Advances in Nonlinear Waves and Symbolic Computation [1 ed.] 9781608766079, 9781606922606

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Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

ADVANCES IN NONLINEAR WAVES AND SYMBOLIC COMPUTATION

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

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Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

ADVANCES IN NONLINEAR WAVES AND SYMBOLIC COMPUTATION

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

ZHENYA YAN

Nova Science Publishers, Inc. New York

Copyright © 2009 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS.

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LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Yan, Zhenya. Advances in nonlinear waves and symbolic computation / Zhenya Yan. p. cm. ISBN 978-1-60876-607-9 (E-Book) 1. Nonlinear waves--Mathematical models. 2. Solitons--Mathematical models. 3. Nonlinear difference equations. 4. Differential equations, Nonlinear. I. Title. QA927.Y36 2009 531'.1133--dc22 2008044453

Published by Nova Science Publishers, Inc. Ô New York

CONTENTS Preface Chapter 1

Chirped Optical Solutions K. Senthilnathan, K. Nakkeeran, K. W. Chow, Qian Li and P. K. A. Wai

Chapter 2

Direct Methods and Symbolic Software for Conservation Laws of Nonlinear Equations Willy Hereman, Paul J. Adams, Holly L. Eklund, Mark S. Hickman and Barend M. Herbst

Chapter 3

Chapter 4

Chapter 5

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vii

Index

1

19

Nonclassical Potential Symmetries for a Nonlinear Thermal Wave Equation M.L. Gandarias

79

Anti-Dark Solitons of the Resonant Nonlinear Schrödinger Equation D. W. C. Lai, K. W. Chow, C. Rogers and Zhenya Yan

99

Similarity Solutions for the Boiti-Leon-Pempinelli Equation with Symbolic Computation Zhuosheng Lü

109

The New Sine-Gordon Expansion Algorithims to Construct Exact Solutions of nonlinear Wave Equations Zhenya Yan

125 149

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PREFACE Since `soliton' was presented by Zabusky and Kruskal in 1965, many nonlinear wave equations, in particular, soliton equations, admit many `good' properties. Many powerful methods have been constructed to study a wide class of nonlinear wave equations. Moreover, with the development of symbolic computation, many difficult and complex problems are impossible to solve by hand, but them now can be carried out in computer with the aid of software. Symbolic computation plays an important role in the study of nonlinear waves. This book is devoted to the study of nonlinear waves and symbolic computation from the following chapters. In Chapter 1, the authors consider the evolution of nonlinear optical optical pulses in some inhomogeneous optical media wherein the pulse propagation is governed by the nonlinear Schrödinger equation with variable dispersion. The Painlevé analysis is applied to obtain the condition for the soliton pulse propagation. Two dispersion profiles satisfying this criterion are the constant dispersion and exponentially decreasing dispersion profiles. In the exponentially varying dispersive media, the authors explain the existence and the formation of chirped optical soliton through the variational equation for the chirp. In addition, they theoretically discuss the generation of exact chirped higher order solitons using the Hirota bilinear method. The authors also demonstrate the implication for optical communications systems in terms of pulse compression by using these exact chirped solitons. Finally, they analyze the interaction scenarios of the chirped higher order solitons. Chapter 2 is devoted to presenting direct methods, algorithms, and symbolic software for the computation of conservation laws of nonlinear partial differential equations (PDEs) and differential-difference equations (DDEs). Our method for PDEs is based on calculus, linear algebra, and variational calculus. First, the authors compute the dilation symmetries of the given nonlinear system. Next, they build a candidate density as a linear combination with undetermined coefficients of terms that are scaling invariant. The variational derivative (Euler operator) is used to derive a linear system for the undetermined coefficients. This system is then analyzed and solved. Finally, the authors compute the flux with the homotopy operator. The method is applied to nonlinear PDEs in (1+1) dimensions with polynomial nonlinearities which include the Korteweg-de Vries (KdV), Boussinesq, and Drinfel'dSokolov-Wilson equations. An adaptation of the method is applied to PDEs with transcendental nonlinearities. Examples include the sine-Gordon, sinh-Gordon, and Liouville

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viii

Zhenya Yan

equations. For equations in laboratory coordinates, the coefficients of the candidate density are undetermined functions which must satisfy a mixed linear system of algebraic and ordinary differential equations. For the computation of conservation laws of nonlinear DDEs the authors use a splitting of the identity operator. This method is more efficient that an approach based on the discrete Euler and homotopy operators. The authors apply the method of undetermined coefficients to the Kac-van Moerbeke, Toda, and Ablowitz-Ladik lattices. To overcome the shortcomings of the undetermined coefficient technique, the authors designed a new method that first calculates the leading order term and then the required terms of lower order. That method, which is no longer restricted to polynomial conservation laws, is applied to discretizations of the KdV and modified KdV equations, and a combination thereof. Additional examples include lattices due to Bogoyavlenskii, Belov--Chaltikian, and Blaszak--Marciniak. The undetermined coefficient methods for PDEs and DDEs have been implemented in Mathematica. The code TransPDEDensityFlux.m computes densities and fluxes of systems of PDEs with or without transcendental nonlinearities. The code DDEDensity\\Flux.m does the same for polynomial nonlinear DDEs. Starting from the leading order terms, the new Maple library discrete computes densities and fluxes of nonlinear DDEs. The software can be used to answer integrability questions and to gain insight in the physical and mathematical properties of nonlinear models. When applied to nonlinear systems with parameters, the software computes the conditions on the parameters for conservation laws to exist. The existence of a hierarchy of conservation laws is a predictor for complete integrability of the system and its solvability with the Inverse Scattering Transform. Chapter 3 is concerned with classes of symmetries for partial differential equations which can be written in a conserved form. These nonclassical potential symmetries are realized as nonclassical symmetries of the associated potential system and are neither classical potential symmetries realized as Lie symmetries of a related auxiliary system nor nonclassical symmetries of the considered equation. Some of these symmetries are carried out for a nonlinear equation which serves as a model for thermal conductivity and thermal waves in a heated plasma, as well as for the Fokker-Planck equation. For some special values of the parameters it happens that the nonlinear thermal wave equation does not admit an infiniteparameter Lie group of contact transformations, so is not linearizable by an invertible mapping, however the associated potential system admits an infinite-parameter Lie group of point transformations and, consequently, the equation is linearized by a non-invertible mapping. In these cases, these nonclassical potential symmetries are realized as nonclassical symmetries of the linearized form. The similarity solutions are also discussed in terms of the linearized form and yield solutions of the nonlinear thermal wave equation which are neither nonclassical solutions nor solutions arising from classical potential symmetries. Chapter 4 considers the special classes of soliton solutions of the resonant nonlinear Schrödinger equation (RNLS), a model relevant in Madelung fluids, reaction diffusion systems, and black hole problems in astrophysics. Just like the conventional nonlinear Schrödinger model, the character of pulses or localized solutions still depends critically on the relative signs of the cubic nonlinear and dispersive terms. When the signs arethe same, 'antidark' solitons, or bright pulses propagating on a continuous wave background, as well as oneand two-soliton solutions, are found. However, when the signs are different, only dark one-

Preface

ix

soliton solution exists and only does so for a very special choice of the parameter. Moreover, some periodic wave solutions are also given in terms of the Jacobi elliptic functions. In Chapter 5, symmetry reductions and similarity solutions for the Boiti--Leon-Pempinelli (BLP) equation are performed. It is shown that the infinitesimals of the BLP equation form an infinite dimensional Lie algebra. The similarity solutions of the equation are obtained by combining the classical Lie symmetry group method with a constructive algorithm, the further extended tanh method. A symbolic computation implementation of the further extended tanh method is also presented whose efficiency is shown by some concrete examples. Chapter 6 develops a new constructive algorithm to new multiple solutions of nonlinear wave equations via the general sine-Gordon reduction equation (SGRE) and its solutions. It is proved that the algorithm is more powerful than many known approaches such as the tanhfunction method, the sine-cosine method, the project Riccati system method, the sinh-Gordon expansion method, the sn- and cn-function method, the extended Jacobi elliptic function method, etc. The algorithm is applied to (i) The KdV-mKdV equation with first-order dispersion term, (ii) The (2+1)-dimensional higher-degree Burgers equation, (iii) The modified Boussinesq equation, (iv) Nonlinear Schrödinger equation with cubic-quintic nonlinearity, (v) The (2+1)-dimensional generalization of mKdV equation and (vi) The (2+1)dimensional break soliton equation. As a consequence, many types of exact solutions are deduced which include solitary wave solutions, doubly periodic solutions, optical solitary wave solutions. These solutions may be useful to explain the corresponding physical phenomena.

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Zhenya Yan, Beijing, P. R. China

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

In: Advances in Nonlinear Waves and Symbolic Computation ISBN 978-1-60692-260-6 c 2009 Nova Science Publishers, Inc. Editor: Zhenya Yan

Chapter 1

C HIRPED O PTICAL S OLITONS K. Senthilnathan 1, K. Nakkeeran2, K. W. Chow3 , Qian Li1 and P. K. A. Wai1 1

Photonics Research Centre and Department of Electronic and Information Engineering The Hong Kong Polytechnic University Hung Hom, Kowloon, Hong Kong 2

3

School of Engineering, Fraser Noble Building, King’s College University of Aberdeen Aberdeen AB24 3UE, UK

Department of Mechanical Engineering, University of Hong Kong Pokfulam, Hong Kong

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Abstract We consider the evolution of nonlinear optical pulses in some inhomogeneous optical media wherein the pulse propagation is governed by the nonlinear Schr¨odinger equation with variable dispersion. The Painlev´e analysis is applied to obtain the condition for the soliton pulse propagation. Two dispersion profiles satisfying this criterion are the constant dispersion and exponentially decreasing dispersion profiles. In the exponentially varying dispersive media, we explain the existence and the formation of chirped optical soliton through the variational equation for the chirp. In addition, we theoretically discuss the generation of exact chirped higher order solitons using the Hirota bilinear method. We also demonstrate the implication for optical communications systems in terms of pulse compression by using these exact chirped solitons. Finally, we analyze the interaction scenarios of the chirped higher order solitons.

1

Introduction

It is well known that soliton is a self-reinforcing solitary wave formed by a balance between nonlinear and dispersive effects in the medium. In the recent past, it has been shown that solitons do exist in many areas of science namely, particle physics, molecular biology, quantum mechanics, geology, meteorology, oceanography, astrophysics, optics and cosmology. But solitons that exist in optics − the so called “optical solitons” − have been drawing

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2

K. Senthilnathan, K. Nakkeeran, K. W. Chow, Qian Li and P. K. A. Wai

greater attention among the scientific community as they seem to be the right candidates for transferring information across the world through optical fibers. In optical fibers, a dynamical balance occurs between the linear effect, anomalous group velocity dispersion (AGVD) which produces negative chirp, and nonlinear effect, self-phase modulation (SPM) which generates positive chirp. The resulting pulse exhibits zero chirp, and is called a soliton [1]– [3]. Soliton pulse propagation in optical fibers is governed by the well-known nonlinear Schr¨odinger equation(NLSE)[1]–[3]. This NLSE model is one of the most important universal nonlinear models in modern science as the NLSE appears in many branches of science and engineering. For instance, if the medium presents linear and nonlinear gains or losses, the NLSE becomes the so-called cubic complex Ginzburg-Landau (CCGL) equation, which is a universal model describing the evolution of the envelope of a pulse in a non-conservative medium. The CCGL equation is an essential model for the study of laser dynamics and especially for fiber lasers. In addition, NLSE plays a significant role in other physical situation like periodic structures in fibers called fiber Bragg grating wherein it successfully governs pulse propagation outside the photonic band gap structure under certain physically valid conditions [4]. The NLSE also appears in the description of the Bose-Einstein Condensate (BEC), a context in which it is often called the Gross-Pitaevskii equation [5]. The NLSE is completely integrable and its N -soliton solutions can be obtained using the standard inverse scattering transform method [6]. In any physical system governed by the NLSE, additional physical effects like dissipation typically will destroy the integrability property. Various mathematical techniques such as perturbation analysis, numerical analysis, variational analysis, etc., have been developed to study the dynamics of such physical systems. In general, NLS equation, being exactly integrable, admits not only exact one (fundamental) soliton, but also multi-soliton solutions. In particular, the initial condition, i.e., initial pulse shape of N sech(t), gives rise to the so called N -soliton solutions where N , an integer ≥1, is known as the order of soliton. This N -soliton solution may be regarded as a nonlinear superposition of N -different solitons [2], [3], [7]. The first order soliton, called fundamental soliton, is a self-maintaining pulse whereas higher order solitons split and narrow, recovering their initial pulse shape at the end of a period. This period is known as the soliton period and is a function of pulse width, dispersion and wavelength. Thus, a fundamental difference between the N -soliton breather and the fundamental soliton (N = 1) soliton is the fact that the breather exhibits oscillatory properties [1]–[3], [7]. The shape-preserving property of fundamental optical solitons has been utilized as information carrying entities in high bit rate optical fiber communication systems. To meet the current bandwidth requirement, extremely short pulses called ultrashort pulses will normally be used. Optical pulse compression technique is one of the most important techniques for the generation of ultrashort optical pulses. Much effort has also been devoted to optical pulse compression techniques owing to their practical utility for shortening the durations of pulses (in the time domain) generated by oscillators and amplifiers. Most of these techniques rely on chirping produced either by SPM in the normal dispersion regime or by combining phase modulation with amplification [4]. Soliton pulse compression and adiabatic pulse compression techniques have been proposed for the compression process. In soliton pulse compression, the compressed pulses suffer from significant pedestal generation, leading to nonlinear interactions between neigh-

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Chirped Optical Solitons

3

boring solitons [4]. Even though the adiabatic compression performs the pulse compression with minimal pedestal, the fiber length required for the compression of pulses broader than 5ps tends to be excessively long. Thus, soliton effect compression and adiabatic compression could not satisfy all the requirements. It has been demonstrated that increasing the input soliton order enhances the compression factor but the compressed pulses suffer from the pedestal as the higher order solitons do not have linear chirp. Therefore, it is of great interest to develop a compression technique capable of achieving both high quality compression and large compression factor. Instead of the chirp free solitons, more recently, attention has been riveted on the generation of chirped solitons. It has been suggested that chirped solitary waves could be compressed more efficiently if the dispersion decreases approximately exponentially [8]. Recently, self-similar analysis has been used to obtain the linearly chirped solitary waves [9]–[11]. These self-similar pulses propagate in the nonlinear media subject to exponential scaling of pulse amplitude and pulse width. As reported in the literature, these chirped solitary pulses possess a strictly linear chirp that leads to efficient compression or amplification. Therefore, they are particularly useful in the design of optical fiber amplifiers, optical pulse compressors, and solitary wave based communication links. Very recently, using the selfsimilar analysis, chirped Bragg solitary pulses have been theoretically generated near the photonic bandgap of a non-uniform fiber Bragg grating, and the possibility of pedestal free Bragg soliton pulse compression is examined [12]. Even though there are some progress in the investigation of the chirped solitary waves, to our knowledge, to date no clear physical meaning for the formation of chirped soliton has been reported. In this work, we describe the existence and the formation of chirped optical soliton in a media with exponentially decreasing dispersion. This exponential decreasing dispersion profile for the governing media has been obtained by the Painlev´e analysis. In other words, the NLSE with variable dispersion coefficient is integrable if and only if the dispersion coefficient varies exponentially. We derive the exact chirped one-soliton solution for the NLSE with exponentially varying dispersion using the Hirota bilinear method. We also perform the variational analysis to explore the dynamics of chirped optical solitons in the exponentially varying dispersive nonlinear systems. The plan of the work is as follows. In Section 2, we describe our theoretical model and also analyze the condition for the soliton pulse propagation using Painlev´e analysis. We theoretically discuss the generation of chirped fundamental optical solitons in Section 3. With the help of variational analysis, we discuss the physical explanation of the formation of chirped optical solitons in Section 4. In Section 5, we present the implementation of the chirped optical soliton in terms of an effective chirp and pedestal free pulse compression technique. Next, we discuss the existence of chirped higher order solitons using Hirota bilinear method in Section 6. We address the evolution and interaction scenarios of the chirped higher order solitons in Section 7. Finally, we conclude in Section 8.

2

Theoretical Model

We consider the NLSE with varying dispersion in the following form, iψz −

β(z) ψtt + γ|ψ|2ψ = 0, 2

(1)

4

K. Senthilnathan, K. Nakkeeran, K. W. Chow, Qian Li and P. K. A. Wai

where ψ is the slowly varying envelope of the axial electrical field, β(z) and γ represent the group-velocity dispersion parameter which varies along the propagation direction and the self-phase modulation parameter respectively. These definitions of the variables and parameters are for the context of envelope soliton propagation in optical fibers, but they will vary for other physical systems governed by the NLSE [5]. The variable coefficient NLSE (1) also governs the dynamics of dispersion-managed optical fiber solitons which is another important type of pulse propagation in fibers useful for high-speed data transmission [2]. Equation (1) has been extensively investigated by many researchers in different contexts [9], [10], [13]–[19]. We now apply the well established Painlev´e analysis to Eq. (1) to derive the parametric condition on β(z) for which the NLSE (1) is completely integrable [20]. To proceed further with the Painlev´e analysis, we introduce a new set of variables a(= ψ) and b(= ψ ∗). By Eq. (1), a and b can be written as β(z) att + γa2b = 0, 2 β(z) −ibz − btt + γb2a = 0. 2 iaz −

(2) (3)

Generalized Laurent series expansion of a and b are, a=

∞ X r=0

ar ϕ

r+µ

, b=

∞ X

br ϕr+δ ,

(4)

r=0

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with a0 , b0 6= 0, where µ and δ are negative integers, ar and br are set of expansion coefficients which are analytic in the neighborhood of the non-characteristic singular manifold ϕ(z, t) = 0. Looking at the leading order, a ≈ a0 ϕµ and b ≈ b0ϕδ are substituted in Eqs. (2) and upon balancing dominant terms, the results obtained are: µ = δ = −1 and a0 b0 = ϕ2t β(z)/γ. Substituting the full Laurent series (4) in Eqs. (2) and considering the leading order terms, the resonances are found to be, r = −1, 0, 3, 4. The resonance at r = −1 represents the arbitrariness of the singularity manifold and r = 0 corresponds to the fact that either a0 or b0 is arbitrary. Collecting and balancing the coefficients of the different powers of ϕ show that a sufficient number of arbitrary functions exists only for the following parametric condition on β(z)   d2β(z) dβ(z) 2 β(z) − = 0. dz 2 dz

(5)

On solving this equation, we have β = β0 exp(−σz),

(6)

where β0 and α are integration constants. Thus the Painlev´e analysis implies that the dispersion must vary in an exponential manner for the system equation (1) to be completely integrable. It should be emphasized that the exponential scaling (for the dispersion profile), obtained here by Painlev´e analysis, is the same as that obtained through self-similar analysis [9]. It is remarkable that we have systematically derived this integrable case through

Chirped Optical Solitons

5

singularity structure analysis. Thus the integrable form of Eq. (1) can be written as (with σ = 2β0α0 ) iψz −

β0 exp(−2β0α0 z) ψtt + γ|ψ|2ψ = 0, 2

(7)

We would now like to point out a conjecture regarding the resonance values derived in the Painlev´e analysis. The resonances r = −1, 0, 3, 4 obtained here for the variable coefficient NLSE (1) are the same as those for the constant coefficient NLSE. Past experience has shown that such coincidences usually imply that the newly derived integrable nonlinear evolution equation could be connected to existing systems of equations. This is in fact true and there is a connection between the variable coefficient NLSE (7) and the conventional NLSE. We consider the gauge transformation ψ → ψ exp(iα0t2 /2); t → t exp(−β0α0 z),

(8)

which maps the exponentially varying dispersion NLSE (7) into the following variable coefficient NLSE:

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iψz −

β0 β0 2 2 β0 ψtt + γ|ψ|2ψ − α0 t ψ − i α0 ψ = 0. 2 2 2

(9)

This variable coefficient NLSE (9) has been analyzed for its integrability through Painlev´e analysis and possesses a transformation connecting it to constant coefficient NLSE [21]. Thus the above mentioned conjecture about the resonance values of the Painlev´e analysis holds good as there is a connection between the integrable dispersion varying NLSE (7) and the conventional constant coefficient NLSE. Equation (9) is also applicable in many physical contexts like averaged DM fiber system, nonlinear compression of chirped solitary waves, quasi-soliton propagation in DM optical fiber and other settings [22]–[24], [8], [25]– [27]. Very recently, 1–, 2– bright and dark solitons, as well as the periodic wave solutions were also calculated for the variable coefficient NLSE by the Hirota bilinear method [28]. Here, it should be emphasized that even though the results on Painlev´e analysis, for the NLSE with the variable coefficient, has been reported in different contexts in the literature, it is fair to say that no clear physical understanding has emerged regarding the existence and the formation of chirped optical solitons in nonlinear optical media. Therefore, in what follows, we address this issue and which is considered to be main theme of this work.

3

Chirped Optical Soliton: Hirota Bilinear Method

Now we present the essential steps for deriving the exact soliton solution of the variable coefficient NLSE (9) using the Hirota bilinear method. First we separate an exponential factor due to the chirp in the bilinear form as ψ = exp(iα0 t2 /2)(g/f ). The bilinear equations are then   β0 iDz − Dt2 − iα0β0 tDt − iα0β0 g · f = 0, 2 β0 2 D f · f = −γ|g|2, 2 t

6

K. Senthilnathan, K. Nakkeeran, K. W. Chow, Qian Li and P. K. A. Wai

where D is the Hirota bilinear operator [29]. Here the main difference with the conventional Hirota method is the usage of a spatially varying wavenumber. This can be handled readily by straightforward calculus and the details are described in our earlier work [28]. This modified Hirota expansion now gives the bright 1–soliton for Eq. (9) as (with χ = α0 β0 z) 1 ψ= exp 2φ where φ=

r



 iα0t2 ∗ + η − η sech (η + η ∗ + ln φ) , 2

(10)

λteχ iλ2e2χ χ −γ e−χ , η = − + + η (0). β0 λ + λ ∗ 2 8α0 2

Here λ and η (0) are complex constants. Now using the 1–soliton solution (10) and the gauge transformation (8), the exact 1–soliton solution for the exponentially decreasing dispersion NLSE (7) can be derived as (for simplicity here we consider λ = 1/τ and η (0) = 0, where τ is a real constant which can represent the initial pulse width): ψ=

s

 2χ  r    −β0 eχ te τ −γ e2χ 2 2χ sech + ln exp iα0t e − i . γ τ τ 2 β0 4α0τ 2

(11)

This soliton solution is similar to that reported earlier in the literature [8], [10], [19]. From the factor χ in the 1–soliton solution (11) it is clear that the soliton pulse intensity and chirp are exponentially increasing and the width is exponentially decreasing at the same rate along the propagation direction. To explain the dynamics of the chirped solitons we make use of the following variational analysis [30].

4

Physical Explanation of Chirped Optical Soliton: Variational Analysis

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In order to explain the formation of chirped optical soliton we make use of the . When we consider the hyperbolic secant ansatz ψ = x1sech



t x2



exp(ix3t2 + ix4),

in the variational analysis of the NLSE (1), the evolution equations for the pulse amplitude (x1 ), width (x2), chirp (x3 ) and phase (x4) could be derived. In particular, the evolution equation of the chirp is established as dx3 2 = 2β(z)x23 + 2 2 dz π x2



1 1 − LD LN L



,

(12)

where LD = −x22 /β(z) and LN L = 1/(γx21), which respectively represent the dispersion and nonlinear lengths. In the following we discuss the possibility of chirp free solitons and chirped solitons in the optical systems governed by the integrable NLSE (7).

Chirped Optical Solitons

4.1

7

Chirp-free solitons

For the case of conventional 1–soliton pulse propagation in the fiber with constant dispersion and nonlinearity, as LD = LN L it is clear that there is no chirp evolution associated with the soliton as the chirp equation (12) becomes zero. For the dispersion varying optical fibers, we derived from the Painlev´e analysis the integrability condition (6) which in fact explains the well known soliton pulse propagation in dispersion decreasing fibers with optical losses. In the case of fibers with optical losses, the intensity of the pulses decreases exponentially and the same holds for the nonlinearity. Thus if the dispersion also decreases in the same way as the nonlinearity the soliton could maintain a constant pulse width intensity product and could propagate without any chirp [13].

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4.2

Chirped solitons

The same integrability condition (6) for a lossless fiber could also support a chirped optical soliton. From the soliton solution (11) we can get the expressions for the pulse width and intensity as x2 = x20 exp(−2χ) and x21 = x210 exp(2χ) (where x10 and x20 are initial pulse amplitude and width, respectively). From the integrability condition the dispersion could decrease exponentially as β(z) = β0 exp(−2χ). Thus the dispersion and nonlinear lengths for the chirped soliton propagation are given by LD = −x220 exp(−2χ)/β0 and LN L = exp(−2χ)/(γx210). Unlike the situation in conventional optical soliton, the dispersion and nonlinear lengths decrease in the same way as the dispersion along the propagation direction in the case of chirped soliton. The dispersion length is equal to the nonlinear length at every point of the fiber. This is possible only because of the evolution of the chirp. When we substitute the expressions for the dispersion, pulse width, LD and LN L in the chirp evolution equation (12) we get dx3 /dz = 2β0 exp(−2χ)x23. If we consider the initial chirp of the soliton pulse as α0 , then the chirp would evolve as x3 = α0 exp(2χ). This is also obvious from the 1–soliton solution (11) derived for the dispersion varying NLSE (7). In the case of conventional optical soliton, the chirp produced by the dispersion is balanced by the chirp produced by the nonlinearity. In the variable coefficient NLSE (7) the dispersion decreases exponentially and the Kerr nonlinearity remains constant. Thus we can say that in the absence of any chirp, the chirp produced by the linear effect is not balanced by the nonlinear effect. Thus it looks like the nonlinear effect overwhelms the linear effect. However, the additional chirp, which is present initially in the pulse (the sign of this initial chirp is the same as that of the chirp produced by the anomalous dispersion fiber) grows exponentially and adds to the chirp generated by the dispersion to balance with the nonlinear chirp for the formation of chirped solitons. Thus these chirped optical solitons are formed basically because of the growth of the chirp in contrast to a zero chirp in case of the conventional solitons.

5

Pulse Compression in terms of Chirped Soliton

As discussed in the introduction, the dispersion varying NLSE (7) governs the nonlinear pulse propagation in dispersion decreasing fibers and also near the photonic band gap of a nonuniform fiber Bragg grating [31], [32]. For the purpose of illustration, fiber Bragg

8

K. Senthilnathan, K. Nakkeeran, K. W. Chow, Qian Li and P. K. A. Wai

grating has been preferred as it possesses the huge amount of grating induced dispersion whose value is six orders of magnitude greater than the conventional telecommunication fiber. The availability of large dispersion allows the soliton dynamics to be studied on length scales of centimeters. Thus, one could realize very short and efficient pulse compressors whose advantages would be the high degree of compression and the high pulse quality. For the demonstration purpose, we assume a nonuniform fiber Bragg grating (NFBG) of length L = 6 cm, which has the initial dispersion β(0) = −33 ps2 cm−1 . The effective core area of the fiber Bragg grating is considered as 20 µm2 and the nonlinear coefficient as 2.3 × 10−16 cm2 W−1 . The initial dispersion monotonically decreases to a final value at the end of the dispersion decreasing NFBG as β(L), which can easily be calculated from (6) as β(z = L) = −4.56 ps2 cm−1 for a given value of the initial chirp α0 = −0.005 THz2 . Having calculated the dispersion at the end points of the grating, the maximum compression factor, in the case of constant nonlinearity and no loss/gain, is determined by the ratio of the input to output dispersion. In this case, the compression factor is 7.24. The evolution of the Bragg soliton (11) with pulse compression is shown in Fig. 1.

4

x 10 5

|ψ|2 [W]

4 3 2 1 0 6 20

4

z [c

10

m]

0

2 -10

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0 -20

t [ps]

Figure 1: 3D plot of bright Bragg solitary pulse compression for the physical parameters β(0) = −33 ps2 cm−1 , α0 = −0.005 THz2, τ = 10 ps and z = 6 cm.

6

Chirped Higher Order Solitons

The results for the conventional solitons (including higher order solitons for NLS), as well as more general arguments, suggest studying the chirped higher order solitons in the exponentially decreasing dispersive media. Thus, in this section, using the chirped higher order solitons, we demonstrate a new type of pulse compression that utilizes a combination of chirp and nonlinear higher order soliton effect. Unlike adiabatic soliton compression, the proposed scheme possesses advantage of an exact solution to the variable coefficient NLS equation for the evolution of chirped higher order solitons. The highly enhanced linear

Chirped Optical Solitons

9

chirp in the two-soliton will be used for achieving the highly compressed pulses in the exponentially decreasing dispersive media. We now proceed to generate the chirped higher order soliton by the Hirota method. The auxiliary functions g and f of chirped two-soliton solution are found to be: g = exp[φ1 ] + exp[φ2] + n1 exp[φ1 + φ2 + φ?2] + n2 exp[φ2 + φ1 + φ?1 ], f

= 1 + m11 exp[φ1 + φ?1] + m12 exp[φ1 + φ?2 ] + m21 exp[φ2 + φ?1 ] + m22 exp[φ2 + φ?2 ] + M exp[φ1 + φ?1 + φ2 + φ?2 ],

where φ1 = th1 (z) + h10(z), φ2 = th2 (z) + h20(z), ir2 exp(σz) σz γ exp(−σz) hn = rn exp(σz), hn0 = n + (n = 1, 2), mij = , σ 2 2(ri + rj? )2 n1 =

γ(r1 − r2)2 exp(−σz) γ(r1 − r2)2 exp(−σz) , n2 = , ? ? 2 2 2(r2 + r1) (r2 + r2) 2(r1? + r1)2 (r1? + r2 )2 γ 2(r1 − r2)2 (r1? − r2? )2 exp(−2σz) M= . 4(r1 + r1?)2 (r1 + r2?)2 (r1? + r2)2(r2 + r2?)2

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Thus, the chirped two soliton is realized with the auxiliary functions g and f along with the relation (ψ = exp(iα0t2 /2)(g/f )). The parameters r1 and r2 are complex. Note that the physical parameters such as ( 1), pulse evolution is totally different from the case of a fundamental soliton [2], [3], [7]. Now, we discuss the physical mechanism of the periodic evolution of the two solitons under the influence of chirp in the exponentially decreasing dispersive media. For all higher order solitons (N > 1), the effect of SPM dominates over the effect of AGVD. The effect of SPM is to generate a positive chirp, and the effect of AGVD is to produce a negative chirp. Owing to the higher intensity of the two solitons, the amount of positive chirp generated by SPM is greater than that of the negative chirp produced by the AGVD. Indeed the negative chirp due to AGVD decreases exponentially and at the same time it is maintained by the exponential growth of the external chirp. The resulting opposite chirps cannot be canceled out completely. As a result, the two soliton pulses acquire positive chirp. The effect of AGVD for a positively chirped pulse is to compress the pulse. Firstly, we consider a situation wherein the two pulses have different amplitudes and velocities. The interaction scenario of the chirped two-soliton pulses with compression in the time domain is shown in Fig. 2. Fig. 3 illustrates the evolution of the chirped twosoliton pulses in the frequency domain. In the time domain, the two soliton pulses undergo compression up to a particular distance at which the resulting pulse possesses a minimum pulse width. This pulse compression occurs mainly because of the initial chirp together with higher order soliton property ( N > 1). In the spectral domain, the pulse spectrum broadens as the pulse compression occurs. From Fig. 3, the spectrum splits to form two peaks. Note that the interaction length between two pulses decreases owing to the exponentially dispersion decreasing media (zint = πt2 /2β(z)). In the conventional two soliton, pulse compression occurs only during the constructive interference between the two pulses whereas in the case of chirped two soliton the two pulses undergo compression during the whole process.

Chirped Optical Solitons

11

Intensity

10

5

0 8

7

6

5

4

z

3

6

3

2

1

0 −6

0 −3 frequency

Figure 3: Spectral broadening and interaction scenario of chirped two soliton pulses in frequency domain when σ = 0.3, r1 = 1.75 + 0.05I and r2 = 1.25 + 0.025I.

50 Intensity

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40 30 20 10 0 8

7

6

5 z

4

3

2

1

0 −12

−8

−4

0

4

8

12

time

Figure 4: Compression of the chirped bright soliton pulse in time domain for the physical parameters σ = 0.3, r1 = 1.5 + 0.05I and r2 = 1.5 + 0.025I.

12

K. Senthilnathan, K. Nakkeeran, K. W. Chow, Qian Li and P. K. A. Wai

Intensity

8 6 4 2 0 8

7

6

5 z

4

3

2

2

1

0 −4

−2

4

0 frequency

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Figure 5: Spectral broadening and interaction scenario of the chirped bright soliton pulse in frequency domain for the physical parameters σ = 0.3, r1 = 1.5 + 0.05I and r2 = 1.5 + 0.025I. Transmission properties of the chirped two-soliton are shown in Fig. 2. The pulses experience compression under the influence of initial frequency chirp and the higher soliton property. During the compression process, they develop peaks at the half period and reexpand to the original pulse shape at the full period. The central peak is dominant, and this feature reveals that the pulse is compressed. This property suggests that all higher order solitons could be used to compress the pulses to a shorter width [35]. From this, one can clearly observe that chirped two solitons can be nonlinearly compressed cleanly and efficiently in an exponentially dispersion decreasing medium. Secondly, we consider two pulses with identical amplitudes but moving with different velocities. Fig. 4 shows the interaction as well as the compression of chirped two-soliton pulses in the time domain when the pulses move with different velocities but with the same amplitude. Fig. 5 represents the spectral broadening of the two chirped soliton pulses in the spectral domain. In the last case, we consider a different physical situation whereby the two pulses possess different amplitudes but they move with the same velocity. Under this condition, the pulses always remain well separated and they further undergo compression in time domain (Fig. 6). However, the two pulses do not undergo any periodic interaction. Fig. 7 illustrates the pulse broadening in the spectral domain. An increasingly large range of frequencies is involved, clearly supporting the notion of broadening. Finally, we have also calculated the compression factor for the three cases, shown in Fig. 8. In Fig. 8, solid line, dashed lines and dot-dashed lines represent the compression factors for Figs. 2, 4 and 6 respectively. Note that the compression factors undergo oscillations owing to the property of the higher order solitons. The compression factor is a

Chirped Optical Solitons

13

50

Intensity

40 30 20 10 0 8

7

6

5

8 4

z

4 3

2

0 1

−4 0

−8

time

Figure 6: Compression of the chirped bright soliton pulse in time domain for the physical parameters σ = 0.3, r1 = 1.25 + 0.05I and r2 = 1.5 + 0.05I.

Intensity

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8 6 4 2 0 8

7

6

5 z

4 4

2 3

2

0 1

−2 0

−4

frequency

Figure 7: Spectral broadening and interaction scenario of chirped two soliton in frequency domain when σ = 0.3, r1 = 1.25 + 0.05I and r2 = 1.5 + 0.05I.

14

K. Senthilnathan, K. Nakkeeran, K. W. Chow, Qian Li and P. K. A. Wai

Compression factor

15

10

5

0

2

4 z

6

8

Figure 8: Comparison of compression factors for the following three cases. The solid line, dashed lines and dot-dashed lines represent the compression factors for Figs. 2, 4 and 6 respectively.

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function of distance since the chirped two-soliton pulses undergo evolution in the exponentially decreasing dispersive media. However, the compression factor does not undergo any oscillation and keeps on increasing exponentially when the amplitudes of the pulses are equal while they differ in velocity. Figure 9 represents the comparison of the compression factor of Fig. 4 with the exponential function ( exp(σz), where σ = 0.3). The solid line represents exponential function and the dashed lines represent the compression factor of Fig. 4. The compression factor qualitatively agrees with the exponential behavior. The exponential behavior in the compression factor could be expected since the width of compressed pulse decreases exponentially while the amplitude increases exponentially. In addition, we have also calculated the peak power (not shown here) which is also found to have increased in accordance with the nonlinear compression process.

8

Conclusion

In conclusion, we have applied the Painlev´e analysis to get the condition for the soliton pulse propagation to the dispersion decreasing fiber media. Two dispersion profiles are feasible candidates, namely constant and exponential dispersion profiles. For the exponentially scaled dispersion the fiber system equation has connection with the constant coefficient NLSE through another well known variable coefficient NLSE. The explicit chirped one soliton solution is generated through the Hirota bilinear method. Furthermore, using the variational equation of the chirp, the formation of the chirped optical soliton is discussed in detail. Then we have presented an application of the chirped optical solitons for pulse

Chirped Optical Solitons

15

Compression factor

15

10

5

0

2

4 z

6

8

Figure 9: The comparison of compression factor of Fig. 4 with the exponential function (exp(σz), where σ = 0.3). The solid line represents exponential function and the dashed lines represent the compression factor of Fig. 4.

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compression either through the dispersion decreasing fiber or NFBG. We believe that the results reported in this work could also be applicable to other physical systems governed by NLSE with varying dispersion. In addition to the chirped fundamental soliton, we have also discussed the chirped two-soliton solution for the exponentially decreasing dispersive media. Besides, we have also calculated the pulse compression factor as well as the bandwidth broadening factor for the compressed pulses. The interaction scenarios of the two-soliton pulses have also been demonstrated under the influence of chirping in the exponentially decreasing dispersive media. Many physical issues e.g., how to achieve the required initial chirp for the pulse, i.e., pre-chirping and de-chirping processes after the compression process will be investigated and reported elsewhere.

Acknowledgement KWC and KN wish to thank The Royal Society for their support in the form of an International Joint Project Grant. KWC and KN are very grateful to Prof. John Watson for his valuable support for this research collaboration. Partial financial support has been provided by the Research Grants Council of the Hong Kong Special Administrative Region, China to KWC through contracts HKU 7123/05E and HKU 7118/07E and to PKAW through contracts PolyU 5289/07E.

16

K. Senthilnathan, K. Nakkeeran, K. W. Chow, Qian Li and P. K. A. Wai

References [1] A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23, 142 (1973). [2] G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 2001). [3] L. F. Mollenauer and J. P. Gordon, Solitons in optical fibers: Fundamentals and Applications (Elsevier Academic Press, San Diego, 2006). [4] G. P. Agrawal, Applications of Nonlinear Fiber Optics (Academic Press, San Diego, 2001). [5] L. V´azquez, L. Streit and V. M. P´erez-Garc´ıa, Eds., Nonlinear Schr¨odinger and KleinGordon systems: Theory and Applications , World Scientific, Singapore (1996). [6] V. E. Zakharov and A. B. Shabat, Sov. Phys.-JETP, 23, 142 (1972). [7] J. R. Taylor, Optical Solitons - Theory and Experiment (Cambridge University Press, Cambridge, 1992). [8] J. D. Moores, Opt. Lett. 21, 555 (1996). [9] V. I. Kruglov, A. C. Peacock and J. D.Harvey, Phys. Rev. Lett. 90, 113902 (2003). [10] V. I. Kruglov, A. C. Peacock and J. D.Harvey, Phys. Rev. E 71, 056619 (2005). [11] M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, Phys. Rev. Lett. 84, 6010 (2000). [12] K. Senthilnathan, P. K. A. Wai and K. Nakkeeran, Optical Fiber Communication Conference (JWA19), California, USA (2007). [13] K. Tajima, Opt. Lett. 12, 54 (1987). [14] H. H. Kuehl, J. Opt. Soc. Am. B 5, 709 (1988).

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[15] P. V. Mamyshev and S. V. Chernikov, Opt. Lett. 15, 1076 (1990). [16] S. V. Chernikov, E. M. Dianov, D. J. Richardson, D. N. Payne, Opt. Lett. 18, 476 (1993). [17] D. Anderson, M. Lisak, B. Malomed, and M. Quiroga-Teixeiro, J. Opt. Soc. Am. B 11, 2380 (1994). [18] V. N. Serkin and A. Hasegawa, Phys. Rev. Lett. 85, 4502 (2000). [19] V. N. Serkin and A. Hasegawa, IEEE J. Sel. Top. Quantum Electron. 8, 418 (2002). [20] M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations, and Inverse Scattering (Cambridge University Press, Cambridge, 1991). [21] P. A. Clarkson, Proc. Royal Soc. Edinburgh 109A, 109 (1988).

