Bilinear Transformation Method (Mathematics in Science and Engineering, 174)

Bilinear Transformation Method

This is Volume 174 in MATHEMATICS IN SCIENCE AND ENGINEERING A Series of Monographs and Textbooks Edited by RICHARD BELLMAN, University of Southern California The complete listing of books in this series is available from the Publisher upon request.

Bilinear Transformation Method Yoshimasa Matsuno Space System Designing Section MHI Ltd. Nagoya Aircrafi Works Nagoya, Japan

1984

ACADEMIC P R E S S , I N C .

(Harcourt Brace Jovanovich, Publishers) Orlando San Diego New York London Toronto Montreal Sydney Tokyo

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Library of Congress Cataloging in Publication Data

Matsuno, Y. (Yoshimasa) Bilinear transformation method. Includes bibliographical references and index. 1. Bilinear transformation method. 2. Evolution equations, Nonlinear--Numerical solutions. 3. Benjamin-Ono equations. I. Title. 1984 5 15.3'5 84-70234 QA374.M34 ISBN 0-12-480480-2 (alk. paper)

PRINTED IN THE UNITED STATE3 OF AMERICA 84858681

9 8 1 6 S 4 3 2 1

Contents

Preface

vii

1 Introduction and Outline 1 . 1 Introduction 1.2 Outline 3

1

2 Introduction to the Bilinear Transformation Method 2.1 2.2 2.3 2.4 2.5 2.6

Bilinearization 6 Exact Solutions 8 Backlund Transformation Conservation Laws 39 Inverse Scattering Method Bibliography 42

30 41

3 The Benjamin-Ono Equation 3.1 Multisoliton Solutions of the Benjamin-Ono Equation 48 3.2 Backlund Transformation and Conservation Laws of the Benjamin-Ono Equation 68 3.3 Asymptotic Solutions of the Benjamin-Ono 79 Equation 3.4 Stability of the Benjamin-Ono Solitons 90 3.5 The Linearized Benjamin-Ono Equation and Its Solution 91 V

vi

Contents

4 Interaction of the Benjamin-Ono Solitons 4.1 Asymptotic Behaviors of the N-Soliton Solution 98 4.2 Interaction of Two Solitons 102

5 The BenjambOno-Related Equations 118 5.1 Higher-Order Benjamin-Ono Equations 5.2 Higher-Order Korteweg-de Vries Equations 126 5.3 The Finite-Depth Fluid Equation and Its Higher-Order Equations 133 5.4 Higher-Order ModiJied Korteweg-de Vries Equations I50 5.5 Backlund Transformations and Inverse Scattering Transforms of Higher-Order Korteweg-de Vries Equations 158

6 Topics Related to the Benjamin-Ono Equation 6.1 The Modijied Benjamin-Ono Equation 6.2 The Derivative Nonlinear Schrodinger Equation 175 6.3 The Perturbed Benjamin-Ono Equation

170 I78

Appendix I: Formulas of the Bilinear Operators Appendix 11: Properties of the Matrices M and A 191 Appendix 111: Properties of the Hilbert Transform Operator 20 1 207 Appendix IV: Proof of (3.274) References

21 1

Author Index

219

Subject Index

22 I

185

Preface

This volume may be divided into two parts. The first part (Chapter 2) is an introduction to the bilinear transformation method. This method is a powerful tool for solving a wide class of nonlinear evolution equations. In the bilinear formalism, the nonlinear evolution equations are first transformed into bilinear equations through dependent variable transformations. These bilinear equations are then used to construct the N-soliton solutions, the Backlund transformations, and an infinite number of conservation laws in a systematic way. As an example, the method is applied to the Korteweg-de Vries equation, which is typical of nonlinear evolution equations. The essential part of the bilinear transformation method may be understood by reading Chapter 2. The second part (Chapters 3-6) is concerned with the study of the mathematical structure of the Benjamin-Ono (BO) and related equations. The bilinear transformation method is employed extensively to analyze these equations. At the same time, the relationship between the soliton theory and the algebraic equations is stressed. In Chapter 3, especially, the mathematical structure of solutions of the BO equation is clarified from the viewpoint of the theory of algebraic equations. The materials treated in this book are current topics, including open problems. However, the contents are presented at an elementary level and are self-contained. Therefore, the maturity assumed of the reader is that of a beginning graduate student in physics or applied mathematics. Some knowledge of such mathematics as elementary partial differential equations and the theory of linear algevii

...

Vlll

Preface

braic equations will be helpful for a full understanding of the materials. Throughout the text, the discussion is restricted to the mathematical aspects of the problem, and accordingly, for the physical background, the reader should refer to appropriate bibliographies, a few of which are listed at the end of the volume. I would like to express sincere thanks to Professor Richard Bellman for his suggestion to write this book. I am also indebted to Professor Akira Nakamura for his continual encouragement and useful discussions.

Introduction and Outline

1.1 Introduction

A history of the development of the mathematics of solitons begins in 1967 with a remarkable discovery by Gardner et al. [l] of an exact method for solving the initial value problem of the Korteweg-de Vries (KdV) equation. They reduced the nonlinear problem to the linear one, which was well known as the Sturm-Liouville eigenvalue problem characterized by the Schrodinger equation, and then discussed the properties of the exact solution describing the interaction of solitons. Then this method, which we shall call the inverse scattering method, was extended to a more general form to be applicable to a wide class of nonlinear evolution equations such as the modified KdV equation, the nonlinear Schrodinger equation, and the Sine-Gordon equation

2

1

Bilinear Equations

Exact Solutions

Section 2.3

*

Backlund Transformation

Introduction and Outline

Section 2.5

Inverse Scattering Transform

Conselvation Laws

Fig. 1.1 Schematic illustration of the bilinear transformation method.

[2-41. The inverse scattering method is now included in several textbooks. (See, for example, Bullough and Caudrey [S], Lamb [6], Ablowitz and Segur [7], and Calogero and Degasperis [8].) In 1971 Hirota [9] developed an ingenious method for obtaining the exact multisoliton solution of the KdV equation and derived an explicit expression of the N-soliton solution. His method consisted of transforming the nonlinear evolution equation into the bilinear equation through the dependent variable transformation. The bilinear equation thus obtained can be solved by employing a perturbation method.s The Backlund transformation [l 1, 123 is another method for finding multisoliton solutions of some class of nonlinear evolution equations. With this method, the multisoliton solutions can be constructed by purely algebraic procedures. Also, this method can be used to derive an infinite number of conservation laws. Hirota [13] introduced new bilinear operators together with their properties and then developed a unified method for constructing the Backlund transformation on the basis of the bilinear equation written in terms of new bilinear operators. At the same time Hirota clarified the relation between the Backlund transformation method and the inverse scattering method using his new formalism. The procedures mentioned above are depicted in Fig. 1.1. This method was shown to be applicable to a large class of nonlinear evolution equations [S, 9, 101.

1.2 Outline

3

1.2 Outline

In Chapter 2 the bilinear transformation method is applied to the KdV equation to illustrate this method. First, the KdV equation is bilinearized through a dependent variable transformation (Section 2.1). Then the methods for obtaining the N-soliton solution, the generalized soliton solution, which may be interpreted as a generalization of the N-soliton solution, and the periodic wave solution are presented (Section 2.2). Starting with the bilinearized KdV equation, the procedures to derive the Backlund transformation (Section 2.3), an infinite number of conservation laws (Section 2.4), and the inverse scattering transform (Section 2.5) are briefly discussed. The final section (Section 2.6) is devoted to the bilinear transformation method bibliography. In Chapter 3 we discuss in detail the mathematical structure of the Benjamin-Ono (BO) equation from the viewpoint of three different methods: the bilinear transformation method, the theory of linear algebraic equation, and the pole expansion method. Physically, the BO equation describes a large class of internal waves in stratified fluids of great depth [14-171 and also governs the propagation of nonlinear Rossby waves in a rotating fluid [ 181. Mathematically, the BO equation is a nonlinear integrodiflerential equation with a dispersion term characterized by the Hilbert transform. Owing to this definite integral term, the solutions have many different properties from those of the well-known KdV type. The N-soliton and N-periodic wave solutions are presented explicitly using the bilinear transformation method (Section 3.1.1). It is then demonstrated that the N-soliton solution can also be obtained by means of two different methods: the theory of linear algebraic equation (Section 3.1.2) and the pole expansion method (Section 3.1.3). The Backlund transformation is constructed in the bilinear form (Section 3.2.1) and is used to derive an infinite number of conservation laws (Section 3.2.2). A method for solving the initial value problem of the BO equation is then developed (Section 3.3.1) and the asymptotic solutions for large values of time are derived using a zero dispersion limit (Section 3.3.2). To apply this method, an initial condition evolving into pure N solitons is presented (Section 3.3.3). The stability of solitons is also discussed in relation to a small perturbation in the initial condition (Section 3.4). Finally, the initial value problem of the linearized BO equation is solved exactly, and asymptotic behaviors of solutions for large values of time are investigated (Section 3.5).

4

I

Introduction and Outline

In Chapter 4 the general nature of the interaction of the BO solitons is investigated by employing the expression of the N-soliton solution (Section 4.1). The interaction of two solitons is then studied in detail (Section 4.2). In Chapter 5 the BO-related equations are bilinearized using the bilinear transformation method and their solutions are presented. The equations treated are the Lax hierarchy of the BO equation [19-2 13, which we shall call the higher-order BO equations (Section 5.1), the higher-order KdV equations [22] (Section 5.2), the finite-depth fluid equation [23-281 and its higher-order equations [21,29] (Section 5.3), and the higher-order modified KdV equations [30] (Section 5.4). The finite-depth fluid equation, which describes long waves in a stratified fluid of finite depth, is especially interesting since it reduces to the BO equation in the deep-water limit and to the KdV equation in the shallow-water limit and it therefore shares many of the properties of the BO and KdV equations. Finally, the Backlund transformations and the inverse scattering transforms of the higher-order KdV equations are constructed in the bilinear forms [31] (Section 5.5). In Chapter 6 we treat other interesting topics related to the BO equation. The modified BO equation [32], which is generated from the Backlund transformation, is bilinearized, and the N-soliton and Nperiodic wave solutions are presented (Section 6.1). The nonlinear Schrodinger equation, which derives from the nonlinear self-modulation problem of the BO equation [33, 341, is then discussed (Section 6.2). Finally, the effect of a small dissipation on the BO equation is considered. The system of equations that govern the time evolutions of the amplitudes and phases of the BO solitons is derived on the basis of the multiple time-scale expansion method (Section 6.3). The Appendixes contain the properties of the bilinear operators together with their proofs (Appendix I*), the properties of the matrix that appears in the expression of the N-soliton solution of the BO equation (Appendix 11), the properties of the Hilbert transform operator (Appendix 111), and a proof of an exact solution of the integral equation that determines the distribution of the amplitudes of the BO solitons (Appendix IV).

