1(n , t ) = {A0 cosh2[A1 (n + n 0 )] + A2 } e−i (ω1 t + ϕ1),

ψ2(n , t ) =

13. ψ1(n , t ) = {A0 nd2[A1 (n + n 0 ), m ] + A2 } e−i (ω1 t + ϕ1),

ψ2(n , t ) = B0 sinh[A1 (n + n 0 )] e−i (ω2 t + ϕ2)

12. ψ1(n , t ) = A0 cosh[A1 (n + n 0 )] e−i (ω1 t + ϕ1),

ψ2(n , t ) = B0

11. ψ1(n , t ) = A0 nd[A1 (n + n 0 ), m ] e−i (ω1 t + ϕ1),

μ2 B02 , ∣ μ1 ∣

ω1 =

ν1 , μ1

ω2 = μ12

ν1 μ2

,

μ2 B02 , ∣ μ1∣

ω1 =

ν1 , μ1

ω2 = μ12

ν1 μ2

,

μ12

ν1 μ2

, μ1 =

−4 , A02

, ω1 =

ν1 , μ1

μ2 > 0, 0 ⩽ m ⩽ 1,

∣ μ1∣ A02 μ2

μ12

ν1 μ2

, μ1 =

−4 , A02

, ω1 =

μ2 > 0 ,

∣ μ1∣ A02 μ2

ν1 , μ1

A1, A2, ϕ1, and ϕ2 are arbitrary real constants, n0 is an arbitrary real integer

ω2 =

A0 = − 2 A2 , B0 =

A1, A2, ϕ1, and ϕ2 are arbitrary real constants, n0 is an arbitrary real integer

ω2 =

A0 = − 2 A2 , B0 =

A1, B0, ϕ1, and ϕ2 are arbitrary real constants, n0 is an arbitrary real integer

μ1 = −1, μ2 = 1,

A0 =

A1, A2, B0, ϕ1, and ϕ2 are arbitrary real constants, n0 is an arbitrary real integer

μ1 = −1, μ2 = 1, 0 ⩽ m ⩽ 1,

A0 =

(8.80)

–

(8.82)

discrete solitary (8.81) wave

–

discrete solitary (8.79) wave

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

8.12 Coupled Ablowitz–Ladik Equation Equation: i ψ 1nt + ψ 1n + 1 + ψ 1n − 1 − 2 ψ 1n + (μ1 ∣ψ 1n∣2 + μ 2 ∣ψ 2n∣2 ) (ψ 1n + 1 + ψ 1n − 1) = 0, 2 μ2 i ψ 2nt + ψ 2n + 1 + ψ 2n − 1 − ψ 2n + (μ1 ∣ψ 1n∣2 + μ 2 ∣ψ 2n∣2 ) (ψ 2n + 1 + ψ 2n − 1) = 0, μ1

(8.83)

where μ1 and μ2 are real constants, ψj = ψj (n, t ) is the complex function profile, j = 1, 2, the integer site index, n, and t are its two independent variables. Solutions: Solution 1. dn(n,t,m)-sn(n,t,m) moving discrete SW (Figure 8.34)

ψ1(n , t ) = A0 dn[A1 (n − v t + n 0), m ] e−i (ω1 t − κ1 n + ϕ1), ψ2(n , t ) = B0

m sn[A1 (n − v t + n 0), m ] e−i (ω2 t − κ 2 n + ϕ2),

(8.84)

where 1 + μ2 B02 ns2(A1, m ) , μ1 cs2(A1, m ) κ2 = sin−1[cn(A1, m ) sin(κ1)], 2 sin(κ1) (1 + μ2 B02 ) , v= A1 cs(A1, m ) 2 2 (1 + μ2 B0 ) cos(κ1) dn(A1, m ) , ω1 = 2 − cn2(A1, m ) 2 2μ 2 (1 + μ2 B0 ) cos(κ2 ) dn(A1, m ) , ω2 = μ 2 − cn(A1, m ) 1

A0 =

Figure 8.34. Moving discrete solitary wave (8.84) at t = 0 . Blue is ψ1 and red is ψ2 with B0 = A1 = 1, μ1 = − 1, μ2 = κ1 = 1, m = 1/2 , n 0 = − 1/3, and ϕ1 = ϕ2 = 0 . Lines are guide to the eye. Animation available online at https://iopscience.iop.org/book/978-0-7503-2428-1.

8-58

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

1 + μ2 B02 ns2(A1, m ) μ1 cs2(A1, m )

> 0,

0 ⩽ m ⩽ 1, A1, B0, κ1, ϕ1, ϕ2, μ1, μ2 , and n0 are arbitrary real constants.

• Reference: [9]. Solution 2. cn(n,t,m)-sn(n,t,m) moving discrete SW

ψ1(n , t ) = A0

m cn[A1 (n − v t + n 0), m ] e−i (ω1 t − κ1 n + ϕ1),

ψ2(n , t ) = B0

m sn[A1 (n − v t + n 0), m ] e−i (ω2 t − κ 2 n + ϕ2),

(8.85)

where 1 + μ2 B02 ns2(A1, m ) , μ1 ds2(A1, m ) κ2 = sin−1[dn(A1, m ) sin(κ1)], 2 sin(κ1) (1 + m μ2 B02 ) , v= A1 ds(A1, m ) 2 2 (1 + m μ2 B0 ) cos(κ1) cn(A1, m ) , ω1 = 2 − dn2(A1, m ) 2 2μ 2 (1 + m μ2 B0 ) cos(κ2 ) cn(A1, m ) , ω2 = μ 2 − dn(A1, m ) 1 2 2 1 + μ2 B0 ns (A1, m ) > 0, μ1 ds2(A1, m )

A0 =

0 ⩽ m ⩽ 1, A1, B0, κ1, ϕ1, ϕ2, μ1, μ2 , and n0 are arbitrary real constants.

• Reference: [9]. Solution 3. sech(n,t)-tanh(n,t) moving discrete bright-dark soliton (Figure 8.35)

ψ1(n , t ) = A0 sech[A1 (n − v t + n 0)] e−i (ω1 t − κ1 n + ϕ1), ψ2(n , t ) = B0 tanh[A1 (n − v t + n 0)] e−i (ω2 t − κ 2 n + ϕ2), where sinh2(A1) + μ2 B02 cosh2(A1) , μ1 1 − κ2 = sin [sech(A1) sin(κ1)], 2 sin(κ1) sinh(A1) (1 + μ2 B02 ) , v= A1 2 ω1 = 2 − 2 (1 + μ2 B0 ) cos(κ1) cosh(A1), 2μ ω2 = μ 2 − 2 (1 + μ2 B02 ) cos(κ2 ),

A0 =

1

sinh2(A1) + μ2 B02 cosh2(A1) μ1

> 0,

8-59

(8.86)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Figure 8.35. Moving discrete bright-dark soliton (8.86). Blue is ψ1 and red is ψ2 with B0 = A1 = μ2 = κ1 = 1, μ1 = 3, and n 0 = ϕ1 = ϕ2 = 0 . (a) at t = −1, (b) at t = 0, and (c) at t = 1. The lines are guides for the eye. Animation available online at https://iopscience.iop.org/book/978-0-7503-2428-1.

A1, B0, κ1, ϕ1, ϕ2, μ1, μ2 , and n0 are arbitrary real constants. • Reference: [9]. Solution 4. dn(n,t,m)-cn(n,t,m) moving discrete SW

ψ1(n , t ) = A0 dn[A1 (n − v t + n 0), m ] e−i (ω1 t − κ1 n + ϕ1), ψ2(n , t ) = B0

m cn[A1 (n − v t + n 0), m ] e−i (ω2 t − κ 2 n + ϕ2),

(8.87)

where 1 − μ2 B02 ds2(A1, m ) , μ1 cs2(A1, m ) sin( κ ) cn( A , m ) 1 1 κ2 = sin−1[ dn( ], A1, m ) 2 sin(κ1) [1 − (1 − m ) μ2 B02 ] , v= A1 cs(A1, m )

A0 =

⎡ cos(κ ) dn(A , m ) ⎤ ω1 = 2 − 2 [1 − (1 − m ) μ2 B02 ]⎣ cn12(A , m1) ⎦, 1 2 μ2 2 ⎡ cos(κ2 ) ⎤ ω2 = μ − 2 [1 − (1 − m ) μ2 B0 ]⎣ cn(A , m ) ⎦, 1 1 1 − μ2 B02 ds2(A1, m ) μ1 cs2(A1, m )

> 0,

0 < m ⩽ 1, A1, B0, κ1, ϕ1, ϕ2, μ1, μ2 , and n0 are arbitrary real constants.

• Reference: [9]. Note: Solutions (5–18) below can be obtained from solutions (1–14) in section 8.10 2μ with the replacements: ω1 = 2 and ω2 = μ 2 . 1

Solution 5. dn(n,m)-sn(n,m) discrete SW (Figure 8.36)

ψ1(n , t ) = A0 dn[A1 (n + n 0), m ] e−i (ω1 t + ϕ1), ψ2(n , t ) = B0

m sn[A1 (n + n 0), m ] e−i (ω2 t + ϕ2),

8-60

(8.88)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Figure 8.36. Discrete solitary wave (8.88). Blue is ψ1 and red is ψ2 with B0 = A1 = μ1 = μ2 = κ1 = 1, m = 1/2 , and n 0 = ϕ1 = ϕ2 = 0 . (a) at t = −1, (b) at t = 0, and (c) at t = 1. The lines are guides for the eye.

where ω1 = 2, 2μ ω2 = μ 2 , 1

μ1 = μ2 =

−1 , A02 μ1 A02 B02

,

0 ⩽ m ⩽ 1, A0, A1, B0, ϕ1, and ϕ2 are arbitrary real constants, n0 is an arbitrary real integer.

• Reference: [9]. Solution 6. cn(n,m)-sn(n,m) discrete SW (Figure 8.37)

ψ1(n , t ) = A0

m cn[A1 (n + n 0), m ] e−i (ω1 t + ϕ1),

ψ2(n , t ) = B0

m sn[A1 (n + n 0), m ] e−i (ω2 t + ϕ2),

where ω1 = 2, 2μ ω2 = μ 2 , 1

μ1 = μ2 =

−1 , m A02 2 μ1 A0 B02

,

0 ⩽ m ⩽ 1, A0, A1, B0, n0, ϕ1, and ϕ2 are arbitrary real constants.

• Reference: [9].

8-61

(8.89)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Figure 8.37. Discrete solitary wave (8.89). Blue is ψ1 and red is ψ2 with B0 = A1 = μ1 = μ2 = κ1 = 1, m = 1/2 , and n 0 = ϕ1 = ϕ2 = 0 . (a) at t = −1, (b) at t = 0, and (c) at t = 1. The lines are guides for the eye.

Solution 7. sech(n)-tanh(n) discrete bright-dark soliton

ψ1(n , t ) = A0 sech[A1 (n + n 0)] e−i (ω1 t + ϕ1), ψ2(n , t ) = B0 tanh[A1 (n + n 0)] e−i (ω2 t + ϕ2),

(8.90)

where ω1 = 2, 2μ ω2 = μ 2 , 1

μ1 = μ2 =

−1 , A02 μ1 A02 B02

,

A0, A1, B0, n0, ϕ1, and ϕ2 are arbitrary real constants. • Reference: [9]. Solution 8. dn2(n,m)-sn(n,m) dn(n,m) discrete SW

ψ1(n , t ) = {A0 dn2[A1 (n + n 0), m ] + A2 } e−i (ω1 t + ϕ1), ψ2(n , t ) = B0

m sn[A1 (n + n 0), m ] dn[A1 (n + n 0), m ] e−i (ω2 t + ϕ2),

where A0 = −2 A2 , ω1 = 2, 2μ ω2 = μ 2 , 1

μ1 = μ2 =

−4 , A02 μ1 A02 B02

,

0 ⩽ m ⩽ 1,

8-62

(8.91)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

A1, A2, B0, n0, ϕ1, and ϕ2 are arbitrary real constants. • Reference: [9]. Solution 9. dn2(n,m)-sn(n,m)cn(n,m) discrete SW

ψ1(n , t ) = {A0 dn2[A1 (n + n 0), m ] + A2 } e−i (ω1 t + ϕ1), ψ2(n , t ) = B0 m sn[A1 (n + n 0), m ] cn[A1 (n + n 0), m ] e−i (ω2 t + ϕ2),

(8.92)

where −2 A A0 = 2 − m2 , ω1 = 2, 2μ ω2 = μ 2 , 1

μ1 = μ2 =

−4 , m2 A02 2 μ1 A0 B02

,

0 ⩽ m ⩽ 1, A1, A2, B0, n0, ϕ1, and ϕ2 are arbitrary real constants.

• Reference: [9]. Solution 10. sech2(n)-sech(n) tanh(n)

ψ1(n , t ) = {A0 sech2[A1 (n + n 0)] + A2 } e−i (ω1 t + ϕ1), ψ2(n , t ) = B0 sech[A1 (n + n 0)] tanh[A1 (n + n 0)] e−i (ω2 t + ϕ2),

(8.93)

where A0 = −2 A2 , ω1 = 2, 2μ ω2 = μ 2 , 1

μ1 = μ2 =

−4 , A02 μ1 A02 B02

,

A1, A2, B0, n0, ϕ1, and ϕ2 are arbitrary real constants. • Reference: [9]. Solution 11. Rational Solution I

ψ1(n , t ) = ψ2(n , t ) =

A0 1 + n2 B0 n 1 + n2

8-63

e−i (ω1 t + ϕ1), (8.94) e−i (ω2 t + ϕ2),

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

where ω1 = 2, 2μ ω2 = μ 2 , 1

μ1 = μ2 =

−1 , A02 μ1 A02 B02

,

A0, B0, ϕ1, and ϕ2 are arbitrary real constants. • Reference: [9]. Solution 12. Rational Solution II

1 + n2 e−i (ω1 t + ϕ1), 1 + n2 + n 4 B0 n2 e−i (ω2 t + ϕ2), 1 + n2 + n 4

ψ1(n , t ) = A0 ψ2(n , t ) =

(8.95)

where ω1 = 2, 2μ ω2 = μ 2 , 1

μ1 = μ2 =

−1 , A02 μ1 A02 B02

,

A0, B0, ϕ1, and ϕ2 are arbitrary real constants. • Reference: [9]. Solution 13. Rational Solution III

ψ1(n , t ) = A0 ψ2(n , t ) =

2 + n2 −i (ω1 t + ϕ ) 1 , e 1 + n2 B0

1 + n2

e−i (ω2 t + ϕ2),

where ω1 = 2, 2μ ω2 = μ 2 , 1

μ1 =

−1 , A02

μ2 = 1, A0, B0, ϕ1, and ϕ2 are arbitrary real constants.

• Reference: [9].

8-64

(8.96)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Solution 14. cos(n)-sin(n)

ψ1(n , t ) = A0 cos[A1 (n + n 0)] e−i (ω1 t + ϕ1), ψ2(n , t ) = B0 sin[A1 (n + n 0)] e−i (ω2 t + ϕ2),

(8.97)

where ω1 = 2, 2μ ω2 = μ 2 , 1

μ1 = μ2 =

−1 , A02 μ1 A02 B02

,

A0, B0, n0, ϕ1, and ϕ2 are arbitrary real constants. • Reference: [9]. Solution 15. nd(n,m)-sd(n,m) discrete SW

ψ1(n , t ) = A0 nd[A1 (n + n 0), m ] e−i (ω1 t + ϕ1), ψ2(n , t ) = B0

m sd[A1 (n + n 0), m ] e−i (ω2 t + ϕ2),

(8.98)

where A0 =

μ2 B02 , ∣ μ1 ∣

ω1 = 2, 2μ ω2 = μ 2 , 1

μ1 = −1, μ2 = 1, 0 ⩽ m ⩽ 1, A1, A2, B0, n0, ϕ1, and ϕ2 are arbitrary real constants.

• Reference: [9]. Solution 16. cosh(n)-sinh(n)

ψ1(n , t ) = A0 cosh[A1 (n + n 0)] e−i (ω1 t + ϕ1), ψ2(n , t ) = B0 sinh[A1 (n + n 0)] e−i (ω2 t + ϕ2), where A0 =

μ2 B02 , ∣ μ1 ∣

ω1 = 2, 2μ ω2 = μ 2 , 1

8-65

(8.99)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

μ1 = −1, μ2 = 1, A1, B0, n0, ϕ1, and ϕ2 are arbitrary real constants.

• Reference: [9]. Solution 17. discrete SW

ψ1(n , t ) = {A0 nd2[A1 (n + n 0), m ] + A2 } e−i (ω1 t + ϕ1), B m sn[A1 (n + n 0), m ] −i (ω2 t + ϕ ) 2 , e ψ2(n , t ) = 0 2 dn [A1 (n + n 0), m ]

(8.100)

where A0 = −2 A2 , B0 =

∣ μ1 ∣ A02 μ2

,

ω1 = 2, 2μ ω2 = μ 2 , 1

μ1 =

−4 , A02

μ2 > 0 , 0 ⩽ m ⩽ 1, A1, A2, n0, ϕ1, and ϕ2 are arbitrary real constants.

• Reference: [9]. Solution 18.

ψ1(n , t ) = {A0 cosh2[A1 (n + n 0)] + A2 } e−i (ω1 t + ϕ1), ψ2(n , t ) = B0 sinh[A1 (n + n 0)] cosh[A1 (n + n 0)] e−i (ω2 t + ϕ2), where A0 = −2 A2 , B0 =

∣ μ1∣ A02 μ2

,

ω1 = 2, 2μ ω2 = μ 2 , 1

μ1 =

−4 , A02

μ2 > 0 , A1, A2, n0, ϕ1, and ϕ2 are arbitrary real constants.

• Reference: [9].

8-66

(8.101)

2 μ2 μ1

8-67

m cn[A1 (n − v t + n 0 ), m ] e−i (ω1 t − κ1 n + ϕ1),

m sn[A1 (n − v t + n 0 ), m ] e−i (ω2 t − κ2 n + ϕ2)

ψ2(n , t ) = B0

m sn[A1 (n − v t + n 0 ), m ] e−i (ω2 t − κ2 n + ϕ2)

2. ψ1(n , t ) = A0

ψ2(n , t ) = B0

μ1 cs2(A1, m )

1 + μ2 B02 ns2(A1, m )

,v=

−

,

cn2(A1, m ) 2 (1 + μ2 B02 ) cos(κ2 ) dn(A1, m ) , cn(A1, m )

0 ⩽ m ⩽ 1,

> 0,

μ1 ds2(A1, m )

1 + μ2 B02 ns2(A1, m )

,v=

−

2

dn(A1, m )

,

, 0 ⩽ m ⩽ 1,

> 0,

A1, B0, κ1, ϕ1, ϕ2 , μ1, μ2 , and n0 are arbitrary real constants

ω2 =

2 μ2 μ1

ω1 = 2 −

1 + μ2 B02 ns2(A1, m )

2 sin(κ1) (1 + m μ2 B02 ) , A1 ds(A1, m )

μ1 ds2(A1, m ) (1 + m μ2 B02 ) cos(κ1) cn(A1, m ) dn2(A1, m ) 2 (1 + m μ2 B02 ) cos(κ2 ) cn(A1, m )

κ2 = sin−1[dn(A1, m ) sin(κ1)],

A0 =

A1, B0, κ1, ϕ1, ϕ2 , μ1, μ2 , and n0 are arbitrary real constants

ω2 =

2 μ2 μ1

ω1 = 2 −

μ1 cs2(A1, m )

1 + μ2 B02 ns2(A1, m )

2 sin(κ1) (1 + μ2 B02 ) , A1 cs(A1, m )

2 (1 + μ2 B02 ) cos(κ1) dn(A1, m )

κ2 = sin−1[cn(A1, m ) sin(κ1)],

A0 =

Conditions

ψ2 n + (μ1 ∣ψ1n∣2 + μ2 ∣ψ2 n∣2 ) (ψ2 n + 1 + ψ2 n − 1) = 0

1. ψ1(n , t ) = A0 dn[A1 (n − v t + n 0 ), m ] e−i (ω1 t − κ1 n + ϕ1),

# Solution

i ψ 2 n t + ψ2 n + 1 + ψ2 n − 1 −

Equation i ψ1nt + ψ1n + 1 + ψ1n − 1 − 2 ψ1n + (μ1 ∣ψ1n∣2 + μ2 ∣ψ2 n∣2 ) (ψ1n + 1 + ψ1n − 1) = 0,

8.13 Summary of Section 8.12

(8.85)

(8.84)

moving discrete solitary wave

moving discrete solitary wave

Eq. #

Name

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

8-68

m cn[A1 (n − v t + n 0 ), m ] e−i (ω2 t − κ2 n + ϕ2)

m sn[A1 (n + n 0 ), m ] e−i (ω2 t + ϕ2)

m cn[A1 (n + n 0 ), m ] e−i (ω1 t + ϕ1),

m sn[A1 (n + n 0 ), m ] e−i (ω2 t + ϕ2)

ψ2(n , t ) = B0

6. ψ1(n , t ) = A0

ψ2(n , t ) = B0

5. ψ1(n , t ) = A0 dn[A1 (n + n 0 ), m ] e−i (ω1 t + ϕ1),

ψ2(n , t ) = B0

4. ψ1(n , t ) = A0 dn[A1 (n − v t + n 0 ), m ] e−i (ω1 t − κ1 n + ϕ1),

ψ2(n , t ) = B0 tanh[A1 (n − v t + n 0 )] e−i (ω2 t − κ2 n + ϕ2)