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17

[22] S. P. Burtsev, V. E. Zakharov, and A. V. Mikhailov, Theor. Math. Phys. 70, 227 (1987). [23] S. Kumar and A. Hasegawa, Opt. Lett. 22, 372 (1997). [24] Y. Kodama, S. Kumar and A. Maruta, Opt. Lett. 22, 1689 (1997). [25] M. L. Quiroga-Teixeiro, D. Anderson, P. A. Anderkson, A. Berntson and M. Lisak, J. Opt. Soc. Am. B 13, 687 (1996). [26] K. Nakkeeran, J. Phys. A: Mathematical and General 34, 5111 (2001). [27] R. Ganapathy, V. C. Kuriakose and K. Porsezian, Opt. Comm. 194, 299 (2001). [28] C. C. Mak, K. W. Chow and K.Nakkeeran, J. Phys. Soc. Jpn. 74, 1449 (2005). [29] R. Hirota, J. Math. Phys. 14, 805 (1974). [30] B. A. Malomed, Progress in Optics, 43, 71 (2003). [31] E. N. Tsoy and C. M. de Sterke, Phys. Rev. E 62, 2882 (2000). [32] G. Lenz and B. J. Eggleton, J. Opt. Soc. Am. B 15, 2979 (1998). [33] J. Satsuma and N. Yajima, Prog. Theor. Phys. (Suppl.) 55, 62 (1974). [34] N. N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear pulses and beams (Chapman and Hall, London, 1997).

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[35] L. F. Mollenauer, R. H. Stolen, J. P. Gordon, and W. J. Tomlinson, Opt. Lett. 8, 289 (1983).

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In: Advances in Nonlinear Waves and Symbolic Computation ISBN 978-1-60692-260-6 c 2009 Nova Science Publishers, Inc. Editor: Zhenya Yan

Chapter 2

D IRECT M ETHODS AND S YMBOLIC S OFTWARE FOR C ONSERVATION L AWS OF N ONLINEAR E QUATIONS Willy Hereman1∗, Paul J. Adams1 , Holly L. Eklund1, Mark S. Hickman2, and Barend M. Herbst3 1

Department of Mathematical and Computer Sciences, Colorado School of Mines, Golden, CO 80401-1887, U.S.A. 2

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3

Department of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand

Department of Mathematical Sciences, Applied Mathematics Division, General Engineering Building, University of Stellenbosch Private Bag X1, Matieland 7602, South Africa

I N M EMORY OF M ARTIN D. K RUSKAL (1925-2006) Courtesy of Rutgers, The State University of New Jersey. Photographer, Nick Romanenko ∗

E-mail: [email protected]

20

W. Hereman, P. J. Adams, H. L. Eklund, M. S. Hickman and B. M. Herbst

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Abstract We present direct methods, algorithms, and symbolic software for the computation of conservation laws of nonlinear partial differential equations (PDEs) and differentialdifference equations (DDEs). Our method for PDEs is based on calculus, linear algebra, and variational calculus. First, we compute the dilation symmetries of the given nonlinear system. Next, we build a candidate density as a linear combination with undetermined coefficients of terms that are scaling invariant. The variational derivative (Euler operator) is used to derive a linear system for the undetermined coefficients. This system is then analyzed and solved. Finally, we compute the flux with the homotopy operator. The method is applied to nonlinear PDEs in (1 + 1) dimensions with polynomial nonlinearities which include the Korteweg-de Vries (KdV), Boussinesq, and Drinfel’d– Sokolov–Wilson equations. An adaptation of the method is applied to PDEs with transcendental nonlinearities. Examples include the sine-Gordon, sinh-Gordon, and Liouville equations. For equations in laboratory coordinates, the coefficients of the candidate density are undetermined functions which must satisfy a mixed linear system of algebraic and ordinary differential equations. For the computation of conservation laws of nonlinear DDEs we use a splitting of the identity operator. This method is more efficient that an approach based on the discrete Euler and homotopy operators. We apply the method of undetermined coefficients to the Kac-van Moerbeke, Toda, and Ablowitz–Ladik lattices. To overcome the shortcomings of the undetermined coefficient technique, we designed a new method that first calculates the leading order term and then the required terms of lower order. That method, which is no longer restricted to polynomial conservation laws, is applied to discretizations of the KdV and modified KdV equations, and a combination thereof. Additional examples include lattices due to Bogoyavlenskii, Belov–Chaltikian, and Blaszak–Marciniak. The undetermined coefficient methods for PDEs and DDEs have been implemented in Mathematica. The code TransPDEDensityFlux.m computes densities and fluxes of systems of PDEs with or without transcendental nonlinearities. The code DDEDensityFlux.m does the same for polynomial nonlinear DDEs. Starting from the leading order terms, the new Maple library discrete computes densities and fluxes of nonlinear DDEs. The software can be used to answer integrability questions and to gain insight in the physical and mathematical properties of nonlinear models. When applied to nonlinear systems with parameters, the software computes the conditions on the parameters for conservation laws to exist. The existence of a hierarchy of conservation laws is a predictor for complete integrability of the system and its solvability with the Inverse Scattering Transform.

1

Introduction

This chapter focuses on symbolic methods to compute polynomial conservation laws of partial differential equations (PDEs) in (1 + 1) dimensions and differential-difference equations (DDEs), which are semi-discrete lattices. For the latter we treat systems where time is continuous and the spatial variable has been discretized. Nonlinear PDEs that admit conservation laws arise in many disciplines of the applied sciences including physical chemistry, fluid mechanics, particle and quantum physics, plasma physics, elasticity, gas dynamics, electromagnetism, magneto-hydro-dynamics, nonlin-

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Direct Methods and Symbolic Software for Conservation Laws

21

ear optics, and the bio-sciences. Conservation laws are fundamental laws of physics that maintain that a certain quantity will not change in time during physical processes. Familiar conservation laws include conservation of momentum, mass (matter), electric charge, or energy. The continuity equation of electromagnetic theory is an example of a conservation law which relates charge to current. In fluid dynamics, the continuity equation expresses conservation of mass, and in quantum mechanics the conservation of probability of the density and flux functions also yields a continuity equation. There are many reasons to compute conserved densities and fluxes of PDEs explicitly. Invariants often lead to new discoveries as was the case in soliton theory. One may want to verify if conserved quantities of physical importance (e.g. momentum, energy, Hamiltonians, entropy, density, charge) are intact after constitutive relations have been added to close a system. For PDEs with arbitrary parameters one may wish to compute conditions on the parameters so that the model admits conserved quantities. Conserved densities also facilitate the study of qualitative properties of PDEs [86], [97], such as recursion operators, bior tri-Hamiltonian structures, and the like. They often guide the choice of solution methods or reveal the nature of special solutions. For example, an infinite sequence of conserved densities is a predictor of the existence of solitons [7] and complete integrability [2] which means that the PDE can be solved with the Inverse Scattering Transform (IST) method [2]. Conserved densities aid in the design of numerical solvers for PDEs [87], [88] and their stability analysis (see references in [23]). Indeed, semi-discretizations that conserve discrete conserved quantities lead to stable numerical schemes that are free of nonlinear instabilities and blowup. While solving DDEs, which arise in nonlinear networks and as semi-discretizations of PDEs, one should check that their conserved quantities indeed remain unchanged as time steps are taken. Computer algebra systems (CAS) like Mathematica, Maple, and REDUCE, can greatly assist the computation of conservation laws of nonlinear PDEs and DDEs. Using CAS interactively, one can make a judicious guess (ansatz) and find a few simple densities and fluxes. Yet, that approach is fruitless for complicated systems with nontrivial conservation laws with increasing complexity. Furthermore, completely integrable equations PDEs [2], [7], [74], [89] and DDEs [10], [75] admit infinitely many independent conservation laws. Computing them is a challenging task. It involves tedious computations which are prone to error if done with pen and paper. Kruskal and collaborators demonstrated the complexities of calculating conservation laws in their seminal papers [67], [78], [79] on the Korteweg-de Vries (KdV) equation from soliton theory [2], [7], [30]. We use this historical example to introduce the method of undetermined coefficients. In the first part of this chapter we cover the symbolic computation of conservation laws of completely integrable PDEs in (1 + 1) dimensions (with independent variables x and t). Our approach [8], [38], [51], [53] uses the concept of dilation (scaling) invariance and the method of undetermined coefficients. Our method proceeds as follows. First, build a candidate density as a linear combination (with undetermined coefficients) of “building blocks” that are homogeneous under the scaling symmetry of the PDE. If no such symmetry exists, construct one by introducing parameters with scaling. Next, use the Euler operator (variational derivative) to derive a linear algebraic system for the undetermined coefficients. After the system is analyzed and solved, use the homotopy operator to compute the flux. When applied to systems with parameters, our codes can determine the conditions on the

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22

W. Hereman, P. J. Adams, H. L. Eklund, M. S. Hickman and B. M. Herbst

parameters so that a sequence of conserved densities exists. The method is applied to nonlinear PDEs in (1 + 1) dimensions with polynomial terms which include the KdV, Boussinesq, and Drinfel’d-Sokolov-Wilson equations. An adaptation of the method is applied to PDEs with transcendental nonlinearities. Examples include the sine-Gordon, sinh-Gordon, and Liouville equations. For equations written in laboratory coordinates, the coefficients of the candidate density are undetermined functions which must satisfy a mixed linear system of algebraic and ordinary differential equations (ODEs). Capitalizing on the analogy between PDEs and DDEs, the second part of this chapter deals with the symbolic computation of conservation laws of nonlinear DDEs [33], [39], [42], [51], [54], [57]. Again, we use scaling symmetries and the method of undetermined coefficients. One could use discrete versions of the Euler operator (to verify exactness) and the homotopy operator (to invert the forward difference). Although these operators might be valuable in theory, they are highly inefficient as tools for the symbolic computation of conservation laws of DDEs. We advocate the use of a “splitting and shifting” technique, which allows us to compute densities and fluxes simultaneously at minimal cost. The undetermined coefficient method for DDEs is illustrated with the Kac-van Moerbeke, Toda, and Ablowitz-Ladik lattices. There is a fundamental difference between the continuous and discrete cases in the way densities are constructed. The total derivative has a weight whereas the shift operator does not. Consequently, a density of a PDE is bounded in order with respect to the space variable. Unfortunately, there is no a priori bound on the number of shifts in the density, unless a leading order analysis is carried out. To overcome this difficulty and other shortcomings of the undetermined coefficient method, we present a new method to compute conserved densities of DDEs. That method no longer uses dilation invariance and is no longer restricted to polynomial conservation laws. Instead of building a candidate density with undetermined coefficients, one first computes the leading order term in the density and, secondly, generates the required terms of lower order. The method is fast and efficient since unnecessary terms are never computed. The new method is illustrated using a modified Volterra lattice as an example. The new method performs exceedingly well when applied to lattices due to Bogoyavlenskii, Belov–Chaltikian, and Blaszak–Marciniak. The new method is also applied to completely integrable discretizations of the KdV and modified KdV (mKdV) equations, and a combination thereof, known as the Gardner equation. Starting from a discretized eigenvalue problem, we first derive the Gardner lattice and then compute conservation laws. There are several methods (see [51]) to compute conservation laws of nonlinear PDEs and DDEs. Some methods use a generating function [2], [7], which requires the knowledge of key pieces of the IST. Another common approach uses the link between conservation laws and symmetries as stated in Noether’s theorem [14], [15], [66], [81]. However, the computation of generalized (variational) symmetries, though algorithmic, is as daunting a task as the direct computation of conservation laws. Most of the more algorithmic methods [12], [13], [20], [25], [63], [101], require the solution of a determining system of ODEs or PDEs. Despite their power, only a few of these methods have been implemented in CAS. We devote a section to symbolic software for the computation of conservation laws. Additional reviews can be found in [38], [51], [101]. Over the past decade, in collaboration with students and researchers, we have de-

Direct Methods and Symbolic Software for Conservation Laws

23

signed and implemented direct algorithms for the computation of conservation laws of nonlinear PDEs and DDEs. We purposely avoid Noether’s theorem, pre-knowledge of symmetries, and a Lagrangian formulation. Neither do we use differential forms or advanced differential-geometric tools. Instead, we concentrate on the undetermined coefficient method for PDEs and DDEs, which uses tools from calculus, linear algebra, and the variational calculus. Therefore, the method is easy to implement in Mathematica and easy to use by scientists and engineers. The code TransPDEDensityFlux.m computes densities and fluxes of systems of PDEs with or without transcendental nonlinearities. The code DDEDensityFlux.m does the same for polynomial nonlinear DDEs. Starting from the leading order terms, the new Maple library discrete computes densities and fluxes of nonlinear DDEs very efficiently. The software can thus be used to answer integrability questions and to gain insight in the physical and mathematical properties of nonlinear models. Our software is in the public domain. The Mathematica packages and notebooks are available at [48] and Hickman’s code in Maple is available at [56]. We are currently working on a comprehensive package to compute conservation laws of PDEs in multiple space dimensions [45], [51], [83].

Part I: Partial Differential Equations in (1 + 1) Dimensions In this first part we cover PDEs in (1 + 1) dimensions, that is, PDEs in one space variable and time. Starting from a historical example, we introduce the concept of dilation invariance and use the method of undetermined coefficients to compute conservation laws of evolution equations. Later on, we adapt the method of undetermined coefficients to cover PDEs with transcendental terms.

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2

The Most Famous Example in Historical Perspective

The story of conservation laws for nonlinear PDEs begins with the discovery of an infinite number of conservation laws of the ubiquitous Korteweg-de Vries equation which models a variety of nonlinear wave phenomena, including shallow water waves [46] and ion-acoustic waves in plasmas [2], [7], [30]. The KdV equation can be recast in dimensionless variables as ut + αuux + u3x = 0, (1) 3

∂u ∂ u where the subscripts denote partial derivatives, i.e. ut = ∂u ∂t , ux = ∂x , and u3x = ∂x3 . The parameter α can be scaled to any real number. Commonly used values are α = ±1 or α = ±6. Equation (1) is an example of a scalar (1 + 1)−dimensional evolution equation,

ut = F (x, t, u, ux, u2x, . . . , unx ),

(2)

of order n in the independent space variable x and of first order in time t. Obviously, the dependent variable is u(x, t). If parameters are present in (2), they will be denoted by lower-case Greek letters. A conservation law of (2) is of the form Dt ρ + Dx J = 0,

(3)

24

W. Hereman, P. J. Adams, H. L. Eklund, M. S. Hickman and B. M. Herbst

which is satisfied for all solutions u(x, t) of the PDE. In physics, ρ is called the conserved density (or charge); J is the associated flux (or current). In general, both are differential functions (functionals), i.e. functions of x, t, u, and partial derivatives of u with respect to x. In (3), Dx denotes the total derivative with respect to x, that is, N

∂J X ∂J Dx J = + u , ∂x ∂ukx (k+1)x

(4)

k=0

where N is the order of J, and Dt is the total derivative with respect to t, defined by M

Dt ρ =

∂ρ ∂ρ X ∂ρ k + ρ0[ut ] = + D ut , ∂t ∂t ∂ukx x

(5)

k=0

where ρ0 [ut] is the Fr´echet derivative of ρ in the direction of ut and M is the order of ρ. The densities ρ(1) = u and ρ(2) = u2 of (1) were long known. In 1965, Whitham [98] had found a third density, ρ(3) = u3 − α3 u2x , which, in the context of water waves, corresponds to Boussinesq’s moment of instability [76]. One can readily verify that   1 2 Dt (u) + Dx (6) αu + u2x = 0, 2   2 3 Dt (u2) + Dx (7) αu − u2x + 2uu2x = 0, 3     3 2 3 4 3 2 6 3 2 2 Dt u − ux +Dx αu −6uux +3u u2x + u2x − ux u3x = 0. (8) α 4 α α Indeed, (6) is the KdV equation written as a conservation law; (7) is obtained after multiplying (1) by 2u; (8) requires more work. Hence, the first three density-flux pairs of (1) are 1 2 αu + u2x, 2 2 = u2 , J (2) = αu3 − u2x + 2uu2x, 3 3 3 3 6 = u3 − u2x , J (3) = αu4 −6uu2x +3u2u2x + u22x − ux u3x. α 4 α α

ρ(1) = u,

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ρ(2) ρ(3)

J (1) =

(9) (10) (11)

Integrals of motion readily follow from the densities. Indeed, assuming that J vanishes at infinity (for example due to sufficiently fast decay of u and its x derivatives), upon integration of (3) with respect to x one obtains that P =

Z∞

ρ dx

(12)

−∞

is constant in time. Such constants of motion also arise when u is periodic, in which case one integrates over the finite period. Depending on the physical setting, the first few

Direct Methods and Symbolic Software for Conservation Laws

25

constants of motion (i.e. integrals (12)) express conservation of mass, momentum, and energy. Martin Kruskal and postdoctoral fellow Norman Zabusky discovered the fourth and fifth densities for the KdV equation [111]. However, they failed in finding a sixth conservation law due to an algebraic mistake in their computations. Kruskal asked Robert Miura, also postdoctoral fellow at the Princeton Plasma Physics Laboratory at New Jersey, to search for further conservation laws of the KdV equation. Miura [78] computed the seventh conservation law. After correcting the mistake mentioned before, he also found the sixth and eventually three additional conservation laws. Rumor [80] has it that in the summer of 1966 Miura went up into the Canadian Rockies and returned from the mountains with the first 10 conservation laws of the KdV equation engraved in his notebook. This biblical metaphor probably does not do justice to Miura’s intense and tedious work with pen and paper. With ten conservation laws in hand, it was conjectured that the KdV equation had an infinite sequence of conservation laws, later proven to be true [67], [79]. Aficionados of explicit formulas can find the first ten densities (and seven of the associated fluxes) in [79] and the eleventh density (with 45 terms) in [67], where a recursion formula is given to generate all further conserved densities. As an aside, in 1966 the first five conserved densities were computed on an IBM 7094 computer with FORMAC, an early CAS. The sixth density could no longer be computed because the available storage space was exceeded. In contrast, using a method of undetermined coefficients, the first eleven densities were computed in 1969 on a AEC CDC-6600 computer in a record time of 2.2 seconds. Due to limitations in handling large integers, the computer could not correctly produce any further densities. Undoubtedly, the discovery of conservation laws played a pivotal role in the comprehensive study of the properties and solutions of nonlinear completely integrable PDEs (like the KdV equation) and the development of the IST (see e.g. [80] for the history). Clifford Gardner, John Greene, Martin Kruskal, and Robert Miura received the 2006 Leroy P. Steele Prize [115], awarded by the American Mathematical Society, for their seminal contribution to research on the KdV equation. In turn, Martin Kruskal has received numerous honors and awards [114] for his fundamental contributions to the understanding of integrable systems and soliton theory. This chapter is dedicated to Martin Kruskal (1925–2006).

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3

The Method of Undetermined Coefficients

We now sketch the method of undetermined coefficients to compute conservation laws [38], [96], which draws on ideas and observations in before mentioned work by Kruskal and collaborators.

3.1

Dilation Invariance of Nonlinear PDEs

Crucial to the computation of conservation laws is that (3) must hold on the PDE. This is achieved by substituting ut (and utx , utxx, etc.) from (1) in the evaluation of (6)–(8) and in all subsequent conservation laws of degree larger than 3. The elimination of all t−derivatives of u in favor of x derivatives has two important consequences: (i) any symmetry of the PDE, in particular, the dilation symmetry, will be adopted by the conservation

26

W. Hereman, P. J. Adams, H. L. Eklund, M. S. Hickman and B. M. Herbst

law, (ii) once Dt is computed and evaluated on the PDE, t becomes a parameter in the computation of the flux. We will first investigate the dilation (scaling) symmetry of evolution equations. The KdV equation is dilation invariant under the scaling symmetry (t, x, u) → (λ−3 t, λ−1x, λ2u),

(13)

where λ is an arbitrary parameter. Indeed, after a change of variables with t˜ = λ−3 t, x ˜= λ−1 x, u ˜ = λ2 u, and cancellation of a common factor λ5, the KdV for u ˜(˜ x, ˜ t) arises. The dilation symmetry of (1) can be expressed as u∼

∂2 , ∂x2

∂ ∂3 ∼ , ∂t ∂x3

(14)

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which means that u corresponds to two x−derivatives and the time derivative corresponds to three x−derivatives. If we define the weight, W, of a variable (or operator) as the exponent ∂ ∂ of λ in (13), then W (x) = −1 or W ( ∂x ) = 1; W (t) = −3 or W ( ∂t ) = 3, and W (u) = 2. All weights of dependent variables and the weights of ∂/∂x, ∂/∂t, are assumed to be non-negative and rational. The rank of a monomial is defined as the total weight of the monomial. Such monomials may involve the independent and dependent variables and the ∂ ∂ operators ∂x , Dx, ∂t , and Dt . Ranks must be positive integers or positive rational numbers. An expression (or equation) is uniform in rank if its monomial terms have equal rank. For example, (1) is uniform in rank since each of the three terms has rank 5. Conversely, if one does not know the dilation symmetry of (1), then it can be readily computed by requiring that (1) is uniform in rank. Indeed, setting W (∂/∂x) = 1 and equating the ranks of the three terms in (1) gives   ∂ W (u) + W = 2W (u) + 1 = W (u) + 3, (15) ∂t which yields W (u) = 2, W (∂/∂t) = 3, and, in turn, confirms (13). So, requiring uniformity in rank of a PDE allows one to compute the weights of the variables (and thus the scaling symmetry) with linear algebra. Dilation symmetries, which are special Lie-point symmetries, are common to many nonlinear PDEs. Needless to say, not every PDE is dilation invariant, but non-uniform PDEs can be made uniform by extending the set of dependent variables with auxiliary parameters with appropriate weights. Upon completion of the computations one can set these parameters to one. In what follows, we set W (∂/∂x) = W (Dx ) = 1 and W (∂/∂t) = W (Dt ). Applied to (6), rank ρ(1) = 2, rank J (1) = 4. Hence, rank (Dt ρ(1)) = rank (Dx J (1)) = 5. Therefore, (6) is uniform of rank 5. In (7), rank ρ(2) = 4 and rank J (2) = 6, consequently, (7) is uniform of rank 7. In (11), each term in ρ(3) has rank 6 and each term in J (3) has rank 8. Consequently, rank (Dt ρ(3)) = rank (Dx J (3) ) = 9, which makes (8) is uniform of rank 9. All densities of (1) are uniform in rank and so are the associated fluxes and the conservation laws. Equation (1) also has density-flux pairs that depend explicitly on t and x; for example,     2 2 2 3 2 2 2 2 ˜ ρ˜ = tu + xu, J = t αu − ux + 2uu2x − x u − u2x + ux . (16) α 3 α α

Direct Methods and Symbolic Software for Conservation Laws

27

Since W (x) = −1 and W (t) = −3, one has rank ρ˜ = 1, and rank J˜ = 3. The methods and algorithms discussed in subsequent sections have been adapted to compute densities and fluxes explicitly dependent on x and t. Instead of addressing this issue in this chapter, we refer the reader to [38], [53].

3.2

The Method of Undetermined Coefficients Applied to a Scalar Nonlinear PDE

We outline how densities and fluxes can be constructed for a scalar evolution equation (2). To keep matters transparent, we illustrate the steps for the KdV equation resulting in ρ(3) of rank R = 6 with associated flux J (3) of rank 8, both listed in (11). The tools needed for the computations will be presented in the next section. • Select the rank R of ρ. Make a list, R, of all monomials in u and its x-derivatives so that each monomial has rank R. This can be done as follows. Starting from the set V of dependent variables (including parameters with weight, when applicable), make a set M of all non-constant monomials of rank R or less (but without x−derivatives). Next, for each term in M, introduce the right number of x−derivatives to adjust the rank of that term. Distribute the x−derivatives, strip off the numerical coefficients, and gather the resulting terms in a set R. For the KdV equation and R = 6, V = {u} and M = {u3 , u2, u}. Since u3 , u2, and u have ranks 6, 4 and 2, respectively, one computes ∂ 0 u3 ∂ 2 u2 ∂ 4u 3 2 = u , = 2u + 2uu , = u4x. (17) 2x x ∂x0 ∂x2 ∂x4 Ignoring numerical coefficients in the right hand sides of the equations in (17), one gets R = {u3 , u2x, uu2x, u4x}. • Remove from R all monomials that are total x−derivatives. Also remove all “equivalent” monomials, i.e. the monomials that differ from another by a total x−derivative, keeping the monomial of lowest order. Call the resulting set S. In our example, u4x must be removed (because u4x = Dx u3x) and uu2x must be removed since uu2x and u2x are equivalent. Indeed, uu2x = Dx(uux ) − u2x . Thus, S = {u3 , u2x}.

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• Linearly combine the monomials in S with constant undetermined coefficients ci to obtain the candidate ρ. Continuing with the example, ρ = c1 u3 + c2 u2x ,

(18)

which is of first order in x. • Using (5), compute Dt ρ. Applied to (18) where M = 1, one gets Dt ρ = (3c1u2 I + 2c2ux Dx )[ut].

(19)

As usual, D0x = I is the identity operator. • Evaluate −Dt ρ on the PDE (2) by replacing ut by F. The result is a differential function E in which t is a parameter. For the KdV equation (1), F = −(αuux +u3x). After reversing the sign, the evaluated form of (19) is E = (3c1u2 I + 2c2ux Dx )(αuux + u3x ) = 3c1 αu3 ux + 2c2αu3x + 2c2αuux u2x + 3c1 u2u3x + 2c2 ux u4x.

(20)

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W. Hereman, P. J. Adams, H. L. Eklund, M. S. Hickman and B. M. Herbst

• To obtain a conservation law, E must be a total derivative. Starting with highest orders, repeatedly integrate E by parts. Doing so, allows one to write E as the sum of a total x−derivative, Dx J, and a non-integrable part (i.e. the obstructing terms). J is the (candidate) flux with rank J = R + W (Dt ) − 1. Integration by parts of (20) gives E = Dx



 3 4 2 2 c1αu + 3c1u u2x + c2αuux + 2c2ux u3x − 6c1uux u2x 4

−2c2u2x u3x + c2 αu3x   3 4 2 2 2 2 = Dx c1αu − 3c1uux + c2 αuux + 3c1u u2x + 2c2ux u3x − c2u2x 4 +(3c1 + c2α)u3x .

(21)

The candidate flux therefore is J=

3 c1 αu4 − 3c1uu2x + c2αuu2x + 3c1u2 u2x + 2c2ux u3x − c2 u22x. 4

(22)

• Equate the coefficients of the obstructing terms to zero. Solve the linear system for the undetermined coefficients ci . In the example (3c1 + c2α)u3x is the only obstructing term which vanishes for c2 = − α3 c1, where c1 is arbitrary. • Substitute the coefficients ci into ρ and J to obtain the final forms of the density and associated flux (with a common arbitrary factor which can be set to 1). Setting c1 = 1 and substituting c2 = − α3 into (18) and (22) yields ρ(3) and J (3) as given in (11).

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Constructing “minimal” densities, i.e. densities which are free of equivalent terms and total derivatives terms, becomes challenging if the rank R is high. Furthermore, integration by parts is cumbersome and prone to mistakes if done by hand. Moreover, it would be advantageous if the integration by parts could be postponed until the undetermined coefficients ci have been computed and substituted in E. Ideally, the computations of the density and the flux should be decoupled. There is a need for computational tools to address these issues, in particular, if one wants to compute conservation laws of systems of evolution equations.

4

Tools from the Calculus of Variations and Differential Geometry

A scalar differential function E of order M is called exact (integrable) if and only if there exists a scalar Rdifferential function J of order M − 1 such that E = Dx J. Obviously, J = D−1 E dx is then the primitive (or integral) of E. Two questions arise: (i) How x E = can one test whether or not E is exact? (ii) If E is exact, how can one compute J without using standard integration by parts? To answer the first question we will use the variational derivative (Euler operator) from the calculus of variations. To perform integration by parts we will use the homotopy operator from differential geometry.

Direct Methods and Symbolic Software for Conservation Laws

4.1

29

The Continuous Variational Derivative (Euler Operator) (0)

The continuous variational derivative, also called the Euler operator of order zero , Lu(x) , for variable u(x) is defined [81] by (0)

Lu(x) E =

M X ∂E (−Dx )k ∂ukx k=0

=

∂E ∂E ∂E ∂E ∂E − Dx + D2x − D3x + · · · + (−1)M DM , (23) x ∂u ∂ux ∂u2x ∂u3x ∂uM x

where E is a differential function in u(x) of order M. A necessary and sufficient condition for a differential function E to be exact is that (0) (0) Lu(x) E ≡ 0. A proof of this statement is given in e.g. [67]. If Lu(x) E 6= 0, then E is not a total x−derivative due to obstructing terms. Application 1. Returning to (20), we now use the variational derivative to determine c1 and c2 so that E of order M = 4 will be exact. Using nothing but differentiations, we readily compute (0)

∂E ∂E ∂E ∂E ∂E − Dx + D2x − D3x + D4x ∂u ∂ux ∂u2x ∂u3x ∂u4x 2 = 9c1αu ux + 2c2αux u2x + 6c1uu3x − Dx(3c1αu3 + 6c2αu2x

Lu(x) E =

+2c2αuu2x + 2c2u4x) + D2x(2c2αuux ) − D3x(3c1u2 ) + D4x (2c2ux ) = (9c1αu2 ux + 2c2 αux u2x + 6c1uu3x ) − (9c1αu2 ux + 14c2αux u2x +2c2αuu3x + 2c2u5x) + (6c2αux u2x + 2c2αuu3x ) −(18c1ux u2x + 6c1uu3x) + (2c2u5x ) = −6(3c1 + c2α)ux u2x.

(24) (0)

Note that the terms in u2 ux , uu3x, and u5x dropped out. Hence, requiring that Lu(x) E ≡ 0

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leads to 3c1 +c2α = 0. Substituting c1 = 1, c2 = − α3 , into (18) yields ρ(3) in (11). Application 2. It is paramount that the candidate density is free of total x−derivatives and equivalent terms. If such terms were present, they could be moved into the flux J, and their coefficients ci would be arbitrary. ρ(1) and ρ(2), are equivalent if and only if ρ(1) + kρ(2) = Dx J, for some J and non-zero scalar k. We write ρ(1) ≡ ρ(2). Clearly ρ is equivalent to any non-zero multiple of itself and ρ ≡ 0 if and only if ρ is exact. Instead of working with different densities, we investigate the equivalence of terms ti in the same density. For example, returning to the set R = {u3 , u2x, uu2x, u4x}, terms t2 = u2x and t3 = uu2x are equivalent because t3 + t2 = uu2x + u2x = Dx (uux ). The variational derivative can be used to detect equivalent and exact terms. Indeed, note (0) (0) that v1 = Lu(x) (uu2x) = 2u2x and v2 = Lu(x) u2x = −2u2x are linearly dependent. Also, (0)

for t4 = u4x = Dx u3x one gets v3 = Lu(x) u4x = 0. To weed out the terms ti in R that are equivalent or total derivatives, it suffices to check the linear independence of their images vi under the Euler operator. One can optimize this procedure by starting from a set R where some of the equivalent and total derivatives terms have been removed a priori. Indeed, in view of (17), one can

30

W. Hereman, P. J. Adams, H. L. Eklund, M. S. Hickman and B. M. Herbst

ignore the highest-order terms (typically the last terms) in each of the right hand sides. Therefore, R = {u3, u2x} and, for this example, no further reduction would be necessary. Various algorithms are possible to construct minimal densities. Details are given in [38], [51].

4.2

The Continuous Homotopy Operator

We now discuss the homotopy operator R [12], [13], [52], [81] which will allow one to reduce the computation of J = D−1 E = E dx to a single integral with respect to an auxiliary x variable denoted by λ (not to be confused with λ in Section 3.1). Hence, the homotopy operator circumvents integration by parts and reduces the inversion of Dx to a problem of single-variable calculus. The homotopy operator [81, p. 372] for variable u(x), acting on an exact expression E of order M, is given by Z1 dλ Hu(x) E = (Iu E) [λu] , (25) λ 0

where the integrand Iu E is given by Iu E =

M k−1 X X k=1

i=0

uix (−Dx )k−(i+1)

!

∂E . ∂ukx

(26)

In (25), (Iu E)[λu] means that in Iu E one replaces u → λu, ux → λux, etc. This is a special case of the homotopy, λ(u(1) − u(0)) + u(0) , between two points, u(0) = (u0 , u0x, u02x, . . ., u0M x) and u(1) = (u1, u1x, u12x, . . . , u1M x), in the jet space. For our purposes we set u(0) = (0, 0, . . ., 0) and u(1) = (u, ux, u2x, · · · , uM x).

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Formula (26) is equivalent to the one in [52], which in turn is equivalent to the formula in terms of higher Euler operators [45], [51]. R Given an exact differential function E of order M one has J = D−1 E dx = x E = Hu(x) E. A proof of this statement can be found in [52]. Application. After substituting c1 = 1 and c2 = − α3 into (20) we obtain the exact expression 6 E = 3αu3 ux − 6u3x − 6uuxu2x + 3u2u3x − ux u4x , (27) α of order M = 4. First, using (26), we compute ! 4 k−1 X X ∂E Iu E = uix (−Dx )k−(i+1) ∂ukx k=1 i=0       ∂E ∂E ∂E + (uxI − uDx ) + (u2xI − ux Dx + uD2x ) = (uI) ∂ux ∂u2x ∂u3x   ∂E +(u3xI − u2x Dx + ux D2x − uD3x) . (28) ∂u4x

Direct Methods and Symbolic Software for Conservation Laws After substitution of (27), one gets   6 3 2 Iu E = (uI) 3αu + 18ux − 6uu2x − u4x + (ux I − uDx )(−6uux) α +(u2xI − ux Dx +

uD2x )(3u2)

= 3αu4 − 18uu2x + 9u2 u2x +

+ (u3xI − u2x Dx +

ux D2x

−

uD3x )

31

  6 − ux α

6 2 12 u − ux u3x, α 2x α

(29)

which has the correct terms of J (3) but incorrect coefficients. Finally, using (25), Z1 dλ J = Hu(x) E = (Iu E)[λu] λ 0

 Z1  6 2 12 3 4 2 2 2 2 = 3αλ u − 18λ uux + 9λ u u2x + λu2x − λuxu3x dλ α α 0

3 3 6 = αu4 − 6uu2x + 3u2u2x + u22x − ux u3x, 4 α α

(30)

which matches J (3) in (11). The crux of the homotopy operator method [12], [13], [26], [81] is that the integration by parts of a differential expression like (27), which involves an arbitrary function u(x) and its x−derivatives, can be reduced to a standard integration of a polynomial in λ.

5

Conservation Laws of Nonlinear Systems of Polynomial PDEs

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Thus far we have dealt with the computation of density-flux pairs of scalar evolution equations, with the KdV equation as the leading example. In this section we show how the method and tools can be generalized to cover systems of evolution equations. We will use the Drinfel’d-Sokolov-Wilson system and the Boussinesq equation to illustrate the steps.

5.1

Tools for Systems of Evolution Equations

For differential functions (like densities and fluxes) of two dependent variables (u, v) and their x−derivatives, the total derivatives are M

Dt ρ =

Dx J

=

M

1 2 X ∂ρ X ∂ρ k ∂ρ k Dx ut + D vt , + ∂t ∂ukx ∂vkx x

∂J + ∂x

k=0 N1 X k=0

k=0 N2 X

∂J u + ∂ukx (k+1)x

k=0

∂J v , ∂vkx (k+1)x

(31)

(32)

where M1 and M2 are the (highest) orders of u and v in ρ, and N1 and N2 are the (highest) orders of u and v in J.

32

W. Hereman, P. J. Adams, H. L. Eklund, M. S. Hickman and B. M. Herbst (0)

(0)

To accommodate two dependent variables, we need Euler operators Lu(x) and Lv(x) for each dependent variable separately. For brevity, we will use vector notation, that is, (0) (0) (0) Lu(x)E = (Lu(x) E, Lv(x)E). Likewise, the homotopy operator in (25) must be replaced by Z1 dλ Hu(x)E = (Iu E + Iv E)[λu] , (33) λ 0

where Iu E =

M1 k−1 X X k=1

and Iv E =

!

i=0

M2 k−1 X X k=1

uix (−Dx )

k−(i+1)

i=0

vix (−Dx )k−(i+1)

!

∂E , ∂ukx

(34)

∂E , ∂vkx

(35)

where M1 , M2 are the orders of E in u, v, respectively. In (33), u → λu, ux → λux , . . ., v → λv, vx → λvx, etc.

5.2

The Drinfel’d–Sokolov–Wilson System: Dilation Invariance and Conservation Laws

We consider a parameterized family of the Drinfel’d–Sokolov–Wilson (DSW) equations

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ut + 3vvx = 0,

vt + 2uvx + αux v + 2v3x = 0,

(36)

where α is a nonzero parameter. The system with α = 1 was first proposed by Drinfel’d and Sokolov [31], [32] and Wilson [99]. It can be obtained [59] as a reduction of the Kadomtsev-Petviashvili equation (i.e. a two-dimensional version of the KdV equation) and is a completely integrable system. In [109], Yao and Li computed conservation laws of (36), where they had introduced four arbitrary coefficients. Using scales on x, t, u and v, all but one coefficients in (36) can be scaled to any real number. Therefore, to cover the entire family of DSW equations it suffices to leave one coefficient arbitrary, e.g. α in front of ux v. To compute the dilation symmetry of (36), we assign weights, W (u) and W (v), to both dependent variables and express that each equation separately must be uniform in rank (i.e. the ranks of the equations in (36) may differ from each other). For the DSW equations (36), one has   ∂ W (u) + W = 2W (v) + 1, ∂t   ∂ W (v) + W = W (u) + W (v) + 1 = W (v) + 3, (37) ∂t which yields W (u) = W (v) = 2, W (∂/∂t) = 3. The DSW system (36) is thus invariant under the scaling symmetry (x, t, u, v) → (λ−1x, λ−3t, λ2u, λ2v),

(38)

Direct Methods and Symbolic Software for Conservation Laws

33

where λ is an arbitrary scaling parameter. The first three density-flux pairs for the DSW equations (36) are 3 2 v , 2 ρ(2) = v, J (2) = 2(uv + v2x ), if α = 2, 3 ρ(3) = (α − 1)u2 + v 2, J (3) = 3(αuv 2 − vx2 + 2vv2x), 2 ρ(1) = u, J (1) =

(39) (40) (41)

Both ρ(1) and ρ(2) have rank 2; their fluxes have rank 4. The pair (ρ(1), J (1)) exists for any α, whereas (ρ(2), J (2)) only exists if α = 2. Density ρ(3) of rank 4 and flux J (3) of rank 6 are valid for any α. At rank R = 6, ρ(4) = (α + 1)(α − 2)u3 −

9 3 27 2 (α + 1)uv 2 − (α − 2)u2x − v . 2 2 2 x

(42)

The corresponding flux (not shown) has 7 terms. At rank R = 8, ρ(5) = u4 −

9 2 2 27 4 9 2 3 2 45 81 2 u v − v − uux + u2x + vux vx + 27uvx2 − v , 2 8 2 4 2 4 2x

(43)

provided α = 1. The corresponding flux (not shown) has 15 terms. There exists a densityflux pair for all even ranks R ≤ 10 provided α = 1, for which (36) is completely integrable.

5.3

Computation of a Conservation Law of the Drinfel’d–Sokolov Wilson System

To illustrate how the presence of a parameter, like α, affects the computation of densities, we compute ρ(1) and ρ(2) of rank R = 2 given in (39) and (40). Step 1: Construct the form of the density The set of dependent variables is V = {u, v}. Both elements are of rank 2 so, no x-derivatives are needed. Thus, M = R = S = {u, v}. Linearly combining the elements in S gives ρ = c1u + c2v. Step 2: Compute the undetermined coefficients ci

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Evaluating E = −Dt ρ = −(c1ut + c2vt ) on (36), yields E = 3c1vvx + c2(2uvx + αux v + 2v3x), (0)

(0)

(44)

(0)

which will be exact if Lu(x) E = (Lu(x) E, Lv(x)E) ≡ (0, 0). Since E is of order M1 = 1 in u and order M2 = 3 in v, (0)

Lu(x) E =

∂E ∂E = 2c2vx − Dx (c2αv) = (2 − α)c2vx , − Dx ∂u ∂ux

(45)

and (0)

∂E ∂E ∂E ∂E + D2x − D3x − Dx ∂v ∂vx ∂v2x ∂v3x = 3c1vx + c2αux − Dx (3c1v + 2c2u) − D3x (2c2) = (α − 2)c2ux.