Appendix I will be very useful for the reader unfamiliar with the bilinear operators.

Introduction to the Bilinear Transformation Method

In this chapter the bilinear transformation method is explicitly illustrated by its application to the Korteweg-de Vries (KdV) equation, the prototype of the nonlinear evolution equation. The KdV equation was first derived by Korteweg and de Vries while developing a theory for shallow-water waves [35] ;it later became clear that many physical systems could be described by the KdV equation [36-381. The method of exact solution, the Backlund transformation, and the inverse scattering transform are based on the bilinearized KdV equation. The Bibliography (Section 2.6) discusses applications of the bilinear transformation method to other nonlinear equations. 5

2 The Bilinear Transformation Method

6

2.1

Bilinearization

Let us consider the KdV equation in the form U,

+ 6~1.4,+ uxXx = 0,

(2.1)

with the boundary condition u + 0 as 1x1 + co. Here u = u(x, t ) is a real function of both time t and space coordinate x, and subscripts denote partial differentiation. What dependent variable transformation could be introduced to transform (2.1) into a more tractable form? A key is the steady-state solution of (2.1), that is, u(x, t ) = (p2/2) sech2(rt/2),

where r j = px

and p and

rjo

- p3t

(2.2)

+ rjo

are arbitrary constants. We can rewrite (2.2) in the form

u(x, t ) = 2p2(eV/’

+ e-q’z)-z

=2

8’ ln(1

+ e“)/dxz.

(2.4)

The functional form of (2.4) suggests the following dependent variable transformation: U ( X ,t ) = 2 az lnf(x, t)/dx2.

(2.5)

Substituting (2.5) into (2.1) and integrating with respect to x, we obtain fx,

f - f x f, + f x x x x f - 4LX f x + 3(fxx)z = 09

with the integration constant being set to zero. Equation (2.6) is the original version of the bilinearized KdV equation derived by Hirota [9]. It can be confirmed by a simple calculation that the functionf = 1 + eq satisfies (2.6). We may regardfas a more fundamental quantity than u in the structure of the KdV equation. We now introduce Hirota’s bilinear operators defined by the following rule [131: D:D:a b

=

(a/&

- d/at’)”(d/dx - a/ax’)”’a(x, t)b(x’, f)lx,=x,

(2.7)

?’=I

where n and m are arbitrary nonnegative integers. Equation (2.6) can be compactly rewritten in terms of these bilinear operators as DAD,

+ D;t)f*f=

0,

(2.8)

7

2.1 Bilinearization

which is a convenient form to use in discussing exact solutions and the Backlund transformation of the KdV equation. The process of transforming the nonlinear evolution equation (2.1) into the form in (2.8) is called bilinearization. If the boundary condition is given by u -P uo (= constant) as 1x1 + co,we introduce the dependent variable transformation

in (2.I), yielding the bilinear equation (DtD,

+ 6uoD: + 0: + c ) f . f =

0,

(2.10)

where c is an integration constant. The bilinearized KdV equation, (2.8) or (2.10), is the starting point for the following sections. Having demonstrated Hirota’s method for bilinearizing the KdV equation, a typical nonlinear evolution equation, the question may arise as to what conditions allow for the bilinearization of a given nonlinear evolution equation. This question has not been answered in general, though many important nonlinear equations in physics and applied mathematics have already been bilinearized [10, 391. The inverse problem, that of reducing a given bilinear equation to a nonlinear equation in the original variable u, is more tractable. For example, consider generalizations of (2.8) and (2.5): D,(D,

+ D:)f.f=

0,

u = 2 8’ Inflax’.

(2.11)

Using appendix formulas (App. 1.3.4) and (App. I.5.3), Eqs. (2.11) are transformed as U,

+ 45~~21,+ 15~,~,, + 1 5 ~ ~ , , +, u,,,,,= 0,

(2.12)

which is called the Sawada-Kotera equation [40]. We can similarly construct a wide variety of nonlinear evolution equations by reducing bilinear equations through dependent variable transformations. The usefulness of the bilinear transformation method is related to the structure of nonlinear evolution equations; this method provides a simple, straightforward way of obtaining various types of exact solutions, which will be demonstrated in the next section.

8

2 The Bilinear Transformation Method

Exact Solutions

2.2 2.2.1

Soliton Solution

We shall first develop a method for obtaining the N-soliton solution of the KdV equation on the basis of the bilinearized KdV equation (2.8) [l 13. Let us expandfformally in powers of an arbitrary parameter E as

f=

c A$, m

(2.13)

fo = 1.

n= 1

(2.14) where we have set

w,, D,)

=

D,W,

+ 0:)

(2.15)

for simplicity. Equating the coefficients E" (n = 1,2, . . .) to zero on the left-hand side of (2.14)yields a hierarchy ofequations forfj(j = 1,2, . . .).# The first is

w ,D J f l . 1 + 1 3

.fl)

=

0,

(2.16)

which reduces to the linear equation (apt

+ a3/ax3)fl = 0,

(2.17)

from (2.15) and properties of the bilinear operators (App. 1.1.l), (App. 1.1.2), and (App. 1.1.4). The simplest solution of (2.17) takes the form fl

=

eqL,

q1

=

plx

+

Q1t

+ VOI,

§Note that the equation derived from the &term is linear with respect tofi.

(2.18)

9

2.2 Exact Solutions

with p , and qol being constants and ZZ, and (2.18) gives 2F(D,, D,)fz . 1

= - F(D,, DJf,

=

-p:. The E' term of (2.14)

.fi = - F(D,, Dx)eq'. eql = 0,

(2.19)

using (App. 1.2.1). Therefore, we may setf, = 0. Similarly, the equations corresponding to the EJ ( j 2 3) terms are satisfied by& = 0 ( j 2 3). This implies that 1 + Eeql is an exact solution of (2.8). If we set &

=

,so-qo1

>

PI=P

(2.20)

in (2.18), the result is equivalent to (2.4), the one-soliton solution of the KdV equation. The exact solution of (2.17) may now be given as f,

= eql

+ eq2,

(2.21)

with

where pj and qoj (j= 1,2) are constants. The equation forfz is given by

(2.23)

where use has been made of formula (App. 1.2.2). From (2.23) .fz is solved as (2.24)

Sincef, # 0, we proceed to the c3 term in (2.14),

10

2 The Bilinear Transformation Method

Substituting (2.21) and (2.24) into (2.25) and using (App. I.2.2), we obtain

x F(D,, Dx)eql

+

"I2

where we have used the relation F(Qj, p j )

=

F(-Qj, -pj)

1

.1

=

=

(2.26)

0,

pj(Qj

+ p j ) = 0,

j = 1,2 (2.27)

[by (2.22)], in arriving at the final value (2.26). Therefore, we can set (j2 4) that = 0 ( j 2 4). = 0; it follows from the equations for The solution thus obtained f3

f = 1 + $1

+

E2f2

corresponds to the two-soliton solution of the KdV equation.*It can be seen from (2.26) that (2.27) is crucial to the proof. Equation (2.27) is Note that E can be set to one by replacing aj with q j - In E (j= 1.2)

11

2.2 Exact Solutions

actually the dispersion relation of the linearized KdV equation (2.29) u, + uxxx = 0, a significant observation. We now proceed to the N-soliton solution. A special solution of (2.17), analogous to the one- and two-soliton solutions, takes the form N

fl

where qj = p j x

and

=

1e”,

(2.30)

j= 1

+ Rjt + qoj,

j = 1 , 2 , . .., N

j = 1 , 2 ,..., N.

Q I. = - p ; ,

(2.31) (2.32)

Setting E = 1, without loss of generality, the N-soliton solution is given as the compact form derived by Hirota [ 5 ] (N)

pjqj

+ j l>Ck ( j p k A j k

(2.33)

with where lp=o, indicates the summation over all possible combinations of p 1 = 0, 1, p 2 = 0, 1,. . . ,pN = 0, 1 and means the summation over all possible combinations of N elements under the condition j > k.O It can be easily confirmed that, for N = 2, Eq. (2.33) reduces to the two-soliton solution (2.28). The somewhat tedious proof of (2.33) is by mathematical induction. We shall demonstrate the proof since this is essential to the bilinear transformation method. Substituting (2.33) into (2.8) and using (App. 1.2.1) yields

c c F( c N

p=O, 1 p’=O, 1

N

(pj

j= 1

- p;)fij,

1

j= 1

(pj

- p;)pj)

where F is given by (2.15). Let the coefficient of the factor

These notations will be used throughout this book.

12

2 The Bilinear Transformation Method

on the left-hand side of (2.35) be G. It follows that N p=o, 1 p'=O, 1

exp

N j=1

1

(N)

1(pjpk + p>&)Ajk

[j>k

(2.36)

9

where the notation cond(p, p') implies that summations over p and p' are performed under the following conditions: pj+p>=l p J. = p ' J. = 1 p J. = p'.J = 0

for j = l , 2 ,..., n, for j = n + 1 , n + 2,..., m, for j = m + l , m + 2,..., N.

(2.37)

aj = pj - pi,

(2.38)

Defining the variable we have (N)

cbjpk

j>k

+ pi&)Ajk N

-

1(P; + pLS2)Ajj

j=1

=$c c + c c + f : i + f f : t T n

n

n

~ = k1= l

m

j=l k=n+l

j=l k=m+l

j=n+ 1 k=n+ 1

j=n+ 1 k=n+ 1

5 f: + 5 f

+ j=m+l

j=m+l k=n+l

5 5

+ j=n+l

5 '(12 + 0jCk)Ajk + 1 1 In)

-

k=l

j=n+l k=l

N

rn

n

i

j=1 k=n+l

j>k

using (2.37) and setting A,

m

=

k=rn+l

m

Ajk

+ 1

](pipk

+ p>dc)Ajk

Ajk,

(2.39)

j,k=n+ 1

0, a consequence of (2.34).