3. ψ1(n , t ) = A0 sech[A1 (n − v t + n 0 )] e−i (ω1 t − κ1 n + ϕ1),

(Continued ) sinh2(A1) + μ2 B02 cosh2(A1) μ1

,

1

> 0,

μ1 cs2(A1, m )

1 − μ2 B02 ds2(A1, m )

, κ2 = sin−1[

sin(κ1) cn(A1, m ) ], dn(A1, m )

2 sin(κ1) [1 − (1 − m ) μ2 B02 ] 1 − μ2 B02 ds2(A1, m ) , A1 cs(A1, m ) μ1 cs2(A1, m )

> 0,

2 μ2 , μ1

μ1 =

−1 , A02

μ2 =

B02

μ1 A02

,

2 μ2 , μ1

μ1 =

−1 , m A02

μ2 =

B02

μ1 A02

,

0 ⩽ m ⩽ 1, A0, A1, B0, ϕ1, and ϕ2 are arbitrary real constants, n0 is an arbitrary real integer

ω1 = 2 , ω2 =

0 ⩽ m ⩽ 1, A0, A1, B0, ϕ1, and ϕ2 are arbitrary real constants, n0 is an arbitrary real integer

ω1 = 2 , ω2 =

A1, B0, κ1, ϕ1, ϕ2 , μ1, μ2 , and n0 are arbitrary real constants

v=

⎡ cos(κ ) dn(A , m) ⎤ ω1 = 2 − 2 [1 − (1 − m ) μ2 B02 ]⎢⎣ cn12(A , m1) ⎥⎦, 1 2 μ2 2 ⎡ cos(κ2 ) ⎤ ω2 = μ − 2 [1 − (1 − m ) μ2 B0 ]⎣ cn(A , m) ⎦, 1 1 0 < m ⩽ 1,

A0 =

A1, B0, κ1, ϕ1, ϕ2 , μ1, μ2 , and n0 are arbitrary real constants

sinh2(A1) + μ2 B02 cosh2(A1) μ1

v=

2 sin(κ1) sinh(A1) (1 + μ2 B02 ) , A1 2 ω1 = 2 − 2 (1 + μ2 B0 ) cos(κ1) cosh(A1), 2μ ω2 = μ 2 − 2 (1 + μ2 B02 ) cos(κ2 ),

κ2 = sin−1[sech(A1) sin(κ1)],

A0 =

discrete solitary wave

discrete solitary wave

moving discrete solitary wave

moving discrete bright-dark soliton

(8.89)

(8.88)

(8.87)

(8.86)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

8-69

2 + n2 −i (ω1 t + ϕ ) 1 , e 1 + n2 B0 e−i (ω2 t + ϕ2) 1 + n2

ψ2(n , t ) =

13. ψ (n , t ) = A 0 1

ψ2(n , t ) =

e−i (ω2 t + ϕ2)

e−i (ω1 t + ϕ1),

1 + n2 e−i (ω1 t + ϕ1), 1 + n2 + n 4 B0 n 2 e−i (ω2 t + ϕ2) 1 + n2 + n 4

1 + n2

12. ψ (n , t ) = A 0 1

ψ2(n , t ) =

1 + n2 B0 n

A0

ψ2(n , t ) = B0 sech[A1 (n + n 0 )] tanh[A1 (n + n 0 )]

e−i (ω2 t + ϕ2)

10. ψ1(n , t ) = {A0 sech2[A1 (n + n 0 )] + A2 } e−i (ω1 t + ϕ1),

ψ2(n , t ) = B0 m sn[A1 (n + n 0 ), m ] cn[A1 (n + n 0 ), m ] e−i (ω2 t + ϕ2)

11. ψ1(n , t ) =

2 μ2 , μ1

μ1 =

−1 , A02

μ2 = B02

μ1 A02

,

μ2 =

μ1 A02 B02

,

A0 = − 2 A2 , ω1 = 2, ω2 =

2 μ2 , μ1

n0 is an arbitrary real integer

μ1 =

−4 , A02

A0, A1, B0, ϕ1, and ϕ2 are arbitrary real constants,

ω1 = 2 , ω2 =

2 μ2 , μ1

μ1 =

−4 , m2 A02

,

2 μ2 , μ1

μ1 =

−4 , A02

2 μ2 , μ1

μ1 =

−1 , A02

μ2 =

B02

μ1 A02

,

2 μ2 , μ1

μ1 =

−1 , A02

μ2 =

B02

μ1 A02

,

2 μ2 , μ1

μ1 =

−1 , A02

μ2 = 1, A0, B0, ϕ1, and ϕ2 are arbitrary real constants

ω1 = 2 , ω2 =

A0, B0, ϕ1, and ϕ2 are arbitrary real constants

ω1 = 2 , ω2 =

A0, B0, ϕ1, and ϕ2 are arbitrary real constants

ω1 = 2 , ω2 =

n0 is an arbitrary real integer

A1, A2, B0, ϕ1, and ϕ2 are arbitrary real constants,

μ2 =

μ1 A02 B02

A0 = − 2 A2 , ω1 = 2, ω2 =

0 ⩽ m ⩽ 1, A1, A2, B0, ϕ1, and ϕ2 are arbitrary real constants, n0 is an arbitrary real integer

,

ω1 = 2, ω2 =

μ1 A02 B02

−2 A2 , 2−m

μ2 =

A0 =

m sn[A1 (n + n 0 ), m ] dn[A1 (n + n 0 ), m ] e−i (ω2 t + ϕ2) 0 ⩽ m ⩽ 1, A1, A2, B0, ϕ1, and ϕ2 are arbitrary real constants, n0 is an arbitrary real integer

9. ψ1(n , t ) = {A0 dn2[A1 (n + n 0 ), m ] + A2 } e−i (ω1 t + ϕ1),

ψ2(n , t ) = B0

8. ψ1(n , t ) = {A0 dn2[A1 (n + n 0 ), m ] + A2 } e−i (ω1 t + ϕ1),

ψ2(n , t ) = B0 tanh[A1 (n + n 0 )] e−i (ω2 t + ϕ2)

7. ψ1(n , t ) = A0 sech[A1 (n + n 0 )] e−i (ω1 t + ϕ1),

–

–

–

–

–

discrete solitary wave

discrete solitary wave

discrete brightdark soliton

(8.96)

(8.95)

(8.94)

(8.93)

(8.92)

(8.91)

(8.90)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

m sd[A1 (n + n 0 ), m ] e−i (ω2 t + ϕ2)

8-70

B0

m sn[A1 (n + n 0 ), m ] dn2[A1 (n + n 0 ), m ]

e−i (ω2 t + ϕ2)

ψ2(n , t ) = B0 sinh[A1 (n + n 0 )] cosh[A1 (n + n 0 )] e−i (ω2 t + ϕ2)

18. ψ1(n , t ) = {A0 cosh2[A1 (n + n 0 )] + A2 } e−i (ω1 t + ϕ1),

ψ2(n , t ) =

17. ψ1(n , t ) = {A0 nd2[A1 (n + n 0 ), m ] + A2 } e−i (ω1 t + ϕ1),

ψ2(n , t ) = B0 sinh[A1 (n + n 0 )] e−i (ω2 t + ϕ2)

16. ψ1(n , t ) = A0 cosh[A1 (n + n 0 )] e−i (ω1 t + ϕ1),

ψ2(n , t ) = B0

15. ψ1(n , t ) = A0 nd[A1 (n + n 0 ), m ] e−i (ω1 t + ϕ1),

ψ2(n , t ) = B0 sin[A1 (n + n 0 )] e−i (ω2 t + ϕ2)

14. ψ1(n , t ) = A0 cos[A1 (n + n 0 )] e−i (ω1 t + ϕ1),

(Continued ) 2 μ2 , μ1

μ1 =

−1 , A02

μ2 = B02

μ1 A02

,

μ2 B02 , ∣ μ1 ∣

ω1 = 2, ω2 =

2 μ2 , μ1

μ2 B02 , ∣ μ1∣

ω1 = 2, ω2 =

2 μ2 , μ1

μ1 = −1, μ2 = 1,

−4 , A02

, ω1 = 2, ω2 =

μ2 > 0, 0 ⩽ m ⩽ 1,

∣ μ1∣ A02 μ2

2 μ2 , μ1

2 μ2 , μ1

∣ μ1∣ A02 , μ2 −4 μ1 = 2 , μ2 A0

> 0,

A1, A2, ϕ1, and ϕ2 are arbitrary real constants, n0 is an arbitrary real integer

ω1 = 2 , ω2 =

A0 = − 2 A2 , B0 =

A1, A2, ϕ1, and ϕ2 are arbitrary real constants, n0 is an arbitrary real integer

μ1 =

A0 = − 2 A2 , B0 =

n0 is an arbitrary real integer

A1, B0, ϕ1, and ϕ2 are arbitrary real constants,

A0 =

n0 is an arbitrary real integer

A1, A2, B0, ϕ1, and ϕ2 are arbitrary real constants,

μ1 = −1, μ2 = 1, 0 ⩽ m ⩽ 1,

A0 =

n0 is an arbitrary real integer

A0, B0, ϕ1, and ϕ2 are arbitrary real constants,

ω1 = 2 , ω2 =

–

discrete solitary wave

–

discrete solitary wave

(8.101)

(8.100)

(8.99)

(8.98)

(8.97)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

8.14 Coupled Saturable Discrete NLSE Equation:

i ψ1nt + ψ1n + 1 + ψ1n − 1 − 2 ψ1n +

ν1 (μ1 ∣ψ1n∣2 + μ2 ∣ψ2 n∣2 ) ψ1n = 0, μ1 (1 + μ1 ∣ψ1n∣2 + μ2 ∣ψ2 n∣2 )

i ψ 2 n t + ψ2 n + 1 + ψ2 n − 1 − 2 ψ2 n ⎡ ⎤ ν μ2 ⎢ν2 − 1 22 + ν2 (μ1 ∣ψ1n∣2 + μ2 ∣ψ2 n∣2 )⎥ ψ2 n ⎢⎣ ⎥⎦ μ1 = 0, + μ2 (1 + μ1 ∣ψ1n∣2 + μ2 ∣ψ2 n∣2 )

(8.102)

where ψj = ψj (n, t ) is the complex function profile, j = 1, 2, the integer site index, n, and t are its two independent variables, μ1, μ2 , ν1, and ν2 are real constants. Solutions: Solution 1. dn(n,m)-sn(n,m) discrete SW

ψ1(n , t ) = A0 dn[A1 (n + n 0), m ] e−i (ω1 t + ϕ1), m sn[A1 (n + n 0), m ] e−i (ω2 t + ϕ2),

ψ2(n , t ) = B0 where A0 =

ν1 2 μ12 dn(A1, m )

B0 =

ν1 cn2(A1, m ) 2 μ1 μ2 dn(A1, m )

A1 = μ1, ω1 = 2 −

ω2 = 2 −

−

1 μ1

−

,

1 μ2

,

ν1 , μ1 ν2 , μ2

μ2 = μ1 cn(A1, m ), ν1 1 > μ, 2 2 μ1 dn(A1, m ) ν1 cn2(A1, m ) 2 μ1 μ2 dn(A1, m )

1

>

1 , μ2

0 ⩽ m ⩽ 1, ν1, ν2 , μ1, n0, ϕ1, and ϕ2 are arbitrary real constants.

• Reference: [9].

8-71

(8.103)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Solution 2. cn(n,m)-sn(n,m) discrete SW

ψ1(n , t ) = A0

m cn[A1 (n + n 0), m ] e−i (ω1 t + ϕ1),

ψ2(n , t ) = B0

m sn[A1 (n + n 0), m ] e−i (ω2 t + ϕ2),

(8.104)

where A0 =

ν1 2 m μ12 cn(A1, m )

B0 =

ν1 dn2(A1, m ) 2 m μ1 μ2 cn(A1, m )

A1 = μ1, ω1 = 2 −

ω2 = 2 −

−

1 m μ1

−

,

1 m μ2

,

ν1 , μ1 ν2 , μ2

μ2 = μ1 dn(A1, m ), ν1 1 > mμ , 2 m μ12 cn(A1, m ) 1 ν1 dn2(A1, m ) 1 > mμ , 2 m μ1 μ2 cn(A1, m ) 2

0 ⩽ m ⩽ 1, ν1, ν2 , μ1, n0, ϕ1, and ϕ2 are arbitrary real constants.

• Reference: [9]. Solution 3. sech(n)-tanh(n) discrete bright-dark soliton

ψ1(n , t ) = A0 sech[A1 (n + n 0)] e−i (ω1 t + ϕ1), ψ2(n , t ) = B0 tanh[A1 (n + n 0)] e−i (ω2 t + ϕ2), where A0 =

ν1 cosh(A1) 2 μ12

B0 =

ν1 2 μ1 μ2 cosh(A1)

A1 = μ1, ω1 = 2 −

ω2 = 2 −

−

1 μ1

−

, 1 μ2

,

ν1 , μ1 ν2 , μ2

μ2 = μ1 sech(A1), ν1 cosh(A1) 1 > μ, 2 μ12 1 ν1 1 > μ, 2 μ1 μ2 cosh(A1) 2

ν1, ν2 , μ1, n0, ϕ1, and ϕ2 are arbitrary real constants.

• Reference: [9].

8-72

(8.105)

8-73

, B0 =

ψ2(n , t ) = B0 tanh[A1 (n + n 0 )] e−i (ω2 t + ϕ2)

>

1 , m μ2

−

ν1, ν2 , μ1, ϕ1, and ϕ2 are arbitrary real constants, n0 is an arbitrary real integer

2 μ12

ν1 cosh(A1)

1 m μ2

0 ⩽ m ⩽ 1,

ν1 1 1 , B0 = 2 μ μ cosh( − μ , A1) μ1 1 2 2 ν ν A1 = μ1, ω1 = 2 − μ1 , ω2 = 2 − μ2 , μ2 = μ1 sech(A1), 1 2 ν1 1 ν1 cosh(A1) 1 , > > μ , 2 μ μ cosh( 2 A1) μ2 2 μ1 1 2 1

A0 =

−

μ2 = μ1 dn(A1, m ),

ν1 dn2(A1, m ) 2 m μ1 μ2 cn(A1, m )

ν1, ν2 , μ1, ϕ1, and ϕ2 are arbitrary real constants, n0 is an arbitrary real integer

1 ν1 − m μ , B0 = 2 m μ12 cn(A1, m ) 1 ν1 ν2 e−i (ω2 t + ϕ2) A1 = μ1, ω1 = 2 − μ1 , ω2 = 2 − μ2 , ν dn2(A , m ) ν1 1 > m μ , 2 m 1μ μ cn(1 A , m) 2 m μ12 cn(A1, m ) 1 1 2 1

m cn[A1 (n + n 0 ), m ] e−i (ω1 t + ϕ1), A = 0

m sn[A1 (n + n 0 ), m ]

>

1 μ1

ν1, ν2 , μ1, ϕ1, and ϕ2 are arbitrary real constants, n0 is an arbitrary real integer

ν1 2 μ12 dn(A1, m )

−

ν1 cn2(A1, m ) 1 − μ , 2 μ1 μ2 dn(A1, m ) 2 ν ν 2 − μ1 , ω2 = 2 − μ2 , μ2 = μ1 cn(A1, m ), 1 2 ν cn2(A , m ) 1 1 , 2 μ1 μ dn(1A , m) > μ , 0 ⩽ m ⩽ 1, μ1 1 1 2 2

=0

Equation

ν1 2 μ12 dn(A1, m )

m sn[A1 (n + n 0 ), m ] e−i (ω2 t + ϕ2) A1 = μ1, ω1 =

A0 =

Conditions

μ2 (1 + μ1 ∣ ψ1n ∣2 +μ2 ∣ ψ2 n ∣2 )

3. ψ1(n , t ) = A0 sech[A1 (n + n 0 )] e−i (ω1 t + ϕ1),

ψ2(n , t ) = B0

2. ψ1(n , t ) = A0

ψ2(n , t ) = B0

= 0,

μ1 (1 + μ1 ∣ ψ1n ∣2 +μ2 ∣ ψ2 n ∣2 ) ⎡ ⎤ ν1 μ 2 ⎢ν2 − 22 + ν2 (μ1 ∣ ψ1n ∣2 +μ2 ∣ ψ2 n ∣2 )⎥ ψ2 n μ1 ⎣ ⎦

1. ψ1(n , t ) = A0 dn[A1 (n + n 0 ), m ] e−i (ω1 t + ϕ1),

# Solution

i ψ 2 n t + ψ2 n + 1 + ψ2 n − 1 − 2 ψ2 n +

i ψ1nt + ψ1n + 1 + ψ1n − 1 − 2 ψ1n +

ν1 (μ1 ∣ ψ1n ∣2 +μ2 ∣ ψ2 n ∣2 ) ψ1n

8.15 Summary of Section 8.14

,

(8.104)

(8.103)

Eq. #

discrete bright-dark soliton (8.105)

discrete solitary wave

discrete solitary wave

Name

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

References [1] Khare A and Saxena A 2015 Periodic and hyperbolic soliton solutions of a number of nonlocal nonlinear equations J. Math. Phys. 56 032104–27 [2] Khare A, Rasmussen K Ø, Samuelsen M R and Saxena A 2005 Exact solutions of the saturable discrete nonlinear Schrödinger equation J. Phys. A: Math. Gen. 38 807–14 [3] Yan Z 2009 Envelope solution profiles of the discrete nonlinear Schrödinger equation with a saturable nonlinearity Appl. Math. Lett. 22 448–52 [4] Khare A and Rasmussen K Ø 2009 Staggered and short-period solutions of the saturable discrete nonlinear Schrödinger equation J. Phys. A: Math. Theor. 42 085002-6 [5] Ankiewicz A, Akhmediev N and Lederer F 2011 Approach to first-order exact solutions of the Ablowitz-Ladik equation Phys. Rev. E 83 056602-6 [6] Hua-Mei L and Feng-Min W 2005 Exact discrete soliton solutions of quintic discrete nonlinear Schrödinger equation Chin. Phys. 14 1069-7 [7] Kevrekidis P G 2009 The Discrete Nonlinear Schrödinger Equation: Mathematical Analysis, Numerical Computations and Physical Perspectives (Springer Tracts in Modern Physics vol 232) (Berlin: Springer) [8] Khare A, Dmitriev S V and Saxena A 2007 Exact moving and stationary solutions of a generalized discrete nonlinear Schrödinger equation J. Phys. A: Math. Theor. 40 11301–17 [9] Khare A and Saxena A 2012 Solutions of several coupled discrete models in terms of Lamé polynomials of order one and two Pramana 78 187–213

8-74

IOP Publishing

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations Usama Al Khawaja and Laila Al Sakkaf

Chapter 9 Nonlocal Nonlinear Schrödinger Equation

A Glance at Chapter 9

doi:10.1088/978-0-7503-2428-1ch9

9-1

ª IOP Publishing Ltd 2020

9-2

0 0

0

0

0

N ¯1 = 0 i Φ1t + a1 Φ1xx + ∑k = 1b1k ∣σk∣2 Φ12 Φ

¯ 1 − g2 Φ2 Φ ¯ 2) Φ1 + g0 (g1 + g2 ) Φ1 − 2 g0 g2 Φ2 = 0, i Φ1t + Φ1xx + (g1 Φ1 Φ ¯ 1 − g2 Φ2 Φ ¯ 2) Φ2 − g0 (g1 + g2 ) Φ2 + 2 g0 g1 Φ1 = 0 i Φ2t + Φ2 xx + (g1 Φ1 Φ

¯ 1 − g2 Φ2 Φ ¯ 2) Φ1 = 0, i Φ1t + Φ1xx + (g1 Φ1 Φ ¯ ¯ 2 ) Φ2 = 0 i Φ2t + Φ2 xx + (g1 Φ1 Φ1 − g2 Φ2 Φ

¯ 1 + a12 Φ2 Φ ¯ 2) Φ1 + 2 (b11 Φ1 Φ ¯ 2 + b12 Φ2 Φ ¯ 1) Φ1 = 0, i Φ1t + Φ1xx + 2 (a11 Φ1 Φ ¯ 1 + a22 Φ2 Φ ¯ 2) Φ2 + 2 (b21 Φ1 Φ ¯ 2 + b22 Φ2 Φ ¯ 1) Φ2 = 0, i Φ2t + Φ2 xx + 2 (a21 Φ1 Φ

¯ 1 + Φ2 Φ ¯ 2) Φ1 + 2 ((a + i b ) Φ1 Φ ¯ 2 + (a − i b ) Φ2 Φ ¯ 1) Φ1 = 0, i Φ1t + Φ1xx − 2(a + b ) (Φ1 Φ ¯ ¯ ¯ ¯ 1) Φ2 = 0 i Φ2t + Φ2 xx − 2(a + b ) (Φ1 Φ1 + Φ2 Φ2) Φ2 + 2 ((a + i b ) Φ1 Φ2 + (a − i b ) Φ2Φ

5

6

7

8

9

4 15

¯n = 0 i Φnt + Φn + 1 + Φn − 1 − 2 Φn + a2 (Φn + 1 + Φn − 1) Φn Φ

¯ n + a3 Φn Φ ¯ n + a 4 Φ2n Φ ¯ 2n (Φn + 1 + Φn − 1) = 0 i Φnt + a1 (Φn + 1 + Φn − 1 − 2 Φn) + a2 Φ2n Φ

12

11

12

Total

(

2

i Φnt + Φn + 1 + Φn − 1 − 2 Φn +

10

)

0

¯1 = 0 i Φ1t + b0 Φ1xx − (c1 + c2 ) Φ12 Φ

4

3

0

¯1 = 0 i Φ1t + b0 Φ1xx + (c1 + c2∣σ∣2 ) Φ12 Φ

3

=0

4

¯ 1 + a2 Φ2 Φ ¯ 2) Φ1 = 0, i Φ1t + Φ1xx + (a1 Φ1 Φ ¯ ¯ 2 ) Φ2 = 0 i Φ2t + Φ2 xx + (b1 Φ1 Φ1 + b2 Φ2 Φ

2

¯n a2 Φ2n Φ ¯n 1 + μ Φn Φ

2

¯ =0 i Φt + a1 Φxx + a2 Φ2 Φ

Solutions

1

Equation

A Statistical View of Chapter 9

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

9.1 Nonlocal NLSE If ψ (x , t ) is a solution of the fundamental NLSE, (2.1),

i ψt + a1 ψxx + a2 ∣ψ ∣2 ψ = 0, then

⎧ ψ (x , t , a1, a2 ), Φ(x , t ; a1, a2 ) = ⎨ ⎩ ψ (x , t , a1, −a2 ),

ψ is an even function in x , ψ is an odd function in x

is a solution of

¯ = 0, i Φt + a1 Φxx + a2 Φ2 Φ

(9.1)

where ψ = ψ (x , t ; a1, a2 ), Φ = Φ(x , t ; a1, a2 ), ¯ = Φ*( −x , t ; a1, a2 ), Φ a1 and a2 are arbitrary real constants. Example 1. Even function: sech(x) bright soliton Given

ψ (ζ, t ) = A0

2 2 a1 ⎡ ⎤ sech[A0 ζ ] e i ⎣a1 A0 (t − t0) + ϕ0⎦ a2

is a solution of (2.1), then

Φ(ζ, t ) = A0

2 2 a1 ⎡ ⎤ sech[A0 ζ ] e i ⎣a1 A0 (t − t0) + ϕ0⎦ a2

is a solution of (9.1), where ζ = x − x0 , a1 a2 > 0, A0, x0, t0, and ϕ0 are arbitrary real constants. Example 2. Odd function: tanh(x) dark soliton Given ⎡ A ⎤ ψ (ζ, t ) = A0 tanh ⎢ 1 ζ ⎥ e−i [A2 (t − t0) + ϕ0] ⎣ a1 ⎦ is a solution of (2.1), where A2 = 2 A12 , a2 = −

2 A12 A02

,

a1 > 0, ζ = x − x0 ,

9-3

(9.2)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

A0, A1, x0, t0, and ϕ0 are arbitrary real constants, then

⎡ A ⎤ Φ(ζ, t ) = A0 tanh ⎢ 1 ζ ⎥ e−i [A2 (t − t0) + ϕ0] ⎣ a1 ⎦

(9.3)

is a solution of (9.1), where A2 = 2 A12 , a2 =

2 A12 A02

,

(notice the change of sign compared to the local case), a1 > 0, ζ = x − x0 , A0, A1, x0, t0, and ϕ0 are arbitrary.