Lv(x) E =

(46)

34

W. Hereman, P. J. Adams, H. L. Eklund, M. S. Hickman and B. M. Herbst

Both (45) and (46) will vanish identically if and only if (α − 2)c2 = 0. This equation (with unknowns c1 and c2 ) is parameterized by α 6= 0. The solution algorithm [38] considers all branches of the solution and possible compatibility conditions. Setting c1 = 1, leads to either (i) c2 = 0 if α 6= 2, or (ii) c2 arbitrary if α = 2. Setting c2 = 1 leads to the compatibility condition, α = 2, and c1 arbitrary. Substituting the solutions into ρ = c1u + c2v gives ρ = u which is valid for any α; and ρ = u + c2v or ρ = c1u + v provided α = 2. In other words, ρ(1) = u is the only density of rank 2 for arbitrary values of α. For α = 2 there exist two independent densities, ρ(1) = u and ρ(2) = v. Step 3: Compute the associated flux J As an example, we compute the flux in (40) associated with ρ(2) = v and α = 2. In this case, c1 = 0, c2 = 1, for which (44) simplifies into E = 2(uvx + ux v + v3x ),

(47)

which is of order M1 = 1 in u and order M2 = 3 in v. Using (34) and (35), we obtain Iu E = (uI)

∂E = (uI)(2v) = 2uv, ∂ux

(48)

and    ∂E ∂E + (v2xI − vx Dx + vD2x ) ∂v2x ∂v3x
2 = (vI)(2u) + v2xI − vx Dx + vDx (2)

Iv E = (vI)



∂E ∂vx



+ (vx I − vDx )



= 2uv + 2v2x .

(49)

Hence, using (33),

J = Hu(x) E =

Z1

dλ (Iu E + Iv E)[λu] = λ

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0

Z1

(4λuv + 2v2x) dλ = 2(uv + v2x ),

(50)

0

which is J (2) in (40). The integration of (47) could easily be done by hand. The homotopy operator method pays off if the expression to be integrated has a large number of terms.

5.4

The Boussinesq Equation: Dilation Invariance and Conservation Laws

The wave equation, u2t − u2x + 3u2x + 3uu2x + αu4x = 0,

(51)

for u(x, t) with real parameter α, was proposed by Boussinesq to describe surface waves in shallow water [2]. For what follows, we rewrite (51) as a system of evolution equations, ut + vx = 0,

vt + ux − 3uux − αu3x = 0,

where v(x, t) is an auxiliary dependent variable.

(52)

Direct Methods and Symbolic Software for Conservation Laws

35

The Boussinesq system (52) is not uniform in rank because the terms ux and αu3x lead to an inconsistent system of weight equations. To circumvent the problem we introduce an auxiliary parameter β with (unknown) weight, and replace (52) by ut + vx = 0,

vt + βux − 3uux − αu3x = 0.

(53)

Requiring uniformity in rank, we obtain (after some algebra) W (u) = 2,

W (v) = 3,

W (β) = 2,

W



∂ ∂t



= 2.

(54)

Therefore, (53) is invariant under the scaling symmetry (x, t, u, v, β) → (λ−1x, λ−2t, λ2u, λ3v, λ2β).

(55)

As the above example shows, a PDE that is not dilation invariant can be made so by extending the set of dependent variables with one or more auxiliary parameters with weights. Upon completion of the computations one can set each of these parameters equal to 1. The Boussinesq equation (51) has infinitely many conservation laws and is completely integrable [2], [7]. The first four density-flux pairs [8] for (53) are ρ(1) = u, J (1) = v, 3 ρ(2) = v, J (2) = βu − u2 − αu2x , 2 1 2 1 1 (3) (3) rho = uv, J = βu − u3 + v 2 + αu2x − αuu2x , 2 2 2 (4) 2 3 2 2 ρ = βu − u + v + αux , J (4) = 2βuv − 3u2 v + 2αux vx − 2αu2xv.

(56) (57) (58)

(59)

These densities are of ranks 2, 3, 5 and 6, respectively. The corresponding fluxes are of one rank higher. After setting β = 1 we obtain the conserved quantities of (52) even though initially this system was not uniform in rank.

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5.5

Computation of a Conservation Law for the Boussinesq System

We show the computation of ρ(4) and J (4) of ranks 6 and 7, respectively. The presence of the auxiliary parameter β with weight complicates matters. At a fixed rank R, conserved densities corresponding to lower ranks might appear in the result. These lower-rank densities are easy to recognize for they are multiplied with arbitrary coefficients ci . Consequently, when parameters with weight are introduced, the densities corresponding to distinct ranks are no longer linearly independent. As the example below will show, densities must be split into independent pieces. Step 1: Construct the form of the density Augment the set of dependent variables with the parameter β (with non-zero weight). Hence, V = {u, v, β}. Construct M = {β 2 u, βu2, βu, βv, u3, u2, u, v 2, v, uv}, which contains all non-constant monomials of (chosen) rank 6 or less (without derivatives). Next,

36

W. Hereman, P. J. Adams, H. L. Eklund, M. S. Hickman and B. M. Herbst

for each term in M, introduce the right number of x-derivatives so that each term has rank 6. For example, ∂ 2 βu ∂ 2 u2 = βu , = 2u2x + 2uu2x, 2x ∂x2 ∂x2

∂ 4u ∂(uv) = vux + uvx , etc.. (60) = u4x , 4 ∂x ∂x

Gather the terms in the right hand sides of the equations in (60) to get  R = β 2 u, βu2, u3, v 2, vux, u2x, βvx , uvx, βu2x, uu2x, v3x, u4x .

(61) (0)

Using (23) and a similar formula for v, for every term ti in R we compute vi = Lu(x)ti = (0)

(0)

(Lu(x) ti , Lv(x) ti ). If vi = (0, 0) then ti is discarded and so is vi . If vi 6= (0, 0) we verify whether or not vi is linearly independent of the non-zero vectors vj , j = 1, 2, · · · , i − 1. If independent, the term ti is kept, otherwise, ti is discarded and so is vi . (0) Upon application of Lu(x) , the first six terms in R lead to linearly independent vectors v1 through v6 . Therefore, t1 through t6 are kept (and so are the corresponding vectors). (0) For t7 = βvx we compute v7 = Lu(x)(βvx ) = (0, 0). So, t7 is discarded and so is v7. For (0)

t8 = uvx we get v8 = Lu(x)(uvx ) = (vx , −ux) = −v5. So, t8 is discarded and so is v8 . Proceeding in a similar fashion, t9 , t10, t11 and t12 are discarded. Thus, R is replaced by  S = β 2 u, βu2, u3, v 2, vux, u2x , (62) which is free of divergences and divergence-equivalent terms. Ignoring the highest-order terms (typically the last terms) in each of the right hand sides of the equations in (60) optimizes the procedure. Indeed, R would have had six instead of twelve terms. Coincidentally, in this example no further eliminations would be needed to obtain S. Next, linearly combine the terms in S to get ρ = c1 β 2 u + c2 βu2 + c3u3 + c4v 2 + c5vux + c6u2x .

(63)

Step 2: Compute the undetermined coefficients ci

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Compute Dtρ. Here, ρ is of order M1 = 1 in u and order M2 = 0 in v. Hence, application of (31) gives ∂ρ ∂ρ ∂ρ Iut + Dx ut + Ivt ∂u ∂ux ∂v = (c1β 2 +2c2βu+3c3u2 )ut +(c5v+2c6ux )utx +(2c4v+c5 ux )vt.

Dt ρ =

(64)

Use (53) to eliminate ut , utx, and vt . Then, E = −Dt ρ evaluates to E = (c1β 2 + 2c2βu + 3c3u2 )vx + (c5v + 2c6 ux )v2x +(2c4v + c5 ux )(βux − 3uux − αu3x ), (0)

(0)

(65)

(0)

which must be exact. Thus, require that Lu(x)E = (Lu(x) E, Lv(x)E) ≡ (0, 0). Group like terms. Set their coefficients equal to zero to obtain the parameterized system β(c2 − c4 ) = 0, c3 + c4 = 0, c5 = 0, αc5 = 0, βc5 = 0, αc4 − c6 = 0,

(66)

Direct Methods and Symbolic Software for Conservation Laws

37

where α 6= 0 and β 6= 0. Investigate the eliminant of the system. Set c1 = 1 and obtain the solution c1 = 1, c2 = c4, c3 = −c4, c5 = 0, c6 = αc4 , (67) which holds without condition on α and β. Substitute (67) into (63) to get ρ = β 2 u + c4(βu2 − u3 + v 2 + αu2x ).

(68)

The density must be split into independent pieces. Indeed, since c4 is arbitrary, set c4 = 0 or c4 = 1, thus splitting (68) into two independent densities ρ = β 2 u ≡ u,

ρ = βu2 − u3 + v 2 + αu2x ,

(69)

which are ρ(1) and ρ(4) in (56)–(59). Step 3: Compute the flux J Compute the flux corresponding to ρ in (69). Substitute (67) into (65). Take the terms in c4 and set c4 = 1. Thus, E = 2βuvx + 2βvux − 3u2 vx − 6uvux + 2αux v2x − 2αvu3x,

(70)

which is of order M1 = 3 in u and order M2 = 2 in v. Using (34) and (35), one readily obtains Iu E = 2βuv − 6u2 v + 2αux vx − 2αu2x v, (71) and Iv E = 2βuv − 3u2 v + 2αux vx − 2αu2x v.

(72)

Hence, using (33), Z1 dλ J = Hu(x) E = (Iu E + Iv E)[λu] λ 0

=

Z1


4βλuv − 9λ2u2 v + 4αλux vx − 4αλu2xv dλ

0

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= 2βuv − 3u2 v + 2αux vx − 2αu2x v,

(73)

which is J (4) in (59). One can set β = 1 at the end of the computations.

6

Conservation Laws of Systems of PDEs with Transcendental Nonlinearities

We now turn to the symbolic computation of conservation laws of certain classes of PDEs with transcendental nonlinearities. We only consider PDEs where the transcendental functions act on one dependent variable u (and not on x−derivatives of u). In contrast to the examples in the previous sections, the candidate density will no longer have constant undetermined coefficients but functional coefficients which depend on the variable u. Furthermore, we consider only PDEs which have one type of nonlinearity. For example, sine, or cosine, or exponential terms are fine but not a mixture of these functions.

38

6.1

W. Hereman, P. J. Adams, H. L. Eklund, M. S. Hickman and B. M. Herbst

The sine-Gordon Equation: Dilation Invariance and Conservation Laws

The sine-Gordon (sG) equation appears in the literature [17], [69] in two different ways: • In light-cone coordinates the sG equation, uxt = sin u, has a mixed derivative term, which complicates matters. We return to this type of equation in Section 7.1. • The sG equation in laboratory coordinates , u2t − u2x = sin u, can be recast as ut + v = 0,

vt + u2x + sin u = 0,

(74)

where v(x, t) is an auxiliary variable. System (74) is amenable to our approach, subject to modifications to accommodate the transcendental nonlinearity. The sG equation describes the propagation of crystal dislocations, superconductivity in a Josephson junction, and ultra-short optical pulse propagation in a resonant medium [69]. In mathematics, the sG equation is long known in the differential geometry of surfaces of constant negative Gaussian curvature [30], [80]. The sine-Gordon equation (74) is not uniform in rank unless we replace it by ut + v = 0,

vt + u2x + α sin u = 0,

(75)

where α is a real parameter with weight. Indeed, substituting the Maclaurin series, sin u = 3 5 u − u3! + u5! − · · · , and requiring uniformity in rank yields   ∂ W (u) + W = W (v), ∂t   ∂ W (v)+W = W (u)+2 = W (α)+W (u) ∂t =W (α)+3W (u) = W (α)+5W (u) = · · · (76) This forces us to set W (u) = 0 and W (α) = 2. Consequently, (75) is scaling invariant under the symmetry

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(x, t, u, v, α) → (λ−1x, λ−1t, λ0u, λ1v, λ2α),

(77)

corresponding to W (∂/∂x) = W (∂/∂t) = 1, W (u) = 0, W (v) = 1, W (α) = 2. The first and second equations in (75) are uniform of ranks 1 and 2, respectively. The first few (of infinitely many) density-flux pairs [8], [29] for the sG equation (75) are ρ(1) = 2α cos u + v 2 + u2x , J (1) = 2vux , ρ

(2)

ρ

(3)

ρ

(4)

= 2vux , J

(2)

2

= −2α cos u + v + 3

= 6αvux cos u + v ux + 2

2

2

vu3x

(78)

u2x ,

(79)

− 8vxu2x ,

2

2

(80) 4

= 2α cos u − 2α sin u + 4αv cos u + v + +6v 2u2x + u4x − 16vx2 − 16u22x,

20αu2x cos u (81)

J (3) and J (4) are not shown due to length. Again, all densities and fluxes are uniform in rank (before α is set equal to 1).

Direct Methods and Symbolic Software for Conservation Laws

6.2

39

Computation of a Conservation Law for the sine-Gordon System

We show how to compute densities ρ(1) and ρ(2), both of rank 2, and their associated fluxes J (1) and J (2) . The candidate density will no longer have constant undetermined coefficients ci but functional coefficients hi (u) which depend on the variable with weight zero [8]. To avoid having to solve PDEs, we tacitly assume that there is only one dependent variable with weight zero. As before, the algorithm proceeds in three steps: Step 1: Construct the form of the density Augment the set of dependent variables with α (with non-zero weight) and replace u by ux (since W (u) = 0). Hence, V = {α, ux, v}. Compute R = {α, v 2, v 2, u2x, vux, u2x} and remove divergences and equivalent terms to get S = {α, v 2, u2x, vux}. The candidate density ρ = αh1 (u) + h2 (u)v 2 + h3 (u)u2x + h4 (u)vux, (82) with undetermined functional coefficients hi (u). Step 2: Compute the functions hi (u) Compute ∂ρ ∂ρ ∂ρ Iut + Dx ut + Ivt ∂u ∂ux ∂v = (αh01 + v 2h02 + u2x h03 + vux h04 )ut + (2uxh3 + vh4)utx

Dt ρ =

+(2vh2 + ux h4 )vt, where h0i denotes

dhi du .

(83)

After replacing ut and vt from (75), E = −Dt ρ becomes

E = (αh01 + v 2 h02 + u2x h03 + vuxh04 )v + (2ux h3 + vh4 )vx +(2vh2 + uxh4 )(α sin u + u2x). (0)

(84) (0)

E must be exact. Therefore, require that Lu(x) E ≡ 0 and Lv(x) E ≡ 0. Set the coefficients of like terms equal to zero to get a mixed linear system of algebraic equations and ODEs: h2 (u) − h3 (u) = 0,

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h004 (u) = 0, h01 (u)

h02 (u) = 0,

h03 (u) = 0,

2h02 (u) − h03 (u) = 0,

+ 2h2 (u) sin u = 0,

h001 (u)

h04 (u) = 0,

h002 (u) = 0,

2h002 (u) − h003 (u) = 0, +

2h02 (u) sin u

(85)

+ 2h2 (u) cos u = 0.

Solve the system [8] and substitute the solution h1 (u) = 2c1 cos u + c3,

h2 (u) = h3 (u) = c1,

h4 (u) = c2,

(86)

(with arbitrary constants ci) into (82) to obtain ρ = c1(2α cos u + v 2 + u2x ) + c2vux + c3α.

(87)

Step 3: Compute the flux J Compute the flux corresponding to ρ in (87). Substitute (86) into (84), to get E = c1 (2uxvx + 2vu2x) + c2(αux sin u + vvx + ux u2x).

(88)

40

W. Hereman, P. J. Adams, H. L. Eklund, M. S. Hickman and B. M. Herbst

Since E = DxJ, one must integrate to obtain J. Using (26) and (35) one gets Iu E = 2c1vux + c2 (αu sin u + u2x ) and Iv E = 2c1vux + c2v 2. Using (33), J

= Hu(x)E =

Z1

(Iu E + Iv E)[λu]

dλ λ

0

=

Z1


4c1λvux + c2 (αu sin(λu) + λv 2 + λu2x) dλ

0

= c1(2vux) + c2



 1 2 1 2 −α cos u + v + ux . 2 2

(89)

Finally, split density (87) and flux (89) into independent pieces (for c1 and c2) to get ρ(1) = 2α cos u + v 2 + u2x , J (1) = 2vux , 1 1 ρ(2) = vux, J (2) = −α cos u + v 2 + u2x . 2 2

(90) (91)

For E in (88), J in (89) can easily be computed by hand [8]. However, the computation of fluxes corresponding to densities of ranks ≥ 2 is cumbersome and requires integration with the homotopy operator.

7

Conservation Laws of Scalar Equations with Transcendental and Mixed Derivative Terms

Our method to compute densities and fluxes of scalar equations with transcendental terms and a mixed derivative term (i.e. uxt ) is an adaptation of the technique shown in Section 5. We only consider single PDEs with one type of transcendental nonlinearity. Since we are no longer dealing with evolution equations, densities and fluxes could dependent on ut , u2t, etc. We do not cover such cases; instead, we refer the reader to [101].

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7.1

The sine-Gordon Equation in Light-Cone Coordinates

In light-cone coordinates (or characteristic coordinates) the sG equation, uxt = sin u,

(92)

has a mixed derivative as well as a transcendental term. A change of variables, Φ = ux, Ψ = −1 + cos u, allows one to replace (92) by Φxt − Φ − ΦΨ = 0,

2Ψ + Ψ2 + Φ2t = 0,

(93)

without transcendental terms. Unfortunately, neither (92) nor (93) can be written as a system of evolution equations. As shown in Section 6.1, to deal with the transcendental nonlinearity, which imposes W (u) = 0, one has to replace (92) by uxt = α sin u,

(94)

Direct Methods and Symbolic Software for Conservation Laws

41

where α is an auxiliary parameter with weight. Indeed, (94) is dilation invariant under the scaling symmetry (x, t, u, α) → (λ−1x, λ−1t, λ0u, λ2α), (95) corresponding to W (∂/∂x) = W (∂/∂t) = 1, W (u) = 0, and W (α) = 2. The density-flux pairs [8], [29] of ranks 2, 4, 6, and 8 (which are independent of ut , u2t, etc.), are ρ(1) = u2x , ρ

(2)

=

(3)

=

ρ

(3)

J = (4) ρ = J (4) =

J (1) = 2α cos u,

u4x u6x

− 4u22x J (2) = 4αu2x cos u, − 20u2xu22x + 8u23x , 2α(3u4x cos u + 8u2x u2x sin u − 4u22x cos u), 5u8x − 280u4xu22x − 112u42x + 224u2xu23x − 64u24x , 8α 5u6x cos u + 40u4xu2x sin u + 20u2x u22x cos u + 16u32x sin u
−16u3x u3x cos u − 48uxu2x u3x sin u + 8u23x cos u .

(96) (97) (98) (99) (100)

There are infinitely many density-flux pairs (all of even rank). Since uxt = (ux )t, one can view (94) as an evolution equation in a new variable, U = ux , and construct densities as linear combinations with constant coefficients of monomials in U and its x−derivatives. As before, each monomial has a (pre-selected) rank. To accommodate the transcendental term(s) one might be incorrectly tempted to linearly combine such monomials with functional coefficients hi (u) instead of constant coefficients ci . For example, however, for rank R = 2, one should take ρ = c1u2x instead of ρ = h1 (u)u2x , because the latter would lead to Dt ρ = h01 utu2x + 2h1 ux u2x and ut cannot be replaced from (94).

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7.2

Examples of Equations with Transcendental Nonlinearities

In this section we consider additional PDEs of the form uxt = f (u), where f (u) has transcendental terms. Using the Painlev´e integrability test, researchers [12] have concluded that the only PDEs of that type that are completely integrable are equivalent to one of the standard forms of the nonlinear Klein-Gordon equation [2], [12]. These standard forms (in light-cone coordinates) include the sine-Gordon equation, uxt = sin u, discussed in Section 6.1, the sinh-Gordon equation uxt = sinh u, the Liouville equation uxt = eu , and the double Liouville equations, uxt = eu ± e−2u . The latter is also referred to in the literature as the Tzetzeica and Mikhailov equations. For each of these equations one can compute conservation laws with the method discussed in Section 7.1. Alternatively, if these equations were transformed into laboratory coordinates, one would apply the method of Section 6.2. The multiple sine-Gordon equations, e.g. uxt = sin u + sin 2u, have only a finite number of conservation laws and are not completely integrable, as supported by other evidence [2]. The sinh-Gordon equation, uxt = sinh u, arises as a special case of the Toda lattice discussed in Section 11.1. It also describes the dynamics of strings in constant curvature space-times [70]. In thermodynamics, the sinh-Gordon equation can be used to calculate partition and correlation functions, and thus support Langevin simulations [64]. In Table 1, we show a few density-flux pairs for the sinh-Gordon equation in light-cone coordinates, uxt = α sinh u. As with the sG equation (92), W (∂/∂x) = W (∂/∂t) = 1, W (u) = 0, and

42

W. Hereman, P. J. Adams, H. L. Eklund, M. S. Hickman and B. M. Herbst

W (α) = 2. The ranks in the first column of Table 1 correspond to the ranks of the densities, which are polynomial in U = ux and its x−derivatives. The sinh-Gordon equation has infinitely many conservation laws and is known to be completely integrable [2]. Table 1: Conservation Laws of the sinh-Gordon equation, uxt = α sinh u Rank

Density (ρ)

Flux (J)

2

u2x

−2α cosh u

4

u4x + 4u22x

−4αu2x cosh u

6

u6x + 20u2x u22x + 8u23x

8

5u8x + 280u4xu22x + 64u24x

−2α[(3u4x + 4u22x) cosh u + 8u2xu2x sinh u]  −8α (5u6x − 20u2xu22x + 16u3x u3x + 8u23x) cosh u  +(40u4x u2x − 16u32x + 48uxu2x u3x) sinh u .

+224u2x u23x − 112u42x

The Liouville equation, uxt = eu , plays an important role in modern field theory [68], e.g. in the theory of strings, where the quantum Liouville field appears as a conformal anomaly [65]. The first few (of infinitely many) density-flux pairs for the Liouville equation in light-cone coordinates, uxt = αeu , are given in Table 2. As before, W (∂/∂x) = W (∂/∂t) = 1, W (u) = 0, and W (α) = 2. The ranks in the table refer to the ranks of the densities. Dodd and Bullough [29] have shown that the Liouville equation has no densities of ranks 3, 5, and 7. As shown in Table 2, there are two densities of rank 6, and three densities of rank 8. Our results agree with those in [29], where one can also find the unique density of rank 9 and the four independent densities of rank 10. The double Liouville equations,

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uxt = eu ± e−2u ,

(101)

arise in the field of “laser-induced vibrational predesorption of molecules physisorbed on insulating substrates.” More precisely, (101) is used to investigate the dynamics of energy flow of excited admolecules on insulating substrates [82]. Double Liouville equations are also relevant in studies of global properties of scalar-vacuum configurations in general relativity and similarly systems in alternative theories of gravity [22]. In Table 3, we show some density-flux pairs of uxt = α(eu − e−2u ). There are no density-flux pairs for ranks 4 and 10. We computed a density-flux pair for rank 12 (not shown due to length). The results for (101) with the plus sign are similar.

Part II: Nonlinear Differential-Difference Equations In the second part of this chapter we discuss two distinct methods to construct conservation laws of nonlinear DDEs. The first method follows closely the technique for PDEs discussed in Part I. It is quite effective for certain classes of DDEs, including the Kac-van Moerbeke and Toda lattices, but far less effective for more complicated lattices, such as the Bogoyavlenskii and the Gardner lattices. The latter examples are treated with a new method based on a leading order analysis proposed by Hickman [55].

Direct Methods and Symbolic Software for Conservation Laws

43

Table 2: Conservation Laws of the Liouville equation, uxt = αeu R

Density (ρ)

2

u2x

−2αeu

4

u4x + 4u22x

6

c1 (u6x − 20u32x − 12u23x)

−4αu2x eu  −α 6c1(u4x − 4u2x u2x − 2u22x)  +c2 u2x (2u2x + u2x) eu  −α 8c1(u6x − 6u4x u2x + 3u2x u22x − 20u32x

+c2 (u2xu22x + u32x + u23x) 8

c1 (u8x − 56u2xu32x − 168u2xu23x −672u2xu23x − 144u24x)

−36u3x u3x − 108ux u2xu3x − 18u23x)

+c2 (u4xu22x + u2x u32x + 5u2xu23x

+c2 (2u4x u2x − u2x u22x + 4u32x + 8u3xu3x

+18u2xu23x + 4u24x ) + c3(u42x

+24ux u2x u3x + 4u23x)

+3u2x u23x + 15u2xu23x + 3u24x)

8

Flux (J)

 +c3 (4u32x + 6u3xu3x + 18uxu2x u3x + 3u23x) eu

Nonlinear DDEs and Conservation Laws

We consider autonomous nonlinear systems of DDEs of the form

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u˙ n = F(un−l , . . . , un−1 , un , un+1 , . . ., un+m ),

(102)

where un and F are vector-valued functions with N components. We only consider DDEs with one discrete variable, denoted by integer n. In many applications, n comes from a discretization of a space variable. The dot stands for differentiation with respect to the continuous variable which frequently is time, t. We assume that F is polynomial with constant coefficients, although this restriction can be waived for the method presented in Section 12. No restrictions are imposed on the degree of nonlinearity of F. If parameters are present in (102), they will be denoted by lower-case Greek letters. F depends on un and a finite number of forward and backward shifts of un . We identify l with the furthest negative shift of any variable in the system, and m with the furthest positive shift of any variable in the system. No restrictions are imposed on the integers l and m, which measure the degree of non-locality in (102). By analogy with Dx , we define the shift operator D by Dun = un+1 . The operator D is often called the up-shift operator or forward- or right-shift operator. Its inverse, D−1 , is the down-shift operator or backward- or left-shift operator, D−1 un = un−1 . The action of the shift operators is extended to functions by acting on their arguments. For example, DF(un−l , . . . , un−1 , un, un+1 , . . . , un+m ) = F(Dun−l , . . . , Dun−1 , Dun, Dun+1 , . . . , Dun+m ) = F(un−l+1 , . . ., un , un+1 , un+2 , . . . , un+m+1 ).

(103)

Following [57], we generate (102) from u˙ 0 = F(u−l , u−l+1 , . . . , u−1, u0, u1, . . . , um−1, um ),

(104)

44

W. Hereman, P. J. Adams, H. L. Eklund, M. S. Hickman and B. M. Herbst Table 3: Conservation Laws of the double Liouville equation, uxt = α(eu − e−2u )

Rank

Density (ρ)

Flux (J)

2

u2x

−α(2eu + e−2u )

4

——

——

6

u6x + 15u2xu22x − 5u32x + 3u23x

8

u8x + 42u4xu22x − 14u2x u32x − 7u42x +21u2xu23x − 21u2xu23x + 3u24x

 −3α (2u4x + 2u2x u2x + u22x)eu +(u4x − 8u2x u2x + 2u22x)e−2u  −α (8u6x + 36u4x u2x − 18u2xu22x − 20u32x +6u3x u3x + 18uxu2x u3x + 3u23x)eu +(4u6x − 72u4xu2x − 18u2x u22x − 4u32x +48u3x u3x − 72ux u2xu3x + 6u23x)e−2 u

10

——



——

where u˙ n = Dn u˙ 0 = Dn F. To further simplify the notation, we denote the zero-shifted dependent variable, u0 , by u. Shifts of u are generated by repeated application of D. For instance, uk = Dk u. The identity operator is denoted by I, where D0u = Iu = u. A conservation law of (104),

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Dt ρ + ∆ J = 0,

(105)

links a conserved density ρ to an associated flux J, where both are scalar functions depending on u and its shifts. In (105), which holds on solutions of (104), Dt is the total derivative with respect to time and ∆ = D − I is the forward difference operator. For readability (in particular, in the examples), the components of u will be denoted by u, v, w, etc. In what follows we consider only autonomous functions, i.e. F, ρ, and J do not explicitly depend on t. The time derivatives are defined in a similar way as in the continuous case, see (5) and (31). We show the discrete analog of (31). For a density ρ(up, up+1, . . . , uq , vr , vr+1 , . . . , vs ), involving two dependent variables (u, v) and their shifts, the time derivative is computed as q s X X ∂ρ ∂ρ u˙ k + v˙ k ∂uk ∂vk k=p k=r   ! q s X ∂ρ X ∂ρ ˙ =  Dk  u˙ + Dk v, ∂uk ∂vk

Dt ρ =

k=p

(106)

k=r

since D and d/dt commute. Obviously, the difference operator extends to functions. For example, ∆J = D J − J for a flux, J.

Direct Methods and Symbolic Software for Conservation Laws

45

A density is trivial if there exists a function ψ so that ρ = ∆ψ. Similar to the continuous case, we say that two densities, ρ(1) and ρ(2), are equivalent if and only if ρ(1) + kρ(2) = ∆ψ, for some ψ and some non-zero scalar k. It is paramount that the density is free of equivalent terms for if such terms were present, they could be moved into the flux J. Instead of working with different densities, we will use the equivalence of monomial terms ti in the same density (of a fixed rank). Compositions of D or D−1 define an equivalence relation (≡) on monomial terms. Simply stated, all shifted terms are equivalent, e.g. u−1 v1 ≡ uv2 ≡ u2 v4 ≡ u−3 v−1 since

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u−1 v1 = uv2 − ∆ (u−1v1 ) = u2v4 − ∆ (u1v3 + uv2 + u−1 v1 ) = u−3 v−1 + ∆ (u−2 v + u−3 v−1 ).

(107)

This equivalence relation holds for any function of the dependent variables, but for the construction of conserved densities we will apply it only to monomials. In the algorithm used in Sections 9.2, 11.2, and 11.4, we will use the following equivalence criterion: two monomial terms, t1 and t2 , are equivalent, t1 ≡ t2 , if and only if t1 = Dr t2 for some integer r. Obviously, if t1 ≡ t2 , then t1 = t2 + ∆J for some polynomial J, which depends on u and its shifts. For example, u−2 u ≡ u−1 u1 because u−2 u = D−1 u−1 u1 . Hence, u−2 u = u−1 u1 + [−u−1 u1 + u−2 u] = u−1 u1 + ∆J, with J = −u−2 u. For efficiency we need a criterion to choose a unique representative from each equivalence class. There are a number of ways to do this. We define the canonical representative as that member that has (i) no negative shifts and (ii) a non-trivial dependence on the local (that is, zero-shifted) variable. For example, uu2 is the canonical representative of the class {. . . , u−2 u, u−1u1 , uu2, u1u3 , . . .}. In the case of e.g. two variables (u and v), u2 v is the canonical representative of the class {. . . , u−1 v−3 , uv−2, u1v−1 , u2v, u3v1 , . . .}. Alternatively, one could choose a variable ordering and then choose the member that depends on the zero-shifted variable of lowest lexicographical order. The code in [48] uses lexicographical ordering of the variables, i.e. u ≺ v ≺ w, etc. Thus, uv−2 (instead of u2 v) is chosen as the canonical representative of {. . . , u−1v−3 , uv−2, u1v−1 , u2v, u3v1, . . . }. It is easy to show [55] that if ρ is a density then Dk ρ is also a density. Hence, using an appropriate “up-shift” all negative shifts in a density can been removed. Thus, without loss of generality, we may assume that a density that depends on q shifts has canonical form ρ(u, u1, . . . , uq ).

9

The Method of Undetermined Coefficients for DDEs

In this section we show how polynomial conservation laws can be computed for a scalar DDE, u˙ = F (u−l , u−l+1 , . . . , u, . . ., um−1, um ). The Kac-van Moerbeke example is used to illustrate the steps.

(108)

46

9.1

W. Hereman, P. J. Adams, H. L. Eklund, M. S. Hickman and B. M. Herbst

A Classic Example: The Kac-van Moerbeke Lattice

The Kac-van Moerbeke (KvM) lattice [60], [62], also known as the Volterra lattice, u˙ n = un (un+1 − un−1 ),

(109)

arises in the study of Langmuir oscillations in plasmas, population dynamics, etc. Eq. (109) appears in the literature in various forms, including R˙ n = 12 (exp(−Rn−1 )−exp(−Rn+1 )), 2 2 − wn−1 ), which relate to (109) by simple transformations [92]. We and w˙ n = wn (wn+1 continue with (109) and, adhering to the simplified notation, write it as u˙ = u(u1 − u−1 ),

(110)

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or, with the conventions adopted above, u˙ = u(Du − D−1 u). Lattice (110) is invariant under the scaling symmetry (t, u) → (λ−1t, λu). Hence, u d corresponds to one derivative with respect to t, i.e. u ∼ dt . In analogy to the continuous case, we define the weight W of a variable as the exponent of λ that multiplies the variable [39], [40]. We assume that shifts of a variable have the same weights, that is, W (u−1 ) = W (u) = W (u1). Weights of dependent variables are nonnegative and rational. The rank of a monomial equals the total weight of the monomial. An expression (or equation) is uniform in rank if all its monomial terms have equal rank. Applied to (110), W (d/dt) = W (Dt ) = 1 and W (u) = 1. Conversely, the scaling symmetry can be computed with linear algebra as follows. Setting W (d/dt) = 1 and requiring that (110) is uniform in rank yields W (u) + 1 = 2W (u). Thus, W (u) = 1, which agrees with the scaling symmetry. Many integrable nonlinear DDEs are scaling (dilation) invariant. If not, they can be made so by extending the set of dependent variables with parameters with weights. Examples of such cases are given in Sections 11.3 and 13.2. The KvM lattice has infinitely many polynomial density-flux pairs. We give the conserved densities of rank R ≤ 4 with associated fluxes (J (4) is omitted due to length): ρ(1) = u, J (1) = −uu−1 , 1 ρ(2) = u2 + uu1 , J (2) = −(u−1 u2 + u−1 uu1), 2 1 ρ(3) = u3 + uu1 (u + u1 + u2 ), 3 (3) J = −(u−1 u3 + 2u−1 u2 u1 + u−1 uu21 + u−1 uu1u2 ), 1 3 ρ(4) = u4 + u3 u1 + u2 u21 + uu21 (u1 + u2 ) 4 2 +uu1 u2 (u + u1 + u2 + u3 ).

(111) (112)

(113)

(114)

In addition to infinitely many polynomial conserved densities, (110) has a non-polynomial density, ρ(0) = ln u with flux J (0) = −(u + u−1 ). We discuss the computation of nonpolynomial densities in Section 12.

Direct Methods and Symbolic Software for Conservation Laws

9.2

47

The Method of Undetermined Coefficients Applied to a Scalar Nonlinear DDE

We outline how densities and fluxes can be constructed for a scalar DDE (108). Using (110) as an example, we compute ρ(3) of rank R = 3 and associated flux J (3) of rank R = 4, both listed in (113). • Select the rank R of ρ. Start from the set V of dependent variables (including parameters with weight, when applicable), and form a set M of all non-constant monomials of rank R or less (without shifts). For each monomial in M introduce the right number of t−derivatives to adjust the rank of that term. Using the DDE, evaluate the t−derivatives, strip off the numerical coefficients, and gather the resulting terms in a set R. For the KvM lattice (110), V = {u} and M = {u3, u2, u}. Since u3 , u2, and u have ranks 3, 2, and 1, respectively, one computes d 0 u3 = u3 , dt0

du2 = 2uu˙ = 2u2 (u1 − u−1 ) = 2u2 u1 − 2u−1 u2 , dt

(115)

and d2 u du˙ d (u(u1 − u−1 )) = = = u(u ˙ 1 − u−1 ) + u(u˙ 1 − u˙ −1 ) 2 dt dt dt = u(u1 − u−1 )2 + u(u1(u2 − u) − u−1 (u − u−2 )) = uu21 − 2u−1 uu1 + u2−1 u + uu1 u2 − u2 u1 − u−1 u2 + u−2 u−1 u, (116) where (110) (and its shifts) has been used to remove the time derivatives. Build R using the terms from the right hand sides of the equations in (115) and ignoring numerical coefficients,  R = u3 , u2u1 , u−1u2 , uu21, u−1uu1 , u2−1u, uu1u2 , u−2u−1 u . (117)

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• Identify the elements in R that belong to the same equivalence classes, replace them by their canonical representatives, and remove all duplicates. Call the resulting set S, which has the building blocks of a candidate density. Continuing with (117), u−2 u−1 u ≡ u−1 uu1 ≡ uu1u2 . Likewise, u−1 u2 ≡ uu21 and u2−1 u ≡ u2 u1 . Thus, S = {u3, u2u1 , uu21, uu1 u2}. • Form an arbitrary linear combination of the elements in S. This is the candidate ρ. Continuing with the example, ρ = c1 u3 + c2 u2 u1 + c3 uu21 + c4 uu1u2 . • Compute q X ∂ρ Dt ρ = u˙ k = ∂uk k=0

q X ∂ρ k D ∂uk

!

u, ˙

(118)

(119)

k=0

where q is the highest shift in ρ. Using (118) where q = 2,   ∂ρ ∂ρ ∂ρ 2 Dtρ = I+ D+ D u˙ ∂u ∂u1 ∂u2  = (3c1u2 + 2c2uu1 + c3 u21 + c4u1 u2 )I  +(c2 u2 + 3c3uu1 + c4uu2 )D+c4uu1 D2 u. ˙

(120)

48

W. Hereman, P. J. Adams, H. L. Eklund, M. S. Hickman and B. M. Herbst

• Evaluate Dt ρ on the DDE (108) by replacing u˙ by F. Call the result E. In (110), F = u(u1 − u−1 ). The evaluated form of (120) is E = (3c1u2 + 2c2uu1 + c3 u21 + c4u1 u2 )u(u1 − u−1 ) + (c4uu1 )u2(u3 − u1 ) +(c2 u2 + 2c3uu1 + c4uu2 )u1(u2 − u) = (3c1 − c2 )u3u1 − 3c1u−1 u3 + 2(c2 − c3 )u2u21 − 2c2u−1 u2u1 +c3 uu31 − c3u−1 uu21 − c4u−1 uu1 u2 + (c2 − c4)u2 u1 u2 +2c3uu21 u2 + c4 uu1 u22 + c4uu1 u2 u3.

(121)

• Set J = 0. Transform E into its canonical form. In doing so modify J so that E + ∆ J remains unchanged. For example in (121), replace −3c1u−1 u3 in E by −3c1uu31 and add −3c1u1 u3 to J since uu31 − [uu31 − u−1 u3 ]. Do the same for all the other terms which are not in canonical form. After grouping like terms, (121) becomes E = (3c1 − c2)u3 u1 + (c3 − 3c1)uu31 + 2(c2 − c3 )u2u21 +2(c3 − c2 )uu21u2 + (c4 − c3 )uu1u22 + (c2 − c4)u2 u1u2 ,

(122)

with J = −(3c1u−1 u3 + 2c2u−1 u2 u1 + c3u−1 uu21 + c4u−1 uu1 u2 ).

(123)

• E is now the obstruction to ρ being a density. Set E = 0 and solve for the undetermined coefficients ci . Thus, 3c1 − c2 = 0, 3c1 − c3 = 0, c2 − c3 = 0, c3 − c4 = 0, c2 − c4 = 0,

(124)

which yields c2 = c3 = c4 = 3c1 , where c1 is arbitrary. • Substitute the solution for the ci into the candidates for ρ and J to obtain the final density and associated flux (up to a common arbitrary factor which can be set to 1 or any other nonzero value). For the example, setting c1 = 13 and substituting c2 = c3 = c4 = 1 into (118) and (123) yields ρ(3) and J (3) as given in (113).