13

2.2 Exact Solutions

Since aj and aktake the values + 1 or - 1 for 1 Ij , k 5 n [by (2.37) and (2.38)], we obtain, from the relations F(R, p) = F( -0,-p) and (2.34),

Substituting (2.38), (2.39), and (2.40) into (2.36) yields

(2.41) where c is a constant that is independent of the summation indices al, a 2 , .. . ,aN.If we can verify the identity

(2.42) for n = 1,2, . . . ,N, then (2.33) is an exact solution of the KdV equation. Using (2.15) and (2.32), (2.42) becomes G(")(pl,p2, * * * , P n )

=0.

(2.43)

We shall now prove (2.43) by mathematical induction. For n = 1, 2, (2.43) obviously holds. If we assume (2.43) to be true up to n - 1, then n

2

G'")(P1,Pz,...,Pn)1p1=~= 2 n p j G j=2

(n-1)

( P Z , P ~ , . . - , P ~ )0,= (2.44)

by induction. Furthermore, PI

G(")(Pl,Pz,...,Pn)Ipl=fp*

=

8 p i n < p : - Pj)

= 0,

j =3

ZG(fl-2)

(P3, P4, * * * 3 P.)

(2.45)

again by induction. Owing to the a summation, G(")is an even function of pl, p2, . . ., pn and invariant under the replacement of pj and Pk for

14

2 The Bilinear Transformation Method

arbitrary j and k. These properties, together with (2.44) and (2.45), lead to the factorization of G(")as G'"'(p1, ~

. ., P A

2 , .

n ~nbf ; - PZ)'G(PI, n

=

j=1

(n)

j>k

. P A (2.46)

~ 2 , .. ?

where G is a polynomial of p l , p2, . . . ,p n . Expression (2.46) shows that the degree of G(")with respect to pl, p 2 , . . . ,p . is at least 2n(n - 1) 2n = 2n2. We see from (2.43) that the degree of G(")is at most n(n - 1) + 4. This is impossible for n 2 2. Hence G(")must be zero identically, completing the proof. We now consider Hirota's theorem [ S ] . Consider the following form of the bilinear equation:

+

F(Dr, D x ) f . f = 0,

(2.47)

where F is a polynomial or exponential function of Dr and D , and satisfies the conditions F(Dr, DJ = F(-Dr, -DJ

F(0,O) = 0.

(2.48) (2.49)

Note that the bilinearized KdV equation (2.8) is a special case of (2.47) using (2.48) and (2.49). Hirota's theorem states that the expression (2.50) with

F(Qj, Pj) =

0,

j = 1,2, ..., N, (2.52)

gives the N-soliton solution of (2.47) provided identity (2.42) holds for n = 1,2,. . . ,N . [Here again we stress the importance of the dispersion relation (2.52).] The proof of Hirota's theorem is of the same form as that presented for the KdV equation.

15

2.2 Exact Solutions

For N = 2 the left-hand side of (2.42) becomes

c c F(a1Q1 + ~ Z Q z , a l P l+ a z P J

a 1 = f l e2=*1

x F(ozQ2 =

F(Q1

- O l Q l , OZPZ - a1Pl)azal

+ 0 2 , P1 + Pz)F(Q,

- F(Q1

- Q,, Pz

-

P1)

- % , P l - P z ) F ( - Q z - Q,, -P1 - P2)

+ Qz,

-P1

+ F(-Q, - Qz,

-P1

- F(-Q1

+ PZ)F(Q, + QbPZ + PA - Pz)F(-Qz + Q,, -P2 + P1) (2.54)

= 0,

using condition (2.48). It is concluded from this fact that the two-soliton solution always exists without further conditions on F. Let us again consider the KdV equation. The N-soliton solution of the KdV equation (2.33) has interesting properties, one of which is its asymptotic behavior for large values of time [41-431. To see this we order the parameters pj (j= 1,2,. . . , N) as 0 < p1 < p2 <

< pN

(2.55)

and transform the reference frame, which moves with the velocity p.' (n = 1, 2 , . . .,or N), giving

5

= x - pit.

(2.56)

Noting that qj = Pj(x - pft) + qoj = pj(p: - pf)t pj5

+

+ qoj,

and using (2.55), in the limit oft +

+ 00

j = 1,2,.

. . , N,t

(2.57)

16

2 The Bilinear Transformation Method

t

+

+a, (2.59) (2.60)

t

+

-co, (2.61)

with

l

tn-= -

1N

A,j.

P n j=n+ 1

(2.62)

In view of (2.34) Pn

- Pj

(2.63)

The functional form of (2.59) [or (2.61)] is exactly the same as that for the one-soliton solution (2.2) except for the phase shift t: (or 5,). In other words, the N-soliton solution evolves asymptotically as t + f00 into localized solitons moving with constant velocities p?, p i , . . . ,p i in the original reference frame. As t + + 00, the trajectory of the nth soliton is shifted by the quantity

relative to that for t + - co.It can be seen from this expression that the interaction between solitons occurs only in a pairwise way. It also follows from (2.64) that N

1 Pnt:

n= 1

N

=

1P n C ,

n= 1

(2.65)

implying the conservation of the total phase shift. Finally, we note that expression (2.33) of the N-soliton solution can be rewritten in determinant form as

f

=

det M,

(2.66)

17

2.2 Exact Solutions

where M is an N x N matrix whose elements are given by for j = k,

(2.67)

Further details of the properties of the N-soliton solution of the KdV equation can be found in the references [41-431. 2.2.2

Generalized Soliton Solution

The N-soliton solution that we have discussed is a special solution of the nonlinear evolution equation. We shall now develop a method for constructing a more general solution (which we shall call a generalized soliton solution) that describes interactions between solitons and ripples. This method of solution, primarily developed by Rosales [44] and Oishi [45-471, is an extension of the bilinear transformation method, and we shall illustrate it in the case of the KdV equation, demonstrating that the generalized soliton solution solves the initial value problem of the KdV equation. As shown in (2.14), if functionsf, ( n = 0, 1, 2, . . .) defined in (2.13) satisfy the system of equations n

with

s=o

(2.70) then thefgiven by (2.13) is an exact solution of the KdV equation.* For n = 1, (2.69) becomes (2.71) W r D,)fl * 1 = 0, where we have used the property (2.72) D,)f* 9 = D,)g *f: Obviously (2.7 1) is identified with the linear equation f1.r +fl,X,X = 0 (2.73) after integration with respect to x. The solution of (2.73) is readily obtained in an integral form as 9

w,,

fl(X9

w,,

t ) = J?P@X

Note that (2.69) is a linear equation forf,.

-

P 3 t ) d7(P)-

(2.74)

18

2 The Bilinear Transformation Method

Here jr dr(p) denotes the contour integral along the contour r which lies in the left half of the complex p plane (that is, Re p < 0) and which goes from p = -im to p = im. If the measure dz(p) introduced in (2.74) is chosen as

J-

OD

/ p P ( P x - p 3 t ) dT(p) =

c ( p r ) exp(iprx - ip:t) dpr

Q)

N

where c(pr) is a real function of pr and poi (j= 1 , 2 , . . . , N) are real constants, then the first term on the right-hand side of (2.75) represents ripples (or dispersive waves) and the second term represents solitons. Therefore, it may be called the generalized soliton solution. For n = 2, using (2.69) and (2.74),

w, 9

0,)fZ

1

1= -2

w, 9

x expC(p1

DXIfl

.fl

+ PZ)X

- (P:

+ P 3 t I dT(P1)

dT(Pz), (2.76)

where use has been made of formula (App. 1.2.1). Noting the relations (2.77) WflD x ) . f 2 . 1 = f z . r x + f2,XXXX' + PL p1 - p 2 ) = (pl - ~ ~ 1 c - p + : p3 + (pq - p2131

WP:

= -3p1pz
(2.76) is solved as

fz

=

/ / ("")'+ r r Pi

P2

(2.78)

~ 2 ) ' ~

I'

exp[i1(pjx - p j t ) ndz(pj). (2.80) j=1

19

2.2 Exact Solutions

In the same way, for general n , f . is determined as [46]

r n

1

”

where R j = -pj” (j= 1, 2, . . . , n). We now show that (2.81) satisfies (2.69). Substituing (2.8 1) into (2.69) yields A

=

n

1w,, D,)L*f,-,

s=o

(2.82)

20

2 The Bilinear Transformation Method

where we have used formula (App. 1.2.1) and introduced the notation ,C, = n!/(n - s)!s!.

(2.83)

Equation (2.82) can be transformed into the form

X

(1

+ al)(l + a z ) . - - ( l + a,)(l

- a,+l)(l -

2”

0,+~)...(1

-

a,)

(2.84)

L=

where means the summation over all possible combinations of al = f 1, az = f 1, . . . ,a, = f 1. The equivalence of (2.82) and (2.84) can be confirmed by noting that there is a contribution to the summation only when a1 = az = ..-= a, = 1 and as+l= a,+z= ... = a, = - 1 and F(D,, D,) = F( -D,,-D,). Since the integrand in (2.84), which we set as I for simplicity, is unchanged by replacing pj with p,,,, and aj with a,,,, (j= 1 , 2 , . . .,n), we obtain

(2.85)

21

2.2 Exact Solutions

where denotes the summation over all possible combinations of s elements taken from n. Using the identity

(2.85 ) becomes

' [-

1sj < k s n

Ic:

x ~ X P

(Pjx 1

-

F(ejRj

ak%,

OjPj

-

akpk)

+ & , P j + Pk)

1'

F(aj

+Qjt>

n d ~ ( p j ) , j= 1

ejek

1

(2.87)

which vanishes due to identity (2.42). Thus, we have shown that f., given by (2.81), satisfies Eq. (2.69). Although we have been concerned with the KdV equation, this method can be applied to a wide class of nonlinear evolution equations. It may be summarized as follows: If the bilinear equation W

t

,D,)f-f

=

0,

t

,

(2.88)

with subsidiary conditions F(Dt D J f . 9 = W 9

F(D,, D,)1

*

D,)g

*f

1=0

(2.89) (2.90)

has an N-soliton solution or, in other words, operator F satisfies identity (2.42), then it also contains the generalized soliton solution given by (2.13) and (2.81), where the dispersion relation in (2.81) must be replaced by

F(CIj, p j ) = 0,

j = 1,2,. .. , n,

(2.91)

22

2 The Bilinear Transformation Method

instead of Rj = -d. The pure N-soliton solution is obtained from (2.81) by defining the measure dz(pj) as m

dz(pj)

=

N

[ 1Csd(Pj - Po,) dpj,

J-m

(2.92)

s=1

where c, and pos (s = 1,2,. . . ,N ) are real constants with poj # P O k for j # k (1 s j , k IN) and 6 is Dirac's delta function. Substitution of (2.92) into (2.8 1) yields m

where ?, = poSJx- p&, t

Note thatf, with n 2 N

+ In

j = 1, 2, . . . , n.