9.2 Nonlocal Coupled NLSE If (ψ1, ψ2 ) is a solution of

( + (b ψ ψ * + b

) ψ *) ψ = 0,

i ψ1t + ψ1xx + a1 ψ1 ψ1* + a2 ψ2 ψ2* ψ1 = 0, i ψ2t + ψ2 xx

1

1

1

2

ψ2

2

(9.4)

2

then

⎧ ψ (x , t , a1, a2 , b1, b2 ), Φ1(x , t ; a1, a2 , b1, b2 ) = ⎨ 1 ⎩ ψ1(x , t , −a1, a2 , −b1, b2 ), ⎧ ψ (x , t , a1, a2 , b1, b2 ), Φ2(x , t ; a1, a2 , b1, b2 ) = ⎨ 2 ⎩ ψ2(x , t , a1, −a2 , b1, −b2 ),

ψ1 is an even function in x , ψ1 is an odd function in x , ψ2 is an even function in x , ψ2 is an odd function in x

is a solution of

¯ 1 + a2 Φ2 Φ ¯ 2) Φ1 = 0, i Φ1t + Φ1xx + (a1 Φ1 Φ ¯ 1 + b 2 Φ2 Φ ¯ 2) Φ2 = 0, i Φ2t + Φ2 xx + (b1 Φ1 Φ where ψ1,2 = ψ1,2(x , t ; a1, a2, b1, b2 ), Φ1,2 = Φ1,2(x , t ; a1, a2, b1, b2 ), ¯ 1,2 = Φ1,2 * ( −x , t ; a1, a2, b1, b2 ), Φ a1, a2, b1, and b2 are arbitrary real constants. Example 1. Even–Odd function: sech(x)-tanh(x) bright-dark soliton Given

ψ1(ζ, t ) = A0 sech[A1 ζ ] e−i [ω1 (t − t0) + ϕ1], ψ2(ζ, t ) = B0 tanh[A1 ζ ] e−i [ω2 (t − t0) + ϕ2]

9-4

(9.5)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

is a solution of (9.4), where ω1 = A12 − a1 A02 , ω2 = 2 A12 − b1 A02 , a1 = b1 =

2 A12 + a2 B02 2

A12

A02 + b2 B02 A02

, ,

ζ = x − x0 , A0 ≠ 0, A1, B0, x0, t0, ϕ1, and ϕ2 are arbitrary real constants, then

Φ1(ζ, t ) = A0 sech[A1 ζ ] e−i [ω1 (t − t0) + ϕ1], Φ2(ζ, t ) = B0 tanh[A1 ζ ] e−i [ω2 (t − t0) + ϕ2]

(9.6)

is a solution of (9.5), where ω1 = A12 − a1 A02 , ω2 = 2 A12 − b1 A02 , a1 = b1 =

2 A12 − a2 B02 2

A12

A02 − b2 B02 A02

,

(changed the sign of a2 ),

,

(changed the sign of b2 ),

ζ = x − x0 , A0 ≠ 0, A1, B0, x0, t0, ϕ1, and ϕ2 are arbitrary real constants.

Example 2. Odd–Odd function: cn(x,m)sn(x,m)-dn(x,m)sn(x,m) solitary wave (SW) Given

ψ1(ζ, t ) = A0 m cn[A1 ζ, m ] sn[A1 ζ, m ] e−i [ω1 (t − t0) + ϕ1], ψ2(ζ, t ) = B0

m dn[A1 ζ, m ] sn[A1 ζ, m ] e−i [ω2 (t − t0) + ϕ2]

is a solution of (9.4), where ω1 = (4 + m ) A12 , ω2 = (1 + 4 m ) A12 , a1 = a2 =

6 A12 A02

(1 − m ) −6 A12

B02 (1 − m )

, ,

b1 = a1, b2 = a2 , 0 ⩽ m < 1, ζ = x − x0 , A0 ≠ 0, B0 ≠ 0,

9-5

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

A1, x0, t0, ϕ1, and ϕ2 are arbitrary real constants, then

Φ1(ζ, t ) = A0 m cn[A1 ζ, m ] sn[A1 ζ, m ] e−i [ω1 (t − t0) + ϕ1], Φ2(ζ, t ) = B0

m dn[A1 ζ, m ] sn[A1 ζ, m ] e−i [ω2 (t − t0) + ϕ2]

(9.7)

is a solution of (9.5), where ω1 = (4 + m ) A12 , ω2 = (1 + 4 m ) A12 , a1 = a2 =

−6 A12 A02

(1 − m ) 6 A12

B02 (1 − m )

,

(changed the sign of a1),

,

(changed the sign of a2 ),

b1 = a1, b2 = a2 , 0 ⩽ m < 1, ζ = x − x0 , A0 ≠ 0, B0 ≠ 0, A1, x0, t0, ϕ1, and ϕ2 are arbitrary real constants.

Example 3. Odd–Even function: sd(x,m)-nd(x,m) SW. Given

ψ1(ζ, t ) = A0

m (1 − m) sd[A1 ζ, m ] e−i [ω1 (t − t0) + ϕ1],

ψ2(ζ, t ) = B0

1 − m nd[A1 ζ, m ] e−i [ω2 (t − t0) + ϕ2]

is a solution of (9.4), where ω1 = −(m − 1) a1 A02 − A12 , ω2 = (m − 2) A12 − (m − 1) b1 A02 , a1 = − b1 = −

−2 A12 + a2 B02 −2

A02 + b2 B02

A12

A02

, ,

0 ⩽ m ⩽ 1, ζ = x − x0 , A0 ≠ 0, A1, B0, x0, t0, ϕ1, and ϕ2 are arbitrary real constants, then

Φ1(ζ, t ) = A0

m (1 − m) sd[A1 ζ, m ] e−i [ω1 (t − t0) + ϕ1],

Φ2(ζ, t ) = B0

1 − m nd[A1 ζ, m ] e−i [ω2 (t − t0) + ϕ2]

is a solution of (9.5), where ω1 = (m − 1) a1 A02 − A12 , ω2 = (m − 2) A12 + (m − 1) b1 A02 ,

(changed the sign of a1), (changed the sign of b1),

9-6

(9.8)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

a1 = b1 =

−2 A12 + a2 B02 −2

A02 + b2 B02

A12

A02

,

(changed the sign of a1),

,

(changed the sign of b1),

0 ⩽ m ⩽ 1, ζ = x − x0 , A0 ≠ 0, A1, B0, x0, t0, ϕ1, and ϕ2 are arbitrary real constants.

Example 4. Even–Even function: sech2(x)-sech2(x). Given ψ1(ζ, t ) = [A0 sech2(A1 ζ ) + A3] e−i [ω1 (t − t0) + ϕ1],

ψ2(ζ, t ) = B0 sech2(A1 ζ ) e−i [ω2 (t − t0) + ϕ2] is a solution of (9.4), then

Φ1(ζ, t ) = {A0 sech2[A1 ζ ] + A3} e−i [ω1 (t − t0) + ϕ1], Φ2(ζ, t ) = B0 sech2[A1 ζ ] e−i [ω2 (t − t0) + ϕ2]

(9.9)

is a solution of (9.5), where −2 A A3 = 3 0 , ω1 = 2 A12 , ω2 = −2 A12 , a1 = a2 =

−9 A12 2 A02 9 A12 2 B02

,

,

b1 = a1, b2 = a2 , ζ = x − x0 , A0 ≠ 0, B0 ≠ 0, A1, x0, t0, ϕ1, and ϕ2 are arbitrary real constants.

9.3 Symmetry Reductions to Scalar Nonlocal NLSE 9.3.1 Symmetry Reduction I From Nonlocal Manakov System to Scalar Nonlocal NLSE The nonlocal CNLSE, (9.5),

¯ 1 + b 2 Φ2 Φ ¯ 2) Φ1 = 0, i Φ1t + b0 Φ1x x + (b1 Φ1 Φ ¯ 1 + c2 Φ2 Φ ¯ 2) Φ2 = 0, i Φ2t + b0 Φ2 xx + (c1 Φ1 Φ transforms to the scalar nonlocal NLSE

¯ 1 = 0, i Φ1t + b0 Φ1xx + (c1 + c2∣σ∣2 ) Φ12 Φ

9-7

(9.10)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

with the replacements: 1. Φ2(x , t ) = σ Φ1(x , t ), 2. b1 = c1 + (c2 − b2 ) ∣σ∣2 , where σ is an arbitrary complex constant. Conclusion: If Φ1(x , t ) is a solution of the nonlocal NLSE

¯ 1 = 0, i Φ1t + a1 Φ1xx + a2 Φ12 Φ then

(Φ1, Φ2) = (Φ1, σ Φ1)

(9.11)

is a solution of the nonlocal CNLSE

¯ 1 + b 2 Φ2 Φ ¯ 2) Φ1 = 0, i Φ1t + b0 Φ1xx + (b1 Φ1 Φ ¯ 1 + c2 Φ2 Φ ¯ 2) Φ2 = 0, i Φ2t + b0 Φ2 xx + (c1 Φ1 Φ with a1 = b0, a2 = c1 + c2∣σ∣2 , b1 = c1 + (c2 − b2 ) ∣σ∣2 . 9.3.2 Symmetry Reduction II From Nonlocal Manakov System to Scalar Nonlocal NLSE The nonlocal CNLSE, (9.5),

¯ 1 + b 2 Φ2 Φ ¯ 2) Φ1 = 0, i Φ1t + b0 Φ1x x + (b1 Φ1 Φ ¯ 1 + c2 Φ2 Φ ¯ 2) Φ2 = 0, i Φ2t − b0 Φ2xx + (c1 Φ1 Φ transforms to the scalar nonlocal NLSE

¯ 1 = 0, i Φ1t + b0 Φ1xx − (c1 + c2 ) Φ12 Φ with the replacements: 1. Φ2(x , t ; b1, b2 , c1, c2 ) ⎧Φ ¯ 1(x , t , b1, b2 , c1, c2 ) = e i ϕ⎨ ¯ 1(x , t , b1, b2 , −c1, −c2 ) ⎩Φ 2. b1 = −(c1 + c2 + b2 ), where ϕ is an arbitrary real constant.

9-8

if Φ1 is an even function in x, if Φ1 is an odd function in x,

(9.12)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Conclusion: I. If Φ1(x , t ) is an even solution of the nonlocal NLSE

¯ 1 = 0, i Φ1t + b0 Φ1xx + a2 Φ12 Φ then

⎡ ⎤ (Φ1, Φ2) = ⎣Φ1(x , t , b1, b2 , c1, c2 ), e i ϕ Φ*1 (x , t , b1, b2 , c1, c2 )⎦ is a solution of the nonlocal CNLSE

¯ 1 + b 2 Φ2 Φ ¯ 2) Φ1 = 0, i Φ1t + b0 Φ1xx + (b1 Φ1 Φ ¯ 1 + c2 Φ2 Φ ¯ 2) Φ2 = 0, i Φ2t − b0 Φ2 xx + (c1 Φ1 Φ with a2 = −(c1 + c2 ). II. If Φ1(x , t ) is an odd solution of the nonlocal NLSE

¯ 1 = 0, i Φ1t + b0 Φ1xx + a2 Φ12 Φ then

⎡ ⎤ (Φ1, Φ2) = ⎣Φ1(x , t , b1, b2 , c1, c2 ), e i ϕ Φ*1 ( − x , t , b1, b2 , − c1, − c2 )⎦

(9.13)

is a solution of the nonlocal CNLSE

¯ 1 + b 2 Φ2 Φ ¯ 2) Φ1 = 0, i Φ1t + b0 Φ1xx + (b1 Φ1 Φ ¯ 1 + c2 Φ2 Φ ¯ 2) Φ2 = 0, i Φ2t − b0 Φ2 xx + (c1 Φ1 Φ with a2 = −(c1 + c2 ). 9.3.3 Symmetry Reduction III From Nonlocal Vector NLSE to Scalar Nonlocal NLSE The generalized nonlocal CNLSE

⎛N ⎞ ¯ k ⎟⎟ Φj = 0, i Φj t + a1 Φj xx + ⎜⎜ ∑ b1k Φk Φ ⎝ k=1 ⎠

j = 1, 2, … , N ,

transforms to the scalar nonlocal NLSE N

i Φ1t + a1 Φ1xx +

∑ b1k ∣σk∣2 Φ12 Φ¯1 = 0, k=1

9-9

(9.14)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

with the replacement:

Φj (x , t ) = σj Φ1(x , t ), where σj are arbitrary complex constants and σ1 = 1. Conclusion: If Φ1(x , t ) is a solution of the nonlocal NLSE

i Φ1t + a1 Φ1xx + a2 Φ12 Φ1 = 0, then

(Φ1, Φ2 , Φ3, … , ΦN ) = (Φ1, σ2 Φ1, σ3 Φ1, … , σN ΦN )

(9.15)

is a solution of the generalized nonlocal CNLSE

i Φj t + a1 Φj xx

⎛N ⎞ ¯ k ⎟⎟ Φj = 0, + ⎜⎜ ∑ b1k Φk Φ ⎝ k=1 ⎠

j = 1, 2, … , N ,

N

with a2 = ∑k = 1b1k ∣σk∣2 .

9.4 Scaling Transformations 9.4.1 Linear and Nonlinear Coupling 9.4.1.1 General Case If (ψ1, ψ2 ) is a solution of

i ψ1t + ψ1xx + (b1 ψ1 ψ¯1 + b2 ψ2 ψ¯2 ) ψ1 = 0, i ψ2t + ψ2 xx + (b1 ψ1 ψ¯1 + b2 ψ2 ψ¯2 ) ψ2 = 0,

(9.16)

then

Φ1(x , t ) =

b1 ψ (x , t ) e i g0 (g1− g2) t + g1 − g2 1

g2 b2 ψ (x , t ) g1 (g2 − g1) 2

e−i g0 (g1− g2) t , Φ2 ( x , t ) =

(9.17)

b1 ψ (x , t ) e i g0 (g1− g2) t + g1 − g2 1 e−i g0 (g1− g2) t

is a solution of

9-10

g1 b2 ψ (x , t ) g2 (g2 − g1) 2

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

¯ 1 − g2 Φ2 Φ ¯ 2) Φ1 + g0 (g1 + g2 ) Φ1 − 2 g0 g2 Φ2 = 0, i Φ1t + Φ1xx + (g1 Φ1 Φ (9.18) ¯ 1 − g2 Φ2 Φ ¯ 2) Φ2 − g0 (g1 + g2 ) Φ2 + 2 g0 g1 Φ1 = 0, i Φ2t + Φ2 xx + (g1 Φ1 Φ where b1 (g1 − g2 ) > 0, b2 g1 g2 (g2 − g1) > 0, b1, b2, g0, g1, and g2 are real constants. 9.4.1.2 Specific Case I: Nonlocal Manakov System to Another Nonlocal Manakov System If (ψ1, ψ2 ) is a solution of

i ψ1t + ψ1xx + (b1 ψ1 ψ¯1 + b2 ψ2 ψ¯2 ) ψ1 = 0, i ψ2t + ψ2 xx + (b1 ψ1 ψ¯1 + b2 ψ2 ψ¯2 ) ψ2 = 0, then

Φ1(x , t ) =

b1 ψ1(x , t ) + g1 − g2

g2 b2 ψ2(x , t ), g1 (g2 − g1)

Φ2 ( x , t ) =

b1 ψ (x , t ) + g1 − g2 1

g1 b2 ψ (x , t ) g2 (g2 − g1) 2

(9.19)

is a solution of

¯ 1 − g2 Φ2 Φ ¯ 2) Φ1 = 0, i Φ1t + Φ1xx + (g1 Φ1 Φ ¯ 1 − g2 Φ2 Φ ¯ 2) Φ2 = 0, i Φ2t + Φ2 xx + (g1 Φ1 Φ

(9.20)

where b1 (g1 − g2 ) > 0, b2 g1 g2 (g2 − g1) > 0, b1, b2, g1 and g2 are real constants. 9.4.1.3 Specific Case II: Nonlocal Manakov System to the Same Nonlocal Manakov System Superposition Principle for a Nonlocal Nonlinear System If (ψ1, ψ2 ) is a solution of (9.16) then

Φ1(x , t ) =

⎡ ⎤ b2 b1 ψ2(x , t )⎥ , ⎢ψ1(x , t ) − ⎦ b1 b1 + b2 ⎣

Φ2 (x , t ) =

b1 [ψ (x , t ) + ψ2(x , t )] b1 + b2 1

is also a solution of (9.16), where b1 + b2 ≠ 0.

9-11

(9.21)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

9.4.2 Complex Coupling 9.4.2.1 General Case If (ψ1, ψ2 ) is a solution of

i ψ1t + ψ1xx + q1 (q2 ψ1 ψ¯1 + q3 ψ2 ψ¯2 ) ψ1 = 0, i ψ2t + ψ2 xx + q1 (q2 ψ1 ψ¯1 + q3 ψ2 ψ¯2 ) ψ2 = 0,

(9.22)

Φ1(x , t ) = c1 ψ1(x , t ) + c2 ψ2(x , t ), Φ2(x , t ) = c3 ψ1(x , t ) + c4 ψ2(x , t )

(9.23)

then

is a solution of ¯ 1 + a12 Φ2 Φ ¯ 2) Φ1 + 2 (b11 Φ1 Φ ¯ 2 + b12 Φ2 Φ ¯ 1) Φ1 = 0, i Φ1t + Φ1xx + 2 (a11 Φ1 Φ (9.24) ¯ ¯ ¯ ¯ 1) Φ2 = 0, i Φ2t + Φ2 xx + 2 (a21 Φ1 Φ1 + a22 Φ2 Φ2) Φ2 + 2 (b21 Φ1 Φ2 + b22 Φ2 Φ

where q1 =

(

),

2 (c1 c4 − c2 c3) c2* c3* − c1* c4* c1* c2 c3 c4* − c1 c2* c3* c4

q2 = (a − i b ) c1* c3 − (a + i b ) c1 c3*, q3 = a (c2 c4* − c2* c4 ) + i b (c2* c4 + c2 c4*), a12 =

(

),

b12 c1* c2* (c2 c3 − c1 c4 ) + b11 c1 c2 c1* c4* − c2* c3* c1 c2* c3* c4 − c1* c2 c3 c4*

(

)

b11 c3* c4* (c2 c3 − c1 c4 ) + b12 c3 c4 c1* c4* − c2* c3*

, a11 = c1 c2* c3* c4 − c1* c2 c3 c4* a22 = a12 , b12 = a − i b, b21 = a + i b, b11 = b21, b22 = b12 , c1 c2* c3* c4 should not be pure real or pure imaginary, c2 c3 − c1 c4 ≠ 0, cj, j = 1, 2, 3, 4 are complex constants, a and b are real constants. 9.4.2.2 Specific Case If (ψ1, ψ2 ) is a solution of

i ψ1t + ψ1xx − 4 (b ψ1 ψ¯1 + a ψ2 ψ¯2 ) ψ1 = 0, i ψ2t + ψ2 xx − 4 (b ψ1 ψ¯1 + a ψ2 ψ¯2 ) ψ2 = 0,

9-12

(9.25)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

then

Φ1(x , t ) = ψ1(x , t ) + ψ2(x , t ), Φ2(x , t ) = ψ1(x , t ) + i ψ2(x , t )

(9.26)

is a solution of

¯ 1 + Φ2 Φ ¯ 2) Φ1 i Φ1t + Φ1xx − 2(a + b) (Φ1 Φ ¯ 2 + (a − i b ) Φ2 Φ ¯ 1] Φ1 = 0, + 2 [(a + i b) Φ1 Φ ¯ ¯ i Φ2t + Φ2 xx − 2(a + b) (Φ1 Φ1 + Φ2 Φ2) Φ2 ¯ 2 + (a − i b ) Φ2 Φ ¯ 1] Φ2 = 0, +2 [(a + i b) Φ1 Φ

(9.27)

where a and b are real constants.