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10

Discrete Euler and Homotopy Operators

For simplicity, we will consider the case of only one discrete (dependent) variable u. First, ˜ = Dl E where l corresponds to the lowest shift we remove negative shifts from E. Thus, E in E. The discrete variational derivative (discrete Euler operator) An expression E is exact if and only if it is a total difference. The following exactness test is well-known [10], [57]: A necessary and sufficient condition for a function E, with positive (0) (0) shifts, to be exact is that Lu E ≡ 0, where Lu is the discrete variational derivative (discrete Euler operator of order zero) [10] defined by ! m X
∂ ∂ L(0) D−k E = I + D−1 + D−2 + D−3 + · · · + D−m E, u E = ∂u ∂u k=0

∂E ∂E ∂E ∂E ∂E = I + D−2 + D−3 + · · · + D−m , + D−1 ∂u ∂u1 ∂u2 ∂u3 ∂um

(125)

Direct Methods and Symbolic Software for Conservation Laws

49

where m is highest shift occurring in E. Application. We return to (121) where l = 1. Therefore, ˜ = DE = (3c1 − c2 )u3u2 − 3c1 uu3 + 2(c2 − c3 )u2u2 − 2c2uu2 u2 E 1

1

1 2

1

+c3 u1u32 − c3uu1 u22 − c4 uu1u2 u3 + (c2 − c4)u21 u2 u3 +2c3u1 u22 u3 + c4 u1u2 u23 + c4 u1u2 u3 u4 ,

(126)

which is free of negative shifts. Applying (125) to (126), where m = 4, gives
∂ ˜ ˜ L(0) I + D−1 + D−2 + D−3 + D−4 E u E = ∂u = 3(3c1 − c2)u2 u1 + 3(c3 − 3c1)u−1 u2 + 2(c2 − c4)uu1 u2 +4(c2 − c3)uu21 + 4(c3 − c2 )u−1uu1 + 2(c3 − c2 )u21u2 +(c3 − 3c1)u31 + (c4 − c3 )u−1 u21 + (c4 − c3)u1u22 +(3c1 −

c2)u3−1

+2(c3 −

c2)u−2 u2−1

+ (c2 −

c4)u2−1 u1

+ 4(c2 −

(127)

c3)u2−1 u

+ 2(c4 − c3)u−2 u−1 u + (c2 − c4 )u2−2 u−1 ,

which, as expected, vanishes identically when c1 = 13 , c2 = c3 = c4 = 1. Due to the large amount of terms generated by the Euler operator, this method for finding the undetermined coefficients is much less efficient than the “splitting and shifting” approach illustrated on the same example in Section 9.2. The discrete homotopy operator As in the continuous case, the discrete homotopy operator reduces the inversion of the difference operator, ∆ = D − I, to a problem of single-variable calculus. Indeed, assuming that E is exact and free of negative shifts, the flux J = ∆−1 E can be computed without “summation by parts”. Instead, one computes a single integral with respect to an auxiliary variable denoted by λ (not to be confused with λ in Section 9.1). Consider an exact expression E (of one variable u), free of negative shifts, and with highest shift m. The discrete homotopy operator is defined [61], [71], [72] by Hu E =

Z1

(Iu E) [λu]

dλ , λ

(128)

0 Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

with Iu E =

m X

m−k X

k=1

i=0

∂ ui ∂ui

!

D

−k

E=

m k X X k=1

i=1

D

−i

!

uk

∂E , ∂uk

(129)

where (Iu E)[λu] means that in Iu E one replaces u → λu, u1 → λu1, u2 → λu2 , etc. The formulas in (129) are equivalent to the one in [52], which in turn is equivalent to the formula in terms of discrete higher Euler operators [51], [54]. Given an exact function E without negative shifts one has J = ∆−1 E = Hu E. A proof can be found in [61], [72]. Application. Upon substitution of c1 = 13 , c2 = c3 = c4 = 1 into (126), we obtain ˜ = DE E = −uu31 − 2uu21 u2 + u1 u32 − uu1 u22 − uu1 u2 u3 = +2u1 u22u3 + u1u2 u23 + u1 u2 u3 u4,

(130)

50

W. Hereman, P. J. Adams, H. L. Eklund, M. S. Hickman and B. M. Herbst

where the highest shift is m = 4. Using (129), ! 4 k X X ˜ ˜ ˜ ∂E ∂E ∂E −i ˜ = Iu E D uk = (D−1 )u1 + (D−1 + D−2 )u2 ∂uk ∂u1 ∂u2 k=1

i=1

˜ ˜ ∂E ∂E + (D−1 + D−2 + D−3 + D−4 )u4 ∂u3 ∂u4 −1 2 3 2 2 2 = D u1(−3uu1 − 4uu1 u2 + u2 − uu2 − uu2 u3 + 2u2u3 + u2u3 + u2u3 u4 ) +(D−1 + D−2 + D−3 )u3

+(D−1 + D−2 )u2 (−2uu21 + 3u1u22 − 2uu1 u2 − uu1 u3 +4u1u2 u3 + u1 u23 + u1 u3 u4 ) +(D−1 + D−2 + D−3 )u3(−uu1 u2 + 2u1 u22 + 2u1u2 u3 + u1 u2 u4 ) +(D−1 + D−2 + D−3 + D−4 )u4 (u1u2 u3 ) = 4(uu31 + 2uu21 u2 + uu1 u22 + uu1 u2 u3),

(131)

which has the correct terms of J (3) but incorrect coefficients. Finally, using (128) Z1 dλ ˜ = − (Iu E)[λu] ˜ J˜ = Hu (−E) λ 0

= −4

Z1


uu31 + 2uu21 u2 + uu1 u22 + uu1 u2 u3 λ3 dλ

0

= −(uu31 + 2uu21 u2 + uu1 u22 + uu1 u2 u3).

(132)

˜ = DE. Hence, Recall that E

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J = D−1 J˜ = −(u−1 u3 + 2u−1 u2u1 + u−1 uu21 + u−1 uu1 u2 ),

(133)

which corresponds to J (3) in (11). The homotopy method is computationally inefficient. Even for a simple example, like (131), the integrand has a large number of terms, most of which eventually cancel. To compute the flux we recommend the “splitting and shifting” approach which was illustrated (on the same example) in Section 9.2. The generalization of the exactness test to an expression E with multiple dependent variables (u, v, · · ·) is straightforward. For example, an expression E of discrete variables (0) (0) (0) u, v and their forward shifts will be exact if and only if Lu E = (Lu E, Lv E) ≡ (0, 0), (0) where Lv is defined analogously to (125). Similar to the continuous case, the homotopy operator formulas (128) and (129) straightforwardly generalize to multiple dependent variables. The reader is referred to [51], [52], [54] for details.

11

Conservation Laws of Nonlinear Systems of DDEs

We use the method discussed in Section 9.2 to compute conservation laws for the Toda and Ablowitz–Ladik lattices. Using the latter lattice, we illustrate a “divide and conquer” strategy, based on multiple scales, which allows on to circumvent difficulties in the computation of densities of DDEs that are not dilation invariant.

Direct Methods and Symbolic Software for Conservation Laws

11.1

51

The Toda Lattice

One of the earliest and most famous examples of completely integrable DDEs is the Toda lattice [93], [94], y¨n = exp (yn−1 − yn ) − exp (yn − yn+1 ), (134) where yn is the displacement from equilibrium of the nth particle with unit mass under an exponential decaying interaction force between nearest neighbors. With the change of variables, un = ˙ny, vn = exp (yn − yn+1 ), lattice (134) can be written in algebraic form u˙ n = vn−1 − vn ,

v˙ n = vn (un − un+1 ).

(135)

Adhering to the simplified notation, we continue with u˙ = v−1 − v,

v˙ = v(u − u1 ).

(136)

As before, we set W (d/dt) = 1, assign unknown weights, W (u), W (v), to the dependent variables, and require that each equation in (136) is uniform in rank. This yields W (u) + 1 = W (v),

W (v) + 1 = W (u) + W (v).

(137)

The solution W (u) = 1, W (v) = 2 reveals that (136) is invariant under the scaling symmetry (t, u, v) → (λ−1t, λu, λ2v), (138)

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where λ is an arbitrary parameter. The Toda lattice has infinitely many conservation laws [44]. The first two density-flux pairs are easy to compute by hand. Here we give the densities of rank R ≤ 4 with associated fluxes, J (4) being omitted due to length: ρ(1) = u, J (1) = v−1 , 1 ρ(2) = u2 + v, J (2) = uv−1 , 2 1 2 ρ(3) = u3 + u(v−1 + v), J (3) = u−1 uv−1 + v−1 , 3 1 1 ρ(4) = u4 + u2 (v−1 + v) + uu1 v + v 2 + vv1. 4 2

11.2

(139) (140) (141) (142)

Computation of a Conservation Law of the Toda Lattice

As an example, we compute density ρ(3) (of rank R = 3) and associated flux J (3) (of rank 4) in (141). Step 1: Construct the form of the density Start from V = {u, v}, i.e. the set of dependent variables with weight. List all monomials in u and v of rank R = 3 or less: M = {u3, u2, uv, u, v}. Next, for each monomial in M, introduce the correct number of t-derivatives so that each term has rank 3. Using (136),

52

W. Hereman, P. J. Adams, H. L. Eklund, M. S. Hickman and B. M. Herbst

compute d0 u3 d0 uv 3 = u , = uv, dt0 dt0 dv du2 = 2uu˙ = 2uv−1 − 2uv, = v˙ = uv − u1 v, dt dt d2 u du˙ d(v−1 − v) = = = u−1 v−1 − uv−1 − uv + u1 v. 2 dt dt dt

(143)

Gather the terms in the right hand sides in (143) to get R = {u3 , uv−1, uv, u−1v−1 , u1v}. Identify members belonging to the same equivalence classes and replace them by their canonical representatives. For example, uv−1 ≡ u1 v. Adhering to lexicographical ordering, we will use uv−1 instead of u1 v. Doing so, replace R by S = {u3 , uv−1, uv}, which has the building blocks of the density. Linearly combine the monomials in S with undetermined coefficients ci to get the candidate density of rank 3 : ρ = c1 u3 + c2 uv−1 + c3 uv.

(144)

Step 2: Compute the undetermined coefficients ci Compute Dt ρ and use (136) to eliminate u˙ and v˙ and their shifts. Thus, E = (3c1 − c2)u2 v−1 + (c3 − 3c1)u2 v + (c3 − c2)v−1 v 2 +c2u−1 uv−1 + c2 v−1 − c3uu1 v − c3v 2 .

(145)

Step 3: Find the associated flux J Transform (145) into canonical form to obtain E = (3c1 −c2)u21 v +(c3 −3c1)u2 v +(c3 −c2 )vv1 +c2 uu1v +c2 v 2 −c3uu1 v −c3v 2 (146) with 2 J = (3c1 − c2)u2v−1 + (c3 − c2)v−1 v + c2 u−1 uv−1 + c2v−1 .

(147)

Set E = 0 to get the linear system Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

3c1 − c2 = 0,

c3 − 3c1 = 0,

c2 − c3 = 0.

(148)

Set c1 = 13 and substitute the solution c1 = 13 , c2 = c3 = 1, into (144) and (147) to obtain ρ(3) and J (3) in (141).

11.3

The Ablowitz–Ladik Lattice

In [4], [5], [6], Ablowitz and Ladik derived and studied the following completely integrable discretization of the nonlinear Schr¨odinger equation: i u˙ n = un+1 − 2un + un−1 ± u∗n un (un+1 + un−1 ),

(149)

where u∗n is the complex conjugate of un . We continue with (149) with the plus sign; the case with the negative sign is analogous. Instead of splitting un into its real and imaginary

Direct Methods and Symbolic Software for Conservation Laws

53

parts, we treat un and vn = u∗n as independent variables and augment (149) with its complex conjugate equation, u˙ n = (un+1 − 2un + un−1 ) + un vn (un+1 + un−1 ), v˙ n = −(vn+1 − 2vn + vn−1 ) − un vn (vn+1 + vn−1 ),

(150)

where i has been absorbed into a scale on t. Since vn = u∗n we have W (vn ) = W (un ). Neither equation in (150) is dilation invariant. To circumvent this problem we introduce an auxiliary parameter α with weight, and replace (150) by u˙ = α(u1 − 2u + u−1 ) + uv(u1 + u−1 ), v˙ = −α(v1 − 2v + v−1 ) − uv(v1 + v−1 ),

(151)

presented in the simplified notation. Both equations in (151) are uniform in rank provided W (u) + 1 = W (α) + W (u) = 2W (u) + W (v), W (v) + 1 = W (α) + W (v) = 2W (v) + W (u),

(152)

which holds when W (u) + W (v) = W (α) = 1. Since W (u) = W (v) we have W (u) = W (v) = 12 , and W (α) = 1, which expresses that (151) is invariant under the scaling symmetry 1 1 (t, u, v, α) → (λ−1 t, λ 2 u, λ 2 v, λα). (153) We give the conserved densities of (151) of ranks 2 through 4. Only the flux of rank 3 associated to ρ(1) is shown. The others are omitted due to length. ρ(1) = α(c1 uv−1 + c2 uv1 ), J

(1)

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ρ(2)

ρ(3)

(154)

= −α [c1(αuv−2 − αu−1 v−1 + u−1 uv−2 v−1 ) + c2(αuv − αu−1 v1 − u−1 uvv1 )] , (155)    1 2 2 = α c1 u v−1 + uu1v−1 v + αuv−2 2   1 2 2 +c2 (156) u v1 + uu1 v1 v2 + αuv2 , 2   1 3 = α c1 u3 v−1 + uu1v−1 v(uv−1 + u1 v + u2v1 ) + αuv−1 (uv−2 + u1 v−1 ) 3  +αuv(u1 v−2 + u2v−1 ) + α2 uv−3 + c2 [+uu1 v1 v2 (uv1 + u1 v2 + u2 v3)]  1 3 3 2 + u v1 + αuv2 (uv1 + u1 v2) + αuv3 (u1v1 + u2 v2 ) + α uv3 , (157) 3

where c1 and c2 are arbitrary constants. Our results confirm those in [6]. Since our method is restricted to polynomial densities we cannot compute the density with a logarithmic term, ρn = (u∗n (un−1 + un+1 ) − 2 ln(1 + un u∗n )) , which corresponds [3], [6] to the Hamiltonian of (149), viz. H = −i

(158) P

n

ρn .

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W. Hereman, P. J. Adams, H. L. Eklund, M. S. Hickman and B. M. Herbst

11.4

Computation of a Conservation Law of the Ablowitz–Ladik Lattice

To make (150) scaling invariant we had to introduce an auxiliary parameter α. This complicates matters in two ways as we will show in this section. First, to compute rather simple conserved densities, like ρ = uv−1 and ρ = uv1 , we will have to select rank R = 2 for which the candidate density has twenty terms. However, eighteen of these terms eventually drop out. Second, conserved densities corresponding to lower ranks might appear in the result. These lower-rank densities are easy to recognize for they are multiplied with arbitrary coefficients ci . Consequently, when parameters with weight are introduced, the densities corresponding to distinct ranks are no longer linearly independent. We compute ρ(1) of rank R = 2 in (154) with associated flux J (1) in (155). Note that ρ(1) and J (1) cannot be computed with the steps below when R = 1. Step 1: Construct the form of the density Augment the set of dependent variables with the parameter α (with non-zero weight). Hence, V = {u, v, α}. Construct M = {u, v, u2, uv, v 2, αu, u3, αv, u2v, uv 2, v 3, αu2, u4, αuv, u3v, αv 2, u2v 2, uv 3, v 4},

(159)

which contains all non-constant monomials of (chosen) rank 2 or less (without shifts). As with the previous examples, for each element in M add the right number of t−derivatives. Use (151) to evaluate the t−derivatives, gather the terms in the right hand sides, introduce the canonical representatives (based on lexicographical ordering), and remove duplicates to get S = {αu2 , u4, αuu1, αuv−1 , αuv, u3v, u2u1v, u2v−1 v, αv 2, u2v 2, uu1v 2, uv−1 v 2 , uv 3, v 4, αuv1, uu21v1 , αvv1, u2vv1, uv 2v1, uu1v12 }.

(160)

Linearly combine the monomials in S with undetermined coefficients ci to get the candidate density of rank 2: ρ = c1 αu2 + c2 u4 + c3 αuu1 + c4 αuv−1 + c5 αuv + c6 u3 v + c7 u2 u1v

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+c8 u2v−1 v + c9 αv 2 + c10 u2v 2 + c11 uu1 v 2 + c12 uv−1 v 2 + c13 uv 3 +c14 v 4 + c15 αuv1 + c16 uu21 v1 + c17 αvv1 + c18 u2 vv1 +c19 uv 2 v1 + c20 uu1v12 .

(161)

Step 2: Compute the undetermined coefficients ci The computations proceed as in the examples in Sections 9.2 and 11.2. Thus, compute Dt ρ and use (151) to eliminate u˙ and v˙ and their shifts. Next, bring the expression E into canonical form to obtain the linear system for the undetermined coefficients ci . Without showing the lengthy computations, one finds that all constants ci = 0, except c4 and c15 which are arbitrary. Substitute the coefficients into (161) to get ρ(1) in (154). Step 3: Find the associated flux J The associated flux comes for free when E is transformed into canonical form. Alternatively, one could apply the homotopy approach for multiple dependent variables [51], [54] to compute the flux. In either case, one gets J (1) in (155).

Direct Methods and Symbolic Software for Conservation Laws

11.5

55

A “Divide and Conquer” Strategy

It should be clear from the example in Section 11.4 that our method is not efficient if the densities are not of the form (111)-(114) for the KvM lattice and (139)-(142) for the Toda lattice. Indeed, the densities for the AL lattice in (154)-(157) are quite different in structure. Therefore, in [33], Eklund presented alternate strategies to deal more efficiently with DDEs, in particular with those involving weighted parameters such as (151). A first alternative is to work with multiple scales by setting either W (Dt) = 0 or W (Dt ) = 1, the latter choice is what we have used thus far. A second possibility is to leave W (Dt ) unspecified and, if needed, introduce extra parameters with weight into the DDE. Let us explore these ideas for the AL lattice (151), which we therefore replace by u˙ = αβ(u1 − 2u + u−1 ) + βuv(u1 + u−1 ), v˙ = −αβ(v1 − 2v + v−1 ) − βuv(v1 + v−1 ),

(162)

where β is a second auxiliary parameter with weight. Requiring uniformity in rank leads to W (u) + W (Dt ) = W (α) + W (β) + W (u) = W (β) + 2W (u) + W (v), W (v) + W (Dt ) = W (α) + W (β) + W (v) = W (β) + 2W (v) + W (u), (163) which are satisfied when W (α) = W (u) + W (v) and W (Dt) = W (u) + W (v) + W (β). Therefore, we can set W (Dt) = a, W (u) = b, and W (β) = c with a, b, c rational numbers so that W (v) = a − b − c and W (α) = a − c are strictly positive. Thus (162) is dilation invariant under a three-parameter family of scaling symmetries, (t, u, v, α, β) → (λ−at, λbu, λa−b−c v, λa−cα, λcβ).

(164)

1 2,

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Scaling symmetry (153) corresponds to the case where a = 1, b = and c = 0, more precisely, β = 1. The fact that (162) is invariant under multiple scales is advantageous. Indeed, one can use the invariance under one scale to construct a candidate density and, subsequently, use additional scale(s) to split ρ into smaller densities. This “divide and conquer” strategy drastically reduces the complexity of the computations. The use of multiple scales has proven to be successful in the computation of conservation laws for PDEs with more than one space variable [51]. Candidate density (161) is uniform of rank 2 under (153) but can be split into smaller pieces, ρi, using (164), even without specifying values for a, b, and c. Indeed, as shown in Table 4, ρ in (161) can be split into ρ1 through ρ5 of distinct ranks under (164). Steps 2 and 3 of the algorithm are then carried out for each of these ρi , (i = 1, · · · , 5) separately. As shown in the table, all but one density lead to a trivial result. The longest density, ρ3, with 8 terms, leads to ρ(1) and J (1) in (154) and (155), respectively.

12

A New Method to Compute Conservation Laws of Nonlinear DDEs

In the continuous case, the total derivative Dx has weight one. Consequently, any density is bounded with respect to the order in x. For example, as shown in Section 3.2 for the KdV

56

W. Hereman, P. J. Adams, H. L. Eklund, M. S. Hickman and B. M. Herbst

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Table 4: Candidate densities for the Ablowitz-Ladik lattice (162). i 1

Rank a + 2b − c

2 3

4b 2(a − c)

4

3a − 2b − 3c

5

4(a − b − c)

Candidate ρi c1αu2 + c3αuu1 + c6u3 v +c7 u2u1 v + c16uu21 v1 c 2 u4 c4αuv−1 + c5 αuv + c8u2 v−1 v +c10u2 v 2 + c11uu1 v 2 + c15αuv1 +c18u2 vv1 + c20uu1 v12 c9αv 2 + c12uv−1 v 2 + c13uv 3 +c17αvv1 + c19uv 2 v1 c14v 4

Final ρi 0

Final Ji 0

0 c4 αuv−1 +c15 αuv1

0 J (1) in (155)

0

0

0

0

case, the candidate density ρ of rank 6 in (18) is of first order (after removing the second and fourth order terms.) Obviously, a density of rank 6 could never have leading order terms of fifth or higher order in x because the rank of such terms would exceed 6. In the discrete case, however, the shift operator D has no weight. Thus, any shift Dk u = uk of a dependent variable u has the same weight as that dependent variable. Simply put, W (uk ) = W (u) for any integer k. Consequently, using the total derivative Dt as a tool to construct a (candidate) density has a major shortcoming: the density may lack terms involving sufficiently high shifts of the variables. As shown in Section 9.2 for the KvM lattice, the candidate density ρ of rank 3 in (118) has leading order term uu1 u2 , i.e. the term with the highest shift (2 in this example). It is a priori not excluded that ρ might have terms involving higher shifts. For example, uu1 u3 has rank 3 and so do infinitely many other cubic terms. Note that for this example we constructed ρ starting from powers of u, viz. u3, u2, u; and, by repeated differentiation, worked our way “down” towards the leading order term uu1 u2 . In this section we outline the key features of a new method which goes in the opposite direction: (i) first compute the leading order term and subsequently (ii) compute the terms involving lower shifts. In step (ii) only the necessary terms are computed, nothing more, nothing less. This method is fast and powerful for it circumvents the use of the dilation invariance and the method of undetermined coefficients. More importantly, the new method is not restricted to densities and fluxes of polynomial form.

12.1

Leading Order Analysis

Consider a density, ρ, that depends on q shifts. Since Dq ρ is also a density, we may, without 2ρ loss of generality, assume that ρ has canonical form ρ(u, u1 , . . . , uq ) with ∂u∂ ∂u 6= 0. q In [55], Hickman derived necessary conditions on this leading term (which, in the system case, is a matrix). First all terms in the candidate density ρ that contribute directly to the flux are removed. Rather than applying the Euler operator on the remaining terms in ρ, the necessary condition [57], ∂ 2g = 0, (165) ∂u ∂uq

Direct Methods and Symbolic Software for Conservation Laws

57

for g to be a total difference is applied to obtain a system of equations for the terms that depend on the maximal shift, uq , in ρ. This system is rewritten as a matrix equation. Solutions to this system will give us the form of the leading term in ρ. We apply a splitting of the identity operator, I = (D − I + I) D−1 = ∆ D−1 + D−1 ,

(166)

to the part, ρ∗, of the candidate ρ that is independent of the variables with the lowest order shift. The first term ∆ D−1 ρ∗ contributes to the flux while the second term D−1 ρ∗ has a strictly lower shift than ρ∗. Applying this split repeatedly we get   I = (Dk − I + I) D−k = ∆ Dk−1 + Dk−2 + · · · + D + I D−k + D−k , (167) where, again, the first term contributes to the flux and the remainder has strictly lower shift. This decomposition is repeatedly applied to terms that do not involve the lowest order shifted variables. Any terms that remain will involve the lowest order shifted variable. These terms yield the constraints on the undetermined coefficients or unknown functions in the density ρ. As shown in [55], the result of this split is that ρ is a density if and only if   l   q X ∂ X j ∂ l j σ=D F D ρ+ D F ρ (168) ∂u ∂u j=0

j=l+1

is a total difference, where the operator F

X ∂ ∂ = Fα α ∂u ∂u α

(169)

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involves a summation over the components in the system of DDEs. As before, in the examples we will use u, v, w, etc. to denote the dependent variables uα . Now, σ depends on the shifted variables u, . . . , uq+L where L = max (l, m). For q > L, (165) gives  2    ∂ 2σ ∂ ρ ∂F T ∂ 2 ρ ∂F L q +D =D = 0, (170) ∂u ∂uq+L ∂u ∂uq ∂u−L ∂uL ∂u ∂uq where T stands for transpose. The system case is treated in detail in [55]. For brevity, we continue with the scalar case. Let ∂F ∂F λ= , µ= , (171) ∂u−L ∂uL then the leading term will satisfy S

∂ 2ρ = 0, ∂u ∂uq

(172)

with S = DL λ DL + Dq µ. We immediately see that if l 6= m then either λ or µ is zero and (172) has no non-trivial solutions. Let q = pL + r with p and r integers, 0 ≤ r < L, and ! p−1 Y c= DkL λ DpL ζ. (173) k=1

58

W. Hereman, P. J. Adams, H. L. Eklund, M. S. Hickman and B. M. Herbst

Then Sc=

p−1 Y


DkL λ DpL λ DLζ + ζ Dr µ .

(174)

k=1

Thus c 6= 0 lies in the kernel of S if and only if λ D L ζ + ζ Dr µ = 0

(175)

has a non-zero solution ζ. Suppose (175) has two non-zero solutions, say, ζ1 and ζ2 . Then,   ζ2 L ζ2 =D . (176) ζ1 ζ1 So, since L 6= 0, ζ2 = aζ1 for some constant a. Therefore, the kernel of S is, at most, one dimensional. This implies that a scalar DDE can have, at most, one conserved density that depends on q shifts for q > L. The leading term, ρ, ˜ will satisfy ∂ 2ρ = c, ∂u ∂uq which, upon integration, yields ρ˜ =

ZZ

c du duq.

(177)

(178)

The density (if it exists) may now be computed by a “split and shift” strategy on this leading term. Starting with ρ = ρ˜. the objective is to successively compute the terms (of lower shift!) that must be added to ρ until Dt ρ ≡ 0. First, Dt ρ is computed and evaluated on the DDE. Next, all terms are shifted so that the resulting expression depends on u (and not on lower shifts of u). Then the leading terms, ξ, in Dt ρ are isolated. Last, we solve

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Dt ρ(1) = ξ + terms of lower shift.

(179)

If (179) has no solution then a density with q shifts does not exist. On the other hand, if (179) has a solution, the “correction” term ρ(1) is then subtracted from ρ and we recompute Dt ρ. By construction, the highest shift that occurs in the result will now be lower than before and we repeat the entire procedure to obtain a new correction term ρ(2). After a finite number of steps, we will either find an that the correction term does not exist (and so the density does not exist) or we will obtain Dt ρ ≡ 0 and ρ will be a density. This algorithm has been coded [56] in Maple. Further details and worked examples will be presented in [58]. For now, we illustrate the algorithm with a simple example.

12.2

A“Modified” Volterra Lattice

Consider the DDE, u˙ = u2 (u2 − u−2 ),

(180)

which is related to the well-known modified Volterra Lattice [11]. Here L = 2 and, using (171), ∂F ∂F λ= = − u2 , µ = = u2 . (181) ∂u−2 ∂u2

Direct Methods and Symbolic Software for Conservation Laws

59

Thus, the condition (175) for a non-trivial density becomes ζ u2r = u2 D2ζ for r = 0, 1. For r = 0, we have ζ = D2 ζ and so we may choose ζ = 1. For r = 1, we have ζ u21 = u2 D2ζ, which has no non-zero solutions. Thus, densities only exist for r = 0. Since q = pL = 2p, with p integer, we conclude that q must be even. In these cases the leading term will satisfy ∂ 2ρ =c= ∂u ∂uq

p−1 Y

u2k

!

= u u2 · · · uq−2 .

(182)

k=1

Therefore, after scaling to remove constants, the leading term is ρ˜ = u2 u2 · · · uq−2 uq . For example, let us compute the density that depends on q = 4 shifts. We set ρ = ρ˜ = 2 u u2 u4. Using (180), Dt ρ = u2 u22u4 (u2 − u−2 ) + 2u u32 u4 (u4 − u) + u u22 u24 (u6 − u2 ) ≡ u u32 u24 − u2 u32 u4.

(183)

The highest shift is u4 and so the leading terms are ξ = u u32 u24 − u2 u2 3 u4 .

(184)

Note that the terms in u6 must cancel by the construction of ρ˜. Next, we note that u4 as a leading term must arise as either u˙ 2 = u22 (u4 − u) or u2 u˙ = u2 u2 (u2 − u−2 ) ≡ u2 u22 − u u22 u4 .

(185)

The quadratic term must arise from (185) (as we have already determined all terms that involve u4 , so, we cannot have a quadratic term given by u4u˙ 2 ). Now, we solve (179) to get 1 ρ(1) = − u2 u22 + · · · . 2

(186)

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Indeed, Dt



1 − u2 u22 2



= −u u22 u˙ − u2 u2 u˙ 2 ≡ u u32 u24 − u2 u32 u4 + lower shift terms = ξ + lower shift terms,

(187)

with ξ in (184). Thus, we update ρ = u2 u2u4 by subtracting ρ(1). This yields, ρ = u2 u2 u4 +

1 2 2 u u2 . 2

(188)

We readily verify that Dt ρ ≡ 0 and, thus, ρ in (188) is the unique density of (180) that depends on 4 shifts.

60

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W. Hereman, P. J. Adams, H. L. Eklund, M. S. Hickman and B. M. Herbst

The Gardner Lattice

In this section we consider the DDE described in [91],   
1 1 1 2 2 2 u˙ = 1 + αh u + βh u u−2 − u−1 + u1 − u2 h3 2 2  α  2 2 + u−1 u−2 + u−1 + u(u−1 − u1) − u1 − u1 u2 2h   β  2 2 + u (u−2 + u) − u1 (u + u2 ) , 2h −1

(189)

which is a completely integrable discretization of the Gardner equation, ut + 6αuux + 6βu2 ux + u3x = 0.

(190)

Therefore, we call (189) the Gardner lattice. Note that (190) is a combination of the KdV equation (β = 0) and the mKdV equation (α = 0). Consequently, (189) includes the completely integrable discretizations of the KdV and mKdV equations as special cases.

13.1

Derivation of the Gardner Lattice

Based on work by Taha [91], we outline the derivation of the Gardner lattice from a discretized version [4] of the eigenvalue problem of Zakharov and Shabat. Consider the discrete system [1], [7], 1 V2,n + Rn (t)V1,n, z = Cn (z)V1,n + Dn (z)V2,n,

V1,n+1 = zV1,n + Qn (t)V2,n,

V2,n+1 =

(191)

V˙ 1,n = An (z)V1,n + Bn (z)V2,n,

V˙ 2,n

(192)

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where, in general, the coefficients An through Dn depend on the potentials, Rn and Qn . The eigenvalue z is assumed to be time-independent. Step 1: Expressing the compatibility conditions, V˙ i,n+1 = DV˙ i,n , for i = 1, 2, and equating the coefficients of V1,n and V2,n, we get [7] Cn Qn − Bn+1 Rn = z(An+1 − An ), 1 Bn Rn − Cn+1 Qn = (Dn+1 − Dn ), z 1 Q˙ n + Dn Qn − An+1 Qn = Bn+1 − zBn , z 1 R˙ n + An Rn − Dn+1 Rn = Cn+1 − zCn . z

(193)

Step 2: Substituting the expansions [91], An =

2 X

z

2j

A(2j) n ,

Dn =

j=−2

Bn =

2 X j=−1

2 X

z 2j Dn(2j),

(194)

z 2j−1 Cn(2j−1) ,

(195)

j=−2

z

2j−1

Bn(2j−1) ,

Cn =

2 X j=−1

Direct Methods and Symbolic Software for Conservation Laws

61

into (193) and (193) and setting the coefficients of like powers of z equal to zero, we (2j) (2j) obtain a system of twenty equations for the eighteen unknowns functions An , Dn with (2j−1) (2j−1) j = −2, −1, 0, 1, 2; and Bn , Cn with j = −1, 0, 1, 2. The simplest equations ±5 arise from the coefficients of the terms in z and z ±4 : .

(4)

(−4)

An+1 − A(4) n = 0,

Dn+1 − Dn(−4) = 0,

(4)

Qn (Dn(4) − An+1 ) + Bn(3) = 0, (−4)

(−3)

(196) (−4)

Rn (A(−4) − Dn+1 ) + Cn(−3) = 0, n

Qn (Dn(−4) − An+1 ) − Bn+1 = 0,

(4)

(3)

Rn (A(4) n − Dn+1 ) − Cn+1 = 0.

(197) (198)

Step 3: We outline the solution process which follows the strategy in [7, Section 2.2a]. (4) (−4) (4) From (196), we conclude that An and Dn are independent of n. Hence, An = A˜(4) and (−4) ˜ (−4) are constants. The tilde will remind us that we are dealing with constants. Dn = D Solving (197) and (198), we get ˜ (−4) − A(−4) )Rn , Bn(3) = (A˜(4) − Dn(4))Qn , Cn(−3) = (D n ˜ (−4) − A(−4) )Qn−1 , C (3) = (A˜(4) − D(4))Rn−1 . Bn(−3) = (D n n n

(199) (200) (−4)

Substituting these results into two of the equations coming from z ±3 , we find that An = (4) ˜ (4) are constants. From equations corresponding to z ±3 , z ±1 , we A˜(−4) and Dn = D obtain (2)

∆A(2) = An+1 − A(2) ˜ n Rn−1 − Qn+1 Rn ), n n = α(Q ∆Dn(−2) = ∆A(−2) = n ∆Dn(2)

=

(−2) ˜ n Qn−1 − Rn+1 Qn ), Dn+1 − Dn(−2) = β(R (−2) ˜ n Rn+1 − Qn−1 Rn ), An+1 − A(−2) = β(Q n (2) Dn+1 − Dn(2) = α(R ˜ nQn+1 − Rn−1 Qn ),

(201) (202) (203) (204)

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˜ (4) and β˜ = D ˜ (−4) − A˜(−4) . Equations (201)–(204) are satisfied for where α ˜ = A˜(4) − D A(2) = A˜(2) − αQ ˜ n Rn−1 , n (−2) (−2) ˜ n−1 Rn , An = A˜ + βQ

˜ n Qn−1 , ˜ (−2) − βR Dn(−2) = D (2) (2) ˜ + αR Dn = D ˜ n−1 Qn ,

(205) (206)

˜ (−2) and A˜(−2) , D ˜ (2) are constants. Next, we solve equations (from the terms where A˜(2), D (±1) (±1) in z ±2 ) for Bn and Cn , yielding ˜ n+α Bn(1) = δQ ˜ Qn+1 − αQ ˜ n (Qn+1 Rn + Qn Rn−1 ), (−1) ˜ n+1 − βR ˜ n (Rn+1 Qn + Rn Qn−1 ), Cn = γ ˜ Rn + βR ˜ n−2 − βQ ˜ n−1 (Rn−1Qn−2 + Rn Qn−1 ), B (−1) = γ ˜ Qn−1 + βQ

(209)

˜ n−1 + αR = δR ˜ n−2 − αR ˜ n−1 (Qn−1 Rn−2 + Qn Rn−1 ),

(210)

n

Cn(1)

(207) (208)

˜ (2) and γ ˜ (−2) − A˜(−2). Next, we solve equations coming from where δ˜ = A˜(2) − D ˜ =D (0) (0) z ±1 , which involve ∆An and ∆Dn . These equations, which are similar in structure to

62

W. Hereman, P. J. Adams, H. L. Eklund, M. S. Hickman and B. M. Herbst

(201)-(204), yield A(0) ˜ (−Qn+1 Rn−1 − Qn Rn−2 + Qn Qn+1 Rn−1 Rn n = α
˜ n−1 Qn , +Qn−1 Qn Rn−2 Rn−1 + Q2 R2 + A˜(0) − δR

(211)

Dn(0) = β˜ (−Rn+1 Qn−1 − Rn Qn−2 + Rn Rn+1 Qn−1 Qn
˜ (0) − γ˜Qn−1 Rn , +Rn−1 Rn Qn−2 Qn−1 + R2n Q2n−1 + D

(212)

n

n−1

˜ (0) are constants. All the difference equations for the unknown coeffiwhere A˜(0) and D cients are now satisfied. We are left with the two DDEs (coming from the terms in z 0 ), h ˜ n+1 + γ ˜ n−2 Q˙ n − κ ˜ Qn + (1 − Qn Rn ) −αQ ˜ n+2 − δQ ˜ Qn−1 + βQ ˜ n−1 (Qn−2 Rn−1 + Qn−1 Rn ) +αQ ˜ n+1 (Qn+2 Rn+1 +Qn+1 Rn ) − βQ i ˜ +αQ ˜ n Qn+1 Rn−1 − βQn Qn−1 Rn+1 = 0, h ˜ n+2 − γ˜Rn+1 + δR ˜ n−1 + αR R˙ n + κ ˜Rn + (1 − Rn Qn ) −βR ˜ n−2 ˜ n+1 (Rn+2 Qn+1 +Rn+1 Qn ) − α +βR ˜ Rn−1 (Rn−2 Qn−1 + Rn−1 Qn ) i ˜ +βRn Rn+1 Qn−1 − αR ˜ n Rn−1 Qn+1 = 0,

(213)

(214)

˜ (0). where κ ˜ = A˜(0) − D Step 4: The terms −˜ κQn in (213) and κ ˜Rn in (214) could be removed with a suitable transformation. Accomplishing the same, we set κ ˜ = 0. Next, we substitute Q n = un ,

Rn = −h2 (α + βun ),

(215)

into (213) and (214). Next, we set β˜ = α ˜ and γ˜ = δ˜ to remove all constant terms. Doing so, (213) and (214) collapse into a single DDE, n ˜ n−1 − un+1 ) u˙ n + (1 + αh2 un + βh2 u2n ) α(u ˜ n−2 − un+2 ) + δ(u

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+ααh ˜ 2 (un−2 un−1 + u2n−1 + un (un−1 −un+1 ) −u2n+1 −un+1 un+2 ) o +β αh ˜ 2(u2n−1 (un−2 +un ) −u2n+1 (un +un+2 )) = 0,

(216)

˜ and δ, ˜ we consider the limit of Step 5: To fix the scale on t, as well as the constants α, ˜ β, (216) for h → 0. Using, limh→0 un (t) = u(x, t), limh→0 u˙ n (t) = ut (x, t), and substituting 1 lim un+m (t) = u(x, t) + mh ux (x, t) + (mh)2u2x (x, t) h→0 2 1 + (mh)3u3x (x, t) + · · · , 6

(217)

into (216), we obtain   1 3 ˜ 2 ˜ ut − 2h(δ + 2α)u ˜ x − 2h (δ + 8α) ˜ αuux + βu ux + u3x + O(h5 ) = 0. 6

(218)

Direct Methods and Symbolic Software for Conservation Laws

63

The term in h disappears when δ˜ = −2α. ˜ Substituting α ˜ = − 12 , δ˜ = 1, into (216) and (218), we get u˙ =

1 + αh2 u + βh2 u2






1 1 u−2 − u−1 + u1 − u2 2 2

  1 + αh2 u−1 u−2 + u2−1 + u(u−1 − u1 ) − u21 − u1 u2 2    1 + βh2 u2−1 (u−2 + u) − u21 (u + u2 ) , 2

(219)

and ut + h3 (6αuux + 6βu2 ux + u3x ) = 0,

(220)

where the O(h5 ) terms were ignored. Using a scale, t → h3 t, we get (189) and (190). Remarks. Step 2 is well suited for a CAS. Solving the system, as outlined in Step 3, is a challenging task, in particular, if attempted with pen and paper. The system consists of two DDEs and eighteen difference equations. None of the CAS has a build-in solver for such mixed systems. A fully automated solution is therefore impossible. We used a feedback mechanism which mimics what one would do by hand: Solve the simplest difference equations; enter that partial solution; let CAS simplify the entire system; repeat the process until all difference equations are satisfied and the two DDEs are simplified as far as possible. Continuing with (213) and (214), steps 4 and 5 were straightforward to implement. The five steps have been implemented in Mathematica [50]. Starting from (191), the code generates the Gardner lattice (219). The code could be modified to assist in the derivation of other completely integrable DDEs, such as discrete versions of the nonlinear Schr¨odinger and sine-Gordon equations.