EC,,,

(2.94)

+ 1 vanishes identically owing to the factor

=

lbj
(F )'. Osj

(2.95)

- POsk -k

so that the infinite sum reduces to the finite one. It can easily be shown that expression (2.93) coincides with the N-soliton solution (2.33) of the KdV equation. Finally, we shall briefly discuss the initial value problem of the KdV equation. We may rewrite the generalized soliton solution of the KdV equation (2.8 1) in the form

(2.96) where A,, is an n x n matrix whose elements are given by
=

2pj/(Pj

+ Pk),

i, k

=

1, 2, * . n. 7

(2.97)

23

2.2 Exact Solutions

The equivalence of (2.81) and (2.96) may be seen by noting the identity of the Gram determinant

n

det An =

2

lgjcksn

( " I . ) P j + Pk

(2.98)

Equation (2.96) may be rewritten as

f= 1

+

f n.

n=l

fm fm x

... fmdet(Qn)fidsj,

x

(2.99)

j= 1

X

where 0,is an n x n matrix with the 0,k ) element given by F(sj Here

+

sk;

t).

We now introduce function D(x, z ; t), which corresponds to the Fredholm first minor by the relation D(x,z;t ) = -F(x

* +z;t)- 1 I f f n.

f det(Q,)fidsj, m

m & n

n=l

x

x

j= 1

X

(2.101)

where 0,is an (n + 1) x (n + 1) matrix given by

Qn =

I: F(x F(s,

+ 2;t ) + 2;t )

F(x F(s,

+ s,; t ) + s,; t )

F(x + s,; t ) . . . F(x F(s, s,; t ) ... F ( s ,

LF(s,

+ z;t)

F(sn

+ s,; t )

F(sn

+

+ s,;

t)

* * *

F(sn

1.

+ s,; t ) + s,; t )

: + s,; t ) ]

(2.102) Differentiating (2.99) with respect to x and comparing with (2.101), yields the useful expression D(x, x; t ) = fXX, t).

(2.103)

To obtain the relation between D andfwe expand det(0,) with respect to the first column. It follows from (2.99) and (2.101) that D(x, 2 ; t ) = - F ( x

+ 2 ;t ) f ( x , t ) -

&

D(x, s; t)F(s + z ; t ) ds. (2.104)

24

2 The Bilinear Transformation Method

If we define the function K(x, z ; t) as K(x, z ; t ) = w , z ; t)/f(x, t),

(2.105)

where f ( x , t) # 0 is assumed, then K(x, z ; t) satisfies the Gel’fandLevitan-Marchenko (GLM) integral equation K(x, z ; t )

+ F(x + Z; t ) + E

K ( x , S; t)F(s + Z ; t ) ds = 0. (2.106)

Inversely, if K(x, z ; t) is a solution of (2.106), then K(x, z ; t) is represented as (2.107) Relation (2.107), combined with (2.5), yields the useful expression U ( X t, ) = ~ [ K ( xX ,; t)],.

(2.108)

Function K(x, z ; t ) is satisfied by the linear equation [47] K,,(x, Z ; t ) - K,,(x, Z ; t )

+ U ( X t)K(x, , Z ;t ) = 0

(2.109)

by applying the uniqueness theorem to the solution of the GLM equation. The procedure for solving the initial value problem of the KdV equation is now established and can be explained as follows: Given initial data u(x, 0), first solve (2.109) under the boundary condition K(x, z ; 0) + 0 as x + z + 00, then introduce this K(x, z ; 0) into (2.106) to obtain F(x;O). Next, determine the measure d z ( p ) from F ( x ; 0) and (2.100). Finally, the desired solution u(x, t) is constructed using (2.13), (2.81), and (2.5). This procedure is illustrated in Fig. 2.1.

2.2.3 Periodic Wave Solution The bilinear transformation method discussed in this chapter is also used to obtain the periodic wave solutions of some nonlinear evolution equations. We now illustrate the method of solution developed by Nakamura [48,49] for the case of the KdV equation.

25

2.2 Exact Solutions (2.100)

F(x; 0 )

dr(p)

(2.13) (2.106)

(2.81)

K(x,z;O)

f(x.t)

A (2.5)

(2.109)

_-_-______

u (x, 0 )

u(x, t) ~

Fig. 1.1 Schematic illustration of the bilinear transformation method.

The bilinear form of the KdV equation appropriate to the periodic problem is written as F(D,, D,)f.f=

(DtD,

+ 6uoDf + D: + c ) f . f =

0,

(2.110)

with the dependent variable transformation u(x, t) = uo

+ 2 8’

lnf(x, t)/8x2,

(2.1 11)

where uo is a constant and c an integration constant generally dependent on time. For a one-periodic wave solution, we take the one-dimensional Riemann theta function in the form

f = ~ ( qz); =

m

C

n=-m

e x p ~ n i n q+ nin’z),

with q = px

i=

+ Rt + q o ,

J-l, (2.1 12) (2.113)

where z is a complex constant satisfying the condition Imz>O

(2.1 14)

and p and R represent the wave number and frequency, respectively, and qo is a phase constant. Substituting (2.1 12) into (2.110) and using

26

2 The Bilinear Transformation Method

formula (App. 1.2.1), we obtain m

Ff.f =

1

n,n’=-m

m

=

1

F(D,, D,) exp(2xinq

x exp[2ni(n =

1

. exp(2xin’q + x i n ” ~ )

F[2xi(n - n‘)Q 2ni(n - n’)p]

n.n’=-m

m

+ xin’z)

+ n’)q + ni(n2 + n”)z]

F[2xi(2n - m)Q 2xi(2n - m)p]

n.m= - m

x exp(2ximq

+ xi[n’ + (n - m)’]z}

m

=

1 F(m)exp(2nimq),

(2.115)

m=-m

where the new summation index m = n F(m) is defined by

+ n‘ has been introduced and

+ (n - m)’]~}. Shifting index n by unity as n = n’ + 1, (2.116) becomes x exp{xi[n’

x exp{xi[n”

+ (m - 2 - n’)’]r}

= F(m - 2) exp[2xi(m

(2.116)

exp[2ni(m - l)z]

- 1)zJ

(2.1 17)

From relation (2.117) we can conclude that, if F(0) and F(1) are zero, then all F s become zero. This implies that (2.1 12) is an exact solution of the bilinear equation (2.1 10). Examining the definition of F(m), the conditions F(0) = &l) = 0 become OD

1 m

1

n=-w

F(4xinQ 4xinp) exp(2xin’z) = 0,

~[2xi(2n- l)n, 2ni(2n - l)p] exp{xi[n’

+ (n - 1)’]}

(2.1 18) = 0.

(2.119)

27

2.2 Exact Solutions

Substituting expression (2.1 10) for F, (2.1 18) and (2.119) reduce to 00

1 (- 16n’n’pR

-

n= -m

96nznzpzu,

+ 256n4n4p4+ c) exp(2ain’z)

= 0,

(2.120) m

1 -

[-4n2(2n - 1)’pR - 24n2(2n - 1)%,

n=--03

x exp{ni[n’

+ (n - I)’]}

=

+ 16n’(Ln

- l)4p4 + c] (2.121)

0,

respectively. Introducing the quantities 03

Ao(7) =

1 exp(2nin2z),

(2.122)

n=-CG

m

Al(7) =

1

exp(2nin2z),

(2.123)

1 (4r1)~exp(2nin2r),

(2.124)

n=-m

(42)’

W

Az(7) =

n=-m m

&(7)

=

1 exp{ni[n’ + (n -

(2.125)

n=-w

-

m

Bl(7) =

1 (4n - 2)’ exp{ni[n2 + (n - l)’]~},

n=-m m

&(7)

=

1

n=-m

(4n - 2)4 exp{ni[n’

+ (n - l)’]~},

(2.126) (2.127)

(2.120) and (2.121) can be written compactly as

+ n4A2p4 + CA, = 0, - n 2 ~ , p R- n 2 ~ o ~ l + pn Z 4 ~ , p 4+ CB, = 0.

-n2A1pR - n’u,A,p’

(2.128) (2.129)

From these two equations, quantities R and c are determined to beo (2.130) (2.13 1) The important role of integration constant constant c, which has been taken to be. zero in previous discussions of both soliton and generalized soliton solutions, is now evident. If we set c = 0 in (2.128) and (2.129), these two equations are obviously incompatible.

28

2 The Bilinear Transformation Method

Expression (2.112), with (2.113) and (2.130), gives the one-periodic wave solution of the KdV equation. The soliton solution can be obtained from the periodic wave solution in an appropriate limiting procedure. To illustrate this we introduce a quantity

q = enrr (2.132) and take a limit q - 0 (or Im 2 + co). The quantities defined in (2.122)-(2.127) are then expanded in powers of q as

+ 2qz + 2q3 + ..., A1 = 32q2 + 128q8 + ..., A , = 512q2 + 8192q8 + - . - , B, = 2q + 2q5 + 2 p + . . ., B, = 8q + 72q5 + 200q13 + ..., B , = 32q + 2592q5 + 2ooOOql3 + . + .. A,

=

1

(2.133) (2.134) (2.135) (2.136) (2.137) (2.138)

Substituting these expressions into (2.130)and (2.131)yields

=

-u,p

+ 81q4 + . * . + 4 2 1 1--30qz P3 6qz+ 9q4 + + 4nZp3, q + 0,

=

0,

q -+ 0.

n = -u

op

(2.139)

(2.140)

By introducing the quantities p” = 2nip,

(2.141)

?i = 2 n i ~= -u,p - p3,

(2.142)

r?o = ?o

+ 2/29

(2.143)

the one-periodic wave solution (2.112) reduces in the limit of q + 0 (or Im 2 + 00) to

f

= =

1 + exp(2niq + ni2) + exp( - 2niq + ni2) + . . . (2.144) 1 + expvx + ?it + e0),

which is simply the one-soliton solution of the KdV equation expressed in the bilinear variable.