9.5 Nonlocal Discrete NLSE with Saturable Nonlinearity If ψ (x , t ) is a solution of the NLSE with saturable nonlinearity, (8.1),

i ψnt + ψn + 1 + ψn − 1 − 2 ψn +

a2 ∣ψn∣2 ψn = 0, 1 + μ ∣ψn∣2

then

⎧ ψ (n , t , a2 , μ) Φn(n , t ; a2 , μ) = ⎨ n ⎩ ψn(n , t , −a2 , −μ)

if ψn is an even function in n , if ψn is an odd function in n

is a solution of

i Φnt + Φn + 1 + Φn − 1 − 2 Φn +

¯n a2 Φ2n Φ = 0, ¯n 1 + μ Φn Φ

where ψn = ψn(n, t ; a2, μ), Φn = Φn(n, t ; a2, μ), ¯ n = Φ*n ( −n, t ; a2, μ), Φ a2 and μ are real constants. 9.5.1 Nonstaggered Solutions Example 1. Even function: sech(n) discrete bright soliton Given

ψ (ζ, t ) = A0 sech[A1 ζ ] e−i [A2 (t − t0) + ϕ0]

9-13

(9.28)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

is a solution of (8.1), then

Φ(ζ, t ) = A0 sech[A1 ζ ] e−i [A2 (t − t0) + ϕ0]

(9.29)

is a solution of (9.28), where sinh(A1) , A0 = μ 2 μ − a2 , μ a2 sech(A1) 2

A2 =

μ= > 0, ζ = n − n 0, A1, t0, n0, and ϕ0 are arbitrary real constants.

Example 2. Odd function: tanh(n) discrete dark soliton Given

ψ (ζ, t ) = A0 tanh[A1 ζ ] e−i [A2 (t − t0) + ϕ0] is a solution of (8.1), where tanh(A ) A0 = −μ 1 , A2 =

−2 μ + a2 , −μ 2 μ sech2(A1),

a2 = ζ = n − n 0, μ < 0, A1, n0, and ϕ0 are arbitrary real constants, then

Φ(ζ, t ) = A0 tanh[A1 ζ ] e−i [A2 (t − t0) + ϕ0] is a solution of (9.28), where tanh(A1) , (changed the sign of μ), A0 = μ A2 =

2 μ − a2 , μ

(changed the sign of μ and a2 ),

sech2(A1),

a2 = 2 μ ζ = n − n 0, μ > 0, A1, n0, and ϕ0 are arbitrary real constants.

9-14

(9.30)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

9.5.2 Staggered Solutions If Φn(n, t ; a2 ) is a nonstaggered solution of (9.28), then

Φn s (n , t , a2 ) = ( −1)n Φ*n (n , t , −a2 ) e−4 i (t − t0),

(9.31)

is a staggered solution of the same equation. Example 1. sech(n) staggered discrete bright soliton. Given

Φ(ζ, t ) = A0 sech[A1 ζ ] e i [A2 (t − t0) + ϕ0] is a nonstaggered solution of (9.28), where sinh(A1) , A0 = μ 2 μ − a2 , μ a2 sech(A1) 2

A2 =

(changed the sign of a2 ),

μ= > 0, (changed the sign of a2 ), , ζ = n − n0 A1, n0, t0, and ϕ0 are arbitrary real constants, then

Φs (ζ, t ) = ( −1)ζ + n 0 A0 sech(A1 ζ ) e i [A2 (t − t0) − 4 (t − t0) + ϕ0]

(9.32)

is a staggered solution of (9.28), where sinh(A1) , A0 = μ 2 μ + a2 , μ a sech(A ) − 2 2 1

A2 =

(changed the sign of a2 ),

μ= > 0, (changed the sign of a2 ), ζ = n − n 0, A1, n0, t0, and ϕ0 are arbitrary real constants.

9.6 Nonlocal Ablowitz–Ladik Equation If ψ (x , t ) is a solution of the Ablowitz–Ladik equation, (8.41),

i ψnt + ψn + 1 + ψn − 1 − 2 ψn + a2 (ψn + 1 + ψn − 1) ∣ψn∣2 = 0, then

⎧ ψ (n , t ; a2 ) Φn(n , t ; a2 ) = ⎨ n ⎩ ψn(n , t ; −a2 )

if ψn is an even function in n, if ψn is an odd function in n

is a solution of

¯ n = 0, i Φnt + Φn + 1 + Φn − 1 − 2 Φn + a2 (Φn + 1 + Φn − 1) Φn Φ

9-15

(9.33)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

where ψn = ψn(n, t ; a2 ), Φn = Φn(n, t ; a2 ), ¯ n = Φ*n ( −n, t ; a2 ), Φ a2 is an arbitrary real constant. Example 1. Even function: sech(n) discrete bright soliton Given

ψ (ζ, t ) = A0 sech[A1 ζ ] e−i [(A2 + 2) (t − t0) + ϕ0] is a solution of (8.41), then Φ(ζ, t ) = A0 sech[A1 ζ ] e−i [(A2 + 2) (t − t0) + ϕ0]

(9.34)

is a solution of (9.33), where A2 = −2 cosh(A1), a2 =

sinh2(A1) , A02

ζ = n − n 0, A0 ≠ 0, A1, n0, t0, and ϕ0 are arbitrary real constants.

Example 2. Odd function: tanh(n) discrete dark soliton Given ψ (ζ, t ) = A0 tanh[A1 ζ ] e−i [(A2 + 2) (t − t0) + ϕ0] is a solution of (8.41), where A2 = −2 sech(A1), a2 = −

tanh2(A1) , A02

ζ = n − n 0, A0 ≠ 0, A1, n0, t0, and ϕ0 are arbitrary real constants, then

Φ(ζ, t ) = A0 tanh[A1 ζ ] e−i [(A2 + 2) (t − t0) + ϕ0] is a solution of (9.33), where A2 = −2 sech(A1), a2 =

tanh2(A1) , A02

(changed the sign of a2 ),

ζ = n − n 0, A0 ≠ 0, A1, n0, t0, and ϕ0 are arbitrary real constants.

9-16

(9.35)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

9.7 Nonlocal Cubic-Quintic Discrete NLSE If ψ (x , t ) is a solution of the cubic-quintic discrete NLSE, (8.54),

i ψnt + a1 (ψn + 1 + ψn − 1 − 2 ψn ) + a2 ∣ψn∣2 ψn + (a3 ∣ψn∣2 + a 4 ∣ψn∣4 )(ψn + 1 + ψn − 1) = 0, then

Φn(n , t ; a1, a2 , a3, a 4 ) ⎧ ψ (n , t ; a1, a2 , a3, a 4 ) =⎨ n ⎩ ψn(n , t ; a1, −a2 , −a3, a 4 )

if ψn is an even function in n , if ψn is an odd function in n

is a solution of

¯ n + (a3 Φn Φ ¯ n + a 4 Φ2n Φ ¯ 2n)(Φn + 1 i Φnt + a1 (Φn + 1 + Φn − 1 − 2 Φn) + a2 Φ2n Φ

(9.36)

+ Φn − 1) = 0, where ψn = ψn(n, t ; a1, a2, a3, a 4 ), Φn = Φn(n, t ; a1, a2, a3, a 4 ), ¯ n = Φ*n ( −n, t ; a1, a2, a3, a 4 ), Φ a1, a2, a3, and a4 are arbitrary real constants. Example 1. Even function: sech(n) I discrete bright soliton Given

ψ (ζ, t ) = A0 sech[A1 ζ ] e i [A2 (t − t0) + ϕ0] is a solution of (8.54), then Φ(ζ, t ) = A0 sech[A1 ζ ] e i [A2 (t − t0) + ϕ0] is a solution of (9.33), where A0 = ±

a3 (a 22 − 4 a1 a 4 ) + a32 − 4 a1 a 4 (a 22 + 4 a1 a 4 )

⎛ −a3 + A1 = sech−1 ⎜ ⎝ A2 = −2 a1 −

2 a4

2 a1

a32 − 4 a1 a 4 a2

(

⎞ ⎟, ⎠

a2 a3 + a32 − 4 a1 a 4 2 a4

,

),

a32 − 4 a 4 a1 ⩾ 0, a1 > 0,

a3 (a 22 − 4 a1 a 4 ) + a32 − 4 a1 a 4 (a 22 + 4 a1 a 4 ) > 0, ζ = n − n 0, t0, n0, and ϕ0 are arbitrary real constants.

9-17

(9.37)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Example 2. Even function: sech(n) II staggered discrete bright soliton Given

ψ (ζ, t ) = ( −1)ζ + n 0 A0 sech[A1 ζ ] e i [A2 (t − t0) + ϕ0] is a solution of (8.54), then

Φ(ζ, t ) = ( −1)ζ + n 0 A0 sech[A1 ζ ] e i [A2 (t − t0) + ϕ0] is a solution of (9.33), where A0 = ±

a3 (a 22 − 4 a1 a 4 ) + a32 − 4 a1 a 4 (a 22 + 4 a1 a 4 ) 2 a4

⎛ a3 − A1 = sech−1 ⎜ ⎝ A2 = −2 a1 −

a32 − 4 a1 a 4 a2

2 a1

,

⎞ ⎟, ⎠

(

a2 a3 + a32 − 4 a1 a 4 2 a4

),

a32 − 4 a 4 a1 ⩾ 0, a1 > 0,

a3 (a 22 − 4 a1 a 4 ) + a32 − 4 a1 a 4 (a 22 + 4 a1 a 4 ) > 0, ζ = n − n 0, t0, n0, and ϕ0 are arbitrary real constants.

Example 3. Odd function: tanh(n) I discrete dark soliton Given

ψ (ζ, t ) = A0 tanh[A1 ζ ] e i [A2 (t − t0) + ϕ0] is a solution of (8.54), where A0 = ±

a2 + a3 + a32 − 4 a1 a 4

⎛ A1 = cosh−1 ⎜⎜ ⎝ A2 = −2 a1 −

−2 a 4

,

a3 + a32 − 4 a1 a 4 −a2

⎞ ⎟⎟, ⎠

(

a2 a3 − a32 − 4 a1 a 4 2 a4

),

a32 − 4 a 4 a1 ⩾ 0,

( (a

a 4 a2 + a3 + a2

3

+

)

a32 − 4 a1 a 4 < 0,

)

a32 − 4 a1 a 4 < 0,

ζ = n − n 0, t0, n0, and ϕ0 are arbitrary real constants, then

9-18

(9.38)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Φ(ζ, t ) = A0 tanh[A1 ζ ] e i [A2 (t − t0) + ϕ0]

(9.39)

is a solution of (9.33), where A0 = ±

−a2 − a3 + a32 − 4 a1 a 4 −2 a 4

⎛ A1 = cosh−1 ⎜⎜ ⎝

,

(changed the sign of a2 and a3),

−a3 + a32 − 4 a1 a 4 a2

A2 = −2 a1 −

⎞ ⎟⎟, ⎠

(

−a2 −a3 − a32 − 4 a1 a 4 2 a4

(changed the sign of a2 and a3),

),

(changed the sign of a2 and a3),

a32 − 4 a 4 a1 ⩾ 0,

( (−a

)

a32 − 4 a1 a 4 < 0,

a 4 −a2 − a3 + a2

3

+

a32

)

− 4 a1 a 4 > 0,

ζ = n − n 0, t0, n0, and ϕ0 are arbitrary real constants.

Example 4. Odd function: tanh(n) II staggered discrete dark soliton Given

ψ (ζ, t ) = ( −1)ζ + n 0 A0 tanh[A1 ζ ] e i [A2 (t − t0) + ϕ0] is a solution of (8.54), where A0 = ±

a2 − a3 − a32 − 4 a1 a 4 2 a4

⎛ A1 = cosh−1 ⎜⎜ ⎝

,

a3 + a32 − 4 a1 a 4 a2

A2 = −2 a1 −

⎞ ⎟⎟, ⎠

(

a2 a3 − a32 − 4 a1 a 4 2 a4

),

a32 − 4 a 4 a1 ⩾ 0,

(a

2

− a3 −

(

a2 a3 +

)

a32 − 4 a1 a 4 > 0,

)

a32 − 4 a1 a 4 > 0,

ζ = n − n 0, t0, n0, and ϕ0 are arbitrary real constants, then

Φ(ζ, t ) = ( −1)ζ + n 0 A0 tanh[A1 ζ ] e i [A2 (t − t0) + ϕ0]

9-19

(9.40)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

is a solution of (9.33), where A0 = ±

−a2 + a3 − a32 − 4 a1 a 4

⎛ A1 = cosh−1 ⎜⎜ ⎝ A2 = −2 a1 − a32

2 a4

−a3 + a32 − 4 a1 a 4 −a2

(

a2 a3 + a32 − 4 a1 a 4 2 a4

(changed the sign of a2 and a3), ⎞ ⎟⎟, ⎠

),

(changed the sign of a2 and a3), (changed the sign of a2 and a3),

− 4 a 4 a1 ⩾ 0,

( (−a

a 4 −a2 + a3 − a2

,

3

+

)

a32 − 4 a1 a 4 > 0,

)

a32 − 4 a1 a 4 < 0,

ζ = n − n 0, t0, n0, and ϕ0 are arbitrary real constants.

9-20

2 a1 a2

9-21

(t − t0 ) + ϕ0⎤⎦

⎤ ζ ⎥⎦ e−i [A2 (t − t0) + ϕ0]

2

A02

2 A12

, a1 > 0 , ζ = x − x0 ,

A0 ≠ 0,

A02

, b1 =

A02

2 A12 − b2 B02

, ζ = x − x0 ,

bright-dark soliton 2 A12 − a2 B02

ω1 = A12 − a1 A02 , ω2 = 2 A12 − b1 A02 ,

1. Φ1(ζ , t ) = A0 sech[A1 ζ ] e−i [ω1 (t − t0) + ϕ1],

a1 =

Name

Conditions

# Example

¯ 1 + a2 Φ2 Φ ¯ 2) Φ1 = 0, i Φ2t + Φ2 xx + (b1 Φ1 Φ ¯ 1 + b 2 Φ2 Φ ¯ 2 ) Φ2 = 0 Equation: i Φ1t + Φ1xx + (a1 Φ1 Φ

if ψ2 is an even function in x, if ψ2 is an odd function in x

if ψ1 is an even function in x, if ψ1 is an odd function in x,

Nonlocal Coupled NLSE

A0, A1, x0, t0, and ϕ0 are arbitrary

A2 = 2 A12 , a2 =

(9.2)

Eq. #

(9.6)

Eq. #

(Continued)

ψ is a solution of the fundamental NLSE (2.1)

dark soliton (9.3)

bright soliton

ζ = x − x0 , a1 a2 > 0 , A0, x0, t0, and ϕ0 are arbitrary real constants

Name

Conditions

if ψ is an even function in x, if ψ is an odd function in x

Nonlocal NLSE

⎧ ψ (x , t , a1, a2, b1, b2 ) Φ1(x , t ; a1, a2, b1, b2 ) = ⎨ 1 ⎩ ψ1(x , t , − a1, a2, − b1, b2 ) Transformation: ⎧ ψ (x , t , a1, a2, b1, b2 ) Φ2(x , t ; a1, a2, b1, b2 ) = ⎨ 2 ⎩ ψ2(x , t , a1, − a2, b1, − b2 )

A1 a1

⎡

sech[A0 ζ ] ei ⎣a1 A0

2. Φ(ζ , t ) = A0 tanh ⎡⎢ ⎣

1. Φ(ζ , t ) = A0

# Example

¯ =0 Equation: i Φt + a1 Φxx + a2 ∣Φ2 Φ

⎧ ψ (x , t , a1, a2 ) Transformation: (Φ(x , t ; a1, a2 ) = ⎨ ⎩ ψ (x , t , a1, − a2 )

9.8 Summary of Chapter 9

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

1 − m nd[A1 ζ , m ]

e−i [ω2 (t − t0) + ϕ2]

m (1 − m ) sd[A1 ζ , m ] e−i [ω1 (t − t0) + ϕ1],

m dn[A1 ζ , m ] sn[A1 ζ , m ]

9-22

6 A12 B02 (1 − m )

, b1 = a1, b2 = a2 , 0 ⩽ m < 1,

A02

−2 A12 + a2 B02

, b1 = A02

−2 A12 + b2 B02

, 0 ⩽ m ⩽ 1,

a2 =

b1 = c1 + (c2 − b2 ) ∣σ∣2

⎧ ⎪ Φ*( − x , t , b1, b2 , c1, c2 ) Transformation: Φ2(x , t ; b1, b2 , c1, c2 ) = ei ϕ⎨ 1 ⎪ ⎩ Φ1*( − x , t , b1, b2 , − c1, − c2 ) b1 = − (c1 + c2 + b2 )

if Φ1 is an odd function in x,

if Φ1 is an even function in x,

Symmetry Reductions II: From Nonlocal Manakov System to Scalar Nonlocal NLSE

¯1 = 0 Equation: i Φ1t + b0 Φ1xx + (c1 + c2∣σ∣2 ) Φ12 Φ

2 A02

−9 A12

,

B0 ≠ 0, A1, x0, t0, ϕ1, and ϕ2 are arbitrary real constants

2

ω1 = 2 A12 , ω2 = − 2 A12 , a1 =

, b1 = a1, b2 = a2 , ζ = x − x0 , A0 ≠ 0,

A12 B02

9

−2 A0 , 3

A3 =

–

(9.9)

solitary wave (9.8)

, solitary wave (9.7)

ζ = x − x0 , A0 ≠ 0, A1, B0, x0, t0, ϕ1, and ϕ2 are arbitrary real constants

a1 =

ω1 = (m − 1) a1 A02 − A12 , ω2 = (m − 2) A12 + (m − 1) b1 A02 ,

Symmetry Reductions I: From Nonlocal Manakov System to Scalar Nonlocal NLSE

Transformation: Φ2(x , t ) = σ Φ1(x , t ) ,

−6 A12 A02 (1 − m )

ζ = x − x0 , A0 ≠ 0 , B0 ≠ 0, A1, x0, t0, ϕ1, and ϕ2 are arbitrary real constants

a2 =

ω1 = (4 + m ) A12 , ω2 = (1 + 4 m ) A12 , a1 =

A1, B0, x0, t0, ϕ1, and ϕ2 are arbitrary real constants

Nonlocal NLSE

Symmetry Reductions to Scalar Nonlocal NLSE

4. Φ1(ζ , t ) = {A0 sech2[A1 ζ ] + A3} e−i [ω1 (t − t0) + ϕ1], Φ2(ζ , t ) = B0 sech2[A1 ζ ] e−i [ω2 (t − t0) + ϕ2]

3. Φ1(ζ , t ) = A0 Φ2(ζ , t ) = B0

Φ2(ζ , t ) = B0

e−i [ω2 (t − t0) + ϕ2]

2. Φ1(ζ , t ) = A0 m cn[A1 ζ , m ] sn[A1 ζ , m ] e−i [ω1 (t − t0) + ϕ1],

Φ2(ζ , t ) = B0 tanh[A1 ζ ] e−i [ω2 (t − t0) + ϕ2]

(Continued )

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

9-23

b1 g1 − g2

Φ2 (x , t ) =

ψ1(x , t ) ei g0 (g1− g2) t +

ψ1(x , t ) ei g0 (g1− g2) t + g1 b2 g2 (g2 − g1)

g2 b2 g1 (g2 − g1)

ψ2(x , t ) e−i g0 (g1− g2) t

ψ2(x , t ) e−i g0 (g1− g2) t ,

¯ 1 − g2 Φ2 Φ ¯ 2) Φ1 + g0 (g1 + g2 ) Φ1 − 2 g0 g2 Φ2 = 0, i Φ2t + Φ2 xx + (g1 Φ1 Φ ¯ 1 − g2 Φ2 Φ ¯ 2 ) Φ2 i Φ1t + Φ1xx + (g1 Φ1 Φ − g0 (g1 + g2 ) Φ2 + 2 g0 g1 Φ1 =0

b1 g1 − g2

Φ1(x , t ) =

Scaling Transformations

Equation:

b1 g1 − g2

Φ2 (x , t ) =

ψ1(x , t ) +

ψ1(x , t ) + g1 b2 g2 (g2 − g1)

g2 b2 g1 (g2 − g1)