13.2

Dilation Invariance of the Gardner Lattice

Since (189) is not uniform in rank we must introduce auxiliary parameters with weight. This can be done in several ways. One of the possibilities is to replace (189) by    1 1 u˙ = γ + αu + βu γ u−2 − u−1 + u1 − u2 2 2  α + u−1 u−2 + u2−1 + u(u−1 − u1) − u21 − u1 u2 2   β 2 2 + u−1 (u−2 + u) − u1 (u + u2 ) , 2

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2




(221)

where we have set h = 1 (by scaling) and introduced a parameter γ. Expressing uniformity of rank, setting W (Dt ) = 1, and solving the linear system for the weights, one finds that W (u) = W (α) = 14 , W (β) = 0, and W (γ) = 12 . So, we do not need a scale on β and (221) is invariant under the scaling symmetry 1

1

1

(t, u, α, γ) → (λ−1t, λ 4 u, λ 4 α, λ 2 γ).

(222)

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W. Hereman, P. J. Adams, H. L. Eklund, M. S. Hickman and B. M. Herbst

For β = 0, (221) reduces to    1 1 u−2 − u−1 + u1 − u2 u˙ = (γ + αu) γ 2 2 o α + u−1 u−2 + u2−1 + u(u−1 − u1) − u21 − u1 u2 , 2

(223)

which is a completely integrable discretization of the KdV equation, ut +6αuux +u3x = 0. Computing the weights, one can set W (α) = 0, which leads to W (u) = W (γ) = 12 . So, 1 1 (223) is invariant under the scaling symmetry (t, u, γ) → (λ−1t, λ 2 u, λ 2 γ). For α = 0, (221) reduces to   
1 1 2 u˙ = γ + βu γ u−2 − u−1 + u1 − u2 2 2   β 2 + u−1 (u−2 + u) − u21(u + u2) , (224) 2 which is a completely integrable discretization of the mKdV equation, ut +6βu2ux +u3x = 0. In this case, one can set W (β) = 0. Then W (u) = 14 and W (γ) = 12 . Thus, (224) is 1 1 invariant under the scaling symmetry (t, u, γ) → (λ−1t, λ 4 u, λ 2 γ).

13.3

Conservation Laws of the Gardner Lattice

One can either apply the method of Section 12 directly to (189) or, alternatively, apply to (221) the technique based on dilation invariance outlined in Sections 9.2, 11.2, and 11.4. In particular, one can use the “divide and conquer” strategy of Section 11.5 to split candidate densities into smaller pieces. Computational details can be found in [33] The results below were computed [58] with the method in Section 12. For q = 0 shifts there are two (nonpolynomial) density-flux pairs: (0)

ρ1

(0)

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J1

= ln (1 + αh2 u + βh2 u2), (225) n 1 α β = − (u−2 − u−1 − u + u1 ) + (2u−2 u − 4u−1u + 2u−1 u1 ) 2 h h
+α2 h u−1 (u−2 + u−1 + u) + u(u−1 + u + u1 ) + αβ h u2−1 u−2
+2u−2 u−1 u + 3u2−1 u + 3u−1 u2 + 2u−1 uu1 + u2 u1
o +2β 2 h u−1 u u−2 u−1 + u−1 u + uu1 , (226)

and (0) ρ2 (0)

J2

! h(α + 2βu) = arctanh p , α2 h2 − 4β n1 1p 2 2 = α h − 4β 2 (2u−1 − u−2 − 2u1 + u2 ) 4 h
+α u−2 u−1 + u2−1 + 2u−1 u + u2 + uu1
o +β u2−1 (u−2 + u) + u2(u−1 + u1) .

(227)

(228)

Direct Methods and Symbolic Software for Conservation Laws The next two (of infinitely many polynomial) densities are α ρ(1) = uu1 + u, β
(2) ρ = uu2 1 + αh2 u1 + βh2 u21 + αh2 uu1 (u + u1 ) 1 α α2 2 α2 2 2 + βh2 u2 u21 + (1 − h )u + h u , 2 β β 2β where the associated fluxes have been omitted due to length.

65

(229)

(230)

Special Cases. We consider two important special cases. The first few densities for (189) with β = 0 are: (0)

ρ1

ρ(1) ρ

(2)

(0)

= ln (1 + αh2 u), ρ2 = u, 1 = u2 + uu1 , 2  2

2

u21

= uu2 (1 + αh u1 ) + αh u

The first few densities for (189) with α = 0 are (0)

ρ1

(0)

= ln (1 + βh2 u2 ),

ρ2 = arctan

ρ(1) = uu1 ,

14 14.1

(231)  1 2 + uu1 + u . 3

p

 βhu ,

ρ(2) = uu2 (1 + βh2 u21) +

(232)

(233) 1 2 2 2 βh u u1. 2

(234)

Additional Examples of Nonlinear DDEs The Bogoyavlenskii Lattice

The Bogoyavlenskii lattice [21] and [90, Eq. (17.1.2)],   p p Y Y u˙ = u  uj − u−j  , j=1

(235)

j=1

is a generalization of the KvM lattice (110). For p = 2, lattice (235) becomes

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u˙ = u(u1u2 − u−1 u−2 ),

(236)

which is invariant under the following scaling symmetry 1

(t, u) → (λ−1 t, λ 2 u).

(237)

Lattice (236) has the following density-flux pairs (of infinitely many): ρ(0) = ln u, ρ

(1)

ρ

(2)

ρ(3)

= u,

J (0) = −(u−1 u−2 + u−1 u + uu1 ), J

(1)

= −uu−1 (u−2 + u1 ),

(2)

= uu1 , J = −u−1 uu1 (u−2 + u + u2 ),   1 = uu1 uu1 + u1 u2 + u2 u3 , 2

J (3) = −u−1 uu1 (u−2 uu1 + u2 u1 + u−2 u1 u2 + 2uu1 u2 + u1 u22 +u−2 u2 u3 + uu2 u3 + u22u3 + u2u3 u4 ).

(238) (239) (240) (241)

(242)

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W. Hereman, P. J. Adams, H. L. Eklund, M. S. Hickman and B. M. Herbst

For (236), we also computed the densities ρ(4) through ρ(9). Every time the rank increases by one, the number of terms in the density increases by a factor three. For example, ρ(9) has 2187 terms and the highest shift is 15.

14.2

The Belov–Chaltikian Lattice

The Belov–Chaltikian lattice [18, Eq. (12)], u˙ = u(u1 − u−1 ) + v−1 − v,

v˙ = v(u2 − u−1 ),

(243)

in invariant under the scaling symmetry (t, u, v) → (λ−1t, λu, λ2v).

(244)

The first few density-flux pairs (of infinitely many) are ρ(1) = u, J (1) = −u−1 u + v−1 , 1 2 ρ(2) = u + uu1 − v, 2 J (2) = −u−1 u2 − u−1 uu1 + uv−1 + u1 v−1 + u−1 v,   1 2 (3) 2 ρ = u u + uu1 + u1 + u1 u2 − v−2 − v−1 − v − v1 , 3

(245)

(246) (247)

where J (3) has been omitted due to length. Our results match these in [84].

14.3

The Blaszak–Marciniak Lattices

In [19], Blaszak and Marciniak used the R matrix approach to derive families of integrable lattices involving three and four fields. Below we consider two cases involving three fields. Examples based on four fields could be dealt with in a similar fashion [113]. The Blaszak-Marciniak three field lattice I [84, Eq. (2)], u˙ = w1 − w−1 ,

v˙ = u−1 w−1 − uw,

w˙ = w(v − v1 ),

(248)

is invariant under the scaling symmetry Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

1

3

(t, u, v, w) → (λ−1 t, λ 2 u, λv, λ 2 w).

(249)

We computed the following density-flux pairs of (248), which is a completely integrable lattice: ρ(0) = ln w, ρ

(1)

ρ

(2)

ρ(3) ρ(4) J (4)

= u,

J (0) = v, J

(1) (2)

= −w−1 − w,

= v, J = u−1 w−1 , 1 2 = J (3) = u−1 vw−1 − w−1 w, v + uw, 2 1 3 = v + uvw + uv1 w − ww1, 3 = w−1 (u−1 v 2 + u−1 uw − vw − v1 w).

(250) (251) (252) (253) (254) (255)

Direct Methods and Symbolic Software for Conservation Laws

67

Our results confirm those in [84] and [112]. The Blaszak–Marciniak three field lattice II [112, Eq. (1.4)], u˙ = v1 − v + u(w−1 − w),

v˙ = v(w−2 − w),

w˙ = u1 − u,

(256)

is invariant under the scaling symmetry (t, u, v, w) → (λ−1t, λ2u, λ3v, λw).

(257)

The first few density-flux pairs for (256), which is completely integrable, are ρ(1) = w, J (1) = −u, 1 ρ(2) = w2 − u, J (2) = v − uw−1, 2 1 2 ρ(3) = w3 + v − uw−1 − uw, J (3) = u−1 u + vw−2 + vw−1 − uw−1 , 3 1 1 2 ρ(4) = w4 + u2 + uu1 + vw−2 + vw−1 − uw−1 4 2 +vw − uw−1 w − uw2 , J

(4)

=

2 −u−2 v − uv − u−1 v1 + vw−2 + 2u−1 uw−1 2 3 +vw−2w−1 + vw−1 − uw−1 + u−1 uw.

(258) (259) (260)

(261) (262)

Our results agree with those in [84] and [112].

15

Software to Compute Conservation Laws for PDEs and DDEs

We first discuss our packages for conservation laws of PDEs and DDEs, followed by a brief summary of symbolic codes developed by other researchers.

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15.1

Our Mathematica and Maple Software

The package TransPDEDensityFlux.m[9] , automates the computation of conservation laws of nonlinear PDEs in (1+1) dimensions. In addition to polynomial PDEs, the software can handle PDEs with transcendental nonlinearities. The results in Sections 3 and 5 were computed with TransPDEDensityFlux.m and cross-checked with the newest version of condens.m, introduced in [38]. We used TransPDEDensityFlux.m to compute the density-flux pairs for the examples in Sections 6 and 7. Details about the algorithm and a discussion of implementation issues can be found in [8]. The code DDEDensityFlux.m [34] was used to compute the conservation laws in Sections 9 and 11. The results were cross-checked with the latest version of diffdens.m , featured in [39]. Using multiple scales, the efficiency of DDEDensityFlux.m was drastically improved. Nonetheless, the algorithms [33] within DDEDensityFlux.m are impractical for finding densities and fluxes of high rank. Therefore, we used the new Maple library discrete [56] to compute the results in Sections 13 and 14. Some of the features of earlier versions of condens.m and diffdens.m were combined into the InvariantsSymmetries.m [39], [41], which allows one to compute

68

W. Hereman, P. J. Adams, H. L. Eklund, M. S. Hickman and B. M. Herbst

generalized symmetries as well as conserved densities (but no fluxes). InvariantsSymmetries.m is available from MathSource, the Mathematica program bank of Wolfram Research, Inc. Our Mathematica packages and notebooks are available at [48] and Hickman’s Maple code is available at [56]. Our Mathematica codes for the continuous and discrete Euler and homotopy operators in one dimension are available at [49]. We are currently designing a comprehensive package to compute conservation laws of PDEs in multiple space dimensions [45], [51], [83]. Our codes have been used in a variety of research projects. For example, condens.m [38] was used by Sakovich and Tsuchida [85], [95] to compute conservation laws of nonlinear Schr¨odinger equations. In [84], Sahadevan and Khousalya use the algorithms of diffdens.m [42] and InvariantsSymmetries.m [39], [41] to compute conserved densities of the Belov–Chaltikian and Blaszak–Marciniak lattices. Ergenc¸ and Karas¨ozen [35] used our software in the design of Poisson integrators for Volterra lattices.

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15.2

Software Packages of Other Researchers

Our Mathematica code for conservation laws of PDEs has been “translated” [108], [109], [110] into a Maple package, called CONSLAW , which only handles PDEs in (1 + 1) dimensions. Based on the concept of dilation invariance and the method of undetermined coefficients, similar software was developed by Deconinck and Nivala [27] and Yang et al. [107]. Our algorithms [38], [42] for DDEs have been adapted to fully-discretized equations [36], [37]. There are several algorithms (see e.g. [101]) to symbolically compute conservation laws of nonlinear PDEs but few have been fully implemented in CAS. Wolf’s package ConLaw [101], [102], [106] computes first integrals of ODEs and conservation laws of PDEs. ConLaw uses the REDUCE package CRACK [103], [104], [105], which contains tools to solve overdetermined systems of PDEs. Wolf’s application packages heavily rely on the capabilities of CRACK, which took years to develop and perfect. Unfortunately, no such package is available in Mathematica. A common approach is to use the link between conservation laws and symmetries as stated in Noether’s theorem [15], [66], [81]. Among the newest software based on that approach is the Maple code GeM [25] by Cheviakov [23], [24], which allows one to compute conservation laws of systems of ODEs and PDEs based on the knowledge of generalized symmetries. However, the computation of such symmetries [47] is as difficult a task as the direct computation of conservation laws for it requires solving systems of overdetermined PDEs with, e.g., the Rif package [47], [100]. Some methods circumvent the existence of a variational principle (required by Noether’s theorem) [12], [20], [101], [106] but they still rely on software to solve ODEs or PDEs. The package DE APPLS [16], [77] also offers commands for constructing conservation laws from (variational) symmetries by Noether’s theorem, but the computation is not fully automated. Likewise, the package Noether [43] in Maple allows one to compute conservation laws from infinitesimal symmetry generators corresponding to (simple) Lagrangians. Based on the formal symmetry approach, Sokolov and Shabat [89], Mikhailov et al.

Direct Methods and Symbolic Software for Conservation Laws

69

[74], [75], and Adler et al. [10] classified completely integrable PDEs and DDEs in (1 + 1) dimensions. Unfortunately, the software used (see [38]) in the classification is obsolete. For completeness, we also mention the packages Jets by Marvan [73] and Vessiot by Anderson [16], [77]. Both are general purpose suites of Maple packages for computations on jet spaces. The commands within Jets and Vessiot use differential forms and advanced concepts from differential geometry. By avoiding differential forms, our codes were readily adaptable to nonlinear DDEs (not covered in Jets and Vessiot). Finally, Deconinck and Nivala [26] developed Maple software for the continuous and discrete homotopy operators. Their code is available at [28].

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16

Summary

We presented methods to symbolically compute conservation laws of nonlinear polynomial and transcendental systems of PDEs in (1 + 1) dimensions and polynomial DDEs in one discrete variable. The first part of this chapter dealt with nonlinear PDEs for which we showed the computation of densities and fluxes in detail. Using the dilation invariance of the given PDE, candidate polynomial densities are constructed as linear combinations with undetermined coefficients of scaling invariant building blocks. For polynomial PDEs, the undetermined coefficients are constants which must satisfy a linear system of algebraic equations. That system will be parameterized by constants appearing in the PDE, if any. For transcendental PDEs, the undetermined coefficients are functions which much satisfy a linear system which is a mixture of algebraic equations and ODEs. The continuous homotopy operator is a powerful, algorithmic tool to compute fluxes explicitly. Indeed, the homotopy operator handles integration by parts which allowed us to invert the total derivative operator. The methods for polynomial PDEs are illustrated with classical examples such as the KdV and Boussinesq equations and the Drinfel’d-SokolovWilson system. The computation of conservation laws of system with transcendental nonlinearities is applied to sine-Gordon, sinh-Gordon, and Liouville equations. In the second part we dealt with the symbolic computation of conservation laws of nonlinear DDEs. Again, we used the scaling symmetries of the DDE and the method of undetermined coefficients to find densities and fluxes. In analogy with the continuous case, to compute the flux one could use the discrete homotopy operator, which handles summation by parts and inverts the forward difference operator. However, in comparison with the “splitting and shifting” technique, the discrete Euler and homotopy operators are inefficient tools for the symbolic computation of conservation laws of DDEs. The undetermined coefficient method is illustrated with classical examples such as the Kac-van Moerbeke, Toda and Ablowitz-Ladik lattices. There is a fundamental difference between the continuous and discrete cases in the way densities (of selected rank) are constructed. The total derivative has a weight but the shift operator does not. Consequently, a density of a PDE is bounded in order (with respect to x). Unfortunately, there is no a priori bound on the number of shifts in the density, unless a leading order analysis is carried out. To overcome this difficulty and other shortcomings of the undetermined coefficient method, we presented a new method to compute conserved densities of DDEs. That method no longer uses dilation invariance

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W. Hereman, P. J. Adams, H. L. Eklund, M. S. Hickman and B. M. Herbst

and is no longer restricted to polynomial conservation laws. Instead of building a candidate density with undetermined coefficients, one first computes the leading order term in the density and, second, generates the correction terms of lower order. The method is fast and efficient since no unnecessary terms are computed. The new method was illustrated using a modified Volterra lattice as an example, and applied to lattices due to Bogoyavlenskii, Belov-Chaltikian, Blaszak-Marciniak, and Gardner. A derivation of the latter lattice was also given.

Acknowledgements This material is based in part upon work supported by the National Science Foundation (NSF) under Grant No. CCF-0830783 and the National Research Foundation (NRF) of South Africa under Grant No. FA2007032500003 Jan Sanders (Free University of Amsterdam) and Bernard Deconinck (University of Washington) are thanked for valuable discussions. Loren “Douglas” Poole is thanked for proof reading the manuscript. Willy Hereman is grateful for the hospitality and support of the Department of Mathematics and Statistics of the University of Canterbury (Christchurch, New Zealand) and the Applied Mathematics Division of the Department of Mathematical Sciences of the University of Stellenbosch (Stellenbosch, South Africa) during his sabbatical visits in AY 20072008.

References [1] M. J. Ablowitz, Nonlinear evolution equations–continuous and discrete. SIAM Rev. 19 (1977), 663–684.

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[2] M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering. London Mathematical Society Lecture Note Series 149, Cambridge University Press, Cambridge, 1991. [3] M. J. Ablowitz and B. M. Herbst, On homoclinic structure and numerically induced chaos for the nonlinear Schr¨odinger equation. SIAM J. Appl. Math. 50 (1990), 339– 351. [4] M. J. Ablowitz and J. F. Ladik, Nonlinear differential-difference equations. J. Math. Phys. 16 (1975), 598–603. [5] M. J. Ablowitz and J. F. Ladik, A nonlinear difference scheme and inverse scattering. Stud. Appl. Math. 55 (1976), 213–229. [6] M. J. Ablowitz and J. F. Ladik, Nonlinear differential-difference equations and Fourier analysis. J. Math. Phys. 17 (1976), 1011–1018. [7] M. J. Ablowitz and H. Segur, Solitons and The Inverse Scattering. SIAM Studies in Applied Mathematics 4, SIAM, Philadelphia, 1981.

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[8] P. J. Adams, Symbolic Computation of Conserved Densities and Fluxes for Systems of Partial Differential Equations with Transcendental Nonlinearities . Master of Science Thesis, Dept. of Mathematical and Computer Sciences, Colorado School of Mines, Golden, Colorado, 2003. [9] P. J. Adams and W. Hereman, TransPDEDensityFlux.m: A Mathematica package for the symbolic computation of conserved densities and fluxes for systems of partial differential equations with transcendental nonlinearities (2002). [10] V. E. Adler, A. B. Shabat, and R. I. Yamilov, Symmetry approach to the integrability problem, Theor. Math. Phys. 125 (2000), 1603–1661. [11] V. E. Adler, S. I. Svinolupov, and R. I. Yamilov, Multi-component Volterra and Toda type integrable equations. Phys. Lett. A 254 (1999), 24–36. [12] S. C. Anco and G. Bluman, Direct construction method for conservation laws of partial differential equations. Part I: Examples of conservation law classifications. Europ. J. Appl. Math. 13 (2002), 545–566. [13] S. C. Anco and G. Bluman, Direct construction method for conservation laws of partial differential equations. Part II: General treatment. Europ. J. Appl. Math. 13 (2002), 567–585. [14] I. M. Anderson, Introduction to the variational bicomplex, Contemp. Math. 132, AMS, Providence, Rhode Island (1992), 51–73. [15] I. M. Anderson, The Variational Bicomplex. Dept. of Mathematics, Utah State University, Logan, Utah (November, 2004) 318 pages. Manuscript of book available at http://www.math.usu.edu/˜fg_mp/Publications/VB/vb.pdf. [16] I. M. Anderson, The Vessiot Package. Dept. of Mathematics, Utah State University, Logan, Utah (November, 2004) 253 pages. Available at http://www.math.usu. edu/˜fg_mp/Pages/SymbolicsPage/VessiotDownloads.html.

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[17] A. Barone, F. Esposito, C. J. Magee, and A. C. Scott, Theory and applications of the sine-Gordon equation. Riv. Nuovo Cim. 1 (1971), 227–267. [18] A. A. Belov and K. D. Chaltikian, Lattice analogues of W −algebras and classical integrable equations. Phys. Lett. B 309 (1993), 268–274. [19] M. Blaszak and K. Marciniak, R matrix approach to lattice integrable systems. J. Math. Phys. 35 (1994), 4661–4682. [20] G. W. Bluman and S. C. Anco, Symmetry and Integration Methods for Differential Equations. Applied Mathematical Sciences 154, Springer Verlag, Berlin, 2002. [21] O. I. Bogoyavlenskii, Some constructions of integrable dynamical systems. Math. USSR Izv. 31 (1988), 47–75. [22] K. A. Bronnikov, Scalar vacuum structure in general relativity and alternate theories: conformal continuations. Acta Phys. Polonica B 11 (2001), 3571–3592.

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In: Advances in Nonlinear Waves and Symbolic Computation ISBN 978-1-60692-260-6 c 2009 Nova Science Publishers, Inc. Editor: Zhenya Yan

Chapter 3

N ONCLASSICAL P OTENTIAL S YMMETRIES FOR A N ONLINEAR T HERMAL WAVE E QUATION M. L. Gandarias∗ Departamento de Matematicas, Universidad de Cadiz PO. BOX 40, 11510 Puerto Real, Cadiz, Spain

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Abstract In this work classes of symmetries for partial differential equations which can be written in a conserved form are found. These nonclassical potential symmetries are realized as nonclassical symmetries of the associated potential system and are neither classical potential symmetries realized as Lie symmetries of a related auxiliary system nor nonclassical symmetries of the considered equation. Some of these symmetries are carried out for a nonlinear equation which serves as a model for thermal conductivity and thermal waves in a heated plasma, as well as for the Fokker–Planck equation. For some special values of the parameters it happens that the nonlinear thermal wave equation does not admit an infinite-parameter Lie group of contact transformations, so is not linearizable by an invertible mapping, however the associated potential system admits an infinite-parameter Lie group of point transformations and, consequently, the equation is linearized by a non-invertible mapping. In these cases, these nonclassical potential symmetries are realized as nonclassical symmetries of the linearized form. The similarity solutions are also discussed in terms of the linearized form and yield solutions of the nonlinear thermal wave equation which are neither nonclassical solutions nor solutions arising from classical potential symmetries.

1

Introduction

Symmetries admitted by a partial differential equation (PDE) are useful for finding invariant solutions, these solutions are obtained by using group invariants to reduce the number of independent variables. The fundamental basis of the technique is that, when a differential equation is invariant under a Lie group of transformations, a symmetry reduction exists which reduces the equation to a lower dimensional equation. The machinery of Lie group ∗

E-mail: [email protected]

80

M. L. Gandarias

theory provides the systematic method to search for these special group-invariant solutions. For PDE’s with two independent variables, as are the nonlinear thermal wave equation or porous medium equation   f (x) m n ut = (u )x + u , (1) m x and, when n = 1 and m = 1, the Fokker–Planck equation ut = uxx + (f (x)u)x,

(2)

a single group reduction transforms the PDE into ordinary differential equations (ODE’s), which are generally easier to solve than the original PDE. Most of the required theory and description of the method can be found in for example [5], [19], [27], [28], [33]. Symmetries admitted by a nonlinear PDE are also useful to discover whether or not the equation can be linearized by an invertible mapping and construct an explicit linearization when one exists. A nonlinear scalar PDE is linearizable by an invertible contact (point) transformation if and only if it admits an infinite-parameter Lie group of contact transformations satisfying specific criteria [4]–[6], [26]. There have been several generalizations of the classical Lie group method for symmetry reductions. Bluman and Cole [7] developed the nonclassical method to study the symmetry reductions of the heat equation. The basic idea of the method is to require that the N order PDE   ∆ = ∆ x, t, u, u(1)(x, t), . . ., u(N )(x, t) = 0, (3) where (x, t) ∈ R2 are the independent variables, u ∈ R is the dependent variable and u(l)(x, t) denote the set of all partial derivatives of l order of u and the invariance surface condition ∂u ∂u ξ +τ − φ = 0, (4) ∂x ∂t which is associated with the vector field

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u = ξ(x, t, u)

∂u ∂u ∂u + τ (x, t, u) + φ(x, t, u) , ∂x ∂t ∂u

(5)

are both invariant under the transformation with infinitesimal generator (5). Since then, a great number of papers have been devoted to the study of nonclassical symmetries of nonlinear PDE’s in both one and several dimensions. An obvious limitation of group-theoretic methods based in local symmetries, in their utility for particular PDE’s, is that there exists PDE’s of physical interest possessing few symmetries or none at all [27]. It turns out that PDE’s can admit nonlocal symmetries whose infinitesimal generators depend on integrals of the dependent variables in some specific manner. It also happens that if a nonlinear scalar PDE does not admit an infinite-parameter Lie group of contact transformations is not linearizable by an invertible contact transformation. However most of the interesting linearizations involve non-invertible transformations, such linearizations can be found by embedding given nonlinear PDE’s in auxiliary systems of PDE’s [5]. Krasil’shchik and Vinogradov [24], [25], [34] gave criteria which must be satisfied by nonlocal symmetries of a PDE when realized as local symmetries of a system of PDE’s

Nonclassical Potential Symmetries for a Nonlinear Thermal Wave Equation

81

which ‘covers’ the given PDE. Akhatov, Gazizov and Ibragimov [1] gave nontrivial examples of nonlocal symmetries generated by heuristic procedures. By using nonlocal symmetries, some exact solutions which are not similarity solutions of (1) for special values of n were obtained by King [22], [23]. In [8] Bluman introduced a method to find a new class of symmetries for a PDE. Suppose a given scalar PDE of second order F (x, t, u, ux, ut, uxx, uxt, utt) = 0,

(6)

where the subscripts denote the partial derivatives of u, can be written as a conservation law D D f (x, t, u, ux, ut) − g(x, t, u, ux, ut) = 0, Dt Dx for some functions f and g of the indicated arguments. Here operators defined by D Dx D Dt

= =

D Dx

and

D Dt

(7) are total derivative

∂ ∂ ∂ ∂ + ux + uxx + uxt + ··· , ∂x ∂u ∂ux ∂ut ∂ ∂ ∂ ∂ + ut + uxt + utt + ··· . ∂t ∂u ∂ux ∂ut

Through the conservation law (7) one can introduce an auxiliary potential variable v and form an auxiliary potential system (system approach) vx = f (x, t, u, ux, ut), vt = g(x, t, u, ux, ut). For many physical equations one can eliminate u from the potential system (8) and form an auxiliary integrated or potential equation (integrated equation approach) G(x, t, v, vx, vt, vxx, vxt, vtt) = 0,

(8)

for some function G of the indicated arguments. Any Lie group of point transformations ∂ ∂ ∂ ∂ (9) + τ (x, t, u, v) + φ(x, t, u, v) + ψ(x, t, u, v) , ∂x ∂t ∂u ∂v admitted by (8) yields a nonlocal symmetry potential symmetry of the given PDE (7) if and only if the following condition is satisfied  2  2  2 ∂ξ ∂τ ∂φ + + 6= 0. (10) ∂v ∂v ∂v

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v = ξ(x, t, u, v)

We point out that if we consider a Lie group of point transformations w = ξ(x, t, v)

∂ ∂ ∂ + τ (x, t, v) + ψ(x, t, v) ∂x ∂t ∂v

(11)

admitted by (8) the condition 

∂ξ ∂v

2

+



∂τ ∂v

2

6= 0

(12)

is a sufficient but not necessary condition in order to yield nonlocal symmetries of (7). The nature of potential symmetries allows one to extend the uses of point symmetries to such nonlocal symmetries. In particular:

82

M. L. Gandarias 1) Invariant solutions of (8), yield solutions of (6) which are not invariant solutions for any local symmetry admitted by (6). 2) If (6) admits a potential symmetry leading to the linearization of (8), then (6) is linearized by a non-invertible mapping.

Bluman [5] gave theorems which give necessary and sufficient conditions under which nonlinear partial differential equations (scalar or systems) can be transformed to linear PDE’s by invertible mappings. In particular such an invertible mapping does not exist if 1) a nonlinear scalar PDE does not admit an infinite-parameter Lie group of contact transformations; 2) a nonlinear system of PDE’s does not admit an infinite-parameter Lie group of point transformations. Suppose (6) cannot be linearized by an invertible mapping but an associated system (8), admits an infinite-parameter Lie group of point transformations which leads to its linearization by an invertible mapping; then (6) is linearized by a non-invertible mapping. In [10] two algorithms were proposed which extend the nonclassical method to a potential system (8) or a potential equation (8): • Algorithm I Nonclassical potential system approach: The nonclassical method is applied to the associated potential system (8). Any Lie group of point transformations with generator (9) admitted by (8) yields a nonlocal symmetry potential symmetry of the given PDE (7) if condition (10) is satisfied.

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• Algorithm II Nonclassical potential equation approach: The nonclassical method is applied to the associated potential equation (8). Any Lie group of point transformations with generator (11) admitted by (8) yields a nonlocal symmetry potential symmetry of the given PDE (7) if condition (12) is satisfied. Algorithm I has been considered in Bluman and Shtelen [9] and Saccomandi [32], Algorithm II has been considered in [18] for a dissipative KdV equation, but neither of these papers exhibited nonclassical potential solutions. The nonclassical symmetries for the Burgers have been considered in [2], [29]. The nonclassical potential symmetries for the Burgers equation have been derived in [16] as nonclassical symmetries of the integrated equation (Algorithm II). In [10] it was pointed out that often the nonclassical method when it is applied to the potential system (Algoritm I) yields a set of undetermined determining equations while the nonclassical method it is much easier to apply to the potential equation (Algorithm II). However we point out that a great disadvantage of Algoritm II is that condition (12) is a sufficient but not necessary condition in order to see if a generator is a nonclassical potential generator or not. In [17] we proposed a modification to the nonclassical potential system approach, in a way in which is easy to apply and we can give a sufficient and necessary condition in order to see if a generator is a nonclassical potential generator. Note that if the generator considered is not a nonclassical potential generator then no new solutions are found, i.e. all such solutions can be obtained from the nonclassical method applied to the given PDE (7).

Nonclassical Potential Symmetries for a Nonlinear Thermal Wave Equation

83

• Modified Algorithm I Modified nonclassical potential system approach: The nonclassical method is applied to the associated potential system (8). Any Lie group of point transformations (9) in which we require that ∂ξ ∂τ ∂ψ = = = 0, ∂u ∂u ∂u

(13)

that is, any generator ∂ ∂ ∂ ∂ + τ (x, t, v) + φ(x, t, u, v) + ψ(x, t, v) , (14) ∂x ∂t ∂u ∂v admitted by (8) yields a nonlocal symmetry potential symmetry of the given PDE (7) if condition (10) is satisfied. In [17] the nonclassical potential symmetries for the Burgers equation were derived as nonclassical symmetries, for which conditions (13) are satisfied, of the potential associated system (Modified Algorithm I). The nonlinear equation v ˆ = ξ(x, t, v)

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ut = (un )xx + g(x)um + f (x)us ux

(15)

corresponds to porous media with sources or thermal evolution with sources and convection. This equation exibits a wide variety of wave phenomena, some of them were studied for f(x)=constant and g(x)=constant by Rosenau and Kamin [31]. The third term on the right hand side of (15) is of convective nature. In the theory of an unsatured porous medium medium, the convective part represents the effect of gravity. The second term on the right side describes volumetric absorption, which in the case of plasma is caused by radiation to which the plasma is transparent. There is no fundamental reason to assume the spatial dependent factors in (15) to be constant. Actually, allowing for their spatial dependence enables to incorporate additional factors into the study which may play an important role. For instance, in a porous medium this may account for stationary factors like mediums contamination with another material or in plasma, this may express the impact that solid impurities coming from the walls, have on the enhacement of the radiation channel. In [14] a group classification problem for equation (15) was solved, by studying those spatial forms which admit the classical symmetry group. Both the symmetry group and the spatial dependence were found through consistent application of the Lie-group formalism. The reductions obtained from the optimal system of subalgebras were derived. In order to find the potential symmetries of (15), we must write this equation in the conserved form; when m = s + 1 and mg(x) = f 0 (x), equation (15) can be written in the conserved form (1). The potential symmetries for (1) were classified in [15] these symmetries were realized as local symmetries of the related auxiliary system (16) and lead to the construction of the corresponding invariant solutions. For some special values of n and m it happens that (1) does not admit an infinite-parameter of contact transformations, so is not linearizable by an invertible mapping, however the associated system (16) admits an infinite-parameter Lie group of point transformations and, consequently, equation (1) is linearized by a noninvertible mapping, in these cases, we will consider the linearized form.

84

M. L. Gandarias

The aim of this work is to obtain nonclassical potential symmetries for the nonlinear thermal wave equation or porous medium equation (1) as well as for the Fokker–Planck equation (2). For the Fokker–Planck equation we apply the modified potential system approach to (2). For the nonlinear thermal wave equation or porous medium equation (6), we may consider the linearized form and apply the modified potential system approach to the associated linear system.

2

Nonclassical Potential Symmetries for the Nonlinear Thermal Wave Equation

In this section we discuss nonclassical potential symmetries of equation (1) by considering a linearized system. The classical potential symmetries of (1) were classified in [15]. These symmetries were realized as local symmetries of the related potential system vx = u, (16) f (x) m u . m For n 6= 0, if (9) is the infinitesimal generator that leaves (16) invariant then we have [15] that vt = (un )x +

ξ = ξ(x, t, v), τ = τ (t), φ = −ξv u2 + (ψv − ξx )u + ψx and ψ = ψ(x, t, v). We can distinguish the following cases depending on n, m and f

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2.1

Case n = −1, m = 1 and f (x) = cx

The nonlinear equation (1) with n = −1, m = 1 and f (x) = cx does not admit an infiniteparameter Lie group of contact transformations; however its associated auxiliary system (16) admits an infinite-parameter Lie group of point transformations with infinitesimal generator ∂ ∂α 2 ∂ X∞ = α(v, t) − u , ∂x ∂v ∂u where α(v, t) is an arbitrary function satisfying the following equation ∂ 2 α ∂α + + cα = 0, ∂v 2 ∂t One can obtain the invertible mapping 1 , u which transforms any solution (w1(z1 , z2), w2(z1 , z2)) of the linear system z1 = v, z2 = t, w1 = x, w2 =

∂w1 = w2 , ∂z1 ∂w2 ∂w1 = − cw1, ∂z1 ∂z2

(17)

(18)

(19)

Nonclassical Potential Symmetries for a Nonlinear Thermal Wave Equation

85

to a solution (u(x, t), v(x, t)) of the nonlinear system (16) and hence to a solution u(x, t) of (1). In order to obtain nonclassical potential symmetries for equation (1) we apply the Modified Algorithm I. The idea is that the associated linear system (19) is augmented with the invariance surface condition ξ

∂w1 ∂w1 +τ − ψ = 0. ∂z1 ∂z2

(20)

Then we require both (19) and (20) to be invariant under the transformations with infinitesimal generator ∂ ∂ ∂ + τ (z1 , z2, w1) + ψ(z1, z2, w1) + ∂z1 ∂z2 ∂w1 ∂ + φ(z1 , z2, w1, w2) . ∂w2

v ˘ = ξ(z1 , z2, w1)

(21)

If the following condition 

∂ξ ∂w1

2

+



∂τ ∂w1

2

+



∂φ ∂w1

2

6= 0

(22)

is satisfied we get a nonclassical potential symmetry of the linear heat equation ∂w2 ∂ 2 w2 = + cw2. ∂z2 ∂z12

(23)

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Applying the nonclassical method to (19), from the determining equations we obtain φ = (ψw1 − ξz1 )w2 + ψz1 ,

(24)

ξ = ξ(z1 , z2),

(25)

ψ = ψ1(z1 , z2)w1 + ψ2 (z1, z2),

(26)

where ξ, ψ1, and ψ2 must be related by ∂ψ1 ∂ 2ξ ∂ξ ∂ξ + −2ξ − = 0, ∂z1 ∂z12 ∂z1 ∂z2 ∂ 2 ψ1 ∂ψ1 ∂ψ1 ∂ξ − − 2c + ψ1 = 0, +2 2 ∂z1 ∂z1 ∂t ∂z1 ∂ 2 ψ2 ∂ψ2 ∂ξ − − cψ2 + +2 ψ2 = 0. ∂z2 ∂z1 ∂z12 −2

(27) (28) (29)

Although we are not able to solve this system in general we can obtain some solutions, setting ξ = ξ(x), ψ1 = ψ1(x) and ψ2 = 0, we obtain   1 dξ 2 ψ1 = − ξ − cξ (30) 2 dz1

86

M. L. Gandarias

and ξ must satisfy the following equation 1 d2 ξ 1 − + 2 2 2ξ dz1 4ξ



dξ dz1

2

+

dξ ξ 2 k2 − − 2 + k1 = 0, dz1 4 ξ

(31)

where k1 = c + k. Setting k1 = k2 = 0, and making the change of variables w2(z1 ) ξ(z1 ) = − z1 , R w2(s)ds

(32)

d2 w − kw = 0. dz12

(33)

equation (31) can be written as

When (22) is satisfied, we have a nonclassical potential symmetry . We can now distinguish the following subcases. Subcase 1.1 : If k = 0 we have w = az1 + b,

ψ1 =

ξ=−

(az1 + b)2 a2 z13 + abz12 + b2z1 3

, (34)

a2k1 z13 + 3abk1z12 + 3b2k1 z1 − 3a2 z1 − 3ab z1 (a2 z12 + 3abz1 + 3b2)

and we obtain the nonclassical symmetry reduction b3 a z12 + 2 b z1 − − z2 , 6a 3 a3 z1 + 3 a 2 b  3  a k1 z13 + 3 a2 b k1 z12 + 2 a b2 k1 z1 + 2 b3 k1 u = h(z) (a z1 + b) exp − . 6 a3 z1 + 6 a2 b Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

z=−

(35)

Here h(z) satisfies d2 h dh + 2c + c2 h = 0, 2 dz dz which has solution h = (k3 z + k4 ) e−c z . Consequently, an exact solution for (19) is
k3 ec z2 a3 z13 + 3 a2 b z12 + 2 a b2 z1 + 2 b3 w1 = − − 6 a2 − (k3 z2 − k4 ) ec z2 (a z1 + b) , w2 =

∂w1 . ∂z1

(36)

Nonclassical Potential Symmetries for a Nonlinear Thermal Wave Equation

87

By (18), we obtain for (16) the following solution
k 3 ec t a 3 v 3 + 3 a 2 b v 2 + 2 a b 2 v + 2 b3 x=− − (k3 t − k4 ) ec t (a v + b) , 6 a2 u=

1 w2

(37)

.