29

2.2 Exact Solutions

The extension of these results to the N-periodic wave solution is straightforward. The solution is expressed in terms of the multidimensional theta function as f ( x , f,

=

%ql, v z , .

=

1...

..

m

n1,

,nN=

9

v N ; T)

exp 00

v j + ni

j= 1

N

C

j,k=l

Tjknjnk

with qj = pjx

+ Qjf + q o j ,

j = 1,2, ..., N,

(2.146)

and p j , Q j , and qoj defined as in the one-periodic wave case. The term ?jk (j# k) represents the effect of interaction between periodic waves and is assumed to satisfy the conditions Tjk

=

Zkj,

Imzjk > 0,

j , k = 1, 2, . . . , N,

(2.147)

1 , 2,..., N.

(2.148)

j,k

=

The relation corresponding to (2.1 15) is now written as

where m

1

=

nl,....npJ=

- 03

"

N

F 2 n i 1 ( 2 n j - mj)Qj, 2ni j= 1

j= 1

(mj

-

1

1(2nj - mj)pj

nj)Zjk(mk - nk)]

By shifting the hth summation index nhby unity, we obtain the relation corresponding to (2.1 17): F(ml,

. ..

9

mN)

=

F(ml?. . . mh- 1 , mh - 27 mh+ 1, 9

* *

9

mN)

(2.15 1)

If relations

F(m1,. . . ,mN) = O

(2.152)

30

2 The Bilinear Transformation Method

hold for all combinations of m , = 0, 1, m, = 0, 1, . . . , mN = 0, 1, expression (2.145) with (2.146) gives the N-periodic wave solutions of the KdV equation. Whether relations (2.152) are sufficient to determine the unknownsdepends on the explicit functional form of F(D,, D J . We can at least say that the unknown quantities included in the problem are frequencies 0,(j = 1,2,. . . ,N ) , interaction terms zjk(j # k);j,k = 1, 2, . . . , N ) , and an integration constant c, the total number of these being N NC, -I- 1 = i(NZ + N + 2). (2.153) The number of equations is given by 2N [from (2.151)]. For N = 1, 2, the number of equations corresponds to that of unknowns, implying the existence of the one- and two-periodic wave solutions. For further details, including a discussion of the extension of these results to other types of nonlinear evolution equations such as the modified KdV equation and the Kadomtsev-Petviashvili (or the two-dimensional KdV) equation, refer to the original publications by Nakamura [48,49].

+

2.3 Backlund Transformation The Backlund transformation is a transformation that relates pairs of solutions of nonlinear evolution equations. It is a powerful method for analyzing some classes of nonlinear evolution equations. The Backlund transformation is used to obtain an N-soliton solution by purely algebraic means. It also provides a method for constructing an infinite number of conservation laws in a systematic way.B In this section we shall derive the Backlund transformation of the KdV equation on the basis of the bilinear transformation method [13]. Letfandf‘ be two solutions of the bilinearized KdV equation (2.8), that is, D,(Dl + Wf.f= 0, (2.154) D,(D, and consider an equation [D,(D,

+ D:)f’

+ Df)f’ .f’= 0.

*f’]ff- f’f’[D,(D,

+ D:)f-f]

(2.155) =

0. (2.156)

An introductorymonograph which treats the Bicklund transformations of nonlinear evolution equations that appear in physics and applied mathematics has been written by Rogers and Shadwick [12].

31

2.3 Backlund Transformation

Obviously, i f f satisfies (2.154), f ‘ also satisfies the same equation. Therefore, (2.156) can be regarded as a relation that connects the pair of solutionsf and f‘, that is, the Backlund transformation of the KdV equation. Using formulas (App. 1.6.2) and (App. 1.6.5), (2.156) is converted to 2D,[(D,

+ 3 1 0 , + mf‘.fl . (ff’)

+ 6D,[(D:

(2.157) - 1)f’ .f] . (D,f.f’) = 0, where an arbitrary parameter 1 has been introduced. Equation (2.157)

is satisfied provided that the following equations hold: (0,

+ 3 1 0 , + 0;)f‘.f= 0,

(2.158)

0: f’*f = 1.y

(2.159)

These constitute the Backlund transformation of the KdV equation in the bilinear formalism. To see the relation between the present bilinear formalism and the Wahlquist and Estabrook formalism [SO], we introduce functions w and w’ as w = 2 a Inflax, WI =

(2.160)

2 a lnf’lax.

From formula (App. I.7.2), with a

=

(DZf’ .f’)ff+ f ’ f ‘ ( D m - ) = (Ef’.f)ff‘+ flf(D:f*f‘)

b

(2.161)

=f a n d c =

-

d = f’,

2(D,f’ *f)(D,f’ *f). (2.162)

Using (2.159)-(2.161), we then have w:

+ w,

= 21 - 3 w ’

- w)’.

(2.163)

Note that w‘ and w satisfy the nonlinear evolution equations

+ 3(w:)’ + w:,, w, + 3(w,)’ + w,,,

w;

= 0,

(2.164)

= 0,

(2.165)

respectively. It follows from (2.158) and formulas (App. 1.3.1)-(App. I. 3.3) that w; - w,

+

+ 3qw:

+ w:,, - w,,, + w,)], + *[(w‘

w,) $[(w‘ - w)(w: -

- w)~], = 0.

(2.166)

32

2 The Bilinear Transformation Method

Equations (2.163) and (2.166) correspond to the Backlund transformation first found by Wahlquist and Estabrook [SO]. The Backlund transformations of the higher-order KdV equations will be derived in Section 5.5. To illustrate the process of generating soliton solutions starting from a known solution, we shall take the simplest case

f = 1,

(2.167)

which is transformed into the original variable u as

u = 2 a2 In f/ax2 = 0,

(2.168)

so that (2.167) can be regarded as a vacuum solution. Substitution of (2.167) into (2.158) and (2.159) yields the linear equations for f and f'as

f; + 31J-k +fk,, = 0, f i x = Af '.

(2.169) (2.170)

A solution that satisfies both (2.169) and (2.170) is f' = e"/2

+ e-91/2

= ,-11/2(1 +

(2.171)

with

1 = k:/4,

(2.172)

where ' IJ . = p .Jx - p j 3 t + ' I 0 j ,

j = 1,2,....

(2.173)

Noting that d2 In e-q1/2/ax2 = 0,

(2.174)

it can be seen that (2.171) gives a one-soliton solution of the KdV equation

2 a2 In f '/ax2 = 2 d2 ln(1 =2

+ e")/dx2

a2 lnfl/ax2 = ul.

(2.175)

A two-soliton solution is generated from (2.158) and (2.159) with f' given by (2.171) and

1 = kt/4,

(2.176)

33

2.3 Backlund Transformation

yielding f=

- p2)[e(91+4z2)/2 - (pl +

+

e-(41+42)/2

1

+ ,-(41-422)/2

p2)[e(41-422)/2

1

Taking phase constants qol and qO2as qol = lnqo2 = In

P2 P2 P2 ~

P2

- P1

(2.178)

- P1

(2.179)

+ P1 + 401, + P 1 + ij02,

with ijol and ijO2 being constants, it can be seen that (2.177) gives a two-soliton solution of the KdV equation [see (2.33)]. This procedure may be extended to generate multisoliton solutions. The relation that connects ( N - 1)-soliton, N-soliton, and ( N + 1)soliton solutions is called the superposition formula and it will now be introduced by following the work of Hirota and Satsuma [Sl]. To derive this formula, the four solutions of (2.159) are taken to be (2.180)

fo = 1, fl

=f1k t ; P l ) ,

(2.181)

12

=f

t ;P 2 ) ,

(2.182)

2 k

fiZ = f 1 2 ( x ,

t ; P 1 , PZ),

(2.183)

where fl and f2 correspond to one-soliton solutions with parameters

p1 and p2, respectively, andf,, to a two-soliton solution with parameters p1 and p2. These are satisfied by

0,

(2.184)

= 0,

(2.185)

(0: - ~ : / 4 ) f z- f 1 2 = 0,

(2.186)

(D: - P f / 4 ) f I

(2.187)

(0: - P:/4)fo (0: - ~

*fi =

W f . fo2 .f12

= 0.

34

2 The Bilinear Transformation Method

Fig. 2.2 Interrelationship among the four solutionsfo,f,,f2, andf,, (=fz,). See text for discussion.

In Fig. 2.2 the commutability relation f 1 2 = fil has been assumed? We will now derive the superposition formula given these conditions. It follows from (2.184) and (2.186) that (DXO . f l ) f 2 f 1 2

--fOfi(DZf2

.f12) =

0 3

(2.188)

which is transformed, using formula (App. I.6.3), into the form

which reduces to

which gives the superposition formula [Sl] fo f 1 2 = C D X ( f 1

*f2)

(2.193)

by formula (App. 1.1.5),where constant c is determined by the form of solutions. The subtraction of (2.191) from (2.189) yields DxC(Dx fo This relation will be proved later.

* f 1 2 ) * f 2f l l

= 0,

(2.194)

35

2.3 Backlund Transformation

or equivalently,

fl f 2

(2.195)

= C D X ( f 0 .f12).

The generalization of (2.193) and (2.195) to ( N - I)-, N - , and ( N + 1)soliton solutions is straightforward. and the results may be expressed as (2.196)

f N - 1f N + 1 = C D X ( f N .3N>, fNfN

= cDx(fN-

1 'fN+ 1).

Here the parameter dependences of the four solutions fNare written explicitly as

(2.197) 1, fN,&,

and

fN+

From these formulas, an N-soliton solution may be expressed in the form [Sl]

where CE=*lis the summation over all possible combinations of = * l , E Z = f l ) ...)EN = *1. The superposition formula (2.193) or (2.196) has been derived assuming the commutability relation .fi2 = f i t (see Fig. 2.2). We shall now verify this relation by employing bilinear formalism. Define function Tl as El

T12

=

(2.203)

CfO'DXfi .f2,

withf,f,f, # 0. It can be shown that T12is related by the Backlund transformation tofl andf,, that is, p,

( 0 2x

-L 4 P2 ) f l

-712

=

O,

(2.204)

p2 5

( 0 2x

-1 4Pl)fZ 2

'T12

=

0.