¯ 1 − g2 Φ2 Φ ¯ 2) Φ1 = 0, i Φ1t + Φ1xx + (g1 Φ1 Φ ¯ ¯ 2 ) Φ2 = 0 i Φ2t + Φ2 xx + (g1 Φ1 Φ1 − g2 Φ2 Φ

Transformation:

b1 g1 − g2

Φ1(x , t ) = ψ2(x , t )

ψ2(x , t ),

Specific Case I: Nonlocal Manakov System to Another Nonlocal Manakov System

Equation:

Transformation:

General Case

Linear and Nonlinear Coupling

¯1 = 0 Equation: i Φ1t + a1 Φ1xx + ∑N b ∣σ ∣2 Φ12 Φ k = 1 1k k

Transformation: Φj (x , t ) = σj Φ1(x , t )

Symmetry Reductions III: From Nonlocal Vector NLSE to Scalar Nonlocal NLSE

¯1 = 0 Equation: i Φ1t + b0 Φ1xx − (c1 + c2 ) Φ12 Φ

(Continued)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

[ψ1(x , t ) + ψ2(x , t )]

b1 b1 + b2

9-24

Φ1(x , t ) = c1 ψ1(x , t ) + c2 ψ2(x , t ), Φ2(x , t ) = c3 ψ1(x , t ) + c4 ψ2(x , t )

Φ2(x , t ) = ψ1(x , t ) + i ψ2(x , t )

⎧ ψ (n , t , a2, μ ) Transformation: Φn(n , t ; a2, μ ) = ⎨ n ⎩ ψn(n , t , − a2, − μ )

if ψn is an even function in n, if ψn is an odd function in n

Nonlocal Discrete NLSE with Saturable Nonlinearity

¯ 1 + Φ2 Φ ¯ 2) Φ1 + 2 [(a + i b ) Φ1 Φ ¯ 2 + (a − i b ) Φ2 Φ ¯ 1] Φ1 = 0, i Φ1t + Φ1xx − 2(a + b ) (Φ1 Φ ¯ ¯ ¯ ¯ 1] Φ2 = 0 i Φ2t + Φ2 xx − 2(a + b ) (Φ1 Φ1 + Φ2 Φ2) Φ2 + 2 [(a + i b ) Φ1 Φ2 + (a − i b ) Φ2Φ

Transformation: Φ1(x , t ) = ψ1(x , t ) + ψ2(x , t ),

Equation:

Nonlocal NLSE

¯ 1 + a12 Φ2 Φ ¯ 2) Φ1 + 2 (b11 Φ1 Φ ¯ 2 + b12 Φ2 Φ ¯ 1) Φ1 = 0, i Φ1t + Φ1xx + 2 (a11 Φ1 Φ ¯ 1 + a22 Φ2 Φ ¯ 2) Φ2 + 2 (b21 Φ1 Φ ¯ 2 + b22 Φ2 Φ ¯ 1) Φ2 = 0 i Φ2t + Φ2 xx + 2 (a21 Φ1 Φ

Specific Case

Equation:

Transformation:

General Case

ψ2(x , t )⎤⎦ ,

Φ2 (x , t ) =

b2 b1

⎡ ⎣ψ1(x , t ) −

b1 b1 + b2

Φ1(x , t ) =

Complex Coupling

Transformation:

Specific Case II: Nonlocal Manakov System to the Same System Superposition Principle for a Nonlocal Nonlinear System

(Continued )

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

¯n a2 Φ2n Φ ¯n 1 + μ Φn Φ

=0

9-25

sinh(A1) , μ

A2 =

2 μ − a2 , μ

μ=

a2 sech(A1) 2

> 0,

tanh(A1) , μ

A2 =

2 μ − a2 , μ

a2 = 2 μ sech2(A1),

sinh(A1) , μ

A2 =

2 μ + a2 , μ

μ=−

a2 sech(A1) 2

> 0,

if ψn is an even function in n, if ψn is an odd function in n

A2 = − 2 cosh(A1), a2 =

1. Φ(ζ , t ) = A0 sech[A1 ζ ] e−i [(A2 + 2) (t − t0) + ϕ0]

, ζ = n − n 0 , A0 ≠ 0,

A0, A1, n0, t0, and ϕ0 are arbitrary real constants

Conditions

# Example

sinh2(A1) A02

Nonlocal Ablowitz–Ladik equation

ζ = n − n0, A1, n0, t0, and ϕ0 are arbitrary real constants

A0 =

μ > 0, ζ = n − n0 , A1, n0, and ϕ0 are arbitrary real constants

A0 =

ζ = n − n0, A1, n0, t0, and ϕ0 are arbitrary real constants

A0 =

Conditions

¯n = 0 Equation: i Φnt + Φn + 1 + Φn − 1 − 2 Φn + a2 (Φn + 1 + Φn − 1) Φn Φ

⎧ ψ (n , t ; a2 ) Transformation: Φn(n , t ; a2 ) = ⎨ n ⎩ ψn(n , t ; − a2 )

3. Φs(ζ , t ) = ( − 1)ζ + n0 A0 sech[A1 ζ ] ei [A2 (t − t0) − 4 (t − t0) + ϕ0]

2. Φ(ζ , t ) = A0 tanh[A1 ζ ] e−i [A2 (t − t0) + ϕ0]

1. Φ(ζ , t ) = A0 sech[A1 ζ ] e−i [A2 (t − t0) + ϕ0]

# Example

Equation: i Φnt + Φn + 1 + Φn − 1 − 2 Φn +

(9.29)

bright soliton

discrete bright soliton

Name

staggered bright soliton

(9.34)

Eq. #

(9.32)

dark soliton (9.30)

Eq. #

Name

(Continued)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

A02

tanh2(A1)

, ζ = n − n 0 , A0 ≠ 0,

)

9-26

2. Φ(ζ , t ) = ( − 1)ζ + n0 A0 sech[A1 ζ ] ei [A2 (t − t0) + ϕ0]

1. Φ(ζ , t ) = A0 sech[A1 ζ ] ei [A2 (t − t0) + ϕ0]

# Example

2 a4 a32

a2 (a3 + a32 − 4 a1 a 4 )

, a32 − 4 a 4 a1 ⩾ 0 ,

,

(

2 a4 a32

3

), a 2 − 4 a a1 a 4 (a 22

a2 a3 + a32 − 4 a1 a 4

⎞ ⎟, ζ = n − n0 , ⎠

2 a1

4

a1 ⩾ 0 ,

,

− 4 a1 a 4 ) + −4 + 4 a1 a 4 ) > 0, a1 > 0 , t0, n0, and ϕ0 are arbitrary real constants

a3 (a 22

A2 = − 2 a1 −

a2

a32 − 4 a1 a 4

2 a4

a3 (a 22 − 4 a1 a 4 ) + a32 − 4 a1 a 4 (a 22 + 4 a1 a 4 )

⎛ a3 − A1 = sech−1 ⎜ ⎝

A0 = ±

a3 (a 22 − 4 a1 a 4 ) + − 4 a1 a 4 (a 22 + 4 a1 a 4 ) > 0, a1 > 0 , t0, n0, and ϕ0 are arbitrary real constants

A2 = − 2 a1 −

a2

⎞ ⎟, ζ = n − n 0 , ⎠

2 a1

a32 − 4 a1 a 4

2 a4

a3 (a 22 − 4 a1 a 4 ) + a32 − 4 a1 a 4 (a 22 + 4 a1 a 4 )

⎛ −a3 + A1 = sech−1 ⎜ ⎝

A0 = ±

Conditions

¯ n + a3 Φn Φ ¯ n + a 4 Φ2n Φ ¯ 2n (Φn + 1 + Φn − 1) = 0 Equation: i Φnt + a1 (Φn + 1 + Φn − 1 − 2 Φn) + a2 Φ2n Φ

(

if ψn is an even function in n, if ψn is an odd function in n

Nonlocal Cubic-Quintic Discrete NLSE

A0, A1, n0, t0, and ϕ0 are arbitrary real constants

A2 = − 2 sech(A1), a2 =

Nonlocal NLSE

⎧ ψ (n , t ; a1, a2, a3, a 4 ) Transformation: Φn(n , t ; a1, a2, a3, a 4 ) = ⎨ n ⎩ ψn(n , t ; a1, − a2, − a3, a 4 )

2. Φ(ζ , t ) = A0 tanh[A1 ζ ] e−i [(A2 + 2) (t − t0) + ϕ0]

(Continued )

staggered discrete bright soliton

discrete bright soliton

Name

(9.38)

(9.37)

Eq. #

discrete dark (9.35) soliton

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

4. Φ(ζ , t ) = ( − 1)ζ + n0 A0 tanh[A1 ζ ] ei [A2 (t − t0) + ϕ0]

3. Φ(ζ , t ) = A0 tanh[A1 ζ ] ei [A2 (t − t0) + ϕ0]

3

)

a32 − 4 a1 a 4 > 0 ,

)

a32 − 4 a1 a 4 < 0,

2 a4

),

⎞ ⎟⎟, ζ = n − n 0 , ⎠

9-27

(

3

),

)

a32 − 4 a1 a 4 < 0 ,

)

a32 − 4 a1 a 4 > 0,

2 a4

⎞ ⎟⎟, ζ = n − n 0 , ⎠

t0, n0, and ϕ0 are arbitrary real constants

a2

a 4 − a2 + a3 −

( (−a +

−a2 a2 a3 + a32 − 4 a1 a 4

a32 − 4 a 4 a1 ⩾ 0,

A2 = − 2 a1 −

,

−a3 + a32 − 4 a1 a 4

2 a4

−a2 + a3 − a32 − 4 a1 a 4

⎛ A1 = cosh−1 ⎜⎜ ⎝

A0 = ±

t0, n0, and ϕ0 are arbitrary real constants

a2

a 4 − a2 − a3 +

( (−a +

(

a2 −a2 −a3 − a32 − 4 a1 a 4

a32 − 4 a 4 a1 ⩾ 0,

A2 = − 2 a1 −

,

−a3 + a32 − 4 a1 a 4

−2 a 4

−a2 − a3 + a32 − 4 a1 a 4

⎛ A1 = cosh−1 ⎜⎜ ⎝

A0 = ±

staggered discrete dark soliton

(9.40)

discrete dark (9.39) soliton

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

IOP Publishing

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations Usama Al Khawaja and Laila Al Sakkaf

Appendix A Derivation of Some Solutions of Chapters 2 and 3

Remark: Throughout this appendix, x0, t0, ϕ0, A0, A1, and A2 are arbitrary real constants.

A.1 Derivation of Some Solutions of Section 2.1 A.1.1 Schematic Representation

Equation i ψt + a1 ψxx + a2 ∣ψ ∣2 ψ = 0 Solutions with real a1 and a2 complex a1 and a2 Case A1′: ψ (x , t ) = A0 ei ϕ(t ) Case A1: ψ (x , t ) = A0 ei ϕ(t ) Solution (2.2) Solution (2.52) Case A2′: ψ (x , t ) = A0 ei ϕ(x ) Case A2: ψ (x , t ) = A0 ei ϕ(x ) Solution (2.3) Solution (2.53) Case A3′: ψ (x , t ) = A(x ) ei ϕ0 Case A3: ψ (x , t ) = A0 ei ϕ(x, t ) Solution (5.59) Solution (2.4) Solution (2.60) Solution (2.61) Case A4′: ψ (x , t ) = A(x ) ei ϕ(t ) Case A4: ψ (x , t ) = A(x ) ei ϕ0 a c≠0 c=0

doi:10.1088/978-0-7503-2428-1ch10

A-1

ª IOP Publishing Ltd 2020

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

(Continued )

Solution (2.24) Solution (2.26) Solution (2.6) Solution (2.28) Case A5: ψ (x , t ) = A(x ) ei ϕ(t ) c≠0 c=0 Solution (2.10) Solution (2.23) Solution (2.8) Solution (2.14) Solution (2.25) Solution (2.12) Solution (2.13) Solution (2.27) Case A6: ψ (x , t ) = A(x ) ei ϕ(x, t ) Solution (2.32) Case A7: ψ (x , t ) = A(t ) ei ϕ(x, t ) Solution (2.5)

Solution (2.54) Case A5′: ψ (x , t ) = A(t ) ei ϕ(x, t ) Solution (2.55) Solution (2.56) Solution (2.57) Solution (2.58)

a

c is an arbitrary constant of integration resulting from integrating the NLSE, as detailed below.

A.1.2 Detailed Derivations A.1.2.1 Real Coefficients The general solution to (2.1) can be written in the polar form:

ψ (x , t ) = Z e i ϕ ,

(A.1)

where Z = Z (x , t ) and ϕ = ϕ(x , t ) are real functions. Substituting (A.1) in the fundamental NLSE (2.1) generates its real and imaginary parts:

Re[Equation (2.1)]:

a2 Z 3 − Z ϕt − a1 Z ϕx2 + a1 Zxx = 0,

(A.2)

Zt + 2 a1 Zx ϕx + a1 Z ϕxx = 0,

(A.3)

and

Im[Equation (2.1)]:

where the subscripts indicate differentiation with respect to x and t. In the following, we take cases for Z and ϕ. Case A1:

Z (x , t ) = A0 ,

(A.4)

ϕ(x , t ) = ϕ(t ).

(A.5)

A-2

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Substituting back into (A.2) and (A.3) leads to the following Re[Equation (2.1)] and Im[Equation (2.1)]:

A03 a2 − A0 ϕ′(t ) = 0,

Re[Equation (2.1)]:

Im[Equation (2.1)]:

0 = 0.

(A.6) (A.7)

Solving (A.6) for ϕ(t ) gives

ϕ(t ) = A02 a2 (t − t0) + ϕ0.

(A.8)

Substituting (A.4) and (A.8) back into (A.1) leads to solution (2.2). Case A2:

Z (x , t ) = A0 ,

(A.9)

ϕ(x , t ) = ϕ(x ).

(A.10)

Substituting back into (A.2) and (A.3) leads to the updated forms of Re[Equation (2.1)] and Im[Equation (2.1)]:

Re[Equation (2.1)]:

A03 a2 − A0 a1 ϕ′2 (x ) = 0,

(A.11)

A0 a1 ϕ″(x ) = 0.

(A.12)

Im[Equation (2.1)]: Solving (A.12) for ϕ(x ) gives

ϕ(x ) = A0

a2 (x − x0) + ϕ0. a1

(A.13)

Substituting (A.9) and (A.13) back into (A.1) leads to solution (2.3). Case A3:

Z (x , t ) = A0 ,

(A.14)

ϕ(x , t ) = ϕ(x , t ),

(A.15)

where A0 is an arbitrary real constant. Substituting back into (A.2) and (A.3) leads to the following Re[Equation (2.1)] and Im[Equation (2.1)]:

Re[Equation (2.1)]:

A03 a2 − A0 ϕt (x , t ) − A0 a1 ϕx2(x , t ) = 0,

Im[Equation (2.1)]:

A0 a1 ϕxx(x , t ) = 0.

(A.16) (A.17)

Solving (A.17) for ϕ(x , t ) leads to

ϕ(x , t ) = s1(t ) + (x − x0) s2(t ),

A-3

(A.18)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

where s1(t ) and s2(t ) are real functions of t. Substituting back into (A.16), collecting coefficients of (x − x0 )0 and (x − x0 ), separately, and equating to zero, we obtain

(x − x0)0 :

A03 a2 − A0 a1 s22(t ) − A0 s1′(t ) = 0, (x − x0): −A0 s2′(t ) = 0.

(A.19) (A.20)

Solving (A.20) for s2(t ) reads

s2(t ) = A1.

(A.21)

Substituting this result in (A.19) and solving for s1(t ), we get

(

)

s1(t ) = A02 a2 − a1 A12 (t − t0) + ϕ0.

(A.22)

The form of ϕ(x , t ) in (A.18) will then be

(

)

ϕ(x , t ) = A1 (x − x0) + A02 a2 − a1 A12 (t − t0) + ϕ0.

(A.23)

Using (A.23) back into (A.1) leads to solution (2.4). Case A4:

Z (x , t ) = A(x ),

(A.24)

ϕ(x , t ) = ϕ0 ,

(A.25)

where A(x ) is a real function to be determined. Substituting back into (A.2) and (A.3) leads to the following new forms of Re[Equation (2.1)] and Im[Equation (2.1)]:

Re[Equation (2.1)]:

a2 A3(x ) + a1 A″(x ) = 0,

Im[Equation (2.1)]:

0 = 0.

(A.26) (A.27)

Employing the separation of variables method using the chain rule A′ (x ) dA ′ (x ) in (A.26), we get the following integral of the independent A″(x ) = dA(x ) variable x:

x − x0 =

∫

1 dA(x ), a2 4 A (x ) c− 2 a1

(A.28)

where c is a real constant of integration. In the following, we take two categories of c. If c ≠ 0: The integration above can be written as

x − x0 =

1 c

∫

1 1−b

A2 (x )

A-4

1 + b A2 (x )

dA(x ),

(A.29)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

where b =

a2 2 c a1 1

. In the following, we take two options for A(x ). π

1. A(x ) = sin(θ ), 0 > θ > 2 . b The integral in (A.29) becomes

x − x0 =

1 cb

cos(θ )

∫

sin2(θ )

1−

1 + sin2(θ )

dθ.

(A.30)

This integration gives

x − x0 =

1 cb

F(θ , −1),

(A.31)

where F gives the elliptic integral of the first kind. Resubstituting θ = sin−1[ b A(x )] and b = 2 ac2a in (A.31) and solving for A(x ) leads to 1

1 1 ⎡ ⎤ ⎛ 2 c a1 ⎞ 4 ⎛ c a2 ⎞ 4 ⎢ A(x ) = ⎜ ⎟ sn ⎢⎜ ⎟ (x − x0), −1⎥⎥ . ⎝ a2 ⎠ ⎝ 2 a1 ⎠ ⎣ ⎦

From (A.32) and (A.25), (A.1) will lead to (2.24), where 1

(A.32)

( ) c a2 2 a1

1 4

→ A0.

π . 2

2. A(x ) = cos(θ ), 0 > θ > b The integral in (A.29) becomes

x − x0 =

1 cb

−sin(θ )

∫

1−

cos2(θ )

1 + cos2(θ )

dθ.

(A.33)

This integration leads to

x − x0 =

⎛ 1⎞ F⎜θ , ⎟ , 2 c b ⎝ 2⎠ −1

(A.34)

where F gives the elliptic integral of the first kind. Resubstituting a θ = cos−1 [ b A(x )] and b = 2 c2a in (A.34) and solving for A(x ) lead to 1

1 1 ⎡ ⎤ ⎛ 2 c a1 ⎞ 4 ⎛ 2 c a2 ⎞ 4 1 ⎢ A(x ) = ⎜ ⎟ cn ⎢⎜ ⎟ (x − x0), ⎥⎥ . ⎝ a2 ⎠ ⎝ ⎠ 2⎦ ⎣ a1

From (A.25) and (A.35), (A.1) will lead to (2.26), where

( ) 2 c a2 a1

(A.35) 1 4

→ A0.

3.

A(x ) = c

2 a1 dn [c (x − x0), 2]. a2

A-5

(A.36)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

From (A.25) and (A.36), (A.1) will lead to (2.28), where c → A0. The procedure of how we get this solution will be shown in the next case, item 4 in Case A5. If c = 0: Solving (A.28) for A(x ), we get

A(x ) =

1 −2 a1 . a2 x − x0

(A.37)

Substituting (A.25) and (A.37) back into (A.1) leads to solution (2.6). Case A5:

Z (x , t ) = A(x ),

(A.38)

ϕ(x , t ) = e−i [λ (t − t0) + ϕ0],

(A.39)

where A(x ) is a real function to be determined and λ is an arbitrary real constant. (2.1) takes the form

λ A(x ) + a1 A″(x ) + a2 A3(x ) = 0. Using the chain rule A″(x ) = independent variable x:

x − x0 =

A′ (x ) dA ′ (x ) , dA(x )

we get the following integral of the

1

∫

λ 2 a A (x ) − 2 A4 (x ) 2 a1 a1

c−

(A.40)

dA(x ),

(A.41)

where c is a real constant of integration. In the following, we take two categories of c. If c ≠ 0: −λ 2 1. From (A.41), with c = 2 a a , x − x0 will be given by 1

2

⎡ a ⎤ −2 a1 tan−1 ⎢ 2 A(x )⎥. ⎣ λ ⎦ λ

x − x0 =

(A.42)

Solving (A.42) for A(x ), we get

A(x ) =

⎡ −λ ⎤ λ tan ⎢ (x − x0)⎥. a2 ⎣ 2 a1 ⎦

(A.43)

Substituting (A.39) and (A.43) back into (A.1) leads to (2.10), where λ → −2 a1 A02. 2. The integral definition of the Jacobi sn(x , m ) elliptic function with modulus m is given by:

x − x0 =

∫0

sn(x, m)

1 [1 − A2 (x )] [1 − m A2 (x )]

A-6

dA(x ).