Subcase 1.2: If k = d2 we have ξ=− ψ1 = −

a ed z1 + b e−d z1 2 −2 d z a2 e2 d z1 − b e2 d 1 2d 2 4dz1 2 4dz1

a ke

−a ce


2

,

+ 2 a b z1

(38)

−4abk3/2z1 e2dz1 −4abcdz1e2dz1 − b2 k+b2c , a2 e4dz1 + 4abdz1e2dz1 − b2

and we obtain the nonclassical reduction a z1 e2 d z1 − b z1 − z2 , 2 a d e2 d z1 + 2 b d   h(z)(ae2dz1 + b) dz1((3d2 + c)ae2dz1 + b(d2 − c)) w1 = exp − . 2d2(ae2dz1 + b) 3d2 + c a 4d2 z=−

(39)

Here h(z) satisfies

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d2 h dh + 2(c + d2 ) + (c + d2 )2h = 0. 2 dz dz Consequently, an exact solution for (19) is

2 2 (k3 z2 −k4 ) ae2dz1 +b e(c+d )z2−dz1 k3 z1 ae2dz1 −b e(c+d )z2−dz1 w1 = − − , c c a 4k+3/4 2a 4k +3/4 d ∂w1 w2 = . ∂z1 By (18), we obtain that a solution for (16) is given by

2 2 (k3 t − k4 ) a e2 d v + b e(c+d ) t−d v k3 v a e2 d v − b e(c+d ) t−d v x=− − , c c a 4 k +3/4 2 a 4 k +3/4 d 1 u= . w2

(40)

(41)

(42)

Subcase 1.3: If k = −d2 then w = asin (dx) + bcos (dx). Although the general form for the infinitesimals are complicated to obtain, we derive some nonclassical reductions Setting b = 0 we have ξ=

4 d sin2 (d z1) 2 d2 sin (2 d z1) , ψ1 = + k1 sin (2 d z1) − 2 d z1 sin (2 d z1) − 2 d z1

(43)

88

M. L. Gandarias

and we obtain the nonclassical symmetry reduction z1 cot (d z1) z= − z2 , 2d

1


c − d 2 z1 = h(z) sin (d z1) exp , 2 d tan (d z1)

(44)

where h(z) satisfies
dh
2 d2 h + d2 − c h = 0. − 2 d2 − c 2 dz dz Consequently, for (19) an exact solution is   k3 z1 cos (d z1 ) c z2 −d2 z2 w1 = e − (k3 z2 − k4 ) sin (d z1) , 2d

(45)

w2 =

∂w1 ∂z1

(46)

and by (18), we obtain for (16) the following solution x=e

c t−d2 t



 k3 v cos (d v) 1 − (k3 t − k4 ) sin (d v) , u = . 2d w2

(47)

Setting a = 0 we have ξ=−

4dcos 2 (dz1) , sin (2dz1) + 2dz1

ψ1 =

2d2sin(2dz1) + k1 sin (2dz1) + 2dz1

(48)

and we obtain the nonclassical symmetry reduction z=−

z1 tan (dz1) (d2 − c)z1sin (dz1) − z2, w1 = 2 h(z) cos (d z1) exp 2d 2dcos (dz1)

(49)

where h(z) satisfies (45). Consequently, for (19), an exact solution is

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w1 = e

c z2 −d2 z2



 k3 z1 sin (d z1) ∂w1 − − 2 (k3 z2 − k4 ) cos (d z1) , w2 = . (50) d ∂z1

By (18), we obtain for (16) the following solution x=e

2.2

c t−d2 t



 k3 v sin (d v) 1 − . − 2 (k3 t − k4 ) cos (d v) , u = d w2

(51)

Case n = −1, m = −1, and f (x) arbitrary

The nonlinear equation (1) with n = −1, m = −1, and f (x) arbitrary does not admit an infinite-parameter Lie group of contact transformations; however its associated auxiliary system (16) admits an infinite-parameter Lie group of point transformations with infinitesimal generator ∂ ∂α ∂ X∞ = α(v, t)eg − eg ( u2 + f uα) , ∂x ∂v ∂u

Nonclassical Potential Symmetries for a Nonlinear Thermal Wave Equation

89

where α(v, t) is an arbitrary function satisfying the linear heat equation ∂α ∂ 2 α + = 0, ∂t ∂v 2 Zx Zx g(x) = f (s) ds, and h(x) = exp(−g(s)) ds.

(52)

One can easily obtain the invertible mapping  w   x  Zx Z Z 1 z1 = v, z2 = t, w1 = exp − f (s) ds dw, w2 = − exp − f (s) ds , (53) u which transforms any solution (w1(z1 , z2), w2(z1 , z2)) of the linear system ∂w1 = w2 , ∂z1 ∂w1 ∂w2 = , ∂z2 ∂z1

(54)

to a solution (u(x, t), v(x, t)) of the nonlinear system (16) and hence to a solution u(x, t) of (1). Now to study the nonclassical symmetries of (54) we require (54) and (20) to be invariant under the infinitesimal generator (21) and in order to obtain potential nonclassical symmetries we require that (22) must be satisfied. In the case τ 6= 0 without loss of generality we may set τ (x, t, u) = 1. The nonclassical method applied to (54) give rise to nonlinear determining equations for the infinitesimals. Solving these equations we obtain

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ξ = ξ(z1 , z2), ψ = ψ1 (z1, z2)w1 + ψ2(z1 , z2) and ξ2 , ψ1 and ψ2 are related by (27)–(29) with c = 0. Although these equations are too complicated to be solved in general some solutions can be obtained. Choosing ξ = ξ(z1 ), ψ1 = ψ1 (z1) and ψ2 = 0 we obtain that ψ1 is given by (30) and ξ2 satisfies (31). Making the change of variables (32) we obtain that (31) can be written as (33) with c = 0. We can now distinguish the following subcases: Subcase 2.1: If k1 = 0 we have that w, ξ and ψ1 are given by (34) and we obtain the nonclassical symmetry reduction z=−

a z12 + 2 b z1 b3 − − z2 , 6a 3 a3 z1 + 3 a2 b

where h(z) satisfies

d2 h dz2

u = h(z) (a z1 + b)

(55)

= 0. An exact solution of (54) is


k3 a3 z13 + 3a2 bz12 + 2ab2z1 + 2b3 w1 = − − (k3 z2 − k4 ) (az1 + b) , 6a2 ∂w1 w2 = ∂z1

(56)

90

M. L. Gandarias

and by (53) we obtain, for (16), the following implicit solution
Zx Rw k3 a3v 3 +3a2 bv 2 +2ab2v+2b3 − f (s) ds e dw = − −(k3 t−k4) (av+b) , 6a2  x  Z 1  u= exp − f (s) ds . w2

(57)

Subcase 2.2 : If k1 = d2 we have that w, ξ and ψ1 are given by (38) and we obtain the nonclassical symmetry reduction z=−

az1 e2dz1 − bz1 − z2 , 2ade2dz1 + 2bd

w1 = h(z)(ae

2dz1

+ b)a

−3d2 /(4d2 )



 dz1 (3d2ae2dz1 + d2b) exp − , 2d2(ae2dz1 + b)

where h(z) satisfies (??) with c = 0. Consequently, an exact solution for (54) is
2
2 (k3 z2 − k4 ) ae2dz1 + b ed z2 −dz1 k3 z1 a e2 d z1 − b ed z2 −dz1 w1 = − − , a3/4 2a3/4d ∂w1 w2 = . ∂z1

(58)

(59)

By (53), we obtain the following implicit solution of (16) Zx

e

−

Rw

f (s) ds



(k3 t−k4) ae2dv +b ek1 t−dv k3v ae2dv −b ek1 t−dv dw = − − , a3/4 2a3/4d   x Z 1  u=− exp − f (s) ds , w2

(60)

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that leads to a solution of (1). Subcase 2.3: If k1 = −d2 . Setting b = 0 we have that w, ξ2 and s1 are given by (43) and we obtain the nonclassical symmetry reduction z=

−d z1 z1 cot (dz1 ) − z2 , w1 = h(z) sin (d z1) e 2 tan(d z1 ) , 2d

where h(z) satisfies d2 h dh − 2 d2 + d4 h = 0, dz 2 dz which has solution

2

h = (k3 z + k4 ) ed z . Consequently, an exact solution for (54) is   k3 z1 cos (d z1) ∂w1 −d2 z2 w1 = e − (k3 z2 − k4) sin (d z1) , w2 = 2d ∂z1

(61) (62)

(63)

Nonclassical Potential Symmetries for a Nonlinear Thermal Wave Equation and by (18), we obtain for (16) the following implicit solution  w    Zx Z k3 v cos (d v) −d2 t   exp − f (s) ds dw = e − (k3 t − k4 ) sin (d v) , 2d  x  Z 1 exp − f (s) ds . u=− w2

91

(64)

Setting a = 0 we have that w, ξ and ψ1 are given by (48) and we obtain the nonclassical symmetry reduction z=−

d z1 tan (d z1) − z2 , w1 = 2 h(z) cos (d z1) e 2 z1 tan (dz1) , 2d

where h(z) satisfies (61) which has solution (62) and so an exact solution for (54) is   k3 z1 sin (d z1) −d2 z2 w1 = e − − 2 (k3 z2 − k4 ) cos (d z1) , d ∂w1 w2 = . ∂z1

(65)

By (18), we obtain for (16) the following implicit solution  w    Zx Z k3 v sin (d v) −d2 t   exp − f (s) ds dw = e − − 2 (k3 t − k4 ) cos (d v) , d   x (66) Z 1 u = − exp − f (s) ds . w2

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We remark that all the nonclassical generators considered (34), (38) and (43), where φ is given by (24), do satisfy condition (10) and are nonclassical potential generators.

3

Nonclassical Symmetries and Nonclassical Potential Symmetries for the Fokker–Planck Equation

The classical symmetries for the Fokker–Planck equation with drift (2) were derived in [5]. The authors found [5] that, besides the infinite dimensional generator, when f (x) satisfies any of these Riccati equations f0 f2 − 2 4 0 f f2 − 2 4

= ax2 + bx + c, = a(x + λ)2 + c +

(67) d , (x + λ)2

then (2) is invariant under a Lie group with four or two parameters respectively.

(68)

92

M. L. Gandarias

The classical potential symmetries were derived by Pucci and Saccomandi in [30] by using the natural potential system vx = u,

(69)

vt = ux + f (x)u.

They found [30] that, besides the infinite dimensional generator, when f (x) satisfies any of these Riccati equations f0 f2 + 2 4 f0 f2 + 2 4

= ax2 + bx + c, = a(x + λ)2 + c +

(70) d , (x + λ)2

(71)

then (2) is invariant under a Lie group with four or two parameters respectively. In order to apply the nonclassical method to the Fokker–Planck equation (2), we require (2) augmented with the invariance surface condition ∂u ∂u +τ − φ = 0, (72) ∂x ∂t to be invariant under the infinitesimal generator (5). In the case τ 6= 0 without loss of generality we may set τ (x, t) = 1. The nonclassical method applied to (2) give rise to four nonlinear determining equations for the infinitesimals. ξ

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∂ 2ξ = 0, ∂u2 ∂ 2φ ∂ 2ξ ∂ξ − 2 +2 − 2 (ξ + f ) = 0, ∂u ∂u∂x ∂u ∂ 2φ ∂ξ ∂ 2ξ ∂ξ ∂ξ ∂ξ −3 f 0 u−2 +2 φ+ − − f 0 ξ = 0, − (2 ξ + f ) 2 ∂u ∂u∂x ∂u ∂x ∂x ∂t ∂φ ∂ξ ∂ 2φ ∂φ ∂φ ∂ξ f0 u − 2 f0 u − f 00 ξu − −f + +2 φ − f 0 φ = 0. 2 ∂u ∂x ∂x ∂x ∂t ∂x Solving these equations we obtain

(73) (74) (75) (76)

ξ = ξ(x, t), φ = φ1 (x, t)u + φ2(x, t), where ξ, φ1 , φ2 and f are related by the following conditions ∂φ1 ∂ 2ξ ∂ξ ∂ξ ∂ξ + − 2ξ −f − − f 0 ξ = 0, (77) ∂x ∂x2 ∂x ∂x ∂t ∂ 2 φ1 ∂ξ ∂φ1 ∂φ1 ∂ξ − −f (78) + +2 φ1 − 2 f 0 − f 00ξ = 0, ∂x2 ∂x ∂t ∂x ∂x ∂ 2 φ2 ∂φ2 ∂φ2 ∂ξ − −f (79) + +2 φ2 − f 0 φ2 = 0. ∂x2 ∂x ∂t ∂x Although we are not able to solve this system in general we can obtain some solutions, setting ξ = ξ(x), φ1 = φ1(x), we obtain   1 dξ 2 φ1 = (80) − ξ − f ξ − k1 2 dx −2

Nonclassical Potential Symmetries for a Nonlinear Thermal Wave Equation

93

and ξ must satisfy the following equation ξ d2 ξ dξ 1 + ξ2 − + 2 2 dx dx 4



dξ dx

2

ξ 4 k1 ξ 2 − − + 4 2



 f2 1 0 2 − f ξ − k2 = 0. 4 2

(81)

These results are completely different to the ones obtained for classical and classical potential symmetries because now for any arbitrary function f (x) ξ can be obtained from (81) and for this choice of ξ(x) we can derive φ1. Setting k1 = k2 = 0 and making the change of variables w2(x) , (82) ξ(x) = − x R 2 w (s) ds then (81) can be written as d2 w − dx2



f0 f2 − 2 4



w = 0.

(83)

As an example we have that f (x) = −2tan (x), does not satisfy (67) nor (68) consequently the corresponding generator ξ(x) =

1 2 , τ = 1, φ = − 2 u cos(x)sin(x) cos (x)

(84)

is a nonclassical generator. Solving the invariance surface condition (4) we obtain the nonclassical reduction cos(x) h(z) z= − t, u = , (85) 2 cos2(x) where h(z) = k1z + k2. To obtain nonclassical potential symmetries for the Fokker–Planck equation, we apply the Modified Algorithm I, and we require (69) and the invariance surface condition

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ξ

∂v ∂v +τ − ψ = 0, ∂x ∂t

(86)

to be invariant under the infinitesimal generator (14). In the case τ 6= 0, without loss of generality, we may set τ (x, t, v) = 1. Applying the nonclassical method to (69) yields φ = (ψv − ξx )u + ψx , ξ = ξ(x, t), ψ = ψ1 (x, t)v + ψ2(x, t),

(87)

where ξ, ψ1, ψ2 and f are related by −2

∂ψ1 ∂ 2ξ ∂ξ ∂ξ ∂ξ + − 2ξ −f − − f 0 ξ = 0, ∂x ∂x2 ∂x ∂x ∂t ∂ 2 ψ1 ∂ψ1 ∂ψ1 ∂ξ − −f + +2 ψ1 = 0, 2 ∂x ∂x ∂t ∂x ∂ 2 ψ2 ∂ψ2 ∂ψ2 ∂ξ − −f + +2 ψ2 = 0. ∂x2 ∂x ∂t ∂x

(88) (89) (90)

94

M. L. Gandarias

Although we are not able to solve this system in general we can obtain some solutions, setting ξ = ξ(x), ψ1 = ψ1(x) and ψ2 = 0, we obtain   1 dξ ψ1 = − ξ2 − f ξ (91) 2 dx and ξ and f must satisfy the following equation  2  2  1 d2 ξ ξ 2 k1 1 0 1 dξ dξ f − − + 2+ + f = 0. + + 2ξ dx2 4ξ 2 dx dx 4 ξ 4 2

(92)

Consequently, we have that for any arbitrary function f (x), by (92) we can obtain ξ and by (91) we get ψ1 . When f (x) does not satisfy (67) nor (68) then these generators are unobtainable by Lie classical symmetries or classical potential symmetries of the natural potential system (69) and if (10) is satisfied, we have a nonclassical potential symmetry . As an example, f (x) = 5tanh(x), (93) yields the following generator ξ(x) = −3coth(x), τ = 1, ψ = (3coth2 (x) − 6cosech 2 (x))v.

(94)

The corresponding reduction is 1 h(z)sinh x z = − log(cosh(x)) − t, v = , 3 cosh2(x) where h(z) satisfies the following ODE h00 − 36h = 0, whose solution is h(z) = k3e6z + k4 e−6z .

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Consequently, an exact solution for (69) and for (2) is given by
v = tanh (x) k3e−6t sech3 (x) + k4e6t cosh (x) ,
u = k3 e−6t 4sech5 (x) − 3sech3 (x) + k4 e6tcosh (x). We remark that (93) does not satisfy (67) nor (68), and (94) with φ given by (87) does satisfy (10), consequently we have derived a nonclassical potential symmetry reduction.

4

Concluding remarks

In this work we have derived nonclassical potential symmetries for PDE’s. If a PDE can be written in a conserved form (7), then a related system (8) and a related integrated equation may be obtained. The ansatz to generate nonclassical solutions of the associated potential system could yield solutions of the original equation which are neither nonclassical solutions of the original equation nor solutions arising from classical potential symmetries. To

Nonclassical Potential Symmetries for a Nonlinear Thermal Wave Equation

95

explain these results, we have proved the existence of families of solutions of the Fokker– Planck equation which are not found by means of potential nor nonclassical symmetry reductions. We have introduced new classes of symmetries for the nonlinear thermal wave equation or porous medium equation (1). The nonlinear equation (1), with n = −1, m = 1 and n = −1, m = −1 does not admit an infinite-parameter of contact transformations, so it is not linearizable by an invertible mapping, however the associated system (16) admits an infinite parameter Lie group of point transformations and consequently, (1) is linearized by a non-invertible mapping. In these cases we have considered the linearized form and the associated integrated linear equation. The ansatz to generate nonclassical solutions of the associated potential system yields new solutions of (1) which are neither nonclassical solutions of (1) nor solutions arising from potential symmetries.

Acknowledgements The support of DGICYT project MTM2006-05031, Junta de Andaluc´ıa group FQM-201 and project P06-FQM-01448 are gratefully acknowledged.

References [1] Akhatov I. Sh., Gazizov R. K. and Ibragimov N. H., J. Soviet. Math., 55, 1401 (1991). [2] Arrigo D. J., Broadbridge P. and Hill J. M., J. Math. Phys., 34, 4692–4703 (1993). [3] Bluman G. W., Potential symmetries and linearization . In Proceedings of NATO Advanced Research Workshop, Kluwer, Exeter, 1992.

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[4] Bluman G. W. and Kumei S., J. Math. Phys., 21, 1019 (1980). [5] Bluman G. W. and Kumei S., Symmetries and Differential Equations (Berlin: Springer, 1989). [6] Bluman G. W. and Kumei S., Euro. J. Appl.Math. , 1, 189–216 (1990). [7] Bluman G. W., Cole J., The general similarity solution of the heat equation. J. Math. Mech., 18, 1025 (1969). [8] Bluman G. W., Reid G. J., Kumei S., New classes of symmetries for partial differential equations. J. Math. Phys. 29, 806–811 (1988). [9] Bluman G. W. and Shtelen V., Mathematics is for Solving Problems. Cook, SLP, Roytburd V Tulin M (eds), SIAM, pp. 105–118. [10] Bluman G. W. and Yan Z., Eur. J. Appl. Math., 16, 235–266 (2005). [11] Clarkson P. A. and Kruskal, J. Math. Phys., 30, 2201–2213 (1989).

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[12] Clarkson P. A., Chaos, Solitons and Fractals , 1995 (to appear). [13] Clarkson P. A. and Mansfield E. L. SIAM J. Appl. Math, 55, 1693–1719 (1994). [14] Gandarias M. L., J. Phys A: Math and General, 29, 607–633 (1996). [15] Gandarias M. L., J. Phys A: Math and General, 29, 5919–5934 (1996). [16] Gandarias M. L., Nonclassical Potential symmetries of the Burgers equation. Symmetry in Nonlinear Mathematical Physics . Kiev: ed. Natl. Acad. Sci. Ukraine. Inst. Math., 130–137 (1997). [17] Gandarias M. L., New Potential symmetries. CRM Proceedings and Lecture Notes ed. American Math. Society. Publ., Providence RI. 25, 161–165 (2000). [18] Gandarias M. L., Nonclassical Potential symmetries of a porous medium equation . I Colloquium Lie Theory. Vigo: ed. Univ. Vigo, 37–45 (2000). [19] Hill, Differential Equations and Group Methods (Boca Raton: CRC Press., 1992). [20] Ibragimov N. H., Handbook of Lie Group Analysis of Differential Equations (Boca Raton: CRC, 1994). [21] Katkov V. L., Zh. Prikl. Mekh. Tekh. Fiz., 6, 105 (1965). [22] King J. R., J. Phys. A: Math. Gen., 24, 5721–5745 (1991). [23] King J. R., J. Phys. A: Math. Gen., 25, 4861–4868 (1992). [24] Krasil’shchik I. S. and Vinogradov A. M., Acta Applic. Math., 2, 79–96 (1984). [25] Krasil’shchik I. S. .and Vinogradov A. M., Acta Applic. Math., 15, 161–209 (1989).

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[26] Kumei S. and Bluman G. W., SIAM J. Appl. Math., 42, 1157–1173 (1982). [27] Olver P. J., Applications of Lie Groups to Differential Equations (Berlin: Springer, 1986). [28] Ovsiannikov L. V., Group Analysis of Differential Equations (New York: Academic Press, 1982). [29] Pucci E., J.Phys. A: Math. Gen., 25, 2631 (1992). [30] Pucci E. and Saccomandi G., Modern group analysis : advanced analytical and computational methods in mathematical physics . Nordfjordeid: ed. N. H. Ibragimov et. all, 1997, 291–298. Kluwer, Dordrecht, 1993. [31] Rosenau P. and Kamin S., Physica 8D, 273–283 (1983). [32] Saccomandi G., Potential symmetries and direct reduction methods of order two. J. Phys. A. 30, 2211–2217 (1997).

Nonclassical Potential Symmetries for a Nonlinear Thermal Wave Equation

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[33] Stephani H., Differential Equations: Their Solution Using Symmetries (Cambridge: Cambridge U P, 1989).

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[34] Vinogradov A. M., Symmetries of Partial Differential Equations (Dordrecht: Kluver, 1989).

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In: Advances in Nonlinear Waves and Symbolic Computation ISBN 978-1-60692-260-6 c 2009 Nova Science Publishers, Inc. Editor: Zhenya Yan

Chapter 4

A NTI -D ARK S OLITONS OF THE R ESONANT ¨ N ONLINEAR S CHR ODINGER E QUATION D. W. C. Lai1 , K. W. Chow1∗, C. Rogers2,3†, and Zhenya Yan 1

Department of Mechanical Engineering, University of Hong Kong Pokfulam, Hong Kong 2

3

4

School of Mathematics, University of New SouthWales Sydney, Australia

Department of Applied Mathematics, Hong Kong Polytechnic University, Kowloon, Hong Kong

Key Laboratory of Mathematics Mechanization,Institute of Systems Science, Chinese Academy of Sciences Beijing 100080, P. R. China

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Abstract In this chapter, special classes of soliton solutions are investigated for the resonant nonlinear Schr¨odinger equation (RNLS), a model relevant in Madelung fluids, reaction diffusion systems, and black hole problems in astrophysics. Just like the conventional nonlinear Schr¨odinger model, the character of pulses or localized solutions still depends critically on the relative signs of the cubic nonlinear and dispersive terms. When the signs are the same, ’anti-dark’ solitons, or bright pulses propagating on a continuous wave background, as well as one- and two-soliton solutions, are found. However, when the signs are different, only dark one-soliton solution exists and only does so for a very special choice of the parameter. Moreover, some periodic wave solutions are also given in terms of the Jacobi elliptic functions. Key words and phrases: Dark solitons; Anti-dark solitons; Hirota bilinear method; Resonant nonlinear Schr¨odinger equation. ∗ †

E-mail: [email protected] E-mail: [email protected]

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D. W. C. Lai, K. W. Chow, C. Rogers, and Z. Y. Yan

Introduction

Recently the dynamics and propagation of dark solitons for the nonlinear Schr¨odinger equations (NLS) and other integrable envelope equations have been studied intensively. In many situations, the solitary pulse may travel as a local maximum, rather than a minimum, on an otherwise constant background continuous wave. Such pulses have sometimes been termed ‘anti-dark’ solitons (ADS) in the literature. One example is the neighborhood of the zero group velocity dispersion of a normally dispersive optical fiber, where direct numerical simulations or perturbation expansions in the presence of third order dispersion demonstrate the existence of such ADS [1], [2]. Indeed third order dispersion has also used in works on quasi-one-dimensional spatiotemporal pulses [3]. Similarly, ADS have also been studied in other settings, e.g. (3+1) dimensional NLS [4], generalized NLS [5], [6] and vector solitons [7]. In other physical settings, a form of ADS can also arise in deep nonlinear Bragg gratings [8]. Interactions of dark-antidark solitons are also relevant in slowly moving, finite amplitude, Lorentzian optical solitons, when the laser frequency is detuned out of the proper range of a dynamical photonic band gap [9]. Finally, such ADS can also be observed in non-instantaneous, nonlinear media experimentally [10]. The goal of the present work is to demonstrate that such anti-dark solitary pulses can also arise in another physical setting, namely, the ‘resonant nonlinear Schr¨odinger equation’ (RNLS). Moreover, we show that it also admits the periodic wave solutions in terms of the Jacobi elliptic functions.

2

Bilinear form of the resonant nonlinear Schr¨odinger equation

The resonant nonlinear Schr¨odinger equation adopts the form [11]

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iΨt + Ψxx −

δ 2 ? 2(|Ψ|)xxΨ Ψ Ψ = , 4 |Ψ|

(1)

where ∗ represents the complex conjugate of the wave function Ψ. This equation arises in the dynamics of Madelung fluids, reaction diffusion systems, transport and black hole problems in astrophysics. The Hirota bilinear method, well established in the theory of nonlinear waves, will be employed. For this purpose, Ψ is first rewritten in terms of the new dependent variables:     1 1 GH G Ψ = exp(R + iS), R = log , S = log , (2) 2 2 F 2 H The bilinear forms of (1) are then (Dt + Dx2 − C)G · F (Dt −

Dx2

= 0,

+ C)H · F

= 0, δ (Dx2 − C)F · F = GH. (3) 4 The symbol D denotes the Hirota bilinear operator [12]. For the existence of dark or anti-dark soliton solutions, the value of C in (3) is required to be: C =−

δρ2 , 4

(4)

Anti-Dark Solitons of the Resonant Nonlinear Schr¨odinger Equation

101

where ρ represents a measure of the amplitude of the continuous or plane wave in the background.

3

Soliton solutions

3.1

One soliton solution

In this case, if we set φ = px − Ωt, then F

= 1 + exp(φ),

G = ρ exp(αx − ω1 t)[1 + a1 exp(φ)], H = ρ exp(βx − ω2 t)[1 + b1 exp(φ)],

(5)

where α = −β, a1 =

Ω − 2αp + p2 1 δρ2 2 , b = , ω = α + , 1 1 Ω − 2αp − p2 a1 4

ω2 = −ω1 ,

(6)

and we obtain the dispersion relation as r

Ω = Ω± = 2αp ± p

p2 +

δρ2 . 2

The corresponding one-soliton solution of the resonant nonlinear Schr¨odinger equation is thus   4p2 1 2 2 |Ψ| = ρ 1 + 2 δρ (1 + cosh(px − Ωt))

(7)

(8)

where

4p2 1 ≥ 0. δρ2 (1 + cosh(px − Ωt)) This condition always holds for δ > 0. For δ < 0, (9) requires that 1+

δ≤−

2p2 < 0. ρ2

(9)

(10)

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However, the dispersion relation in (7) imposes the constraint: δ≥−

2p2 . ρ2

(11)

Hence, on combining inequalities (11) and (12), one deduces that a special value of δ which permits the present analytical treatment is δ=−

2p2 < 0. ρ2

(12)

This negative δ corresponds to a conventional, single hump dark-solitary wave (Figures 1 and 2). However, for positive δ, an ’anti-dark solitary wave’ on a nonzero background is obtained (Figures 2a and 2b). The sign of δ will thus dramatically change the nature of soliton solution.

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D. W. C. Lai, K. W. Chow, C. Rogers, and Z. Y. Yan

1.2

1

|Ψ|

2

0.8

0.6

0.4

0.2

0 −20

−15

−10

−5

0 x

5

10

15

20

Figure 1: Dark soliton of Equation (1): |Ψ|2 versus x with α = 1, δ = -2, ρ = 1 and p = 1 for t = 0.

3.2

Two-soliton solution

For δ > 0, the forms of F, G and H leading to a two-soliton solution are F = 1 + exp(φ1) + exp(φ2 ) + f12 exp(φ1 + φ2 ), (13) h i G = ρ exp(αx−ω1t) 1+a1 exp(φ1 )+a2 exp(φ2)+f12a1 a2 exp(φ1 +φ2 ) , (14) h i H = ρ exp(βx−ω2t) 1+b1 exp(φ1)+b2 exp(φ2)+f12b1b2 exp(φ1 +φ2 ) , (15) where α = −β, φj = pj x − Ωj t, j = 1, 2, and

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aj

=

f12 =

Ωj − 2αpj + p2j

2 , bj =

Ωj − 2αpj − pj

1 δρ2 , ω 1 = α2 + , ω2 = −ω1 , aj 4

(p2 Ω1 − p1Ω2 )2 − p21p22 (p1 − p2)2 . (p2 Ω1 − p1Ω2 )2 − p21p22 (p1 + p2)2

(16) (17)

The corresponding dispersion relations are Ωj± = 2αpj ± pj

r

p2j +

δρ2 . 2

(18)

The validity of these solutions is verified independently by direct differentiation with the computer algebra software MATHEMATICA. Anti-dark two-soliton solutions exist if opposite signs for Ω1 and Ω2 are chosen, (i.e., Ω1+ and Ω2− , or Ω1− and Ω2+ ). Figure 3 illustrates the time evolution of two anti-dark

Anti-Dark Solitons of the Resonant Nonlinear Schr¨odinger Equation

103

1.2

1

|Ψ|

2

0.8

0.6

0.4

0.2

0 −20

−15

−10

−5

0 x

5

10

15

20

Figure 2: |Ψ|2 versus x with α = 1, δ = -2, ρ = 1 and p = 1 for t = 2. solitary waves. It shows two solitons with different amplitudes propagating on a non-zero background. Strong nonlinear interaction is observed at time t = 0. On the other hand, if the signs of Ω are the same (for positive δ), there will be some constraints for the existence of two-soliton solutions. More precisely, these constraints are as follows:

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Case I. (Ω1+ and Ω2+ are chosen): r δρ2 2 δρ2 δ(p21 + p22 ) (p21 + )(p2 + ) > p 1 p2 + . 2 2 4 Case II. (Ω1− and Ω2− are chosen): r δρ2 2 δρ2 δ(p21 + p22 ) (p21 + )(p2 + ) < −p1 p2 + . 2 2 4

(19)

(20)

For δ < 0, dark 2-soliton solution does not exist for the resonant nonlinear Schr¨odinger equation.

4

Periodic wave solutions

In this section, we study the periodic wave solutions of (1) in terms of the Jacobi elliptic functions. We make the travelling wave transformation Ψ(x, t) = ψ(ξ) exp(iη), ξ = kx − λt, η = αx + βt,

(21)

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D. W. C. Lai, K. W. Chow, C. Rogers, and Z. Y. Yan

3

2.5

|Ψ|

2

2

1.5

1

0.5

0 −20

−15

−10

−5

0 x

5

10

15

20

Figure 3: Anti-dark soliton of Equation (1): |Ψ|2 versus x with α = 1, δ = 2, ρ = 1 and p = 1 for t = 0.

3

2.5

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|Ψ|

2

2

1.5

1

0.5

0 −20

−15

−10

−5

0 x

5

10

15

20

Figure 4: Anti-dark soliton of Equation (1): |Ψ|2 versus x with α = 1, δ = 2, ρ = 1 and p = 1 for t = 2.

Anti-Dark Solitons of the Resonant Nonlinear Schr¨odinger Equation

105

8

7

6

|Ψ|

2

5

4

3

2

1

0 −30

−20

−10

0 x

10

20

30

Figure 5: Two anti-dark solitons of Equation (1): |Ψ|2 versus x with α = 1, δ = 2, ρ = 1 and p1 = 1, p2 = 2, and the choice of Ω1+ and Ω2− for t = -5.

8

7

6

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|Ψ|

2

5

4

3

2

1

0 −30

−20

−10

0 x

10

20

30

Figure 6: Two anti-dark solitons of Equation (1): |Ψ|2 versus x with α = 1, δ = 2, ρ = 1 and p1 = 1, p2 = 2, and the choice of Ω1+ and Ω2− for t = 0.

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D. W. C. Lai, K. W. Chow, C. Rogers, and Z. Y. Yan

8

7

6

|Ψ|

2

5

4

3

2

1

0 −30

−20

−10

0 x

10

20

30

(c) Figure 7: Two anti-dark solitons of Equation (1): |Ψ|2 versus x with α = 1, δ = 2, ρ = 1 and p1 = 1, p2 = 2, and the choice of Ω1+ and Ω2− for t = 3. where k, λ, α, β are real constants, and ψ(ξ) is a real function. The substitution of (21) into (1) yields the ordinary differential equation k2

d2 ψ(ξ) δ + (α2 + β)ψ(ξ) + ψ 3(ξ) = 0, 2 dξ 4

(22)

with the constraint condition

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λ = 2kα.

(23)

With the aid of Maple, we extend the constructive method [13]–[15] to (22) such that periodic wave solutions of (1) are given by Case 1. When δ < 0, r Ψ1 = 2mk

n o δ − sn(kx − λt, m) exp i αx + (k2 + m2 k2 − α2 )t . 2

(24)

Case 2. When δ > 0, r

n o δ Ψ2 = 2mk cn(kx − λt, m) exp i αx + (k2 − 2m2k2 − α2 )t , 2 r n o δ Ψ3 = 2k dn(kx − λt, m) exp i αx − (2k2 − m2 k2 + α2 )t . 2

(25) (26)

Anti-Dark Solitons of the Resonant Nonlinear Schr¨odinger Equation In the limit case (m → 1), we have the dark and bright solitary wave solutions r n o δ Ψ4 = 2k − tanh(kx − λt) exp i αx + (2k2 − α2 )t (δ < 0), 2 r n o δ Ψ5 = 2k sech(kx − λt) exp i αx − (k2 + α2 )t (δ > 0). 2

5

107

(27) (28)

Conclusions

Multiple bright soliton solutions with zero background for the resonant nonlinear Schr¨odinger equation have been derived earlier in the literature [11]. Here we identify the form of antidark and dark soliton solutions with nonzero background for this nonlinear evolution equation. Moreover, the periodic wave solutions are also obtained in terms of the Jacobi elliptic functions.

Acknowledgement Partial financial support has been provided by the Research Grants Council contracts HKU 7123/05E and HKU 7118/07E.

References [1] V. V. Afanasjev, Y. S. Kivshar, Physical Review A 44 (1991) R1446. [2] G. Huang, M. G. Velarde, Physical Review E 54 (1996) 3048. [3] D. J. Frantzeskakis, K. Hizanidis, B. A. Malomed, C. Polymilis, Physics Letters A 248 (1998) 203. [4] D. J. Frantzeskakis, B. A. Malomed, Physics Letters A 264 (1999) 179.

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[5] H. E. Nistazakis, D. J. Frantzeskakis, P. S. Balourdos, A. Tsigopoulos, B. A. Malomed, Physics Letters A 278 (2000) 68. [6] F. G. Bass, V. V. Konotop, S. A. Puzenko, Physical Review A 46 (1992) 4185. [7] D. J. Frantzeskakis, Physics Letters A 285 (2001) 363. [8] H. Alatas, A. A. Iskandar, M. O. Tjia, T. P. Valkering, Physical Review E 73 (2006) 066606. [9] C. Conti, S. Trillo, Physical Review E 64 (2001) 036617. [10] T. H. Coskun, D. N. Christodoulides, Y. R. Kim, Z. Chen, M. Soljacic, M. Segev, Physical Review Letters 84 (2000) 2374. [11] O. K. Pashaev, J. H. Lee, Journal of Nonlinear Mathematical Physics Supplement 8 (2001) 230.

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[12] A. Nakamura, Journal of the Physical Society of Japan 47 (1979) 1701. [13] Z. Y. Yan, Computers Physics Communication, 153 (2003) 145. [14] Z. Y. Yan, Chaos, Solitons and Fractals, 16 (2003) 759. [15] Z. Y. Yan, Communication in Theoretical Physics, 38 (2002) 143. Z. Y. Yan, Communication in Theoretical Physics, 38 (2002) 400.

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Z. Y. Yan, Communication in Theoretical Physics, 39 (2002) 144.

In: Advances in Nonlinear Waves and Symbolic Computation ISBN 978-1-60692-260-6 c 2009 Nova Science Publishers, Inc. Editor: Zhenya Yan

Chapter 5

S IMILARITY S OLUTIONS FOR THE B OITI –L EON –P EMPINELLI E QUATION WITH S YMBOLIC C OMPUTATION Zhuosheng Lu¨ ∗ School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, P.R. China

Abstract In this paper, symmetry reductions and similarity solutions for the Boiti–Leon– Pempinelli (BLP) equation are performed. It is shown that the infinitesimals of the BLP equation form an infinite dimensional Lie algebra. The similarity solutions of the equation are obtained by combining the classical Lie symmetry group method with a constructive algorithm, the further extended tanh method. A symbolic computation implementation of the further extended tanh method is also presented whose efficiency is shown by some concrete examples.

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Key words and phrases: Boiti–Leon–Pempinelli equation, symmetry reduction, similarity solution, the further extended tanh method, symbolic computation.

1

Introduction

Nonlinear evolution equations (NLEEs) are a special class of differential equations arising naturally in many fields of science including chemical physics, plasma physics, fluids dynamics, electrical circuits, optical fibers, etc.. Due to its scientific importance, the discovery of any explicit solutions for the NLEEs whatsoever is of great interest and hence the research on solving NLEEs has long been an utmost important task. One of the most efficient solving methods is the classical Lie symmetry group method firstly introduced by Sophus Lie [1], [2]. This method often reduces the NLEEs to lower dimensional differential equations which may be solved explicitly to obtain the similarity solutions for the ∗

E.mail: [email protected]

110

Z. S. Lu¨

original NLEEs. In the past few decades, there have been considerable developments in the Lie symmetry group method. A number of research papers have devoted to the subject and several generalizations of the classical symmetry group method have been proposed [3]–[8]. Other famous solving methods include the inverse scattering method, the B¨acklund transformation technique, the Painlev´e analysis approach, etc.. There is also much current interest in developing the symbolic computation algorithms for solving the NLEEs. Various constructive algorithms such as the tanh method [9], the tanh-sech method [10], the sinecosine method [11], the Jacobi elliptic function expansion method [12], the extended tanh method [13], [14], etc. have been established. In Refs.[15], [16], we further extended the tanh method and showed that our extended method could be applied to a wide class of NLEEs to obtain abundant types of explicit solutions, including the solitary solutions, the soliton-like solutions, the periodic solutions, the multi-soliton solutions, the rational solutions, and so on. In general, the NLEEs are often very difficult to be solved explicitly. In this paper, we will solve the following Boiti–Leon–Pempinelli (BLP) equation [17] uty = (u2 − ux )xy + 2 vxxx, vt = vxx + 2 u vx,

(1)

by combing the classical Lie symmetry group method with the further extended tanh method. As the further extended tanh method involves tedious calculations which are difficult or even unable to be handled traditionally using the pen & paper, here in this paper, we will also present a symbolic computation implementation of the method. This paper is organized as follows. In Section 2, we seek symmetry reductions of the BLP equation (1). In Section 3, we give a brief review on the further extended tanh method and apply it to the symmetry reductions of the BLP equation (1) to get some similarity solutions for the equation (1). In Section 4, we perform an implementation of the further extended tanh method in Maple. Some illustrative examples are also presented which show the efficiency of the program. We conclude the paper in the last Section.

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2

Symmetry reductions for the BLP equation

The BLP equation (1) appeared for the first time in Ref. [17] which was obtained following the idea of Boiti, Leon and Pempinelli [18]. Its integro-differential equation reduces either to the sine-Gordon equation or to the sinh-Gordon equation in a one-dimensional limit. In Ref. [17] it is shown that the considered equation is Painlev´e integrable and Hamiltonian. Considering its relationship with the physically important sine- and sinh-Gordon equation, in this paper, we would like to seek explicit exact solutions for the BLP equation (1). To apply the Lie classical method to BLP equation (1), we consider one parameter Lie group of infinitesimal transformation in (x, y, t, u, v), given by x ˜ = x + ε ξ + O(ε2 ), y˜ = y + ε ζ + O(ε2 ), t˜ = t + ε τ + O(ε2), u ˜ = u + ε η + O(ε2 ), v˜ = v + ε ψ + O(ε2 ),

(2)

Similarity Solutions for the Boiti–Leon–Pempinelli Equation

111

where ξ, ζ, τ , η, ψ are functions of (x, y, t, u, v) and ε is the group parameter. The associated infinitesimal generator is L = ξ ∂ x + ζ ∂y + τ ∂t + η ∂u + ψ ∂ v .