(2.205)

so if (2.184)and (2.185)are satisfied then (2.203)becomes a new solution.

36

2 The Bilinear Transformation Method

To show (2.204) we use (2.184), (2.189, (App. I.8.2), and (2.203) to give

. f l l f 2 - c(D: - iPp:)fo . f 2 l f l = - 2f0, fl .fz) + f o ( D x fl .f2)x + KP2 - P 3 f O fl f 2 = - 2c - Yo. Jl z + c - YOU0 f12)x + KPZ’ - P 3 f o fl f 2 =f0c-~-’~f0,xJ12 - f o J 1 2 , J + KP2 - P3flf21

0 = [(D2x - 1 4Pl)fO X@X

+ i ( P 2 - P:)flf21.

(2.206)

=f0[-~-1~xf0*J12

Sincef, # 0 we obtain from (2.206) Dxfo

312

=

(2.207)

ic(Pp: - P?)flf2.

It follows that p1f0f2

= [(O:

-id)fi

= DXC(D,fl

‘f121fofz - f ~ . f i z ( ~ : - i d ) f o ’ f 2 ( D x f o .f12>1 = 0,

-fz) .fo J12 + f l f 2

*

(2.208)

where use has been made of (App. I.6.3), (App. 1.1.5), (2.203), and (2.207). Noting (2.185) and fofi # 0, we readily obtain (2.204). A similar argument leads to (2.205) and therefore the commutability relationf,, = fzl =f12 has been proved. So far we have been concerned only with the space part of the Backlund transformation. The commutability relation is also satisfied for the time part of the Backlund transformation, that is, if

cot + id& + D3fO *fl = 0, [D,

+ ipP:D, + D:lfo

*fz

=

(2.209) (2.210)

0,

then Q1 E Q2

Cot + *PP:D,

+ D 3 f i .Ti2 = 0, Cot + ipP:DX+ D 3 f 2 .Ti2 = 0,

(2.211) (2.212)

wherer12is defined by (2.203). Relations (2.211) and (2.212) imply the commutability relationf, = f2 = 2 . The somewhat tedious proof of (2.211) and (2.212) can be carried out by using the formulas for the bilinear operators. To show (2.21l), consider a quantity

rl

Qifof2

E

C(Dt

+ $ d D X+ Wfi .J12Ifof2 + 2P2 D x + D3fO * f 2

- fl J12(D,

3

(2.213)

37

2.3 Backlund Transformation

where the second term on the right-hand side has been added since it is equal to zero by (2.210). Using (App. 1.6.1) and (App. I.6.4), (2.213) becomes

+ $P$ D x + iD;>fifz .fo Jiz $D~[(D:fi *fz)*fo312 + ~ ( Dfix .fz) . (Dxfo *Ti2 ) + f l f Z ( 0 3 0 .JlZ)l. (2.214)

Q ~ f fo i = (Dt

*

Substituting (2.203) and (2.207) into (2.214) and using (App. 1.6.5) yields Q1fOfz

=

+ SP?+ + $D3fifz .(Dxfi .f2) + i C D x ( D 3 - 1 -fz) . f l f Z + $Dxf1f2 .(Dffo

cCDt

P

W

X

312).

(2.215)

Sincef, andf, are solutions of the bilinearized KdV equations, (2.216) DX(4 + D3I-l *fl= 0, which give CDx(D,

(2.2 17)

DAD, + D 3 f 2 *fz = 0,

+ mfi . f l l f Z f i -flflDx(D, + D:>fz

.f2

= 0.

(2.218)

Using (App. 1.6.5) and (App. I.6.2), (2.218) is transformed into D3DXfl .fz) *flfz = D x fl fz . (4 fl *fz)*

(2.219)

It follows by substituting (2.219) into the first term on the right-hand side of (2.215) that Qifofz = W

x f i f z . {CDr + h?+

+ ( 3 / 2 c N f O-Jl1.

P

N

X

-

+Elf1.fz

(2.220)

However, from (2.209), (2.210), (2.184), and (2.185) we have 0 = C(0, + 2 P W X + D3fO .fllfZ - [(Of + 2Pp:Dx + D3fO .fZlfl + 3 w : - *PP:)fO - f 1 l x f z - 3 C E - iPp:)fo *fZlXf1. (2.221)

Using formulas (App. 1.8.1), (App. I.8.3), and (App. 1.8.4) together with (2.203) and (2.207), (2.221) is converted to 0

=

-f0[0,+

%P? + P:)D, - @3f1 .fz + (3/C)f0(Dxf0 . ~ I z ) ,

+ (3/2c)fO(fo J; dXx + 3P: + P W X - +D;31fl * f Z

- ( 6 / ~ ) f o . ~ ~f(1f2o) =

=

-foL-D, + (3/c)(-+.fo,xxL2

+fO.Xf12.X

-f0{[~,+ 3 ~+:P%,

- +fO~I2.XX~fO

- W1fl

.f2+ (3/2c)~;f, .Tlz}.

(2.222)

38

2 The Bilinear Transformation Method

Comparing (2.222) with (2.220) shows

(2.223) or Qi

(2.224)

=O,

since fo fi # 0. Hence (2.211) has been verified. A similar argument leads to the proof of (2.212, and therefore we have completed the proof of the commutability relation for the time part of the Backlund transformation. Finally, we shall consider an important structure of the Backlund transformation. We have seen that the Backlund transformation enables us to obtain new solutions starting with a known solution, such as a vacuum solution. It also provides a new nonlinear evolution equation which usually has a structure similar to the original one. In the case of the KdV equation, this new equation is derived from (2.158) and (2.159) as follows: Define a new dependent variable u by

.a

u = I-ln-. ax

f'

f

(2.225)

Then (2.158) and (2.159) reduce to a single equation V,

+ ~ A u ,+ 60~0,+ u,,,= 0,

(2.226)

using formulas (App. 1.3.1), (App. I.3.2), and (App. 1.3.3). Equation (2.226) is the modified KdV equation. A new nonlinear evolution equation, which we call the second modified KdV equation, is generated from the Backlund transformation of the modified KdV equation when the procedure used in the case of the KdV equation is repeated. Nakamura and Hirota [52] derived the multisoliton solution for the second modified KdV equation. Nakamura [53] generalized this procedure and derived the third modified KdV equation generated from the Backlund transformation of the second modified KdV equation. Nakamura also conjectured a procedure to obtain the nth (n 2 4) modified KdV equation by employing the bilinear transformation method and clarified the structure of the infinite chain process of the Backlund transformation of the KdV equation.

2.4

39

Conservation Laws

2.4 Conservation Laws Some class of nonlinear evolution equations is characterized by the existence of an infinite number of conservation laws. A conservation law associated with a nonlinear evolution equation such as the KdV equation is expressed in the form

T,

+ x, = 0,

(2.227)

where T is the conserved density and X the flux of T. If T is a polynomial in u and its x derivatives, it is called a polynomial-conserved density. Of course, T may include a nonlocal quantity such as the Hilbert transform (see Chapter 3). Integrating (2.227) with respect to x from - co to co and setting the boundary condition X + 0 as 1x1 + co yields (2.228)

T dx = 0,

dt

which implies that JZm T dx is a constant of motion. An infinite number of polynomial conservation laws will be derived starting with the Backlund transformation of the KdV equation presented in Section 2.3. A procedure used to derive an infinite number of conservation laws is as follows: Define a function W as

w = w' - w,

(2.229)

and substitute (2.229) into (2.163) and (2.166) to obtain

w, + 2u = 2A - +w=,

W

+ 3Aw, + W, + $[W(W, + 2~)], + 8W3)>,= 0,

(2.230) (2.231)

where we have used a relation W,

= 2 d2 In f

(2.232)

p x Z = u..

Using (2.230), (2.231) may be rewritten as

W + 6AW,

+ W,

- $W'W,

=

0.

(2.233)

Equation (2.231) has the form of (2.227), therefore W is a conserved density. To derive conservation laws explicitly, expand W as m= 1

(2.234)

40

2 The Bilinear Transformation Method

substitute (2.234) into (2.230), and then compare the q-"' terms on both sides of (2.230). The result is expressed in the form of a recursion formula as

f 1 -- -u, =

fm+l

(2.235 )

1m-1

1

-

-,fm,x

4

1 f,f,-,,

m 2 1.

s= 1

(2.236)

The first fewf, are (2.237)

f2 - 1ZUX, f3 --

(2.2 38)

-LU 4 x x - z U1 9 2

f4

- 18 U X X X + &,, -

f5

= & -,u,

(2.239)

- $uu,,

- &u,)2

- ku3.

(2.240)

It may be seen from these expressions that f2, (m = 1, 2, . . .) vanish when integrated with respect to x from - 00 to and that only odd termsf2,+ yield meaningful results. Equation (2.230) is the well-known Riccati equation and may be linearized by an appropriate transformation. Introducing a function R as W

=

-(1 - R,)/R

(2.241)

and substituting (2.241) into (2.230) yield -2RR,,

+ R:

- 4(u

- 1)R2 = 1.

(2.242)

Differentiating (2.242) with respect to x and dividing by - 2R, we obtain a linear equation for R as R,,

+ 4 ( -~ 1)R, + 2u,R

=

0.

(2.243)

It may be verified from (2.233) and (2.241) that R is also a conserved density, that is!

ddt JrnR(x, t ) d x = 0.

(2.244)

Note the relation

U: + 6AW' + W,,,

- tw'w, + R a/ax)[R,

= (1/R2)(1 - R ,

+ 61R, + R,,,

- t(Rf/R), + t(l/R)J.

41

2.5 Inverse Scattering Method

Expanding R in inverse powers of A as R

m

=

RJ A m + 1 / 2 ,

R 0 -- -12

(2.245)

m=O

and substituting (2.245) into (2.243), we obtain a recursive formula for R, m 2 0. (2.246) Rm+ 1.x = &Rm.xxx + 4uRrn.x + 2uxRm), Formula (2.246) includes all the information about conservation laws and is very important to the study of the properties of conservation laws. Further details are related by Gel’fand and Dikii [54]. 2.5

Inverse Scattering Method

In this section the inverse scattering formalism of the KdV equation, first developed by Gardner et al. [11, will be derived from the Backlund transformation expressed in terms of the bilinear operators. Following Hirota [131, we introduce wave function J/ by the relation

f‘ = 4%

(2.247)

where f and f ’ satisfy (2.158) and (2.159). We then divide (2.158) by f’fand use formulas (App. 1.3.1)-(App. 1.3.3) to obtain -(In

+ 3 w n *)x + (In *)xx + 3(ln * M l n + C(ln *),I3 = 0,

*)xx

*)t

+ 2(ln f)xxl

(2.248)

where we have used the identity (2.249) (In f‘f),, = (In f’f2/flxx = (In *)xx + 2(ln f),,. Differentiating with respect to t and x and using (2.5), that is, u = 2(ln f),,, we obtain the time evolution of II/ as +t

+ 3(u + A)*, + *xxx

(2.250)

= 0.