(A.44)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Equating this integration with (A.41) gives

c−

λ 2 a A (x ) − 2 A4 (x ) − c [1 − c1 A2 (x )] [1 − c1 m A2 (x )] = 0, 2 a1 a1

(A.45)

where c1 is a required real constant to be determined. Equating the coefficients of A2 (x ) and A4 (x ) to zero, separately, and solving for c and c1, we get

c1 =

−a2 (1 + m) , 2mλ

c=

− 2 m λ2 . a1 a2 (1 + m)2

(A.46)

Solving

x − x0 =

∫0

sn(x, m)

1 c [1 − c1

A2 (x )]

[1 − c1 m A2 (x )]

dA(x ),

(A.47)

for A(x ) with the expressions of c an c1 in (A.46), we get

⎡ ⎤ 2mλ λ sn ⎢ (x − x0), m⎥. −a2 (1 + m) ⎣ a1 (1 + m) ⎦

A(x ) =

(A.48)

Substituting (A.39) and (A.48) back into (A.1) leads to solution (2.23), where λ → A02 . If we take the limit of (A.48) when λ → c a1 (1 + m ) and replace m → −1, we return back to (A.32). For the special case of m = 1, equation (A.48) reads

A(x ) =

⎡ λ ⎤ −λ tanh ⎢ ( x − x0 ) ⎥ , a2 ⎣ 2 a1 ⎦

(A.49)

which leads to solution (2.14). 3. The integral definition of the Jacobi elliptic function, cn(x , m ), with modulus m is given by: 1

x − x0 =

∫cn(x,m)

1 [1 −

A2 (x )]

[1 − m A2 (x )]

By pulling out a factor of 1 −1 m and replacing m → integral definition can be transformed into 1

x − x0 =

∫cn(x,m)

m m−1

dA(x ).

and x − x0 →

1 [1 −

A2 (x )]

[1 − m + m A2 (x )]

dA(x ).

(A.50) x − x0 1−m

, this

(A.51)

Equating the above integral with (A.41), we get

c−

λ 2 a A (x ) − 2 A4 (x ) − c [1 − c1 A2 (x )] [1 − m + c1 m A2 (x )] = 0, (A.52) 2 a1 a1

A-7

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

where c1 is a real constant to be determined. Equating the coefficients of A2 (x ) and A4 (x ) to zero, separately, and solving for c and c1, we obtain

c1 =

a2 (1 − 2 m) , 2mλ

c=

2 m λ2 . a1 a2 (1 − 2 m)2

(A.53)

Solving 1

x − x0 =

1

∫cn(x, m)

c [1 − c1

A2 (x )]

[1 − m + c1 m A2 (x )]

dA(x ),

(A.54)

for A(x ) with the help of (A.53), we find

A(x ) =

⎡ λ (1 − m) 2mλ m ⎤ ⎥, (x − x0), cd⎢ m − 1⎦ a2 (1 − 2 m) ⎣ a1 (1 − 2 m)

(A.55)

where cd(x , m ) is a Jacobi elliptic function with modulus m. This is equivalent to

⎡ ⎤ 2mλ λ cn ⎢ (x − x0), m⎥. a2 (1 − 2 m) ⎣ a1 (1 − 2 m) ⎦

A(x ) =

(A.56)

Substituting (A.39) and (A.56) back into (A.1) leads to the solution in (2.25), where λ → −A02 . 1 If we take the limit of (A.56) when λ → c 2 a1 (1 − 2 m ) and replace m → 2 , we return back to (A.35). For the special case of m = 1, equation (A.56) reads

A(x ) =

⎡ −λ ⎤ −2 λ sech ⎢ ( x − x0 ) ⎥ , a2 ⎦ ⎣ a1

(A.57)

which leads to (2.12). 4. The integral definition of the Jacobi elliptic function, dn(x , m ), with modulus m is given by: 1

x − x0 =

∫dn(xm, )

1 [1 −

A2 (x )]

[A2 (x ) − 1 + m ]

dA(x ).

(A.58)

Equating the above integration with the integral in (A.41)

c−

λ 2 a A (x ) − 2 A4 (x ) − c [1 − c1 A2 (x )] [c1 A2 (x ) − 1 + m ] = 0, 2 a1 a1

(A.59)

where c1 is a real constant to be determined. Equating the coefficients of A2 (x ) and A4 (x ) to zero, and solving for c and c1, we get

c1 =

a2 (m − 2) , 2λ

c=

A-8

2 λ2 . a1 a2 (m − 2)2

(A.60)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Solving 1

x − x0 =

∫dn(x,m)

1 c [1 − c1

A2 (x )]

[c1 A2 (x ) − 1 + m ]

dA(x ),

(A.61)

for A(x ) with the help of (A.60), we obtain

A(x ) =

⎡ λ (m − 1) 1 ⎤ 2λ ⎥, (x − x0), cd⎢ 1 − m⎦ a2 (m − 2) ⎣ a1 (m − 2)

(A.62)

where cd(x , m ) is a Jacobi elliptic function with modulus m, which is equivalent to

A(x ) =

⎡ ⎤ 2λ λ dn ⎢ (x − x0), m⎥. a2 (m − 2) ⎣ a1 (m − 2) ⎦

(A.63)

Substituting (A.39) and (A.63) back into (A.1) leads to (2.27), where λ → −A02 . If we take the limit of (A.63) when λ → c 2 a1 (m − 2) and replace m → 2, we get (A.36). For the special case of m = 1, equation (A.63) reads ⎡ −λ ⎤ −2 λ A(x ) = sech ⎢ ( x − x0 ) ⎥ , (A.64) a2 ⎣ a1 ⎦ which again leads to (2.12). If c = 0: From (A.41), x − x0 is given by

−a1 ⎛⎜ ln[A(x )] − ln 2 λ + λ ⎝ Solving (A.65) for A(x ), we get

{

x − x0 =

A(x ) =

−λ x 2 a1

−4 λ e 2 λ a2 e

2λ

4 λ x2 a1

[a2 A2 (x ) + 2 λ ]

.

}).

(A.65)

(A.66)

−1

In the following, we represent the different possible cases of A(x ) depending upon the sign of each of a1, a2, and λ. 1. a1 > 0, a2 < 0, and λ > 0: ⎡ λ ⎤ A(x ) = 2 λ sec ⎢ (x − x0)⎥. (A.67) ⎣ a1 ⎦ Substituting (A.39) and (A.67) back into (A.1) leads to (2.8), where λ → a1 A02 . 2. a1 < 0, a2 > 0, and λ > 0:

A(x ) = 2 λ csch[

λ (x − x0)]. a1

(A.68)

Substituting (A.39) and (A.68) back into (A.1) leads to (2.13), where λ → −a1 A02 . Other cases produce either imaginary or repeated solutions.

A-9

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Case A6:

Z (x , t ) = A(x ),

(A.69)

ϕ(x , t ) = ϕ(x , t ),

(A.70)

where A(x ) is a real function to be determined. Substituting back into (A.2) and (A.3) leads to the following forms of Re[Equation (2.1)] and Im[Equation (2.1)]: Re[Equation (2.1)]:

a2 A3(x ) + a1 A″(x ) − A(x ) ⎡⎣ϕt (x , t ) + a1 ϕx2(x , t )⎤⎦ = 0,

Im[Equation (2.1)]:

a1 [2 A′(x ) ϕx(x , t ) + A(x ) ϕxx(x , t )] = 0.

(A.71)

(A.72)

Solving (A.72) for ϕ(x , t ) will give

ϕ(x , t ) =

∫ As21((tx)) dx + s2(t ),

(A.73)

where s1(t ) and s2(t ) are two real functions of t to be determined. Taking a special case of these two functions, s1(t ) = λ 0 and s2(t ) = λ2 (t − t0 ), equation (A.71) becomes

Re[Equation (2.1)]:

a1 λ02 − λ2 A(x ) + a1 A″(x ) = 0, A3(x )

a2 A3(x ) −

(A.74)

where λ 0 and λ2 are arbitrary real constants. Solving (A.74) for A(x ), we get

A(x ) =

⎡ −a2 m2 ⎤ (x − x0), m⎥ , R3 + m1 sn2 ⎢ 2 a1 ⎣ ⎦ R2 − R3 , Rj, j R1 − R3 λ2 x 2 + a2 x 3,

where sn is a Jacobi elliptic function of the modulus m =

(A.75) = 1, 2, 3, are the

three roots of Y (x ) = 2 a1 λ 02 − 2 a1 λ1 x − 2 m1 = R1 − R3, m2 = R1 − R3, and λ1 is an arbitrary real constant. From (A.73), ϕ(x , t ) takes the form ϕ(x, t ) ⎧R − R ⎡ −a m ⎤ ⎫ ⎡ −a m ⎤ 2 2 2 2 2 2 λ 0 Π⎨ 3 , am⎢ (x − x 0 ), m⎥, m⎬ dn ⎢ (x − x 0 ), m⎥ 2 a1 2 a1 ⎣ ⎦ ⎭ ⎣ ⎦ ⎩ R3 ⎪

=

⎪

R3

− a 2 m2 a1

⎡ −a m ⎤ 2 2 1 − m sn ⎢ (x − x 0 ), m⎥ 2 a1 ⎣ ⎦ 2

+ λ2 (t − t0 ),

A-10

(A.76)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

where dn is the Jacobi elliptic function of the modulus m, Π is the incomplete elliptic integral, and am is the amplitude for Jacobi elliptic functions. Substituting (A.75) and (A.76) back into (A.1) leads to (2.32). Case A7:

Z (x , t ) = A(t ),

(A.77)

ϕ(x , t ) = ϕ(x , t ),

(A.78)

where A(t ) is a real function to be determined. Substituting back into (A.2) and (A.3) leads to the following new forms of Re[Equation (2.1)] and Im[Equation (2.1)]:

A(t ) ⎡⎣a2 A2 (t ) − ϕt (x , t ) − a1 ϕx2(x , t )⎤⎦ = 0,

Re[Equation (2.1)]:

Im[Equation (2.1)]:

A′(t ) + a1 A(t ) ϕxx(x , t ) = 0.

(A.79) (A.80)

Solving (A.80) for ϕ(x , t ) gives

ϕ(x , t ) = s1(t ) + s2(t ) (x − x0) −

(x − x0)2 A′(t ) , 2 a1 A(t )

(A.81)

where s1(t ) and s2(t ) are real functions to be determined. Using (A.81) in (A.79), we get

Re[Equation (2.1)]: A(t )⎡⎣ a2 A2 (t ) − a1 s22(t ) − s1′(t )⎤⎦ + ⎡⎣2 s2(t ) A′(t ) − s2′(t ) A(t )⎤⎦ (x − x0) ⎡ A(t ) A″(t ) − 3 A′2 (t ) ⎤ ⎥ (x − x0)2 = 0. +⎢ ⎢⎣ ⎥⎦ 2 a1 A(t )

(A.82)

Collecting coefficients of (x − x0 )0, (x − x0 ), and (x − x0 )2 , separately, and equating to zero, we obtain

(x − x0)0 : (x − x0)1: ( x − x0 ) 2 :

a2 A2 (t ) − a1 s22(t ) − s1′(t ) = 0,

(A.83)

2 s2(t ) A′(t ) − s2′(t ) A(t ) = 0,

(A.84)

A(t ) A″(t ) − 3 A′2 (t ) = 0. 2 a1 A(t )

(A.85)

Solving for A(t ), s1(t ), and s2(t ), we get

A(t ) =

A0 , A1 + 2 (t − t0)

A-11

(A.86)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

s1(t ) =

a1 A22 1 + a2 A02 ln[A1 + 2 (t − t0)] + ϕ0 , 2 A1 + 4 (t − t0) 2

(A.87)

A2 . A1 + 2 (t − t0)

(A.88)

s2 (t ) =

Substituting (A.86), (A.87), and (A.88) in (A.81) and then back into (A.1) leads to (2.5).

A.1.2.2 Complex Coefficients For complex parameters, we define a1 = a1r + i a1i and a2 = a2r + i a2i , where a1r , a1i , a2r , and a1i are real constants. The real and imaginary parts of (2.1) are given by

Re[Equation (2.1)]: a2r Z 3 − Z ϕt − 2 a1i Zx ϕx − a1r Z ϕx2 + a1r Zxx − a1i Z ϕxx = 0,

(A.89)

and

Im[Equation (2.1)]: a2i Z 3 + Zt + 2 a1r Zx ϕx − a1i Z ϕx2 + a1i Zxx + a1r Z ϕxx = 0,

(A.90)

respectively. In the following, we take cases for Z and ϕ. Case A1′:

Z (x , t ) = A0 ,

(A.91)

ϕ(x , t ) = ϕ(t ).

(A.92)

Substituting back into (A.89) and (A.90) leads to the following Re[Equation (2.1)] and Im[Equation (2.1)]:

Re[Equation (2.1)]:

A03 a2r − A0 ϕ′(t ) = 0,

(A.93)

A03 a2i = 0.

(A.94)

Im[Equation (2.1)]: Solving (A.93) for ϕ(t ) gives

ϕ(t ) = A02 a2r (t − t0) + ϕ0.

(A.95)

Using (A.91) and (A.95) in (A.1) leads to solution (2.52). Case A2′:

Z (x , t ) = A0 ,

(A.96)

ϕ(x , t ) = ϕ(x ).

(A.97)

A-12

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Substituting back into (A.89) and (A.90) leads to the updated forms of Re[Equation (2.1)] and Im[Equation (2.1)]:

Re[Equation (2.1)]:

A03 a2r − A0 a1r ϕ′2 (x ) − A0 a1i ϕ″(x ) = 0,

(A.98)

Im[Equation (2.1)]:

A03 a2i − A0 a1i ϕ′2 (x ) + A0 a1r ϕ ″(x ) = 0.

(A.99)

Multiplying (A.98) by a1r and (A.99) by a1i and taking the summation of the resulting equations leads to

(

)

A03 (a1i a2i + a1r a2r ) − A0 a12i + a12r ϕ′2 (x ) = 0.

(A.100)

Solving (A.100) for ϕ(x ) gives

ϕ(x ) = ±

A02 (a1i a2i + a1r a2r ) (x − x0) + ϕ0. a12i + a12r

(A.101)

Resubstituting (A.101) into (A.98) and (A.99), equating the two resulting equations, and solving for a1i , we find

a1i =

a1r a2i . a2r

(A.102)

Substituting (A.102) back into (A.101) and then into (A.1) with (A.96) leads to solution (2.53). Case A3′:

Z (x , t ) = A(x ),

(A.103)

ϕ(x , t ) = ϕ0 ,

(A.104)

where A(x ) is a real function to be determined. Substituting back into (A.89) and (A.90) leads to the following Re[Equation (2.1)] and Im[Equation (2.1)]:

Re[Equation (2.1)]:

a2r A3(x ) + a1r A″(x ) = 0,

(A.105)

Im[Equation (2.1)]:

a2i A3(x ) + a1i A″(x ) = 0.

(A.106)

Solving the above two equations separately by following the steps used with real a1 and a2 in Case A4, we get two expressions for A(x ). For each of the two solutions to a a satisfy (2.1), a condition arises, namely a1i = 1ar 2i . The resulting solutions are (2.59), 2r (2.60), and (2.61). Case A4′:

Z (x , t ) = A(t ),

(A.107)

ϕ(x , t ) = ϕ0 ,

(A.108)

A-13

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

where A(t ) is a real function to be determined. Substituting back into (A.89) and (A.90) leads to the following new forms of Re[Equation (2.1)] and Im[Equation (2.1)]:

a2r A3(t ) = 0,

(A.109)

a2i A3(t ) + A′(t ) = 0.

(A.110)

Re[Equation (2.1)]: Im[Equation (2.1)]: Solving (A.110) for A(t )

A(t ) =

1 2 a2i (t − t0)

.

(A.111)

Substituting (A.108) and (A.111) back into (A.1) leads to solution (2.54). Case A5′:

Z (x , t ) = A(t ),

(A.112)

ϕ(x , t ) = ϕ(x , t ),

(A.113)

where A(t ) is a real function to be determined. Substituting back into (A.89) and (A.90) leads to the following Re[Equation (2.1)] and Im[Equation (2.1)]:

Re[Equation (2.1)]: A(t ) ⎡⎣a2r A2 (t ) − ϕt (x , t ) − a1r ϕx2(x , t ) − a1i ϕxx(x , t )⎤⎦ = 0,

(A.114)

Im[Equation (2.1)]: a2i A3(t ) + A′(t ) + A(t ) ⎡⎣a1r ϕxx(x , t ) − a1i ϕx2(x , t )⎤⎦ = 0.

(A.115)

Multiplying (A.114) by a1r and (A.115) by a1i and taking the summation of the resulting equations, we get (a1i a2i + a1r a2r ) A3(t ) + a1i A′(t ) − A(t ) [a1r ϕt(x , t ) + (a12i + a12r ) ϕx2(x , t )] = 0. (A.116)

Solving (A.116) for ϕ(x , t ) ϕ(x , t ) = A 0 (x − x 0 ) ⎡ (A.117) a1i A′(t ) ⎤ 1 2 2 2 + ⎢(a1i a2i + a1r a2r ) A2 (t ) − A0 a1i + a1r + ⎥dt + ϕ 0 a1r A (t ) ⎦ ⎣

(

∫

A-14

)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

and resubstituting in both (A.114) and (A.115), equating the two resulting equations, and solving for a1i , we get

a1i =

a2i A3(t ) + A′(t ) . A02 A(t )

(A.118)

In the following, we take four cases for a1i and a2i , simultaneously. 1. a1i ≠ 0 and a2i ≠ 0: Solving (A.118) for A(t ) 2

a1i A02 e a1i A0

A(t ) =

(t − t 0 ) 2

−1 + a2i e 2 a1i A0

,

(A.119)

(t − t 0 )

2. a1i = 0 and a2i ≠ 0: Solving (A.118) for A(t )

A(t ) =

1 , 2 a2i (t − t0)

(A.120)

3. a1i ≠ 0 and a2i = −a1r : Solving (A.118) for A(t )

A(t ) =

a1r A02 2

−a2i + e 2 a1r A0

(t − t 0 )

,

(A.121)

4. a1i = −a1r and a2i = 0: Solving (A.118) for A(t ) 2

A(t ) = A0 e−a1r A1

(t − t0).

(A.122)

We then use the resulting four expressions of A(t ) individually in (A.117) to find the corresponding ϕ(x , t ) for each case. Substituting A(t ) and ϕ(x , t ) for each case into (A.1) leads to solutions (2.57), (2.55), (2.56), and (2.58), respectively.

A.2 Derivation of Some Solutions of Section 3.1 A.2.1 Schematic Representation

Equation i ψt + a1 ψxx + a2 ∣ψ ∣n ψ = 0 Solutions Case B1: ψ (x , t ) = A0 ei ϕ(t ) Solution (3.6) Case B2: ψ (x , t ) = A0 ei ϕ(x ) Solution (3.7)

A-15

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

(Continued )

Case B3: ψ (x , t ) = A0 ei ϕ(x, t ) Solution (3.8) Case B4: ψ (x , t ) = A(x ) ei ϕ0 c=0 Solution (3.10)

c≠0 Solution (3.15) Case B5: ψ (x , t ) = A(x ) ei ϕ(t )

c=0

c≠0 Solution (3.16) Solution (3.17)

Solution (3.13) Case B6: ψ (x , t ) = A(x ) ei ϕ(x, t )

Solution (3.18) Case B7: ψ (x , t ) = A(t ) ei ϕ(x, t ) Solution (3.9)

A.2.2 Detailed Derivations The general solution to (3.1) can be written in the polar form:

ψ (x , t ) = Z e i ϕ ,

(A.123)

where Z = Z (x , t ) and ϕ = ϕ(x , t ) are real functions. Substituting (A.123) in (3.1) generates its real and imaginary parts:

Re[Equation (3.1)]:

a2 Z n + 1 − Z ϕt − a1 Z ϕx2 + a1 Zxx = 0,

(A.124)

and

Im[Equation (3.1)]:

Zt + 2 a1 Zx ϕx + a1 Z ϕxx = 0.

(A.125)

In the following, we take cases for Z and ϕ. Case B1:

Z (x , t ) = A0 ,

(A.126)

ϕ(x , t ) = ϕ(t ).

(A.127)

Substituting back into (A.124) and (A.125) leads to the following Re[Equation (3.1)] and Im[Equation (3.1)]:

Re[Equation (3.1)]:

A0 ⎡⎣ A0n a2 − ϕ′(t )⎤⎦ = 0,

Im[Equation (3.1)]:

A-16

0 = 0.

(A.128) (A.129)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Solving (A.128) for ϕ(t ) gives

ϕ(t ) = A0n a2 (t − t0) + ϕ0.

(A.130)

Substituting (A.126) and (A.130) back into (A.123) leads to solution (3.6). Case B2:

Z (x , t ) = A0 ,

(A.131)

ϕ(x , t ) = ϕ(x ).

(A.132)

Substituting back into (A.124) and (A.125) leads to the following Re[Equation (3.1)] and Im[Equation (3.1)]:

Re[Equation (3.1)]:

A0 ⎡⎣ A0n a2 − a1 ϕ′2 (x )⎤⎦ = 0,

(A.133)

A0 a1 ϕ″(x ) = 0.

(A.134)

Im[Equation (3.1)]: Solving (A.134) for ϕ(x ) gives

ϕ(x ) = A0n /2

a2 (x − x0) + ϕ0. a1

(A.135)

Substituting (A.131) and (A.135) back into (A.123) leads to the continuous wave solution (3.7). Case B3:

Z (x , t ) = A0 ,

(A.136)

ϕ(x , t ) = ϕ(x , t ).

(A.137)

Substituting back into (A.124) and (A.125) leads to the following Re[Equation (3.1)] and Im[Equation (3.1)]:

Re[Equation (3.1)]:

A0 ⎡⎣A0n a2 − ϕt (x , t ) − a1 ϕx2(x , t )⎤⎦ = 0,

Im[Equation (3.1)]:

A0 a1 ϕxx(x , t ) = 0.