(3)

Requiring that (1) is invariant under (2) yields the following twenty seven determine equations for the infinitesimals ξ, ζ, τ , η and ψ: ξu = ξv = ξy = 0, ζx = ζt = ζu = ζv = 0, τx = τy = τu = τv = 0, ηv = ηy = ηuu = 0, ψu = ψvv = 0, τt − 2 ξx = 0, ψvx − ξxx = 0,

ψv − ηu − ξx + ζy = 0,

2 ηu + 2 ξx − ζy − ψv = 0, 2 ψxxx + 2 u ηxy − ηyt − ηxxy = 0, ζy − ηu + τt + ψv − 3 ξx = 0,

3 ψvxx − ξxxx = 0,

2 u ψx + ψxx − ψt = 0,

(4)

2 ηx + 2 u ηux − ηut − ηuxx = 0,

2 ψvx − 2u ξx − ξxx + ξt + 2 u τt + 2 η = 0,

2 u ηu − 2 u ζy − 2 u ψv − 2 ηxu + 2 η + ξt + ξxx + 4 u ξx = 0. The determine equations (4) are obtained using a Mathematica program SYMLIE, which is an innovation of the Mathematica program LIGRMAIN [19]. Solving (4) we have: ξ = 2 α 0(t) x + 2 β(t), ζ = θ(y), τ = 4 α(t), η = −2 α 0 (t) u − α 00(t) x − β 0 (t), ψ = −θ 0 (y) v + ρ(y),

(5)

in which α(t), β(t), θ(y) and ρ(y) are arbitrary functions and prime denotes differentiation. The corresponding infinitesimal generators are as follows I1 (α) = 2 α 0 x ∂x + 4 α ∂t − (2 α 0 u + α 00 x) ∂u , I2(β) = 2 β ∂x − β 0 ∂u ,

I3 (θ) = θ ∂y − θ 0 v ∂v ,

I4 (ρ) = ρ ∂v .

(6)

I1 (α), I2(β), I3 (θ) and I4 (ρ) form an infinite dimensional Lie algebra and the non-zero relations between them are [ I1(α), I2 (β) ] = 2 I2(2 α β 0 − α 0 β), [ I3(α), I4 (γ) ] = I4(θ 0 ρ + θ ρ 0),

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[ I1(α1 ), I1 (α2) ] = 4 I1( α1 α 02 − α 01 α2 ),

(7)

[ I3(θ1), I3 (θ2) ] = I3(θ1 θ 02 − θ 01 θ2 ). In the following, we present three special symmetry reductions of the equation (1). Let α(t) = 1we can obtain the symmetry reduction Z U (X, T ) = u + β, V = θ v − ρ d y, Z Z dy X = x−2 β d t, T = t − , θ

(8)

where U (X, T ) and V (X, T ) satisfies UT T − 2 UT UX − 2 U UT X + UX X T + 2 VX X X = 0, VX X − VT + 2 U VX = 0.

(9)

112

Z. S. Lu¨ For α = 0 the symmetry reduction corresponding to the infinitesimal generator L is Z Z β0 U (X, T ) = u + d y, V = θ v − ρ d y, θ Z (10) β X = x−2 d y, T = t, θ

with U (X, T ), V (X, T ) satisfying 2 2 β UX T + β 00 − 4 β UX − 4 β U UX X + 2 β UX X X + 2 VX X X = 0,

VX X − VT + 2 U VX = 0,

(11)

in which β(t) = β(T ) and prime denotes differentiation with respect to T . As for α = t > 0 and β = 0, the BLP equation (1) possesses the symmetry reduction Z √ U (X, T ) = u t, V = θ v − ρ d y, (12) R x − 4 dθy √ X= , T = te , t in which U (X, T ) and V (X, T ) satisfies −2 X T UX T + 4 T 2 UT T + 2 T UT − −8 T 2 UT UX − 8 T U UX T + 4 T UX X T + 2 VX X X = 0,

(13)

2 VX X + X VX − 2 T VT + 4 U VX = 0. One may further apply the Lie symmetry group method to the equations (9), (11) and (13) to reduce these equations to ordinary differential equations. Here we omit the procedure.

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3

Similarity solutions

The explicit solutions of NLEEs often model natural phenomena and facilitate the testing of numerical methods as well as aid in the stability analysis. In the upper section, we have obtained some (1 + 1)-dimensional symmetry reductions of the (2 + 1)-dimensional BLP equation (1). Further applying the Lie symmetry group method on those reductions, one may get some explicit similarity solutions for the equation (1). But the procedure would be complicated and time consuming. In this section, instead of using the symmetry group method, we will apply a constructive method, the further extended tanh method [15], [16], to the (1+1)-dimensional reductions (9), (11) and (13) to seek explicit similarity solutions for the BLP equation (1).

3.1

Review on the further extended tanh method

Here we give a review on the further extended tanh method. For a given system of NLEEs with the independent variables x = (x1, x2, . . . , xn ) and the dependent variables u = (u1, u2, . . . , um ), we seek its solutions in the form ui =

Ni X j=0

a ij (x) φj (ω(x)) (1 ≤ i ≤ m),

(14)

Similarity Solutions for the Boiti–Leon–Pempinelli Equation

113

with φ satisfying φ 0 = δ + φ2 ,

(15)

where δ is a constant and prime denotes differentiation with respect to ω. To determine ui explicitly, we take the following four steps. Step 1. Determine Ni (1 ≤ i ≤ m) by balancing the highest nonlinear terms with the highest-order partial differential terms in the given NLEEs. (See Ref.[13] for detail.) Step 2. Substituting (14) and (15) into the given NLEEs and collecting the coefficients of the polynomials of φ, then setting each coefficient to zero to derive a set of partial differential equations of aij (x) and ω(x). We call this set of equations determine equations . Step 3. Solving the system of partial differential equations obtained in Step 2 for aij (x) and ω(x). Step 4. As the Riccati equation (15) possesses the solutions  √ √  −δ tanh( −δω) δ < 0, − √ √ φ= δ tan( δ ω) δ > 0,   −1/ω δ = 0,

(16)

substituting aij (x), ω(x) and (16) into (14) to obtain the explicit solutions for the given NLEEs. A Maple implementation (entitled VCTanh) of the further extended tanh method is now available, whose applications will be described detailly in the next Section.

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3.2

Similarity solutions derived from equation (9)

In this subsection, we present some exact solutions of the equations (9) obtained by using the Maple package VCTanh. The solutions can be used to further determine some explicit similarity solutions for the BLP equation (1). When δ < 0, the equation (9) possess the exact solutions √  √ U1 = (2 c3 X + c1) −δ tanh −δ (c3 (2 T + X 2) + c1 X + c2 ) , √  √ V1 = c4 − 2 c3 −δ tanh −δ (c3 (2 T + X 2) + c1 X + c2) , √  √ U2 = c1 + c2 −δ tanh −δ (c2 T + c2 X + c3) , √  √ V2 = c4 − 2 c1 c2 −δ tanh −δ (c2 T + c2 X + c3) ;

(17)

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while for δ > 0, the equations (9) admit the solutions √  √ U3 = −(2 c3 X + c1) δ tan δ (c3 (2 T + X 2) + c1 X + c2) , √  √ V3 = c4 + 2 c3 δ tan δ (c3 (2 T + X 2) + c1 X + c2 ) , √  √ δ (c2 T + c2 X + c3) , U4 = c1 − c2 δ tan √  √ V4 = c4 + 2 c1 c2 δ tan δ (c2 T + c2 X + c3 ) .

(18)

in which c1 , c2 , c3 and c4 are arbitrary constants. The equations (9) also possess the following rational solutions 2 c 3 X + c1 , c3 (2 T + X 2) + c1 X + c2 2 c3 V5 = c4 − , c3 (2 T + X 2) + c1 X + c2 c2 U 6 = c1 + , c2 T + c 2 X + c 3 2 c 1 c2 V6 = c4 − , c2 T + c 2 X + c 3 U5 =

(19)

where c1 , c2, c3 and c4 are arbitrary constants. With the aid of VCTanh, we can also obtain the following multi-soliton solution for the equations (9) N P

U7 = κ +

i=1

a0 +

ai bi exp(bi X + (2 κ bi + b2i ) T ) N P

, ai exp(bi X + (2 κ bi +

i=1 N P

V7 = µ −

(20)

ai (2 κ bi +

i=1

a0 +

b2i ) T )

N P

b2i )

exp(bi X + (2 κ bi +

, ai exp(bi X + (2 κ bi +

i=1 Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

b2i ) T )

b2i ) T )

in which N is arbitrary positive integer, κ, µ, ai , bi (0 ≤ i ≤ N ) are nonzero constants. Therefor, substituting (8) into (17) ∼ (20), we can obtain the explicit similarity solutions for the BLP equation (1).

3.3

Similarity solutions derived from equation (11)

The symmetry reduction (11) can also be solved using the package VCTanh to obtain the following solutions √  F 0(T ) X + G 0(T ) √ −δ (F (T ) X + G(T )) , + −δ F (T ) tanh 2 F (T ) √  √ V1 = c1 − 2 c2 −δ tanh −δ (F (T ) X + G(T )) ,

U1 =

(21)

Similarity Solutions for the Boiti–Leon–Pempinelli Equation

115

where δ < 0 and √  F 0 (T ) X + G 0(T ) √ − δ F (T ) tan δ (F (T ) X + G(T )) , 2 F (T ) √  √ V2 = c1 + 2 c2 δ tan δ (F (T ) X + G(T )) ,

U2 =

(22)

in which δ > 0. Both in (21) and (22), c1 , c2 are arbitrary constants, G(T ) is arbitrary c2 . function of T and F (T ) = β(T ) We omit the rational solutions for the equations (11). Substituting (10) into (21) and (22) yields the exact similarity solutions for the BLP equation (1).

3.4 Similarity solutions derived from equation (13) Solving the symmetry reduction (13) using VCTanh yields the following exact solutions −2 F 00(X) − F 0(X) X + 2 c1 + 4 F 0(X)   √ √ 0 + −δ F (X) tanh −δ(F (X) + c1 ln(T )) , √  √ V1 = c2 − 4 c1 −δ tanh −δ(F (X) + c1 ln(T )) ,

(23)

−2 F 00(X) − F 0(X) X + 2 c1 − 4 F 0(X)   √ √ − δ F 0(X) tan δ(F (X) + c1 ln(T )) , √  √ V2 = c2 + 4 c1 δ tan δ(F (X) + c1 ln(T )) ,

(24)

U1 =

where δ < 0 and

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U2 =

with δ > 0. The parameters c1 , c2 in (23) and (24) are arbitrary constants, and F (X) is nonconstant function of X. The explicit similarity solutions for the BLP equation (1) can be obtained by substituting (12) into (23) and (24). We have successfully obtained some exact solutions for the equations (11) and (13) by using the further extended tanh method. In fact, those two equations are all variable coefficient NLEEs which can not be solved by using the traditional tanh method [9] or the extended tanh method [13]. In reference [16], some explicit exact solutions for the BLP equation (1) have been obtained by using the further extended tanh method. But the similarity solutions of equation (1) obtained in this paper can not be obtained by merely using the further extended tanh method.

4

Maple implementation of the further extended tanh method

The further extended tanh method, as given in Section 3, is highly algorithmic but involves a large amount of tedious algebra and calculus. Hence the use of symbolic computation is

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Z. S. Lu¨

no doubt necessary to carry out the lengthy but straightforward calculations. In this Section, we present a Maple package VCTanh (Variable Coefficient Tanh-method), which is a fully implementation of the method. The package VCTanh contains 6 sub-procedures: altdif(), dif(), step1(), setp2(), step3() and setp4(). Brief descriptions of the sub-procedures are listed below. • The sub-procedures altdif() and dif() are used to simplify the inputs. • The sub-procedures step1(), setp2(), step3() and setp4() are implementations of the corresponding steps of the algorithm. The advantages of our package can be listed as: (1) The inputs are more simpler compared with that of other related software packages. (2) Our code handles NLEEs with any number of dependent and independent variables. (3) Our code works for variable coefficient NLEEs as well as multi-parameter equations. If necessary, it generates automatically the parameter constraints. (4) The users may freely choose the option of calculating travelling wave solutions or non-travelling wave solutions. In what follows, we will show the use and efficiency of VCTanh by some concrete examples. Example 1. Take the Burgers equation ut + κ u ux + µ ux x = 0,

(25)

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with κ, µ be nonzero constants, as an example to show the use of VCTanh. Load VCTanh and press “Enter” to run the program. > libname:=00 D: \\ program \\00, libname: # (specify directory) with(VCTanh): # ( read in package) Main(); # ( run the program) The prompt information and corresponding inputs are displayed as follows. Please input the independent variables: x,t; Please input the dependent variables: u; Please input the evolution equations: diff(u,t) +kappa*u*diff(u,x)+mu*diff(u,x,x); Would you like to get travelling wave solutions (input 1) or non-travelling wave solutions (input 2)? 1; Would you like the parameter(s) in the inputted equation(s) be constrained or not (y\n) ? n; In this case, VCTanh will generate automatically the solutions of the inputted equation and the outputs are as follows. We solve the equation ∂ ∂ ∂2 u=0 u+ κu u+µ ∂t ∂x ∂ x2 for travelling wave solutions. The solutions are listed as follows: √ √ 2 µ −δ tanh( −δ (x − κ λ0 t + σ2)) u = λ0 + κ

Similarity Solutions for the Boiti–Leon–Pempinelli Equation

117

with δ 0) 3

with µ1 be arbitrary constant. This example shows that VCTanh can also manipulate ODEs.

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Example 4. Non-travelling wave solutions of the (2 + 1)-dimensional dispersive long wave equation [22] 1 uy t + vx x + (u2)x y = 0, 2 (27) vt + (u v + u + ux y )x = 0. > libname:=00 D: \\ program \\00, libname: with(VCTanh): Main(); Please input the independent variables: x,y,t; Please input the dependent variables: u,v; Please input the evolution equations: diff(u,y,t)+diff(v,x,x)+1/2*diff(uˆ 2,x,y), diff(v,t)+diff(u*v+u+diff(u,x,y),x); Would you like to get travelling wave solutions (input 1) or non-travelling wave solutions (input 2)? 2; Are there any other constraints? Please input them in a list: {mu[2]=-2*diff(omega, y)*diff(omega,x), kappa[1]=2*diff(omega,x),mu[1]=-2*diff(omega,x,y), diff(omega,t)=diff(omega,x,x), -kappa[0]*diff(omega,x),mu[0] =-(1+2*diff(omega,y) diff(omega,x)*delta+diff(kappa[0],y)), diff(kappa[0],y)=0};  2   2     ∂ ∂ ∂ ∂3 −4 δ ω ω + ω ω =0 ∂x∂y ∂x2 ∂x ∂x2∂y − − − − − − − − − − − − − − − − − − − − − − − − − − − − −−  2   2     ∂ ∂ ∂ ∂3 4δ ω ω + ω ω =0 ∂x∂y ∂x2 ∂x ∂x2∂y − − − − − − − − − − − − − − − − − − − − − − − − − − − − −− Would you like to solve the determine equations using Maple (y\n)? y; We solve the equation  2   2      2  ∂ ∂ ∂ ∂ ∂ v + u + u u +u u =0 2 ∂y ∂t ∂x ∂y ∂x ∂x ∂y           ∂ ∂ ∂ ∂ ∂3 u =0 v + u v+u v + u + ∂t ∂x ∂x ∂x ∂y ∂x2

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Z. S. Lu¨

for non-travelling wave solutions. The solutions are listed as follows: 

 ∂2 ∂ d   + F 2(x, t) F 3(t) − F 2(x, t) ∂t dt ∂ ∂x2 u=− +2 F 2 (x, t) φ ∂ ∂x ∂x F 2 (x, t)       d ∂ d ∂ v = −1 − 2 F 4 (y) F 2 (x, t) δ − 2 F 4 (y) F 2 (x, t) φ2 dy ∂x dy ∂x when , δ < 0 √  √ φ = − −δ tanh −δ ( F 2(x, t) + F 4(y) + F 3(t)) √ φ=

δ tan

√

when , 0 < δ δ ( F 2(x, t) + F 4(y) + F 3(t))



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when , δ = 0 1 φ=− F 2(x, t) + F 4(y) + F 3(t) − − − − − − − − − − − − − − − − − − − − − − − − − − − − −−

d d dt F 4(t) x + dt F 5(t) u=− + 2 F 3(y) F 4(t)φ F 4(t)   d v = −1 − 2 F 3(y) ( F 4(t))2 δ F 3(y) x− dy   d −2 F 3(y) F 4(t) δ F 3(y) F 5(t) dy   d d −2 F 3(y) F 4(t) δ F 6(y) − 2 F 3(y) F 4(t) φ+ dy dy   d 2 +(−2 F 3(y) ( F 4(t)) F 3(y) x− dy   d d −2 F 3(y) F 4(t) F 3(y) F 5(t) − 2 F 3(y) F 4(t) F 6(y))φ2 dy dy when , δ < 0 √  √ φ = − −δ tanh −δ (( F 4(t) x + F 5(t)) F 3(y) + F 6(y)) √ φ=

δ tan

√

when , 0 < δ

 δ (( F 4(t) x + F 5(t)) F 3(y) + F 6(y))

when , δ = 0 1 φ=− ( F 4(t) x + F 5(t)) F 3(y) + F 6(y) − − − − − − − − − − − − − − − − − − − − − − − − − − − − −− We got 6 solutions.

Similarity Solutions for the Boiti–Leon–Pempinelli Equation

123

According to the algorithm, in order to obtain non-trivial solutions of the considered equation, all the listed constraints must be satisfied except for the restriction diff(kappa[0], y) = 0. Besides this restriction, all other constraints come form the determine equations. The upper solutions are automatically generated by VCTanh. We should point out that the arbitrary function F 3(t) in the first three solutions may be omitted since F 2(x, t) is also arbitrary. Functions of the form F i (•) are all arbitrary. We have applied VCTanh to a large variety of NLEEs which confirmed the validity and efficiency of the program.

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5

Conclusion and discussion

In this paper, with the aid of symbolic computation, we pursued symmetry reduction and similarity solutions of the BLP equation (1). We showed that the infinitesimal vector fields of the equation form an infinite dimensional Lie algebra. Using the classical Lie symmetries, we reduced the BLP equation (1) to some (1+1)-dimensional partial differential equations. Generally, even if the symmetry reductions of a given NLEE were obtained, they are difficult to be solved explicitly. In this paper, we applied the further extended tanh method to some symmetry reductions of the BLP equation (1), and successfully obtained some explicit similarity solutions for the equation. To the author’s knowledge, such solutions have not been reported in other literatures. The further extended tanh method has proven to be an effective method for finding nontravelling wave solutions for the NLEEs. Here in this paper, we also presented a Maple implementation (entitled VCTanh) of the algorithm. Validity of VCTanh was shown by some examples, including the (1+1)-dimensional NLEE, the high dimensional NLEE, the nonlinear ordinary differential equation and the (2+1)-dimensional coupled NLEEs. When pursuing travelling wave solutions, VCTanh performs the computations automatically from start to finish without human intervention. However, due to the lack of mature algorithms for solving the determine equations, human intervention is often necessary when seeking non-travelling wave solutions. Searching explicit solutions of NLEEs is an important and genuinely difficult subject in mathematical physics. Till now various effective solving methods have been established. To solve the equations more effectively, one may combine two or more different methods as shown in this paper.

References [1] G. W. Bluman, S. Kumei, Symmetries and Differential Equations (Springer, Berlin, 1989). [2] P. J. Olver, Application of Lie Groups to Differential Equations (Springer, New York, 1991). [3] G. W. Bluman, J. D. Cole, J. Math. Mech. 18 (1969) 1025.

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[4] L. V. Ovsiannikov, Group Analysis of Differential Equations (Academic Press, New York, 1982). [5] D. Levi, P. Winternitz, J. Phys. A. 22 (1989) 2915. [6] G. Gaeta, J. Phys. A. 23 (1990) 3643. [7] E. M. Vorob’ev, Acta Appl. Math. 24 (1991) 1. [8] P. A. Clarkson, M. D. Kruskal, J. Math. Phys. 30 (1989) 2201. [9] H. B. Lan, K. L. Wang, J. Phys. A. 23 (1990) 3923. [10] Z. Y. Yan, H. Q. Zhang, Phys. Lett. A. 285 (2001) 355. [11] C. T. Yan, Phys. Lett. A 224 (1996) 77. [12] S. K. Liu, et. al., Phys. Lett. A. 289 (2001) 69. [13] E. G. Fan, Phys. Lett. A. 277 (2000) 212. [14] Z. Y. Yan, Commun. Theor. Phys., 38 (2002) 143; 38 (2002) 400; 39 (2002) 144. [15] Z. S. Lu¨ , H. Q. Zhang, Phys. Lett. A. 307 (2003) 269. [16] Z. S. Lu¨ , H. Q. Zhang, Chaos, Solitons and Fract. 19 (2004) 527. [17] T. I. Garagash, Theor. Math. Phys. 100 (1994) 1075. [18] M. Boiti, J. J-P. Leon, F. Pempinelli, Inv. Probl. 3 (1987) 37. [19] Temuerchaolu, Chin. J. Comput. Phys. 14 (1997) 375 (in Chinese). [20] Z. S. Lu¨ , Commun. Theor. Phys. 44 (2005) 987. [21] D. Baldwin, et. al., J. Symbol. Comput. 37 (2004) 669.

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[22] W. X. Ma, Phys. Lett. A. 319 (2003) 325.

In: Advances in Nonlinear Waves and Symbolic Computation ISBN 978-1-60692-260-6 c 2009 Nova Science Publishers, Inc. Editor: Zhenya Yan

Chapter 6

T HE N EW S INE -G ORDON E XPANSION A LGORITHMS TO C ONSTRUCT E XACT S OLUTIONS OF N ONLINEAR WAVE E QUATIONS Zhenya Yan∗ Key Laboratory of Mathematics Mechanization, Institute of Systems Science Chinese Academy of Sciences, Beijing 100080, P. R. China

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Abstract A new constructive algorithm is presented to study new multiple solutions of nonlinear wave equations via the general sine-Gordon reduction equation (SGRE) and its solutions. It is proved that the algorithm is more powerful than many known approaches such as the tanh-function method, the sine-cosine method, the project Riccati system method, the sinh-Gordon expansion method, the sn- and cn-function method, the extended Jacobi elliptic function method, etc. The algorithm is applied to (i) The KdV-mKdV equation with first-order dispersion term, (ii) The (2+1)-dimensional higher-degree Burgers equation, (iii) The modified Boussinesq equation, (iv) Nonlinear Schrodinger equation with cubic-quintic nonlinearity, (v) The (2+1)-dimensional generalization of mKdV equation and (vi) The (2+1)-dimensional break soliton equation. As a consequence, many types of exact solutions are deduced which include solitary wave solutions, doubly periodic solutions, optical solitary wave solutions. These solutions may be useful to explain the corresponding physical phenomena.

1

Introduction and Proposition

The investigation of exact solutions, in particular solitary waves and doubly periodic solutions, plays a prime role in nonlinear science. Many powerful approaches have been presented to study exact solutions of nonlinear wave equations, such as the invese scattering transformation [1], Backlund transformation [2], Darboux transformation [3], Hirota bilinear method [4], etc. Particularly there exist other constructive approaches which use the ∗

E-mail: [email protected]

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Z. Y. Yan

solutions of simple nonlinear differential equations to construct the corresponding solutions of complicated nonlinear differential equations. For instance, (a) The Riccati equation w0 (ξ) = b + w2(ξ) possesses the following solutions[5], [6]:  √ √ √ √  −bξ or −b coth −bξ, b < 0 w = √−b tanh √ √ √ w = b tan bξ or − b cot bξ, b>0   w = −1/ξ, b=0 which were used to construct solutions of many nonlinear wave equations [5], [6]. (b) The coupled system of projective Riccati equation [7] σ 0 (ξ) = −σ(ξ)τ (ξ), τ 0 (ξ) = −τ 2 (ξ) − µ/Kσ(ξ) + 1 is of the solution σ(ξ) =

K sinh(ξ) , τ (ξ) = cosh(ξ) + µ cosh(ξ) + µ

which were applied to deduce solutions of many many other nonlinear wave equations [8], [9]. (c) The sinh-Gordon reduction equation [2], [10] w0(ξ) = [sinh2 w(ξ) − m2 + 1]1/2 has the doubly periodic solution

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sinh[w(ξ)] = cs(ξ; m), cosh[w(ξ)] = ns(ξ; m), which were applied to deduce doubly periodic solutions of many many other nonlinear wave equations [10], [11], etc. The two keys in this type of approaches are to find these proper and simple nonlinear differential equations and their more solutions and to establish powerful transformations (bridges) between these simple nonlinear differential equations and those complicated nonlinear differential equations. The later is different and more important. For example, the polynomial transformation in u and its generalization are related to the Riccati equation, and the polynomial transformation in sinhi w coshj w(sinh2 w − m2 + 1)s/2 is related to the above-mentioned the sinh-Gordon reduction equation. Though there have existed many powerful methods to seek solutions of nonlinear wave equations, it is still an interesting subject to develop more powerful and unified algorithms which can be used to find not only known solutions, but new solutions. In this paper we will further consider more types of solutions of the general sine-Gordon reduction equation, and use its solutions and a new transformation to construct more types of solutions of other nonlinear wave equations arising from nonlinear science. The famous sine-Gordon (sG) equation ∂ 2φ = α sin φ. ∂x∂t

(1.1)

The New Sine-Gordon Expansion Algorithms

127

appears in many branches of nonlinear science [2], where α is a constant. Under the travelling wave transformation φ(x, t) = φ(ξ), ξ = k(x − λt), where k and λ are the wave number and wave speed, respectively, (1.1) reduces to an ordinary differential equation h i1 dw(ξ) 2 = ± a + b sin2 w(ξ) , dξ

(1.2)

where b = − kα2 λ , w = φ2 and a is an integration constant. Since a, b are arbitrary constants, thus (1.1) may possess different types of solutions for different a, b. We have the following proposition which is useful to construct new solutions: Proposition. Equation (1.2) has the following solutions for different parameters a and b: Case A. Let a = 1, b = −1. Then (1.2) reduces to the first-order ODE dw(ξ) = cos w(ξ). dξ

(1.3)

sin[w(ξ)] = tanh(ξ), or cos[w(ξ)] = sech(ξ)

(1.4a)

sin[w(ξ)] = coth(ξ), or cos[w(ξ)] = icsch(ξ), i2 = −1.

(1.4b)

which has the solutions:

and

Case B. Let a = 0, b = 1. Then (1.2) reduces to the first-order ODE dw(ξ) = sin w(ξ), dξ

(1.5)

sin[w(ξ)] = sech(ξ), or cos[w(ξ)] = − tanh(ξ)

(1.6a)

sin[w(ξ)] = icsch(ξ), or cos[w(ξ)] = − coth(ξ), i2 = −1.

(1.6b)

which has the solutions:

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and

Case C. Let a = 1, b = −m2 . Then (1.3) reduces to the first-order ODE h i1 dw(ξ) 2 = µ 1 − m2 sin2 w(ξ) , dξ

(1.7)

where m is the modulus of Jacobi elliptic functions, which has the following four sets of solutions: sin[w(ξ)] = sn(ξ; m), or cos[w(ξ)] = cn(ξ; m), (1.8) sin[w(ξ)] = cd(ξ; m), or cos[w(ξ)] = m0sd(ξ; m),

(1.9)

sin[w(ξ)] = ns(ξ; m), or cos[w(ξ)] = ics(ξ; m),

(1.10)

sin[w(ξ)] = m−1 dc(ξ; m), or cos[w(ξ)] = im0m−1 nc(ξ; m),

(1.11)

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where m0 is called the complementary modulus and m02 + m2 = 1. In addition sn(ξ; m) and cn(ξ; m) have the properties dsn(ξ; m)/(dξ) = cn(ξ; m)dn(ξ; m), dcn(ξ; m)/(dξ) = −sn(ξ; m)dn(ξ; m), 2

(1.12)

2

sn (ξ; m) + cn (ξ; m) = 1. Case D. Let a = m2 , b = −1. Then (1.3) reduces to the first-order ODE h i1 dw(ξ) 2 = µ m2 − sin2 w(ξ) , (1.13) dξ which has the following four sets of solutions: sin[w(ξ)] = msn(ξ; m), or cos[w(ξ)] = dn(ξ; m),

(1.14)

sin[w(ξ)] = mcd(ξ; m), or cos[w(ξ)] = m0 nd(ξ; m),

(1.15)

sin[w(ξ)] = ns(ξ; m), or cos[w(ξ)] = ics(ξ; m),

(1.16)

0

sin[w(ξ)] = dc(ξ; m), or cos[w(ξ)] = im sc(ξ; m),

(1.17)

Case E. Let a = −1, b = 1 − m2 . Then (1.3) reduces to the first-order ODE h i1 dw(ξ) 2 = µ − 1 + (1 − m2) sin2 w(ξ) , dξ

(1.18)

which has the following two sets of solutions: sin[w(ξ)] = m0−1 dn(ξ; m), or cos[w(ξ)] = imm0−1cn(ξ; m),

(1.19)

sin[w(ξ)] = mnd(ξ; m), or cos[w(ξ)] = imsd(ξ; m).

(1.20)

Case F. Let a = −m2 , b = −(1 − m2 ). Then (1.3) reduces to the first-order ODE

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h i1 2 dw(ξ)/(dξ) = µ − m2 − (1 − m2 ) sin2 w(ξ) ,

(1.21)

which has the following two sets of solutions: sin[w(ξ)] = nc(ξ; m), or cos[w(ξ)] = isc(ξ; m),

(1.22)

sin[w(ξ)] = m0−1 ds(ξ; m), or cos[w(ξ)] = im0−1cs(ξ; m),

(1.23)

Case H. Let a = 1, b = 0. Then (1.3) reduces to the first-order ODE dw(ξ) = ±1, dξ

(1.24)

which has the following the set of solutions: sin[w(ξ)] = ± sin(ξ), or cos[w(ξ)] = ± cos(ξ),

(1.25)

The New Sine-Gordon Expansion Algorithms

129

Case G. Let a = −1, b = 0. Then (1.3) reduces to the first-order ODE dw(ξ) = ±i, dξ

(1.26)

which has the following the set of solutions: sin[w(ξ)] = ±i sinh(ξ), or cos[w(ξ)] = ± cosh(ξ),

(1.25)

Remark 1. One may obtain other types of solutions of (1.3) for other cases of a, b. Some authors have used the solutions (1.6a) of equation (1.5) or the solution (1.14) of equation (1.13) and the same transformation u(ξ) = A0 +

n X

sinj−1 w[Aj sin w + Bj cos w]

(1.24)

j=1

to seek solitary wave solutions in the form [12], [13] u(ξ) = A0 +

n X

tanhj−1 ξ[Aj tanh ξ + Bj sechξ]

(1.25)

j=1

and doubly periodic solutions in the form [14] u(ξ) = A0 +

n X

snj−1 (ξ; m)[Aj msn(ξ; m) + Bj dn(ξ; m)]

(1.26)

j=1

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of nonlinear wave equations, respectively. It is easy to see that (1.5) and (1.13) are two special cases of the general sine-Gordon reduction equation (1.2), and that solutions (1.6a) and (1.14) are special solutions of (1.5) and (1.13), respectively. As the above-mentioned Proposition is said, we also find other many types of solutions of (1.2). Therefore we can also use these solutions and the transformation (1.24) to seek more types of solitary wave solutions, singular solitary wave solutions, and doubly periodic solutions. In this paper we do not limit the transformation (1.24). We will develop a new transformation to seek more types of solutions of nonlinear wave equations. The rest of this paper is arranged as follows: In Section 2 we develop the Algorithm I which is used to solve nonlinear ODEs with constant coefficients and nonlinear PDEs that can reduce to nonlinear ODEs with constant coefficients, and Algorithm II I which is directly used for nonlinear ODEs and nonlinear PDEs. In Sections 3 and 4, we apply the Algorithms I and II to construct solutions of the following nonlinear wave equations: (i) The KdV-mKdV equation with first-order dispersion term (ii) The (2+1)-dimensional generalized Burgers equation (iii) The modified Boussinesq equation (iv) Nonlinear Schrodinger equation with cubic-quintic nonlinearity (v) The (2+1)-dimensional generalization of mKdV equation (vi) The (2+1)-dimensional break soliton equation. As a consequence, many types of exact solutions are deduced which include solitary wave solutions, doubly periodic solutions, optical solitary wave solutions, etc.

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2 2.1

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The New Sine-Gordon Equation Expansion Algorithm Algorithm I-Travelling wave solutions

For a given nonlinear partial differential equation (PDE), say, in two variables x, t F (u, ut, ux , uxx, uxt, utt, . . . ) = 0,

(2.1)

we seek its travelling wave solution, if available, in the form u(x, t) = U (ξ), ξ = k(x−λt). By using the new variable w = w(ξ) satisfying the sine-Gordon reduction equation (1.2), we assume that (2.1) has the solution in the form U (ξ) = U (w(ξ)) = = A0 +

n X i=1

  sini−1 w(ξ) A sin w(ξ) + B cos w(ξ) . i i [R + P sin w(ξ) + Q cos w(ξ)]i

(2.2)

where n, Ai (i = 0, 1, . . . , n), Bj (j = 1, 2, . . ., n), R2 + P 2 + Q2 6= 0 are parameters to be determined later. According to (1.2) and (2.2), we define a degree of the function U (w) in (2.2) as D(U (w)) = n, thus we have the general formula     s d u(w) q p D u (w) = np + q(n + s). (2.3) dξ s Therefore we can use the formula (2.3) to determine the parameter n by balancing the highest order derivative term with nonlinear terms in (2.1). The algorithm is summed up as follows:

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Step 1: (Reduce nonlinear PDE into nonlinear ODE with constant coefficients) We reduce the given nonlinear partial differential equation (2.1) to an ordinary differential equation (ODE), if available, using the travelling wave transformation u(x, t) = U (ξ), ξ = k(x − λt). We know that every term in (2.2) is of the form  −s T = sini w(ξ) cosj w(ξ) R + P sin w(ξ) + Q cos w(ξ) , where i, j, s are non-negative integers. Therefore we have: Case i. When w(ξ) satisfies (1.3), h Tξ = cos w iR sini−1 w cosj+1 w + (i − s)P sini w cosj+1 w+ +(s − j)Q sini+1 w cosj w + iQ sini−1 w cosj+2 w − jR sini+1 w cosj−1 w− i −s−1 −jP sini+2 w cosj−1 w R + P sin w + Q cos w , (2.4) Case ii. When w(ξ) satisfies (1.5), h Tξ = sin w iR sini−1 w cosj+1 w + (i − s)P sini w cosj+1 w+

The New Sine-Gordon Expansion Algorithms

131

+(s − j)Q sini+1 w cosj w + iQ sini−1 w cosj+2 w − jR sini+1 w cosj−1 w− i −s−1 −jP sini+2 w cosj−1 w R + P sin w + Q cos w . (2.5) Case iii. When w(ξ) does not satisfy (1.7), (1.13), (1.18) and (1.21), h Tξ = w0 iR sini−1 w cosj+1 w + (i − s)P sini w cosj+1 w+ +(s − j)Q sini+1 w cosj w + iQ sini−1 w cosj+2 w − jR sini+1 w cosj−1 w−  −s−1 −jP sini+2 w cosj−1 w R + P sin w + Q cos w . (2.6) Therefore it is easy to see that the higher-order derivatives of T (ξ) is a polynomial in T w0i(i = 0, 1). Step 2: (Determine the parameter n) According to (2.3), we can determine the parameter n in (2.2) by balancing the highest order derivative terms and nonlinear terms and thus give the formal solution (2.2) (Remark: (i) If n is not a positive integers, then we firstly make the new transformation U (ξ) = V (ξ)n , and then we perform the second step again. (ii) If n = 0, then we stop the Algorithm.) Step 3: (Generate the polynomial equation in w0s sini w cosj w) Substitute (2.2) with the fixed parameter n along with (1.2) into the obtained ODE to yield the equation 1 X N X 1 Cijs (R, P, Q, Al, Bl , k, λ)w0s sini w cosj w = 0. (2.7) [R+P sin w+Q cos w]M s,i=0 j=1

Since R2 + P 2 + Q2 6= 0, thus we have the polynomial equation 1 X N X

Cijs (R, P, Q, Al, Bl , k, λ)w0s sini w cosj w = 0.

(2.8)

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s,i=0 j=1

Step 4: (Obtain a set of algebraic equations) Set the coefficients of w0s sini w cosj w to zero to get a set of algebraic equations Cijs = 0 with respect to the unknowns k, λ, R, P , Q, Aj (j = 0, 1, . . ., n) and Bj (j = 1, 2, . . ., n). Step 5: (Arrive at exact solutions) Solve the set of algebraic equations, which may not be consistent, finally derive the solitary wave solutions, singular solitary wave solutions and doubly periodic solutions of the given nonlinear equations (2.1) by using u(x, t) = U (ξ), ξ = k(x − λt) and the known solutions and (1.1) shown in Proposition. Remark 2. The algorithm method is constructive and the terms in this algorithm are limit. Therefore bases on the symbolic computation (Maple) and Wu’s elimination method, the whole procedure can be carried out in computer.

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Remark 3. (i) Let R = 1, P = Q = Bj = 0, and w = w(ξ) satisfies (1.3). Then the mentioned-above Algorithm I is equivalent to the famous tanh-method[15]. (ii) Let R = 1, P = Q = 0, and w = w(ξ) satisfies (1.3) or (1.5), the mentioned-above Algorithm I simplifies to be the sine-Gordon method [12], [13] and the Riccati expansion method [5], [6]. (iii) When P = 0 and w = w(ξ) satisfies (1.3), our algorithm is equivalent to the projective Riccati equation method [8], [9]. (iv) When R = 1, P = Q = 0, and w = w(ξ) satisfies (1.13), our algorithm becomes the Fu’s method [14]. Therefore our algorithm is an extension of many methods.

2.2

Algorithm II-Non-travelling wave solutions

It is easy to see that our Algorithm I is only used for those nonlinear ODEs with constant coefficients or nonlinear partial differential equations that can be reduced to the corresponding nonlinear ODEs with constant coefficients using some simple transformations, otherwise the Algorithm I will fail to work. In order to avoid the disadvantage of the Algorithm I, an alternative Algorithm II is presented as follows: Remark 4. (i) Let R = 1, P = Q = Bj = Ai = 0 (i = 0, 1, 2, . . ., n−1; j = 1, 2, . . ., n), and w = w(ξ) satisfies (1.25). Then the mentioned-above Algorithm I is equivalent to the sine-cosine method. If we set ξ → ψ(x, t), then (1.2) becomes q dw(ψ(x, t))/(dψ(x, t)) = µ a + b sin2 w(ψ(x, t)), (2.9) where ψ(x, t) is a smooth function of x, t. For different a, b we know that (2.9) has the similar solutions to (1.2). For the given nonlinear partial differential equation (2.1), we do not need to firstly make the travelling wave transformation u(x, t) = U (ξ), ξ = k(x−λt) to reduce (2.1) to an ODE with constant coefficients. We directly assume that (2.1) has the generalized formal solution u(x, t) = U (w(ψ)) = = A0 (x, t) +

m X sinj−1 w(ψ)[Aj (x, t) sin w(ψ) + Bj (x, t) cos w(ψ)]

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j=1

[R(x, t) + P (x, t) sin w(ψ) + Q(x, t) cos w(ψ)]j

,

(2.10)

where w(ψ) satisfies (2.9), Aj (x, t)0s, Bj (x, t)0s, ψ(x, t), R(x, t)2 +P (x, t)2 +Q(x, t)2 6≡ 0 are functions to be determined later. Similar to the steps mentioned in Algorithm I. Substituting (2.10) with (2.9) into (2.1) yields a set of differential equations w.r.t Aj (x, t)0s, Bj (x, t)0s, ψ(x, t), R(x, t), P (x, t) and Q(x, t). By solving the set of differential equations, if available, and using (2.10) and solutions of (2.9), we may obtain non-travelling wave solutions. Remark 5. If we take ψ(x, t) to be the form ψ(x, t) = ξ = k(x − λt) (k, λ constants) and Aj (x, t)0s, Bj (x, t)0s, ψ(x, t), R(x, t), P (x, t) and Q(x, t) are all constants, then the Algorithm II reduces to the Algorithm I. The Algorithm II is an extension of Algorithm I. But if we can obtain the case that the function ψ(x, t) is not of the linearly combined form of x and t or Aj (x, t)0s, Bj (x, t)0s, ψ(x, t), R(x, t), P (x, t) and Q(x, t) are all not constants, then we will have new non-travelling wave solutions beyond Algorithm I.