The space evolution of # is derived similarly from (2.159) using (App. 1.3.2) and (2.5) as (In $Ixx + 2(lnf),,

+ [(In

=

4

(2.251a)

or *xx

+ u+

=

A*.

(2.251b)

42

2 The Bilinear Transformation Method

Substituting A+ from (2.251b) into (2.250) yields another expression of the time evolution of $ as *I

=

--4+XXX - 6 4 , - 3ux*.

(2.252)

Equations (2.251) and (2.252) are the basis for the inverse scattering transform of the KdV equation [l].O

2.6

Bibliography

The bilinear transformation method has been illustrated by applying it to the KdV equation, a typical nonlinear evolution equation. From the bilinearized KdV equation various exact solutions, the Backlund transformations, an infinite number of conservation laws, and the inverse scattering transform have been derived in a systematic way; their interrelation is shown in Fig. 1.1. We believe that the essential part of the bilinear transformation method has been fully discussed. However, other references describe the applications of this method to other nonlinear evolution equations. Therefore, we shall describe some of the references related to the bilinear transformation method. The original concept of the bilinear transformation method was described by Hirota [9], who used the bilinearized KdV equation (2.6) to derive an N-soliton solution of the KdV equation. The method has been applied to other nonlinear evolution equations, including the modified KdV equation [55] U,

+ 6 ~ ~+0 u,,,,= 0;

(2.253)

the Sine-Gordon equation [56] uXI = sin u ;

(2.254)

a nonlinear wave equation with envelope-soliton solutions [57]

The method of solution using (2.251) and (2.252) has been fully discussed [43] and we shall not go into detail here. Refer to textbooks listed in the references [5-81 for details of the inverse scattering method.

43

2.6 Bibliography

where a, B, y, and 6 are real constants which satisfy a relation aP = y 6 ; the Boussinesq equation [58] -

utr

uxxx

-

3(u2)xx - uxxxx

=

0;

(2.256)

the model equation for shallow-water waves [59] U, -

u,,, - ~ U U+, 3u,

Jxm

U, dx'

+ U, = 0;

(2.257)

the cylindrical KdV equation [60,61] U,

+ 6uu, + u,,, + u/2t = 0;

(2.258)

and the derivative nonlinear Schrodinger equation [62] iu,

+ fluxx + i6'u*uux + 6u*uu

= 0,

(2.259)

where 8, a', and 6 are real constants and * denotes complex conjugate. The bilinear transformation method has also been applied to obtain exact solutions for certain classes of nonlinear integrodiflerential equations. The first example is the Benjamin-Ono (BO) equation

+ 4uu, + Hu,, = 0,

u,

(2.260)

where H is the Hilbert transform operator defined by

(2.261) The first bilinearization of (2.260) and the N-soliton solution using the bilinear transformation method was described in Ref. [63]. Additionally, the bilinearized BO equation was used to obtain the periodic wave solution in Ref. [64]. The mathematical structure of the BO equation will be studied in Chapter 3. The second example is the finite-depth fluid equation U,

with

J-

+ ~ U U ,+ Gu,, = 0,

(2.262)

W

Gu(x, t ) = (1/2d)P

m

[coth n(y - x)/2d - sgn(y - x)]u(y, t ) dy,

(2.263)

where d is the fluid depth. It should be noted that (2.262) reduces to the BO equation in the deep-water limit d + 00 and to the KdV equation

44

2 The Bilinear Transformation Method

in the shallow-water limit d -+ 0. The bilinearization of (2.261) and the N-soliton solution were given in Refs. [27,65]. For the development of bilinearization of soliton equations the reader is referred to Ref. [39]. These equations are concerned with the one-dimensional system. The two-dimensional nonlinear evolution equations can be treated similarly. The two-dimensional Sine-Gordon equation [66] u,,

+ uyy - u,, = sin u,

(2.264)

the two-dimensional KdV (or Kadomtsev-Petviashvili) equation [67]

4, + 12(uu,),

+ ~,,f,,12uyy= 0,

(2.265)

and the two-dimensional nonlinear Schrodinger equation [68] iu,

+ flu,, + p)uyy+ Gu*uu

=

0,

(2.266)

with flyp', and 6 being real constants, are typical examples. Other important classes of nonlinear evolution equations to which the bilinear transformation method has been successfully applied are nonlinear differential-difference and nonlinear partial difference equations. A famous example is the Toda equation [69]

d2 ln(1

+ K)/dtZ = K V l + V,+, - 2K,

(2.267)

which was discussed in detail in Ref. [70]. The self-dual nonlinear network equation [71] (2.268) (2.269) also belongs to a class of nonlinear differential-difference equations which can be bilinearized through the dependent variable transformation. Another type of nonlinear network equation describes a Volterra system

d ln(z-

' + V,)/dt = I n -

d In@-'

-

I,,

+ In)/dt = V, - V,, ',

(2.270) (2.27 1)

with z being a characteristic parameter of the network, was bilinearized and an N-soliton solution was presented in Ref. [72]. Nonlinear partial difference equations, where the difference analogues of the

45

2.6 Bibliography

KdV, Toda, Sine-Gordon, Liouville, two-wave interaction, Riccati, and Burgers equations are treated, were discussed in Refs. [73-771. Hirota [78] proposed the discrete analogue of a generalized Toda equation (zleDI

+ z2eD2 + z 3 P 3 ) f . f =

0,

(2.272)

where zi ( i = 1,2,3) are arbitrary constants and Di (i = 1,2,3) linear combinations of the bilinear operators D,,D,,D,,D,,etc., and showed that it reduces to various types of nonlinear evolution equations by appropriate choice of ziand D i . A new formulation of the Backlund transformation was presented by Hirota using his bilinear transformation method [131. The bilinear operators defined in the form of (2.7) were first introduced in the same paper. As noted in the last part of Section 2.3, new nonlinear evolution equations are generated from the Backlund transformation of a given nonlinear evolution equation. A variety of nonlinear network equations generated from the Backlund transformation for the Toda equation (2.267) were presented in Refs. [11, 791 together with their N-soliton solutions. The Boussinesq equation (2.256) [80], the KdV equation (2.1) [52, 531, the BO equation (2.260) [32], the finite-depth fluid equation, and the Sine-Gordon equation (2.254) [81, 821 were also employed to generate a new class of nonlinear evolution equations. The Lax hierarchy of nonlinear evolution equations is also a very important class of nonlinear evolution equations. First introduced by Lax [2] in the study of the structure of the KdV equation, they are called the higher-order equations in this book. A systematic method for bilinearizing the higher-order KdV equations was developed by Matsuno [22] and was applied to the higher-order equations for the modified KdV equation (2.253) [30], the nonlinear Schrodinger equation (2.255) with a = y = 0 [30], the BO equation (2.260) [19-211, and the finite-depth fluid equation (2.263) [21,29]. The method of bilinearization for higher-order equations will be presented in Chapter 5. Finally, we shall mention some references concerning various types of solutions of nonlinear evolution equations. The soliton and periodic wave solutions are typical of those that are nonsingular for time and space variables. The generalized soliton solutions discussed in Section 2 . 2 . 2 are combinations of soliton and ripple solutions, the latter being the form of dispersive waves. Along with these solutions, singular solutions also exist. The rational solutions of the KdV equation with decay -2/x2 as 1x1 00 are typical. The bilinear transformation

46

2 The Bilinear Transformation Method

method was used to obtain rational solutions of the KdV equation [83, 841. Another class of solutions, similarity-type decay-mode (or ripplon) solutions, also exist in two-dimensional nonlinear systems; the Backlund transformation in the bilinear formalism was applied to obtain ripplon solutions for the two-dimensional KdV equation [85, 861, the two-dimensional nonlinear Schrodinger equation [87], and the two-dimensional Toda equation [88].

The Benjamin-Ono Equation

This chapter is concerned with the mathematical structure of the Benjamin-Ono equation. Three different methods are presented for obtaining the N-soliton solution. First, the bilinear transformation method is used to obtain the N-soliton and the N-periodic wave solutions of the BO equation. Second, it is shown that the N-soliton solution is derived from the system of N linear algebraic equations. Third, the pole expansion method is applied to the BO equation to obtain the N-soliton solution. The Backlund transformation of the BO equation is then constructed on the basis of the bilinear transformation method, and it is employed to derive an infinite number of conserved quantities and the inverse scattering transform of the BO equation. A method for solving the initial value problem of the BO equation is then developed, and the 41

48

3 The Benjamin-Ono Equation

properties of solutions are investigated using a zero dispersion limit. Finally, the stability of che BO solitons and the linearized BO equation are briefly discussed.