(A.138) (A.139)

Solving (A.139) for ϕ(x , t ) gives

ϕ(x , t ) = s1(t ) + (x − x0) s2(t ),

(A.140)

where s1(t ) and s2(t ) are real functions of t. Substituting back into (A.138), collecting coefficients of (x − x0 )0 and (x − x0 ), separately, and equating to zero, we obtain:

(x − x0)0 :

A0 ⎡⎣ A0n a2 − a1 s22(t ) − s1′(t )⎤⎦ = 0, (x − x0): −A0 s2′(t ) = 0.

A-17

(A.141) (A.142)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Solving (A.142) for s2(t ) reads

s2(t ) = A1.

(A.143)

Substituting this result in (A.141) and solving for s1(t ), we get

(

)

s1(t ) = A0n a2 − A12 a1 (t − t0) + ϕ0.

(A.144)

Then, ϕ(x , t ) in (A.140) becomes

(

)

ϕ(x , t ) = A1 (x − x0) + A0n a2 − A12 a1 (t − t0) + ϕ0.

(A.145)

Using (A.145) back into (A.123) leads to the continuous wave solution (3.8). Case B4:

Z (x , t ) = A(x ),

(A.146)

ϕ(x , t ) = ϕ0 ,

(A.147)

where A(x ) is a real function to be determined. Substituting back into (A.124) and (A.125) leads to the following Re[Equation (3.1)] and Im[Equation (3.1)]:

Re[Equation (3.1)]:

a2 A1 + n (x ) + a1 A″(x ) = 0,

Im[Equation (3.1)]: By employing the chain rule A″(x ) = integral of the independent variable x:

x − x0 =

∫

0 = 0.

A′ (x ) dA′ (x ) dA(x )

(A.148) (A.149)

in (A.148), we get the following

1 dA(x ), 2 a2 2 n + A (x ) c− (n + 2) a1

(A.150)

where c is the real constant of integration. In the following, we take two categories of c. If c ≠ 0: The integration above reads

x − x0 =

⎡1 1 A(x ) n + 3 2 a2 An + 2 (x ) ⎤ , , ⎥ = Y [A(x )] 2F1⎢ , ⎣ 2 n + 2 n + 2 a1 A0 (n + 2) ⎦ A0 ψ (x , t ) = A(x ) e i ϕ0,

(A.151) (A.152)

where

Y [A(x )] =

⎡1 1 A(x ) n + 3 2 a2 An + 2 (x ) ⎤ , , ⎥, 2F1⎢ , ⎣ 2 n + 2 n + 2 a1 A0 (n + 2) ⎦ A0

A-18

(A.153)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

which is formally solved as

A(x ) = Y −1(x − x0).

(A.154)

Here, 2F1 is the hypergeometric function and Y −1 indicates the inverse operator of the function Y [A(x )], and hence, we infer the solution expressed in (3.15). If c = 0: Solving (A.150) for A(x ) 2

⎡ ⎤n ⎢ ⎥ ⎢ ⎥ 1 A(x ) = ⎢ ⎥ . −a2 n2 ⎢ ( x − x0 ) ⎥ ⎢⎣ 2 a1 (2 + n ) ⎦⎥

(A.155)

Substituting (A.147) and (A.155) back into (A.123) leads to (3.10). Case B5:

Z (x , t ) = A(x ),

(A.156)

ϕ(x , t ) = e i [λ (t − t0) + ϕ0],

(A.157)

where A(x ) is a real function to be determined and λ , t0, and ϕ0 are arbitrary real constants. Equation(3.1) becomes

−λ A(x ) + a1 A″(x ) + a2 An + 1(x ) = 0. Using the chain rule A″(x ) = independent variable x:

x − x0 =

∫

A′ (x ) dA′ (x ) , d A(x )

(A.158)

we get the following integral of the

1 dA(x ), 2 a2 λ 2 n + 2 A (x ) c+ A (x ) − (n + 2) a1 a1

(A.159)

where c is real constant of integration. In the following, we take two categories of c. If c ≠ 0: 1. From (A.159), with n = 1, x − x0 reads x − x0 2 (R3 − R2 ) =

3 a1 [R1 − A(x )] [R3 − A(x )] [A(x ) − R2 ] ⎧ −1 ⎡ R3 − A(x ) ⎤ R2 − R3 ⎫ ⎥, ⎬ F⎨sin ⎢ (R1 − R3 ) (R2 − R3 ) 2 ⎣ R3 − R2 ⎦ R1 − R3 ⎭ ⎩ ⎪

⎪

⎪

⎪

−2 a2 A3 (x ) + 3 a1 c + 3 λ A2 (x ) 3λx

a x3

(A.160) ,

where Rj, j = 1, 2, 3, are the three roots of Y (x ) = 3 c + a − 2a , and F is the 1 1 elliptic integral of the first kind. By solving (A.160) for A(x ), we obtain

A-19

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

⎡ − a (R − R ) R − R3 ⎤ 2 1 3 ⎥, (x − x0), 2 A(x ) = R3 + (R2 − R3) sn2 ⎢ 6 a1 R1 − R3 ⎦ ⎣

(A.161)

where sn is the Jacobi elliptic function. Substituting (A.157) and (A.161) back into (A.123), we get (3.16), where RR2 −− RR3 → m, c → A1, and λ → A0. 1 3 2. From (A.159), with n = 4, x − x0 is given by x − x0 =−

⎧ ⎡ A2 (x ) (R − R ) ⎤ R (R − R ) ⎫ ⎪ 9 a1 3 1 ⎥ 3 1 2 ⎪ ⎬, F⎨sin−1 ⎢ , ⎪ a2 R2 (R1 − R3) ⎪ ⎢⎣ R3 [A2 (x ) − R1] ⎥⎦ R2 (R1 − R3) ⎭ ⎩ 2

(A.162)

3

where Rj, j = 1, 2, 3, are the three roots of m 0 = 3 c + 3 λa x − 2 aa2 x , and F gives 1 1 the elliptic integral of the first kind. By solving (A.162) for A(x ), we obtain

A(x ) = −

⎡ a R (R − R − 3) R (R − R2 ) ⎤ ⎥ (x − x 0 ), 3 1 R1 sn ⎢ 2 2 1 3 a1 R2 (R1 − R3) ⎦ ⎣ ⎡ a R (R − R − 3) R1 − R3 R (R − R2 ) ⎤ ⎥ (x − x 0 ), 3 1 + sn2 ⎢ 2 2 1 R3 3 a1 R2 (R1 − R3) ⎦ ⎣

, (A.163)

where sn is the Jacobi elliptic function. Substituting (A.157) and (A.163) back into R (R − R ) (A.123), we get (3.17), where R3 (R1 − R2 ) → m, c → A1, and λ → A0. 2 1 3 If c = 0: Solving (A.159) for A(x ) with c = 0, we get 1

n ⎧ ⎤⎫ ⎡ n2 λ ⎪ λ (2 + n ) ⎪ 2 ⎥ ⎢ sech ( x − x0 ) ⎬ . A(x ) = ⎨ ⎪ ⎪ ⎦⎥⎭ ⎣⎢ 4 a1 ⎩ 2 a2

(A.164)

Substituting (A.157) and (A.163) back into (A.123), we get (3.13), where λ →

4 a1 A02 . n2

Case B6:

Z (x , t ) = A(x ),

(A.165)

ϕ(x , t ) = ϕ(x , t ),

(A.166)

where A(x ) is a real function to be determined. Substituting back into (A.124) and (A.125) leads to the following Re[Equation (3.1)] and Im[Equation (3.1)]:

Re[Equation (3.1)]: a2 A1 + n (x ) + a1 A″(x ) − A(x ) ⎡⎣ϕt (x , t ) + a1 ϕx2(x , t )⎤⎦ = 0, Im[Equation (3.1)]:

a1 [2 A′(x ) ϕx(x , t ) + A(x ) ϕxx(x , t )] = 0.

A-20

(A.167)

(A.168)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Solving (A.168) for ϕ(x , t )

ϕ(x , t ) =

∫ As21((tx)) dx + s2(t ),

(A.169)

where s1(t ) and s2(t ) are two real functions to be determined. Taking a special case of these two functions, s1(t ) = λ 0 and s2(t ) = λ2 (t − t0 ), the equation (A.167) becomes

Re[Equation (3.1)]:

a2 A1 + n (x ) −

a1 λ02 − λ2 A(x ) + a1 A″(x ) = 0, A3(x )

A′ (x ) dA′ (x ) , dA(x )

where λ 0 and λ2 are arbitrary real constants. Using the chain rule A″(x ) = we get the following integral of the independent variable x:

x − x0 =

∫

1 a λ2 2 a2 λ 2 An + 2 (x ) − 12 0 c+ A (x ) − (n + 2) a1 A (x ) a1

(A.170)

dA(x ). (A.171)

Solving (A.171) for A(x ) with n = 4, we get A(x )

=

⎡ − a (R − R ) (R − R ) (R2 − R3) (R1 − R 4 ) ⎤ 2 1 3 2 4 ⎥ R1 (R2 − R 4 ) + R2 (R 4 − R1) sn2 ⎢ (x − x0), 3 a1 (R1 − R3) (R2 − R 4 ) ⎦ ⎣ ⎡ − a (R − R ) (R − R ) (R2 − R3) (R1 − R 4 ) ⎤ 2 1 3 2 4 ⎥ −R2 + R 4 + (R1 − R 4 ) sn2 ⎢ (x − x0), 3 a1 (R1 − R3) (R2 − R 4 ) ⎦ ⎣

,

(A.172)

and hence, from (A.169), ϕ(x , t ) will read ϕ(x, t ) ⎛ −3 a (R − R )2 ⎧ R (R − R ) ⎡ −a m ⎤ ⎫ ⎡ −a m ⎤⎞ 1 1 2 4 2 2 2 2 A0 ⎜⎜ Π⎨ 2 1 , am⎢ (x − x0), m⎥, m⎬ dn ⎢ (x − x0), m⎥⎟⎟ a m R ( R − R ) 3 a 3 a ⎣ ⎦ ⎣ ⎦⎠ 2 2 4 1 1 ⎩ 1 2 ⎭ ⎝ = ⎡ −a m ⎤ 2 2 m2 + (R3 − R2) (R1 − R 4 ) sn2 ⎢ (x − x0), m⎥ 3 a1 ⎣ ⎦ R1 R2 m2 A + 0 (x − x0) + A2 (t − t0) + ϕ0, R2 m1 , Rj, m2 4 a2 x , m1

where m =

⎪

⎪

⎪

⎪

(A.173)

j = 1, 2, 3, 4, are the four roots of Y (x ) = 3 a12 λ 02 − 3 a1 c x−

= (R2 − R3) (R1 − R 4 ), and m2 = (R1 − R3) (R2 − R 4 ). Here, sn 3 λ2 x 2 + and dn are Jacobi elliptic functions, Π is the incomplete elliptic integral, am is the amplitude for Jacobi elliptic functions. Substituting (A.172) and (A.173) back into (A.123) leads to (3.18), where λ 0 → A0, c → A1 and λ2 → A2 .

Case B7:

Z (x , t ) = A(t ),

A-21

(A.174)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

ϕ(x , t ) = ϕ(x , t ),

(A.175)

where A(t ) is a real function to be determined. Substituting back into (A.124) and (A.125) leads to the following forms of Re[Equation (3.1)] and Im[Equation (3.1)]:

A(t ) ⎡⎣a2 An (t ) − ϕt (x , t ) − a1 ϕx2(x , t )⎤⎦ = 0,

Re[Equation (3.1)]:

Im[Equation (3.1)]:

A′(t ) + a1 A(t ) ϕxx(x , t ) = 0.

(A.176) (A.177)

Solving (A.177) for ϕ(x , t ) reads

ϕ(x , t ) = s1(t ) + s2(t ) (x − x0) −

(x − x0)2 A′(t ) , 2 a1 A(t )

(A.178)

where s1(t ) and s2(t ) are real functions to be determined. Using the later expression of ϕ(x , t ) in (A.176) gives Re[Equation (3.1)]: A(t )[a2 An (t ) − a1 s 22(t ) − s1′(t )] + [2 s2(t ) A′(t ) − s 2′(t ) A(t )] (x − x 0 ) ⎡ A(t ) A″(t ) − 3 A′2 (t ) ⎤ ⎥ (x − x 0 )2 = 0. +⎢ ⎢⎣ ⎥⎦ 2 a1 A(t )

(A.179)

Collecting coefficients of (x − x0 )0, (x − x0 ), and (x − x0 )2 , separately, and equating to zero, we obtain:

(x − x0)0 : (x − x0)1: ( x − x0 ) 2 :

a2 An (t ) − a1 s22(t ) − s1′(t ) = 0,

(A.180)

2 s2(t ) A′(t ) − s2′(t ) A(t ) = 0,

(A.181)

A(t ) A″(t ) − 3 A′2 (t ) = 0. 2 a1 A(t )

(A.182)

Solving for A(t ), s1(t ), and s2(t ), we get

A(t ) =

A0 , A1 + 2 (t − t0)

⎤n a1 A22 a2 (A1 + 2 t ) ⎡⎢ A0 ⎥ + ϕ0 , s1(t ) = + 2 A1 + 4 (t − t0) (2 − n ) ⎢⎣ A1 + 2 (t − t0) ⎥⎦ s2 (t ) =

A2 . A1 + 2 (t − t0)

(A.183)

(A.184)

(A.185)

Substituting (A.183), (A.184), and (A.185) in (A.178) and then back into (A.123) leads to (3.9).

A-22

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

A.3 Derivation of Some Solutions of Section 3.3 A.3.1 Schematic Representation

Solution Solution Solution Solution

Equation i ψt + a1 ψxx + a2 ∣ψ ∣n ψ + a3 ∣ψ ∣m ψ = 0 Solutions Case C1: ψ (x , t ) = A0 ei ϕ(t ) (3.20) Case C2: ψ (x , t ) = A0 ei ϕ(x ) (3.21) Case C3: ψ (x , t ) = A0 ei ϕ(x, t ) (3.22) Case C4: ψ (x , t ) = A(t ) ei ϕ(x, t ) (3.23)

A.3.2 Detailed Derivations The general solution to equation (3.19) can be written in the polar form:

ψ (x , t ) = Z e i ϕ ,

(A.186)

where Z = Z (x , t ) and ϕ = ϕ(x , t ) are real functions. Substituting (A.186) in (3.19) generates its real and imaginary parts: Re[Equation (3.19)]:

a3 Z m + 1 + a2 Z n + 1 − Z ϕt − a1 Z ϕx2 + a1 Zxx = 0, (A.187)

and

Im[Equation (3.19)]:

Zt + 2 a1 Zx ϕx + a1 Z ϕxx = 0.

(A.188)

In the following, we take cases for Z and ϕ: Case C1:

Z (x , t ) = A0 ,

(A.189)

ϕ(x , t ) = ϕ(t ).

(A.190)

Substituting back into (A.187) and (A.188) leads to the following Re[Equation (3.19)] and Im[Equation (3.19)]:

Re[Equation (3.19)]:

A0 ⎡⎣ A0n a2 + A0m a3 − ϕ′(t )⎤⎦ = 0,

Im[Equation (3.19)]:

A-23

0 = 0.

(A.191) (A.192)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Solving (A.191) for ϕ(t ) gives

ϕ(t ) = ⎡⎣ A0n a2 + A0m a3⎤⎦ (t − t0) + ϕ0.

(A.193)

Substituting (A.189) and (A.193) back into (A.186) leads to solution (3.20). Case C2:

Z (x , t ) = A0 ,

(A.194)

ϕ(x , t ) = ϕ(x ).

(A.195)

Substituting (A.194) and (A.195) back into (A.187) and (A.188) leads to the following Re[Equation (3.19)] and Im[Equation (3.19)]:

Re[Equation (3.19)]:

A0 ⎡⎣ A0n a2 + A0m a3 − a1 ϕ′2 (x )⎤⎦ = 0,

(A.196)

A0 a1 ϕ″(x ) = 0.

(A.197)

Im[Equation (3.19)]: Solving (A.197) for ϕ(x ) gives

⎛ ϕ(x ) = ±⎜A0n /2 ⎝

a2 + A0m /2 a1

a3 ⎞ ⎟ (x − x0) + ϕ0. a1 ⎠

(A.198)

Substituting (A.194) and (A.198) back into (A.186) leads to solution (3.21). Case C3:

Z (x , t ) = A0 ,

(A.199)

ϕ(x , t ) = ϕ(x , t ).

(A.200)

Substituting back into (A.199) and (A.188) leads to the following Re[Equation (3.19)] and Im[Equation (3.19)]: Re[Equation (3.19)]:

A0 ⎡⎣A0n a2 + A0m a3 − ϕt (x , t ) − a1 ϕx2(x , t )⎤⎦ = 0, (A.201)

Im[Equation (3.19)]:

A0 a1 ϕxx(x , t ) = 0.

(A.202)

Solving (A.201) for ϕ(x , t ) gives

ϕ(x , t ) = s1(t ) + (x − x0) s2(t ),

(A.203)

where s1(t ) and s2(t ) are real functions of t. Substituting back into (A.201), collecting coefficients of (x − x0 )0 and (x − x0 ), separately, and equating to zero, we obtain:

(x − x0)0 :

A0 ⎡⎣ A0n a2 + A0m a3 − a1 s22(t ) − s1′(t )⎤⎦ = 0, (x − x0): −A0 s2′(t ) = 0.

A-24

(A.204) (A.205)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Solving (A.205) for s2(t ) gives

s2(t ) = A1.

(A.206)

Substituting this result in (A.204) and solving for s1(t ), we get

s1(t ) = ⎡⎣ A0n a2 + A0m a3 − A12 a1⎤⎦ (t − t0) + ϕ0.

(A.207)

The final form of ϕ(x , t ) in (A.203) will be

(

)

ϕ(x , t ) = A1 (x − x0) + A0n a2 + A0m a3 − a1 A12 (t − t0) + ϕ0.

(A.208)

Using (A.208) back into (A.186) leads to solution (3.22). Case C4:

Z (x , t ) = A(t ),

(A.209)

ϕ(x , t ) = ϕ(x , t ),

(A.210)

where A(t ) is a real function to be determined. Substituting back into (A.209) and (A.188) leads to the following Re[Equation (3.19)] and Im[Equation (3.19)]:

Re[Equation (3.19)]: A(t ) ⎡⎣a2 An (t ) + a3 Am (t ) − ϕt (x , t ) − a1 ϕx2(x , t )⎤⎦ = 0, Im[Equation (3.19)]:

A′(t ) + a1 A(t ) ϕxx(x , t ) = 0.

(A.211) (A.212)

Solving (A.212) for ϕ(x , t )

ϕ(x , t ) = s1(t ) + s2(t ) (x − x0) −

(x − x0)2 A′(t ) , 2 a1 A(t )

(A.213)

where s1(t ) and s2(t ) are real functions to be determined. Using the expression of ϕ(x , t ) in (A.211), we get Re[Equation (3.19)]: A(t )[a2 An (t ) + a3 Am (t ) − a1 s 22(t ) − s1′(t )] + [2 s 2(t ) A′(t ) − s 2′(t ) A(t )] (x − x0) ⎡ A(t ) A″(t ) − 3 A′2 (t ) ⎤ +⎢ ⎥ (x − x0) 2 = 0. 2 a1 A(t ) ⎣ ⎦

(A.214)

Collecting coefficients of (x − x0 )0, (x − x0 ), and (x − x0 )2 , separately, and equating to zero, we obtain:

(x − x0)0 :

a2 An (t ) + a3 Am (t ) − a1 s22(t ) − s1′(t ) = 0,

(x − x0)1:

2 s2(t ) A′(t ) − s2′(t ) A(t ) = 0,

A-25

(A.215) (A.216)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

( x − x0 ) 2 :

A(t ) A″(t ) − 3 A′2 (t ) = 0. 2 a1 A(t )

(A.217)

Solving for A(t ), s1(t ), and s2(t ), we get

A(t ) =

A0 A1 + 2 (t − t0)

,

(A.218)

⎤n a1 A22 a2 [A1 + 2 (t − t0)] ⎡⎢ A0 ⎥ s1(t ) = + ⎢⎣ A1 + 2 (t − t0) ⎥⎦ 2 A1 + 4 (t − t0) (2 − n )

(A.219)

⎤m a3 [A1 + 2 (t − t0)] ⎡ A0 ⎢ ⎥ + ϕ0 , + (2 − m) ⎢⎣ A1 + 2 (t − t0) ⎥⎦

(A.220)

s2 (t ) =

A2 . A1 + 2 (t − t0)

(A.221)

Substituting (A.218), (A.220), and (A.221) in (A.213) and then back into (A.186) leads to (3.23).

A-26

IOP Publishing

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations Usama Al Khawaja and Laila Al Sakkaf

Appendix B Darboux Transformation: Single Soliton and Breather Solutions

B.1 Darboux Transformation The fundamental nonlinear Schrödinger equation (NLSE) to be solved is

i ut +

1 uxx + ∣u∣2 u = 0, 2

u = u(x , t ).