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133

Remark 6. We know that the Algorithm I transforms (2.1) into a set of nonlinear algebraic equations with respect to unknown parameters. But the Algorithm II may transform (2.1) into a set of nonlinear partial differential equations with respect to unknown variables. In general, solving the set of nonlinear partial differential equations more difficult than the set of nonlinear algebraic equations. Thus the Algorithm II is more complicated to be used than the Algorithm I.

3

The Applications of Algorithm I

3.1 The KdV-mKdV equation with first-order dispersion term Consider the KdV-mKdV equation with first-order dispersion term ut + γux + (2α + 3βu)uux + uxxx = 0

(3.1)

where α, β, γ are constants. In the following we consider Jacobi elliptic function solutions of (3.1) in the case β 6= 0 using the Algorithm I. According to Step 1, under the travelling wave transformation u(x, t) = u(ξ), ξ = k(x − λt), (3.1) reduces to (γ − λ)

du du du d3 u + 2αu + 3βu2 + k2 3 = 0. dξ dξ dξ dξ

(3.2)

According to Step 2, we know that n = 1 in (2.2) and thus suppose that (3.3) has the solution in the form u(ξ) = A0 +

A1 sin w(ξ) B1 cos w(ξ) + , R + P sin w(ξ) + Q cos w(ξ) R + P sin w(ξ) + Q cos w(ξ)

(3.3)

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and w(ξ) satisfying (1.2). Since it is shown that (3.2) possesses doubly periodic solutions, thus we choose the special equations of (1.2) satisfied by w(ξ) are (1.7), (1.13), (1.18) and (1.21). In the following we mainly consider the case: Transformations (3.3) and (1.13). 3.1.1 6.3.1.1 Transformations (3.3) and (1.8) With the aid of Maple, substituting (3.3) into (3.2) along with (1.8), we have the polynomial of w0s sini w cosj w. Setting their coefficients to zero yields a set of algebraic equations. By solving the set of algebraic equations, we can fix these parameters. Therefore we have the following doubly-periodic solutions of (3.1) from (3.3) and (1.14): Case A. In the case P = 0, we have the following doubly periodic solutions of (3.1):

Family A1. u1 =

A1 msn[k(x − λt); m] + B1 dn[k(x − λt); m] + A0 , R + Qdn[k(x − λt); m]

(3.4)

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where A1 = (2β)−1/2(Q2 − Q2 m2 − R2)1/2,  1/2 B1 = kR−1 (2β)−1/2 Q4 − Q4m2 − 2Q2 R2 + Q2 m2R2 + R4 , h i 1 2 2 2 2 3 2 3 A0 = 1 − 2αR B + k(3kQR − 3kQR m + 3kQ m − 3kQ ) , 1 6βR2  1 λ= 3Q2 βk2 m4 − 3βQ2k2 + 6Q2γβm2 − 6γQ2β− 6β(−Q2 + Q2m2 + R2 )  − 2Q2 α2 m2 + 2α2 Q2 + 6γR2β − 2α2 R2 + 3k2R2 β − 6k2R2 m2β . Note that when m → 1, we have the solitary wave solution of (3.1) from (3.4) u01 =

A1 tanh[k(x − λt)] + B1 sech[k(x − λt)] + A0 , R + Qsech[k(x − λt)]

(3.5)

In the case R > |Q|, the solitary wave solution is a regular one, otherwise the solution (3.5) is singular one. When R > |Q|, k(x − λt) → ∞, u01 → ±A1 /R, while when R > |Q|, k(x − λt) → 0, u01 → B1 /(R + Q).

Family A2. u2 =

A1 msn[k(x − λt); m] + B1 dn[k(x − λt); m] + A0 , R + Qdn[k(x − λt); m]

(3.6)

where A1 = mRk(2 − m2)1/2(β(m2 − 3))−1/2,

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B1 = m2 Rk(m2 − 3)−1 β −1/2(2 − m2 )1/2, Q = R(m2 − 3)−1/2(3 − 2m2 )1/2,   1 2 4 2 2 2 2 2 2 2 λ= 6βk m −4α m −15βm k +12γβm +8α +15βk −24γβ , 12β(m2 −2) h 1 A0 = (2αm2 − 6α)β −1/2(2 − m2 )1/2(m2 − 3)1/2(3 − 2m2)1/2 6β(3 − m2) i − 6km4 + 15km2 − 9k .

Family A3. u± 3 =

m2 Rk(−2β)−1/2sn[k(x − λt); m] α − , R ± Rdn[k(x − λt); m] 3β

(3.7)

where λ = (−6βk2 + 6γβ − 2α2 + 3βm2 k2 )/(6β).

Family A4. n o u4 = k(2β)−1/2 imsn[k(x − λt); m] + dn[k(x − λt); m] − α/(3β),

(3.8)

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135

where λ = (3βk2 − 2α2 + 6γβ − 6βm2k2 )/(6β).

Family A5. u5 =

B1 dn[k(x − λt); m] + A0 , R + Qdn[k(x − λt); m]

(3.9)

where  1/2 2kR−1 β −1/2 Q4 − Q4 m2 − 2Q2 R2 + Q2m2 R2 + R4 , h i 1 3 3 2 2 2 2 2 A0 = k(−6kQ + 6kQ m + 6kQR − 3kQR m ) − αB R , 1 3βR2B1 1 λ= × 6β(−Q4 + Q4 m2 + 2Q2R2 − Q2 m2 R2 − R4)  × 2α2 Q4 + 2α2 R4 + 12γR2βQ2 − 6γR4β + 6γβQ4m2 − 6γR2βQ2 m2 − √

B1 =

− 2α2Q4 m2 + 2α2 R2 Q2m2 + 6k2 R4 m2β − 4α2 R2Q2 − 6γβQ4− − 12k2R4 β − 6βQ4 k2 m4 − 3βQ2k2 R2 m4 − 24βQ2k2 R2m2 +  + 18βQ4k2 m2 + 24βQ2k2 R2 − 12βQ4k2 .

Family A6.

p u6 = km −β/2sn[k(x − λt); m] − α/(3β),

(3.10)

where λ = (−3βk2 − α2 + 3γβ − 3βm2k2 )/(3β).

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Family A7. p u7 = km 2(1 − m2)/βsd[k(x − λt); m] − α/(3β),

(3.11)

where λ = (−3γβ − α2 − 3βk2 + 6βm2k2 )/(3β). Case B. In the case Q = 0, we have the following solutions of (3.1):

Family B1. u1 =

A1 msn[k(x − λt); m] + B1 dn[k(x − λt); m] + A0 , R + P msn[k(x − λt); m]

(3.12)

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where

q 1 2βB12 + k2 P 2 m2 , k A1 = B1 (2βB12 + k2 P 2 m2 )−1/2(−2βB12 + k2 P 2 − k2 P 2 m2)1/2, 1 h A0 = − 2αB1 + 3k2 P m2 (2βB12 + k2P 2 m2)−1/2× 6B1 β i × (−2βB12 + k2P 2 − k2 P 2 m2 )1/2 , 1  λ= − 9k4 P 2 m4 + 9k4 P 2 m2 − 12k2 m2βB12 + 12B12 β  + 6k2βB12 + 12γβB12 − 4α2 B12 . R=

Family B2. u± 2 =

±B1 msn[k(x − λt); m] + B1 dn[k(x − λt); m] + A0 , R + 2Rmsn[k(x − λt); m]

(3.13)

where

p R = B1 /k β/(1 − 2m2 ), h i. p A0 = ± 3βm2 k (1 − 2m2)/β + 2αm2 − α (3(1 − 2m2)β),   1 4 2 2 2 2 2 2 2 2 λ= k −4α m +12γβm −6k m β −6γβ +2α −3k β . 6βm 6β(2m2 −1)

Family B3. u± 3

=

k

p

(m2 − 1)/(2β)dn[k(x − λt); m] α − , 1 ± msn[k(x − λt); m] 3β

(3.14)

where λ = (3k2β + 6γβ + 3k2 m2 β − 2α2 )/(6β).

Family B4.

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u4 = where

A1msn[k(x − λt); m] + A0 , R + P msn[k(x − λt); m]

p A1 = kR−1 β −1 (−2R4 − 2P 4 m2 + 2R2P 2 + 2P 2 R2 m2), 1 1 A0 = − αβ −1 − (Rβ)−1(3kR2P + 3kP R2 m2 − 6kP 3 m2)× 3 3 p 4 × β(−2R − 2P 4 m2 + 2R2P 2 + 2P 2 R2m2 )−1 ,  λ = − 2α2R4 − 6k2R4 β + 2α2 R2 P 2 − 6k2 R4 m2β + 2α2 R2P 2 m2 − − 2α2 P 4 m2 − 6γR2βP 2 + 6γβP 4m2 − 6γR2βP 2 m2 − 6k2P 4 m4 β+ + 6γR4β − 6k2 P 4 m2β − 3k2 P 2 R2m4 β + 30k2R2 P 2 m2β− . −1 − 3k2 R2P 2 β 6(R4 + P 4 m2 − R2P 2 − P 2 R2 m2)β .

(3.15)

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137

Family B5. u5 = √

kdn[k(x − λt); m] α − , 2β(1 ± msn[k(x − λt); m] 3β

(3.16)

where λ = (3k2 β + 6γβ + 3k2m2 β − 2α2)(3β).

Family B6. u6 = k

p

−β/2ds[k(x − λt); m] − α/(3β),

(3.17)

where λ = (3γβ − α2 + 6k2 m2β − 3k2β)/(3β). Case C. In the case R = 0, we have the solution u= where

ZQ−1 msn[k(x − λt); m] + B1 dn[k(x − λt); m] + A0 , P msn[k(x − λt); m] + Qdn[k(x − λt); m]

(3.18)

h A0 = −[3βQ(Z − P B1 )]−1 3βP B12 + αP B1 Q − (Qα + 3βB1 )Z+ i + 6k2 P 3 m2 + 6k2 Q2P m2 − 3k2 Q2P , h i−1 λ = 6β(−Q4 + Q4 m2 + P 4 m2 + 2Q2 P 2 m2 − Q2 P 2 ) ×  × − 6γβQ4 + 12P 4 k2 m4β − 6P 4 βk2 m2 + 24P 2 k2Q2 m4 β− − 24k2 P 2 m2βQ2 − 18k2Q4 m2β + 12k2Q4 m4 β + 6γβQ4m2+ + 6γP 4βm2 + 12γP 2βQ2 m2 − 6γP 2βQ2 − 2α2Q4 m2 − 2α2 P 4 m2 −  − 4α2 P 2 Q2 m2 − 3k2 P 2 βQ2 + 6k2 Q4β + 2α2 Q4 + 2α2 P 2 Q2

with Z being the solution of algebraic equation βZ 2 − 2βP B1 Z − 2k2Q4 + βP 2 B12 + 2k2 Q4m2 + 2k2 P 4 m2 +

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+4k2 P 2 m2 Q2 − 2k2 P 2 Q2 = 0.

(3.19)

Similarly, we also obtain other types of doubly-periodic solutions of (3.1) from (3.3) and solutions (1.15)–(1.17) of equation (1.13). We omit them here. 3.1.2 6.3.1.2 Transformations (3.3) and (1.7) Similar to Section 3.1a, with the aid of Maple, substituting (3.3) into (3.2) along with (1.7), we have the polynomial equation of w0s sini w cosj w. Setting their coefficients to zero yields a set of algebraic equations. By solving the set of algebraic equations, we can fix these parameters. Therefore we also get many types of doubly-periodic solutions of (3.1) from (3.3) and solutions (1.8)–(1.11) of (1.7). Here we only two families of solutions as follows: B1 cn[k(x − λt); m] u1 = (3.20) + A0 , ±P + P sn[k(x − λt); m] + Qcn[k(x − λt); m]

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where h i1/2 B1 = kP −1 (−2m2Q2 P 2 + m2P 4 − P 4 − 2Q2P 2 − Q4 + Q4 m2)/(2β) , A0 = −α/(3β) − 1/(6βP )(−3km2P 2 Q − 3kP 2 Q − 3kQ3 + 3kQ3m2 )× i−1/2 h , × (−2m2Q2 P 2 + m2P 4 − P 4 − 2Q2P 2 − Q4 + Q4 m2 )/(2β)  λ = 1/(6β) − 2 α2Q4 m2 + 3 k2m4 P 4 β − 12 γ P 2 β m2 Q2 + 6 γ β Q4m2 + + 4 α2P 2 m2 Q2 + 2 α2Q4 + 2 α2 P 4 + 6 γ P 4 β m2 − 12 γ P 2 β Q2 − − 6 γ β Q4 − 6 k2 m4P 2 Q2 β + 3 k2Q4m4 β − 6 k2P 2 Q2 β+ + 24 k2m2 P 2 Q2 β − 3 k2Q4β − 3 k2P 4 β − 6 γ P 4 β − 2 α2P 4 m2 +  −1 + 4 α 2 P 2 Q2 − 2 m 2 Q2 P 2 + m 2 P 4 − P 4 − 2 Q 2 P 2 − Q 4 + Q4 m2 and u2 =

A1sn[k(x − λt); m] + A0 , ±Q + P sn[k(x − λt); m] + Qcn[k(x − λt); m]

(3.21)

where h i1/2 A1 = kQ−1 (4m2Q2P 2 − 2Q2P 2 − Q4 − P 4 )/(2β) , A0 = −α/(3β) − 1/(6βQ)(6km2Q2P − 3kQ2 P − 3kP 3 )× h i−1/2 × (4m2Q2 P 2 − 2Q2P 2 − Q4 − P 4 )/(2β) ,  λ = 1/(6β) − 6 k2P 2 Q2β − 3 k2P 4 β − 8 α2P 2 m2 Q2 + 4 α2 P 2 Q2+ + 2 α2P 4 − 3 k2Q4 β − 6 γ β Q4 + 24 γ P 2 β m2Q2 + 12 k2m4 P 2 Q2 β−

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− 12 k2m2P 2 Q2β + 6 k2m2 P 4 β + 6 k2Q4m2 β − 12 γ P 2 β Q2 −  − 6 γ P 4 β + 2 α2Q4 (−4m2Q2 P 2 + 2Q2P 2 + Q4 + P 4 )−1 . Moreover, we also get other types of solutions of (3.1) using transformation (3.3), (1.18) and (1.21).

3.2 (2+1)-dimensional generalized Burgers equation We consider the (2+1)-dimensional generalized Burgers equation (ut + αur ux − uxx )x + uyy = 0,

(3.22)

where α, r 6= 0 are parameters. When α = r = 1, (3.22) reduces to the (2+1)-dimensional Burgers equation which has been shown to not pass the standard Painleve test[16]. Let u(x, y, t) = u(ξ), ξ = k(x + ly + λt). Then (3.22) reduces to (λ + l2)u00 + αur u00 + αrur−1 u02 − ku000 = 0.

(3.23)

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139

According to Step 2, we know that we set u = v 1/r . Thus we have from (3.23) r(λ + l2)vv 0 + αrv 2 v 0 − k(1 − r)v 02 − krvv 00 = 0.

(3.24)

Using Step 2 again, we know that n = 1 in (2.2) and thus suppose that (3.24) has the solution in the form B1 cos w(ξ) A1 sin w(ξ) + , (3.25) v(ξ) = A0 + R + P sin w(ξ) + Q cos w(ξ) R + P sin w(ξ) + Q cos w(ξ) with w(ξ) satisfying (1.3). With the aid of Maple, substituting (3.25) into (3.24) along with (1.3), we have the polynomial of sini w cosj w. Setting their coefficients to zero yields a set of algebraic equations. By solving the set of algebraic equations, we can fix these parameters. Therefore we have the following solitary wave solution solutions of (3.22) from (3.25) and solution (1.4a) of equation (1.3) and u = v 1/r :  1/r A1 tanh k(x + ly + λt) u 1 = A0 + , (3.26) R + P tanh k(x + ly + λt) ± Rsechk(x + ly + λt) where A1 = 2RA0 (αrA0 + kr + k)/(k(r + 1)), P = −R(2αrA0 + kr + k)/(k(r + 1)), λ = −(rl2 − k)/r. " #1/r A1 A21 tanh k(x + ly + λt) + A1 B1 sechk(x + ly + λt) p u2 = ± + , R A1 R + R A21 + B12 sechk(x + ly + λt)

(3.27)

where k = −2αrA1 /(R(1 + r)), λ = −l2R + l2Rr − 2αA1 )/(R(1 + r)).   A1 tanh k(x + ly + λt) + B1 sechk(x + ly + λt) 1/r u 3 = A0 + , R + P tanh k(x + ly + λt)

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where

(3.28)

p p A1 = B1 /R P 2 − R2 , A0 = B1 /R (P − R)/(P + R),

k = 2αrB1 /(1 + r)(P 2 − R2 )−1/2, λ = 2αB1 /(1 + r)(P 2 − R2)−1/2 − l2. 1/r  A0 A1 tanh k(x + ly + λt) u4 = A 0 + , (3.29) ±(A0 P + A1 ) + P tanh k(x + ly + λt) where k = −(A0 P + A1 )A0αr/(2rA0P + 2A + 0P + rA1 + A1 ),  λ = − 2αA20 P + 2l2A0 P + 2P rl2A0 + 2αA0A1 +  + l2A1 + rl2A1 /(2rA0P + 2A0 P + rA1 + A1 ).     k(r + 1) k(r + 1) 1/r 1 2 u5 = − . (3.30) tanh k x + ly − (−2k + rl )t − αr r αr In addition, we also get other types of solutions according to (3.25) and solution (1.4b) of equation (1.3). It is shown that (3.22) does not possess doubly periodic solutions.

140

Z. Y. Yan

3.3 The modified Boussinesq equation We consider the modified Boussinesq equation[17,10]   3 2 2 ut = v − α u , 2 x

(3.31)

vt = −3α2 (uxx − uv + α2 u3)x , where α is a constant. Recently, we have give three Jacobi elliptic functions of (3.31) using the sinh-Gordon equation expansion method [10]. In the following we will apply our Algorithm I to (3.31). The travelling wave transformation u = u(ξ), v = v(ξ), ξ = k(x + λt) transforms (3.31) into the set of ODEs 3 λu0 = v 0 − α2 (u2 )0, 2 (3.32) 0 2 000 λv = −3α [u − (uv)0 + α2 (u3)0 ]. Suppose that (3.32) has the solution using Step 2 u(ξ) = A0 +

A1 sin w(ξ) B1 cos w(ξ) + , (3.33a) R + P sin w(ξ) + Q cos w(ξ) R + P sin w(ξ) + Q cos w(ξ)

v(ξ) = a0 + +

a1 sin w(ξ) b1 cos w(ξ) + + R + P sin w(ξ) + Q cos w(ξ) R + P sin w(ξ) + Q cos w(ξ)

a2 sin w(ξ) cos w(ξ) b2 cos2 w(ξ) + , 2 [R + P sin w(ξ) + Q cos w(ξ)] [R + P sin w(ξ) + Q cos w(ξ)]2

(3.33b)

where ai , bi , Ai , Bi , R, P , Q are parameters to be determined and w(ξ) satisfies (1.7). The substitution of (3.33a,b) into (3.32) yields the following new doubly periodic solutions of (3.31): cn[k(x + λt); m] λ − 2, R ± Rsn[k(x + λt); m] + Qcn[k(x + λt); m] 3α

(3.34a)

3α2 cn2 [k(x + λt); m] λ2 − , 2{R ± Rsn[k(x + λt); m] + Qcn[k(x + λt); m]}2 6α2

(3.34b)

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u1 =

v1 = where

p R = α2 /(mλ) (1 − m4 )/2,

p √ k = λ/α 1/(2 + 2m2 ), Q = α2 (1 + m2)/( 2λm). cn[k(x + λt); m] λ − 2, P sn[k(x + λt); m] + Qcn[k(x + λt); m] 3α

(3.35a)

3α2cn2 [k(x + λt); m] λ2 − , 2{P sn[k(x + λt); m] + Qcn[k(x + λt); m]}2 6α2

(3.35b)

u2 = v2 =

The New Sine-Gordon Expansion Algorithms

141

where p P = 2α2 /(m2λ) −m4 + 3m2 − 2, p √ Q = 2α2 (m2 − 2)/(m2λ), k = λ/(α 8 − 4m2 ). 2λm u3 = p sn α2 2(1 + m2 ) 3λ2m2 v3 = 2 sn2 α (1 + m2 )

"

"

# λ λ p (x + λt); m − 2 , 3α α 2(1 + m2)

# λ λ2 p (x + λt); m − 2 , 6α α 2(1 + m2)

(3.36a)

(3.36b)

In particular, when m → 1, we have solitary wave solutions of (3.31) from (3.36a,b)   λ λ λ = 2 tanh (x + λt) − 2 , α 2α 3α

(3.37a)

  3λ2 λ 4λ2 2 = − 2 sech (x + λt) + 2 , 2α 2α 3α

(3.37b)

u03

v30

If the w(ξ) in (3.33a,b) satisfies (1.13), then we have the following doubly periodic solutions of (3.31) dn[k(x + λt); m] λ − , R ± Rmsn[k(x + λt); m] + Qdn[k(x + λt); m] 3α2

(3.38a)

3α2 dn2[k(x + λt); m] λ2 − , 2{R ± mRsn[k(x + λt); m] + Qdn[k(x + λt); m]}2 6α2

(3.38b)

u4 =

v4 = where

p R = α2 /(mλ) (1 − m4 )/2,

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p √ k = λ/α 1/(2 + 2m2 ), Q = α2 (1 + m2)/( 2λm). u5 =

v5 =

dn[k(x + λt); m] λ − , mP sn[k(x + λt); m] + Qdn[k(x + λt); m] 3α2

3α2 dn2[k(x + λt); m] λ2 − , 2{mP sn[k(x + λt); m] + Qdn[k(x + λt); m]}2 6α2

(3.39a)

(3.39b)

where p P = 2α2/λ −2m4 + 3m2 − 1, p √ Q = 2α2 (2m2 − 1)/λ, k = λ/(α 8 − 4m2 ). Using other special cases of (1.2), we may also obtain other types of solutions of (3.31).

142

Z. Y. Yan

3.4 Nonlinear Schr¨odinger equation with cubic-quintic nonlinearity We consider the nonlinear Schr¨odinger equation with cubic-quintic nonlinearity [18] iqz + qtt + 2|q|2q + γ|q|4q + iγ1qttt + iγ2(|q|2q)t + iγ3(|q|4q)t = 0.

(3.40)

where γ, γ1, γ2 and γ3 are constants and 2γ3 = γγ2, which describes the effects of quintic nonlinear terms. (3.40) contain the the group velocity dispersion, self-phase modulation, third order dispersion, cubic and quintic terms. When the pulse widths become greater than 100 fs, one can neglect the last three terms. Hong [19] gave its bright and dark solitary wave solutions. Recently, we used a direct transformation to obtain many types of solitary wave solutions of (3.40). Here we mainly consider its doubly periodic solutions. First of all, we make the gauge transformation in the form u(z, t) = u(ξ) exp(iη), ξ = k(t + λz), η = αt + βz,

(3.41)

where k, λ, α, β are constants. The substitution of (3.41) into (3.40) yields a complex ODE of u(ξ) and its real and imaginary parts read, respectively, (k2 − 3αr1k2 )u00 + (−β − α2 + α2 r1)u + (2 − αr2 )u3 + (r − αr3)u5 = 0, r1k2 u000 + (λ + 2α − 3r1α2 )u0 + 3r2u2u0 + 5r3u4 u0 = 0.

(3.42a) (3.42b)

It is easy to see that (3.42a,b) can reduce to the same equation r1k2 u00 + (λ + 2α − 3r1α2 )u + r2 u3 + r3u5 = 0,

(3.43)

under these conditions α=

r2 − 2r1 1 , β = −α2 + α3 r1 + (αr2 − 2)(λ + 2α − 3r1α2 ). 2r1r2 r2

(3.44)

To seek doubly periodic solutions of (3.43), we make the transformation u(ξ) = v 1/2. Then (3.43) reduces to

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r1 k2(−1/4v 02 + 1/2vv 00) + (λ + 2α − 3r1α2 )v 2 + r2v 3 + r3v 4 = 0.

(3.45)

Suppose that (3.45) has the solution using Step 2 u(ξ) = A0 +

A1 sin w(ξ) B1 cos w(ξ) + . R + P sin w(ξ) + Q cos w(ξ) R + P sin w(ξ) + Q cos w(ξ)

(3.46)

Using (3.46) and (1.7) we get the doubly periodic solutions of (3.40)   3r2 3r2 1/2 u1(t, z) = ± sn[k(t + λz); m] − exp[i(αt + βz)], 8r3 8r3

(3.47)

where k=

r2 p −3/(r1r3), λ = (192α2r3m2 r1 − 128αr3m2 − 3r22 + 15r22m2 )/(64r3m2). 4m

The New Sine-Gordon Expansion Algorithms   3r2 m 3r2 1/2 u2 (t, z) = ± sn[k(t + λz); m] − exp[i(αt + βz)], 8r3 8r3

143 (3.48)

where p k = r2 /4 −3/(r1r3 ), λ = (−3r22 m2 + 192α2r1r3 − 128αr3 + 15r22)/(64r3).   3r2 3r2 1/2 u3(t, z) = ± cn[k(t + λz); m] − exp[i(αt + βz)], 8r3 8r3

(3.49)

where k=

u4 =

r2 p 3/(r1r3 ), λ = (12r22m2 − 128αr3m2 + 192α2r3m2 r1 + 3r22)/(64r3m2). 4m 

2r2(9m2 − 1)sn[k(t + λz); m] r2(9m2 − 1) − r3(15m2 + 1){sn[k(t + λz); m] ± 3} 2r3(15m2 + 1)

1/2

exp[i(αt + βz)], (3.50)

where p k = r2 /(15m2 + 1) (6 − 54m2)/(r1r3), λ = (189 m4r2 2 + 2700 m4α2 r1r3 − 1800 m4α r3 + 360 α2r1 r3m2 − 240 α r3m2+ + 6 r22 m2 − 3 r22 + 12 α2r1 r3 − 8 α r3)/[4r3(15m2 + 1)2]. u5 =



cn[k(t + λz); m] 3r2(1 + 8m2 ) − R + Qcn[k(t + λz); m] 2r3(−1 + 16m2)

1/2

exp[i(αt + βz)],

(3.51)

where k = r2

p

(6 + 48m2)/(r1r3 )/(16m2 − 1),

Q = −r3 (1 + 8m2)/[2r2(−1 + 16m2)], p R = (16m2 − 1)r3 1 − m2 /[6(1 + 8m2)mr2].

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Using (3.46) and (1.13) we get other doubly periodic solutions of (3.40) u6 (t, z) =

(

)1/2 √ 3r2 1 − m2 3r2 ds[k(t + λz); m] − exp[i(αt + βz)], 8r3 8r3

(3.52)

where k = r2

p

3/[16r1r3(m2 − 1)],

λ = 12r22m2 − 15r22 − 128αm2r3 + 128αr3+ + 192α2r1m2 r3 − 192α2r1r3/[64r3(m2 − 1)]. u7 =



dn[k(t + λz); m] 3r2(9m2 − 8) + ±3Q + Qdn[k(t + λz); m] 2r3(15m2 − 16)

1/2

exp[i(αt + βz)],

(3.53)

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Z. Y. Yan

where Q = −r3 (15m2 − 16)/[2r2(9m2 − 8)], p k = r2/(15m2 − 16) (48 − 54m2)/(r1r3), λ = (189 m4r22 − 1800 m4r3α + 2700 m4r3α2 r1 − 384 r22 m2 + 3840 α r3m2 − − 5760 α2r1r3m2 + 3072 α2r1r3 + 192 r22 − 2048 α r3)/[4r3(15m2 − 16)2]. Particularly, when m → 1, we know that the solutio (3.53) reduces to u07

=



3r2 sech[k(t + λz)] − ±3Q + Qsech[k(t + λz)] 2r3

1/2

exp[i(αt + βz)],

(3.54)

where Q = r3/(2r2), k = −r2

p

−6/(r1r3 ),

λ = (189r22 − 1800r3α + 2700r3α2 r1 − 384 r22 + 3840 α r3− −5760 α2r1r3 + 3072 α2r1r3 + 192 r22 − 2048 α r3)/(4r3).

3.5 The (2+1)-dimensional generalization of mKdV equation We study the (2+1)-dimensional generalization of mKdV equation [20] ut = α[(uyy + 3uxx)y + 6u2 uy ] + (ψxu)y + ux ψy , ψyy − ψxx = 6(u2)x ,

(3.55)

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where α is a constant, which was derived from an auxiliary linear system using the binary Darboux transformation[20]. According to our Algorithm I, we can obtain the following doubly periodic solutions of (3.55): Zξ A1 sn(ξ; m) 6 A21 sn2 (ξ; m) u1 = , ψ1 = dξ, (3.56) 1 ± cn(ξ; m) k(l2 − 1) [1 ± cn(ξ; m)]2 where ξ = k(x + ly + λt), A1 = k(16 + 4αl2 − 4α)−1/2(−αl4 + 2αl2 − 3α)1/2, λ = −(−l2 + 2l2m2 − 3 + 6m2 )αk2 l/2. B1 cn(ξ; m) 6 u2 = , ψ2 = 2 1 ± sn(ξ; m) k(l − 1)

Zξ

A21 cn2 (ξ; m) dξ, [1 ± sn(ξ; m)]2

(3.57)

where ξ = k(x + ly + λt), λ = (l2 + l2m2 + 3 + 3m2))αk2l/2, B1 = k(16 + 4αl2 − 4α)−1/2(−3αm2 + 2αl2m2 + αl4 − αl4m2 + 2αl2 − 3α)1/2.

The New Sine-Gordon Expansion Algorithms u3 =

mkcn(ξ; m) , R ± Rsn(ξ; m) + Qcn(ξ; m)

6 ψ3 = k(l2 − 1)

Zξ

m2 k2cn2 (ξ; m) dξ, [R ± Rsn(ξ; m) + Qcn(ξ; m)]2

145 (3.58a)

(3.58b)

where ξ = k(x + ly + λt), λ = −αk2 l3m2 − αk2 l3 − 3αk2 lm2 − 3αk2 l, Q = (αl4 + 2αl2 − 3α)−1/2(αm2 + αl2 m2 − 4m2 − α + αl2 + 4)1/2, R = (αl4 + 2αl2 − 3α)−1/2(−4m2 + 4 − αl2 m2 + αl2 − αm2 + α)1/2. u4 =

mksn(ξ; m) , R + P sn(ξ; m) ± Rcn(ξ; m)

6 ψ4 = 2 k(l − 1)

Zξ

m2 k2 sn2 (ξ; m) dξ, [R + P sn(ξ; m) ± Rcn(ξ; m)]2

(3.59a)

(3.59b)

where ξ = k(x + ly + λt), λ = 2αk2 l3m2 − αk2 l3 − 3αk2 l + 6αk2 lm2, P = (−2αl2 − αl4 + αl4m2 + 2αl2m2 + 3α − 3αm2 )−1/2× ×(−αl2 − 4 + α + 2αl2m2 + 8m2 − 2αm2 )1/2, R = −(−2αl2 − αl4 + αl4 m2 + 2αl2m2 + 3α − 3αm2)−1/2(−α + αl2 + 4)1/2.

4

The Application of Algorithm II

4.1 (2+1)-dimensional break soliton equation Consider the (2+1)-dimensional break soliton equation [21]

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ut + βuxxy + 4βuuy + 4βu∂x−1 uy = 0.

(4.1)

where ∂x = ∂/∂x , ∂x−1 ∂x = ∂x ∂x−1 = 1. To eliminate the integral sign, we see the solution of (4.1) in the form u(x, y, t) = u(ψ), ψ = kx + f (y, t), where k is a constants and f (y, t) is a function of y, t to be determined. Thus (4.1) reduces to du d3 u du ft + βk2 fy 3 + 8βfy u = 0. (4.2) dψ dψ dψ According to the Algorithm II, we know that (4.2) has the solution u = a0(x, y, t) +

+

a1 (x, y, t) sin w(ψ) + b1(x, y, t) cos w(ψ) + R(x, t) + P (x, t) sin w(ψ(x, t)) + Q(x, t) cos w(ψ)

b2(x, y, t) sin w(ψ) cos w(ψ) + a2 (x, y, t) sin2 w(ψ) , [R(x, t) + P (x, t) sin w(ψ(x, t)) + Q(x, t) cos w(ψ)]2

(4.3)

146

Z. Y. Yan

where w(ψ) = w(ψ(x, y, t) satisfies q dw(ψ) = µ 1 − m2 sin2 w(ψ), dψ

(4.4)

The substitution of (4.3) into (4.2) with (4.4) yields a system of differential equations with respect to unknown k, f , a0, a1 , b1, a2 , b2. It may be difficult to solve directly the system. Let a1 = b1 = 0, a0 = const., a2 = const., b2 = const., ft = λfy , λ = const, (4.5) Then we can obtain the following solutions of (4.1) 3 1 u1 = − bk2m2 sn2 [(kx + f (y + λt)); m] + (4bk2m2 + 4bk2 − λ), 2 8

(4.6)

3bk2sn2 [(kx + f (y + λt)); m] 1 + (4bk2m2 − 2bk2 − λ), 2 8{1 ± cn[(kx + f (y + λt)); m]} 8

(4.7)

u1 = −

3 1 u3 = bk2 m2{sn2 (ψ; m) ± isn2 (ψ; m)cn(ψ; m)} + (bk2m2 + 4bk2 − λ), 2 8 u4 =

3 2 2 1 bk (m − 1)sc2[(kx + f (y + λt)); m] + (4bk2m2 − 84bk2 − λ), 2 8

(4.8) (4.9)

where f (y + λt) is an arbitrary smooth function of y + λt. If f (y+λt) is a linear function of y+λt, then we can obtain the corresponding travelling wave doubly periodic solutions of (4.1) from (4.6)–(4.9). Particularly in the case m → 1 we have solitary-like wave solution of (4.1) from (4.6) as 3 1 u = − bk2 tanh2 [kx + f (y + λt)] + (8bk2 − λ), 2 8

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5

(4.10)

Conclusions

In summary, based on the sine-Gordon reduction equation, we have presented two new and powerful algorithms and apply the algorithms to seek for more types of exact solutions of some nonlinear wave equations. Our algorithms are also applied to other many types of nonlinear wave equations.

Acknowledgement This work is supported by the National Natural Science Foundation of China (No. 10401039), the National Key Basic Research Project of China (No. 2004CB318000) and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

The New Sine-Gordon Expansion Algorithms

147

References [1] C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura, Phys. Rev. Lett., 19 (1967), 1095. [2] M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering (Cambridge University Press, Cambridge, 1991). [3] V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons (SpringerVerlag, Berlin, 1991). [4] R. Hirota, Phys. Rev. Lett., 27 (1971), 1192. [5] E. G. Fan, Phys. Lett. A, 277 (2000), 212. [6] Z. Y. Yan, Phys. Lett. A, 285 (2001), 355. [7] R. Anderson, J. Harnad and P. Winternitz, Physica D, 4 (1982), 164. [8] R. Conte and M. Musette, J. Phys. A, 25 (1992), 5609. [9] Z. Y. Yan, Chaos Solitons and Fractals, 16 (2003), 759. [10] Z. Y. Yan, J. Phys. A, 36 (2003), 1916. [11] Z. Y. Yan, Chaos Solitons and Fractals, 16 (2003), 291. [12] C. T. Yan, Phys. Lett. A, 224 (1996), 77. [13] Z. Y. Yan and H. Q. Zhang, Phys. Lett.A, 252(1999), 251. [14] Z. Fu et al., Phys. Lett. A, 299 (2002), 507. [15] W. Malfleit, Am. J. Phys., 60 (1992), 650. [16] G. M. Webb and G. P. Zank, J. Phys. A, 23(1990), 5465.

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[17] P. A. Clarkson (ed.) Symmetries and Integrability of differential Equations (Cambridge University Press, Cambridge, 1999). [18] R. Radhakrishnan, et al., Phys. Rev. E, 60 (1999), 3314. [19] W. P. Hong, Opt. Commun., 194 (2001), 217. [20] K. Imai and K. Nozaki, J. Phys. Soc. Jpn., 65 (1996), 53. [21] Z. Y. Yan and H.Q. Zhang, Comput. Math. Appl., 43 (2002), 1409.

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Index discrete, 32 exact, 10, 32, 34

generalization of mKdV equation, 134 break soliton equation , 135 generalized Burgers equation, 128

flux, 5, 27 Fokker–Planck equation, 65

Ablowitz-Ladik lattice, 36

Gardner lattice, 43 group velocity dispersion, 2

Belov–Chaltikian lattice, 49 bilinear method, 5, 86 bilinear operator, 6, 87 Blaszak-Marciniak lattice, 50 Bogoyavlenskii lattice, 49 Boussinesq equation, 16

homotopy operator continuous, 11, 13 discrete, 33 invariance, 7, 29, 39

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canonical, 28 conservation law, 67 continuous, 5 discrete, 27

Jacobi elliptic function, 90, 123 Kac-van Moerbeke (KvM) lattice, 29 KdV-mKdV equation, 123 Korteweg-de Vries (KdV) equation, 5

density, 5, 27 derivative Fr´echet, 5 partial, 66 total, 5, 28 variational, 10, 32 difference operator, 27 dilation invariant, 7, 29 symmetry, 7, 29 double Liouville equation, 22 doubly periodic solution, 115, 124 Drinfel’d–Sokolov–Wilson (DSW) equations, 14

Laurent series, 4 linearization, 66 Liouville equation, 22 method undetermined coefficients, 7, 8, 29 modified Boussinesq equation, 130 modified Volterra lattice, 42 multiple scales, 38, 39 nonclassical method, 68 nonclassical potential equation approach, 68 nonclassical potential system approach, 68 nonlinear Schr¨odinger equation, 2 cubic-quintic nonlinearity, 131 resonant, 86 varying dispersion, 4

equivalence criterion, 28 equivalence relation, 28 equivalent, 11, 28 Euler operator continuous, 10, 13 149

150

Index

Painlev´e analysis, 3, 4 periodic wave solution, 90 porous medium equation, 66 potential equation, 67, 68 potential system, 65, 67

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rank, 7, 29 Riccati equation projective, 116 Riccati equation, 116 scaling symmetry, 7, 29 self-phase modulation, 2 shift operator down-shift, 27 up-shift, 27 similarity solutions, ix, 97 sine-Gordon (sG) equation, 19, 116 in laboratory coordinates, 20 in light-cone coordinates, 20 sine-Gordon equation expansion algorithms, 120 Algorithm I, 120 Algorithm II, 122 sinh-Gordon equation, 22 reduction equation, 1116 software condens.m, 51 Conlaw, 52 CONSLAW, 51 CRACK, 52 DDEDensityFlux.m, 51 diffdens.m, 51 discrete, 51 GeM, 52 InvariantsSymmetries.m, 51 Jets, 52 Noether, 52 TransPDEDensityFlux.m, 51 Vessiot, 52 solitary wave, 1, 115 soliton, 1 bright, 5 chirp-free, 7 chirped, 6, 7 chirped higher order, 8

chirped two-soliton, 9 dark, 5 fundamental, 2 optical, 2 two-soliton, 88 soliton anti-dark, 89 symmetry, 65 nonclassical, 65, 68 nonclassical potential, 65, 70 nonlocal, 68 potential , 67 tanh method, 101, 122 Toda lattice, 34 transcendental nonlinearities, 19, 21 travelling wave transformation, 130 uniform in rank, 8, 29 variational analysis, 6 Volterra lattice, 29 weight, 7, 29