(B.1)

Darboux transformation applies only for linear differential equations. Therefore, this equation is associated with a linear system as follows. Consider the field

⎛ ψ1 ψ2 ⎞ Φ=⎜ ⎟, ⎝ ϕ1 ϕ2 ⎠

(B.2)

with all components being complex functions of x and t. Consider the linear (Zakharov–Shabat) system of differential equations in this field

Φx = U · Φ + J · Φ · Λ ,

(B.3)

Φt = V · Φ + i U · Φ · Λ + i J · Φ · Λ2 ,

(B.4)

where

U=

doi:10.1088/978-0-7503-2428-1ch11

⎛ 0 u⎞ ⎜ ⎟, ⎝−u* 0 ⎠

B-1

(B.5)

ª IOP Publishing Ltd 2020

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

V=

2 ux ⎞ i ⎛⎜∣u∣ ⎟, 2 ⎜⎝ u x* −∣u∣2 ⎟⎠

(B.6)

⎛ ⎞ J = ⎜1 0 ⎟ , ⎝ 0 −1⎠

(B.7)

⎛λ 0 ⎞ Λ = ⎜ 1 ⎟. ⎝ 0 λ2 ⎠

(B.8)

The arbitrary complex constants, λ1,2, are called the spectral parameters. We refer to (B.3, B.4) as the Lax Pair (LP). It refers also sometimes to the matrices U and V. Link between NLSE and LP The compatability condition

Φxt = Φtx ,

(B.9)

requires

Ut − Vx + [U , V ] = 0,

(B.10)

where [U , V ] is the commutator of U and V. Substituting for U and V from (B.5, B.6), the compatibility condition reads

⎛ ⎛ ⎞ ⎛ ⎞ ⎞ 1 1 − i ϕ2⎜i ut + uxx + ∣u∣2 u⎟ ⎟ ⎜ − i ϕ1⎜i ut + uxx + ∣u∣2 u⎟ ⎝ ⎠ ⎝ ⎠ ⎟ 2 2 ⎜ = 0. ⎜ ⎛ ⎞ ⎛ ⎞⎟ 1 * 2 u*⎟ − i ψ ⎜ − i u * + 1 u * + ∣u∣2 u*⎟ * ⎜ − i ψ − i u + u + ∣ u ∣ ⎜ ⎟ t xx t xx 1 2 ⎝ ⎠ ⎝ ⎠⎠ 2 2 ⎝

(B.11)

Clearly, the compatibility condition is nothing but the NLSE and its complex conjugate; it requires that u is a solution to the NLSE and u* is a solution to the complex conjugate of the NLSE. This is the link between the NLSE and the linear system. Seed Solution For a given (seed) solution of the NLSE, namely u0, the linear system will have a solution Φ0. Darboux Transformation The Darboux Transformation is defined as

Φ[1] = Φ · Λ − σ Φ ,

(B.12)

σ = Φ0 · Λ · Φ0−1.

(B.13)

where

B-2

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

Here, Φ0 is a seed solution of the linear system for a given seed solution of the NLSE, Φ denotes any solution of the linear system, and Φ[1] is the transformed (new) solution of the linear system. We request that the LP, (B.3, B.4), is covariant under the Darboux transformation

Φ[1]x = U [1] · Φ[1] + J · Φ[1] · Λ ,

(B.14)

Φ[1]t = V [1] · Φ[1] + i U [1] · Φ[1] · Λ + i J · Φ[1] · Λ2 .

(B.15)

By substituting in this system for Φ[1] from (B.12) and using (B.5, B.6), the transformed LP U[1] and V [1], must satisfy

U [1] = U0 + [J , σ ],

(B.16)

V [1] = V0 + i [U0, σ ],

(B.17)

where U0 and V0 are the LP in terms of the seed solution. It should be noted that J and Λ are constant matrices and do not change under the Darboux transformation. The new solution, u[1], is obtained from the last equation by noting that

⎛ 0 u[1]⎞ U [1] = ⎜ ⎟ ⎝−u[1]* 0 ⎠ ⎛ 2(λ1 − λ2 )ψ1ψ2 ⎞ 0 ⎜ ⎟ ⎛ 0 u0⎞ φ1ψ2 − φ2ψ1 ⎟ ⎜ ⎜ ⎟ =⎜ ⎟+ ⎟, ⎝−u 0* 0 ⎠ ⎜ 2(λ1 − λ2 )φ1φ2 0 ⎜ ⎟ ⎝ φ1ψ2 − φ2ψ1 ⎠ V [1] =

2 u[1]x ⎞ i ⎛⎜∣u[1]∣ ⎟, 2 ⎜⎝ u[1]*x − ∣u[1]∣2 ⎟⎠

(B.18)

(B.19)

lead to a covariant compatibility condition

U [1]t − V [1]x + [U [1], V [1]] = 0.

(B.20)

This means that u[1] is a solution of the NLSE

i u[1]t +

1 u[1]xx + ∣u[1]∣2 u[1] = 0. 2

(B.21)

Thus, from the solution u to the NLSE, we obtained a new solution u[1] to the same NLSE. The new solution is extracted from (B.18) as

u[1] = u 0 +

2(λ1 − λ2 )ψ1 ψ2 , φ1 ψ2 − φ2 ψ1

B-3

(B.22)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

together with its complex conjugate

u[1]* = u 0* −

2(λ1 − λ2 )φ1 φ2 . φ1 ψ2 − φ2 ψ1

(B.23)

The second term on the right hand side of the last two equations is called the Darboux dressing. Symmetry Reduction For a general seed the linear system reads x-equations:

ψ1x − u ϕ1 − λ1 ψ1 = 0,

(B.24)

ψ2 x − u ϕ2 − λ2 ψ2 = 0,

(B.25)

ϕ1x + u* ψ1 + λ1 ϕ1 = 0,

(B.26)

ϕ2 x + u* ψ2 + λ2 ϕ2 = 0.

(B.27)

t-equations:

⎛1 ⎞ i ψ1t + ⎜ ∣u∣2 + λ12⎟ψ1 + ⎝2 ⎠

⎛1 ⎞ ⎜ ux + λ1 u⎟ϕ1 = 0, ⎝2 ⎠

(B.28)

⎛1 ⎞ ⎛1 ⎞ i ψ2t + ⎜ ∣u∣2 + λ22⎟ψ2 + ⎜ ux + λ2 u⎟ϕ2 = 0, ⎝2 ⎠ ⎝2 ⎠

(B.29)

⎛1 ⎞ ⎛1 ⎞ i ϕ1t − ⎜ ∣u∣2 + λ12⎟ϕ1 + ⎜ u x* − λ1 u*⎟ψ1 = 0, ⎝2 ⎠ ⎝2 ⎠

(B.30)

⎛1 ⎞ ⎛1 ⎞ i ϕ2t − ⎜ ∣u∣2 + λ22⎟ϕ2 + ⎜ u x* − λ2 u*⎟ψ2 = 0. ⎝2 ⎠ ⎝2 ⎠

(B.31)

Symmetry reduction: With the relations:

ϕ2* = ψ1,

(B.32)

ψ2* = −ϕ1,

(B.33)

λ2* = −λ1,

(B.34)

the linear system of eight equations reduces to four equations

ψ1x − u ϕ1 − λ1 ψ1 = 0,

(B.35)

ϕ1x + u* ψ1 + λ1 ϕ1 = 0,

(B.36)

B-4

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

⎛1 ⎞ ⎛1 ⎞ i ψ1t + ⎜ ∣u∣2 + λ12⎟ψ1 + ⎜ ux + λ1 u⎟ϕ1 = 0, ⎝2 ⎠ ⎝2 ⎠ ⎛1 ⎞ i ϕ1t − ⎜ ∣u∣2 + λ12⎟ϕ1 + ⎝2 ⎠

(B.37)

⎛1 ⎞ ⎜ u x* − λ1 u*⎟ψ1 = 0. ⎝2 ⎠

(B.38)

The new solution (B.22) then takes the form

2(λ1 + λ1*)ψ1 ϕ1*

u[1] = u +

∣ϕ1∣2 + ∣ψ1∣2

(B.39)

,

and

u[1]* = u* +

2(λ1 + λ1*)ϕ1 ψ1* ∣ϕ1∣2 + ∣ψ1∣2

(B.40)

.

B.1.1 Bright Soliton Solution: Zero Seed For u = 0, the linear system simplifies to

ψ1x − λ1 ψ1 = 0,

(B.41)

ϕ1x + λ1 ϕ1 = 0,

(B.42)

i ψ1t + λ12 ψ1 = 0,

(B.43)

i ϕ1t − λ12 ϕ1 = 0,

(B.44)

with a general solution 2

ψ1 = c1 e λ1 x + i λ1 t ,

(B.45)

2

ϕ1 = c2 e−λ1 x − i λ1 t .

(B.46)

And upon employing the symmetry reduction, we get 2 ψ2 = −c2* e−λ1* x + i λ1* t ,

(B.47)

2 ϕ2 = c1* e λ1* x − i λ1* t .

(B.48)

The new solution is then given by

u[1] =

2 2 2(λ1 + λ1*)c1 e λ1 x + i λ1 t c2* e−λ1* x + i λ1* t 2

2

2

2

∣c2∣2 e−(λ1*+ λ1)x − i (λ1 − λ1* )t + ∣c1∣2 e (λ1*+ λ1)x + i(λ1 − λ1* )t

B-5

,

(B.49)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

which, without loss of generality, simplifies to ⎡ ⎤ i 1 (α 2 − v 2)t + v x + ϕ0⎦⎥

u[1] = α sech [α(x − x0 − v t )] e ⎣⎢ 2

,

(B.50)

where α = 2λ1r , v = 2λ1i , c2 /c1 = e 2λ1r x0 + i ϕ0 , and ϕ0 is an arbitrary real constant. Remark: This is the bright soliton solution (2.12) characterized by four arbitrary parameters: initial position x0, initial speed v, amplitude (or inverse width) α, and arbitrary global phase ϕ0.

B.1.2 Generalized Breather Solution for Focusing and Defocusing Nonlinearity: CW Seed Here, we derive the generalized breather solution of the fundamental NLSE

i ut +

1 uxx − c ∣u∣2 u = 0, 2

(B.51)

where c = 1( −1) corresponds to the defocusing (focusing) case. The corresponding LP takes the form of (B.3, B.4) where

U=

V=

⎛ 0 u⎞ ⎟, −c ⎜ ⎝− u* 0 ⎠

(B.52)

2 −c ux ⎞ i ⎛⎜ −c ∣u∣ ⎟, 2 ⎜⎝ −c u x* −c∣u∣2 ⎟⎠

(B.53)

and J and Λ are given by (B.7, B.8). The linear system (B.3, B.4) reads explicitly x-equations:

−i

c u ϕ1 − λ1 ψ1 + ψ1x = 0,

(B.54)

−i

c u ϕ2 − λ2 ψ2 + ψ2 x = 0,

(B.55)

λ1 ϕ1 + i

c u* ψ1 + ϕ1x = 0,

(B.56)

λ2 ϕ2 + i

c u* ψ2 + ϕ2 x = 0.

(B.57)

t-equations:

⎛ ⎞ 1 1 −i λ12 ψ1 + ⎜ c λ1 ϕ1 + i c u* ψ1⎟u + ψ1t + ⎝ ⎠ 2 2

c ϕ1 ux = 0,

(B.58)

⎛ ⎞ 1 1 −i λ22 ψ1 + ⎜ c λ2 ϕ2 + i c u* ψ2⎟u + ψ2t + ⎝ ⎠ 2 2

c ϕ2 ux = 0,

(B.59)

B-6

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

1 i (2λ12 − c ∣u∣2 )ϕ1 − 2

c λ1 u* ψ1 + ϕ1t +

1 2

c ψ1 u x* = 0,

(B.60)

1 i (2λ22 − c ∣u∣2 )ϕ2 − 2

c λ2 u* ψ2 + ϕ2t +

1 2

c ψ2 u x* = 0.

(B.61)

Symmetry reduction: Requiring the complex conjugate of (B.54) and (B.55) to be identical with (B.57) and (B.56), respectively, is possible with

ϕ2 = ψ1*,

(B.62)

ψ2 = c ϕ1*,

(B.63)

λ2 = −λ1*,

(B.64)

where the system of eight equations reduces to four equations

−

−i

c u ϕ1 − λ1 ψ1 + ψ1x = 0,

(B.65)

−i

c u* ψ1 − λ1 ϕ1 − ϕ1x = 0,

(B.66)

1 i 2λ12 − c ∣u∣2 ψ1 + 2

(

)

1 i 2λ12 − c ∣u∣2 ϕ1 − 2

(

)

c λ1 u ϕ1 + ψ1t +

1 2

c ϕ1 ux = 0,

(B.67)

c λ1 u* ψ1 + ϕ1t +

1 2

c ψ1 u x* = 0.

(B.68)

For the CW seed 2

u0 = A ei A

t

(B.69)

with arbitrary real constant A, the general solution of the reduced linear system (B.65–B.68) is

ψ1(x , t ) = [(A c c1 i + c2λ1)sinh((x + i λ1 t )ω ) + c2 ω cosh((x + i λ1 t )ω )] (B.70) 1 2 1 × e− 2 i A c t , ω ϕ1(x , t ) = [ − (A c c2 i + c1λ1)sinh((x + i λ1 t )ω) + c1 ω cosh((x + i λ1 t )ω )] (B.71) 1 2 1 × e− 2 i A c t , ω

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Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

where

ω=

A2 c + λ12 ,

(B.72)

and c1 and c2 are arbitrary constants. The seed solution of the linear system thus reads

⎛ ⎞ ψ1 c ϕ1*⎟ ⎜ Φ0 = , ⎜ ⎟ ⎝ ϕ1 ψ1* ⎠

(B.73)

where the symmetry reductions (B.62–B.64) have been taken into account and ψ1 and ϕ1 are given by (B.70) and (B.71), respectively. The new solution, given formally by (B.16) and upon using the symmetry reduction conditions, now reads

u[1] = u 0 − 4i

c λ1r

ψ1 ϕ1* c ∣ϕ1∣2 − ∣ψ1∣2

,

(B.74)

where λ1r is the real part of λ1. The breather solution can be viewed at as a combination of two generalized solitons where each soliton has a localization component and an oscillatory component, both in x and t. This is verified by rewriting the breather solution as ⎡ ⎤ cos(ζ1 − i χ1)cos(ζ2 + i χ 2 ) 8λ1r c 3/2 2 u[1] = ⎢1 − i ⎥ × A e i A t , (B.75) A cos(2ζ1) − c cos(2ζ2 ) + cosh(2χ1) − c cosh(2χ 2 ) ⎦ ⎣

where

ζ1,2 = κ X1,2 − Ω T1,2,

χ1,2 =

X1,2 T1,2 − , α τ

(B.76)

(B.77)

X1,2 = x − x01,02 ,

(B.78)

T1,2 = t − T01,02,

(B.79)

κ = Im[ω ],

(B.80)

Ω = −Re[λ1 ω ],

(B.81)

α=

1 , Re[ω ]

B-8

(B.82)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

τ=

x02 = x01 +

1 , Im[λ1 ω ]

(B.83)

⎡ log q ⎤ ⎡ log q ⎤⎞ 1 ⎛ + λ1r Re ⎢ ⎜λ1i Im ⎢ ⎟, ⎥ ⎣ ω ⎥⎦⎠ ⎣ ω ⎦ 2λ1r ⎝

(B.84)

⎡ log q ⎤ 1 Im ⎢ . ⎣ ω ⎥⎦ 2λ1r

(B.85)

t02 = t01 +

In addition to the arbitrary CW amplitude, A, there are four arbitrary parameters: x01, t01, λ1r , and λ1r : x01: sets the reference for x, t01: sets the reference for t, λ1r and λ1i set: α: width of localization in x, τ: width of localization in t, κ: frequency of oscillation in x, Ω: frequency of oscillation in t.

Note that x02 and t02, which correspond to the second generalized soliton, are not arbitrary as they are given in terms of the above four arbitrary parameters. Furthermore, since the four parameters α, τ, κ, and Ω, are given in terms of λ1r , and λ1i , only two out these four parameters are to be considered arbitrary while the other two are not. Any two of the four parameters can be chosen to be arbitrary.

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IOP Publishing

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations Usama Al Khawaja and Laila Al Sakkaf

Appendix C Derivation of the Similarity Transformations in Chapter 5

C.1 Function Coefficients Given the generalized NLSE (5.64)

i Φt + b1(x , t ) Φxx + b2(x , t ) ∣Φ∣2 Φ + [b3r(x , t ) + i b3i (x , t )] Φ = 0,

(C.1)

we aim at transforming it to the fundamental NLSE

p(x , t )(i ψT + a1 ψXX + a2 ∣ψ ∣2 ψ ) = 0,

(C.2)

with the scaling (similarity) transformation

Φ(x , t ) = A(x , t ) e i B(x, t ) ψ [X (x , t ), T (x , t )],

(C.3)

where a1,2 are arbitrary real constants and b1,2(x , t ), b3r, i (x , t ), and p(x , t ) are arbitrary real functions. The unknown functions X (x , t ), T (x , t ), A(x , t ), and B (x , t ) are assumed to be real and need to be determined in terms of the function coefficients of the generalized NLSE and the constant coefficients of the fundamental NLSE. It is essential to have the function p(x , t ) since otherwise the transformation will restrict the coefficients b1,2(x , t ) to only time-dependent ones and consequently the potential b3r, i (x , t ) will be real and quadratic in x. Substituting (C.3) in (C.1) and requesting the result to take the form of the fundamental NLSE (C.2) gives

A b1 p Tx2 = 0, (from ψTT ),

(C.4)

e i Bp[2b1 Ax Tx + A (i Tt + b1 (2i Bx Tx + Txx ))] = i , (from ψT ),

(C.5)

e i Bp b1 A Xx2 = a1, (from ψXX ),

(C.6)

doi:10.1088/978-0-7503-2428-1ch12

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ª IOP Publishing Ltd 2020

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

e i Bp b2 A3 = a2 , (from ∣ψ ∣2 ψ ),

(C.7)

e i Bp [2b1 Ax Xx + A (i Xt + b1(2i Bx Xx + Xxx ))] = 0, (from ψX ),

(C.8)

e i Bp ⎡⎣ i At + b1 (2i Ax Bx + Axx ) + A i b3i + b3r − Bt − b1 Bx2 − i Bxx ⎤⎦ = 0, (from ψ ).

(

(

))

(C.9)

The solution of this system is given by (6.43–6.50), and

p(r , t ) =

ei B

1 . A g1′

(C.10)

As a hint on the procedure of solving this system, one starts with solving (C.4) for T, then (C.5) for p, then (C.6) and (C.7) for b1 and b2, respectively, then the real part of (C.8) gives A while the imaginary part gives B, and finally, the real part of (C.9) is solved for b3r and the imaginary part of (C.9) is solved for b3i .

C.2 Solution-Dependent Transformation This is very similar to the previous case. The only difference is in the procedure of solving (C.8) and (C.9). While in the previous section we set the coefficient of ψ and ψX to zero separately, here we require the sum of the two terms to be zero, namely

[2b1 Ax Xx + A (i Xt + b1(2i Bx Xx + Xxx ))] ψ + ⎡⎣i At + b1 (2i Ax Bx + Axx )

(C.11)

+ A i b3i + b3r − Bt − b1 Bx2 − i Bxx ⎤⎦ ψX = 0.

(

(

))

The two procedures should both be valid since they lead to satisfying the generalized and fundamental NLSE. Solving the real part of the last equation for b3r and the imaginary part for b3i gives (5.83) and (5.84), respectively. The solutions of b1,2 , X, T, A, and B remain the same as in the previous case.

C.3 Similarity Transformation for the NLSE in (N + 1)-Dimensions Given the generalized NLSE in N-spacial dimensions and polar coordinates

⎛ N−1 ⎞ i Φt + b1(r , t ) ⎜Φrr + Φr⎟ + b2(x , t ) ∣Φ∣2 Φ ⎝ ⎠ r + [b3r(r , t ) + i b3i (r , t )] Φ = 0,

(C.12)

we aim at transforming it to the fundamental NLSE

p(r , t )(i ψT + a1 ψRR + a2 ∣ψ ∣2 ψ ) = 0,

C-2

(C.13)

Handbook of Exact Solutions to the Nonlinear Schro¨dinger Equations

with the similarity transformation

Φ(r , t ) = A(r , t ) e i B(r, t ) ψ [R(r , t ), T (r , t )],

(C.14)

where a1,2 are arbitrary real constants and b1,2(r, t ), b3r (r, t ), b3i (r, t ), and p(r, t ) are arbitrary real functions. The unknown functions R(r, t ), T (r, t ), A(r, t ), and B (r, t ) are assumed to be real and need to be determined in terms of the function coefficients of the generalized NLSE and the constant coefficients of the fundamental NLSE. Substituting (C.14) in (C.12) and requesting the result to take the form of the fundamental NLSE (C.13) gives

A b1 p Tr2 = 0, (from ψTT ),

(C.15)

1 iB e p[2r b1 Ar Tr r + A (i r Tt + b1 ((N − 1 + 2i r Br ) Tr + r Trr ))] = i , (from ψT ),

(C.16)

e i Bp b1 A R r2 = a1, (from ψRR ),

(C.17)

e i Bp b2 A3 = a2 , (from ∣ψ ∣2 ψ ),

(C.18)

1 iB e p [2r b1 Ar Rr r + A (i r Rt + b1((N − 1 + 2ir Br ) Rr + r Rrr ))] = 0, (from ψR ),

(C.19)

1 iB ⎡ e p ⎢⎣ i r At + b1 ((N − 1 + 2i r Br ) Ar r + r Arr ) + i A (r b3i + (N − 1) b1 Br ⎤ + i r −b3r + Bt + b1 Br2 + r b1 Brr ⎥ = 0, (from ψ ). ⎦

(C.20)

(

)

)

The solution of this system is given by (5.20–5.26). The procedure of solving the system is similar to that described in Section C.1.

C-3