3.1 Multisoliton Solutions of the Benjamin-Ono Equation 3.1.1

Derivation of the N-Soliton Solution by the Bilinear Transformation Method

The BO equation describes a large class of internal waves in a stratified fluid of great depth [15-171, and it also governs the propagation of nonlinear Rossby waves in a rotating fluid [l8]. The BO equation may be written in the form u,

+ 4uu, + Hu,,

=

(3.1)

0,

where H denotes the Hilbert transform operator defined by (3.2) The Hilbert transform has a dispersive effect in the BO equation. Note that the Hilbert transform is a definite integral, which differs from an integral term that appears in the model equation for shallow-water waves [59], and it makes the properties of these solutions very different from those of the well-known KdV type. Nevertheless, as will be shown in this chapter, the BO equation shares many of the properties of the KdV and related nonlinear partial differential equations. In this section the N-soliton solution of the BO equation is derived by three different methods: the bilinear transformation method [63,64] ; the theory of linear algebraic equation [89]; and the pole expansion method [90,91]. We shall now employ the bilinear transformation method discussed in Chapter 2. The one-soliton solution of (3.1) has a Lorentzian profile and is expressed as u,(x, t ) =

a

a2(x - at - x0)'

+ 1'

(3.3)

3.1

49

Multisoliton Solutions

where a (> 0) and x o are the amplitude and phase, respectively, of the soliton. To infer the form of a dependent variable transformation that enables us to transform (3.1) into a bilinear equation, we deform the one-soliton solution (3.3) as Us(&

a

t) = -

- at - x o ) + i

i a a(x - _ _ In i

a

2 ax

+

a a(x - at - xo) - i

1,

i = G ,

x0) i a(x - at - x o ) - i

2 ax

---

-

In

- at -

+ +

-i(x - a t - x 0 ) l/a i(x - at - x 0 ) l/a

where

fi = i(x

- at -

xo)

+ l/a,

(3.5)

and * denotes the complex conjugate. For the N-soliton case, we can expect the solution to be represented by a superposition of the onesoliton solution (3.3) in the limit of large values oft. In this limit the N-soliton solution may be represented in the form

where

+ l/aj,

j = 1,2,..., N ,

(3.7)

e j = ~ - a j t - ~ o j , j = 1,2,..., N.

(3.8)

fi

= i9,

with

Here aj (>0) and x o j are the amplitude and phase, respectively, of the jth soliton, and it is assumed that a j # a, forj # k. Examining (3.6), we

50

3 The Benjamin-Ono Equation

write the dependent variable transformation in the form i a j-*(X, t ) u(x, t ) = - - In 2ax f(x, t) '

(3.9)

~

n Cx N

f(x,t) a

j= 1

Im x,(t) > 0,

-

j

(3.10)

x,W17

=

1,2,. . . , N ,

(3.1 1)

where x j (j= 1 , 2 , . . . , N) are complex functions of time t whose imaginary parts are positive. Using the formulas

(3.12)

H [ l / ( x - x j ) ] = - i / ( x - xi)

- xi*)] = i/(x - x r ) ,

H[l/(x

(3.13)

which are consequences of (3.1l), we obtain i a f* Hu(x, t ) = - H - In 2 ax f

= -i H Z N 2

j=1

-_ =

(y--) 1 1 x-x*

x-xi

1

i a 2 ax Wf

*f1.

(3.14)

Substituting (3.9) and (3.14) into (3.1) yields

Integrating (3.15) with respect to x and differentiating we obtain [63]

P Y J - (fLf

(3.16) 2 f X + fxxf') = where an integration constant is assumed to be zero. Equation (3.16) can be rewritten in terms of the bilinear operators introduced in Chapter 2 as i(f?f

-

-

iD, f * - f = D: f*-f,

which is the bilinearized BO equation.

0 7

(3.17)

3.1

51

Multisoliton Solutions

The one-soliton solution of (3.17) is given by (3.9, that is, = ie,

fl

+ l/al,

(3.18)

which can be verified by direct substitution. For the two-soliton solution, we assume f, is of the form

f,

+ c l e l + c,e, + b,

= c,e,e,

(3.19)

where cl, c, c 3 ,and b are unknown constants. Since (3.17) is invariant under a scaling f + cf (c constant), then constant c3 in (3.19) may be arbitrarily chosen. Setting c3

f, becomes

j,

=

-e1e2

= -1,

(3.20)

+ cle, + c,e, + b.

(3.21)

It follows by direct calculation that

+ i(c, - c:)a18t - ia,(c,ct - c:cz - b + b*)dl - ia,(c,cf - clcf - b* + b)& - i(a,cf + a,cf)b + i(alc, + azc2)b*

i D t f t . f 2 = i(c, - cf)a,8:

and

~ f f rf,.

=

+ e: + (c, + c;)e, + (cl + c:)e, + b + b* + (cl + cz)(c: + c t ) ] .

(3.22)

-2ce:

(3.23)

By comparing O f , e:, 8 1, 0, and constant terms on both sides of (3.17), the equations required to determine the unknown constants c,, c,, and b may be derived. The equations i(cl - c:)az

=

-2,

(3.24)

- ct)al

=

-2,

(3.25)

i(c,

+ b*) = 2(c, + c?), ia,(c:c, - clct - b* + b) = 2(c1 + c):, -i(alc: + a,ct)b + i(alcl + azc,)b* = -2Cb + b* + + c~)(c: + c;)] ia,(clct - c:cz

-

b

(

~

1

(3.26) (3.27) (3.28)

52

3 The Benjamin-Ono Equation

are satisfied by the constants c1-= ib2,

(3.29)

i/al,

(3.30)

c2 =

(3.3 1) which, substituted into (3.21), give the two-soliton solution f2 =

Note that f2

-e1e2 + i

(!:+ !:)+ ("' -

-

-

a1a2 a1 - a2 is expressed as the determinant ~

+

(3.32)

(3.33) Repeating the same procedure, the three-soliton solution f3 is found explicitly as f3

= -iele2e3

=

-

ie1 + l/a1 2/bl - a21 2/@1 - a3) 2/(a2 - a l ) ie2 l/a2 2/(a2 - a3) , ie3 + l/a3 m 3 - a l l 2/@3 - a21

+

(3.34)

where (3.35) In general, the N-soliton solution fN is given compactly as [63] (3.36) where M is an N x N matrix whose elements are given by (3.37) (3.38) It may be confirmed by direct calculation that fN given by (3.36) satisfies the bilinearized BO equation (3.17). However, this can be shown more easily by taking the long-wave limit of the N-periodic wave

3.1

53

Multisoliton Solutions

solution of the BO equation, as will be demonstrated later in this section. It is now necessary to verify that the N-soliton solution (3.36) satisfies assumption (3.11) used in the process of deriving (3.17). We write the equation of motion of x,,(t) in the form d dt

,=I

x, - x,

+?+,x, ,=I

-

n = 1 , 2,..., N,

x:

(3.39) which is derived by substituting (3.9) with (3.10) into (3.1) and setting the coefficient of [x - x , ( ~ ) ] - ~ as zero [90,91]. Taking the imaginary part of (3.39), we obtain the time evolution of the imaginary part of x, as d Im x,(t)/dt

=

n = 1, 2, .. .,N,

G,(t) Im x,(t),

(3.40)

where

TI N

Gn(t)

=

(s+ n)

Re(x, - x,) Im x, ([Re(x, - x,)l2 + [Im(x, - x,)I2

Integration of (3.40) with respect to t yields Im x,(t) = Im x,(tO) exp

(3.42)

where to is an initial time. It can be seen from (3.42)that conditions (3.1 1) are satisfied if they hold at some time t o ,since G,(t) is a regular function of t and decays as t - 3 when t + f 00 as shown in Chapter 4. In this case it is convenient to take to = -moo. Then from (4.27) the asymptotic form of x, for large negative values of time is given by

+

x,(t) = ant + xOn i/u,

+ O(t-

l),

t

+

- 00.

(3.43)

Therefore Im x,(-

00)

=

l/u, > 0,

which implies (3.1 1) by (3.42).

n = 1,2,. . . ,N,

(3.44)

54

3 The Benjamin-Ono Equation

We shall now proceed to the periodic wave case. Instead of (3.9) and (3. lo), the appropriate dependent variable transformation is,

i a f’(x, t) u(x, t ) = - - In 2 ax f ( x , t i ’

(3.45)

m

(3.46)

f ’(x, t ) =

n m

j= 1

(3.47)

cx - x ; ( m

Im x,@) > 0,

j = 1, 2, . . . ,

(3.48)

Im xi@) c 0,

.j = 1, 2, . . . ,

(3.49)

where x j and xi are complex functions o f t whose imaginary parts are positive or negative, respectively. Since f and f‘ are represented by the form of an infinite product, it may be shown that (3.50) using (3.12), (3.13), (3.48), and (3.49). Substituting (3.45) and (3.50) into (3.1) yields the bilinearized BO equation iD, f ’ . f

(3.51)

Dz f ’ . f,

=

which has the same form as (3.17). Equation (3.51) can be solved by means of a perturbation method, which was used in deriving the N soliton solution of the KdV equation (see Section 2.2). The one-periodic wave solution is obtained by taking

+ exp(it, - 4d, f = 1 + exp(itl + 4&,

f ’= 1 with 51

=

+

k , ( x - a1t - ~ 0 1 )

a , = k , coth 4,,

t\O),

(3.52) (3.53) (3.54) (3.55)

where k , , a,, and xol are real constants and t\’) is an arbitrary phase constant. To satisfy conditions (3.48) or (3.49) it is necessary that

4lh

’0.

(3.56)

55

Multisoliton Solutions

3.1

Introducing (3.52) and (3.53) into (3.49, we obtain the one-periodic wave solution expressed in the original variable u as u=

(k 1/21 tanh 4 1 1 + sech 41cos t1‘

(3.57)

This form coincides with the periodic wave solution presented by Benjamin [l5] and Ono [17]. The one-soliton solution is derived from (3.57) in the long-wave limit. To show this, keep a, and xol finite, choose tio)= n, and take the long-wave limit k, -+ 0. Substituting the expansions cos 5 1 = cos(k18, + n) = - 1 + $k:8: + O(kf), (3.58) sech 41 = 1 - &k,/al)’

+ O(kf)

(3.59)

[Eq. (3.59) is a result of (3.55)] into (3.57) we obtain = al/C(alw

+ 11,

(3.60) in the limit of k, -+ 0, which is the one-soliton solution of the BO equation (3.3). The N-periodic wave solution of (3.51) may be constructed by the method presented in Section 2.2 and is expressed as [64] 24

with

tj = k,<x - ajt - xOj) + t?), aj = kj coth

(3.63)

$j,

4j/kj > 0, 8’’ = [(aj - aJ2 - (kj - kr)2]/[(aj - al)2 - (kj +

(3-64) (3.65) kJ2], (3.66)

denotes the summation over all possible combinations where lr=o, of p1 = 0, 1, p 2 = O , l , . . .,pN = O,1, and 1 i T ) k means the summation under the condition j c k. The value of u generated from (3.61) and (3.62) is generally a complex quantity. However, a real u is assured by choosing (Y) as

tio’= tr!eal+ i 1AjJ2, j#k

j = 1,2, . . . , N,

(3.67)

56

3 The Benjamin-Ono Equation


where (j = 1,2, . . . ,N) are real constants. Indeed, introducing (3.67) into (3.62) yields

f ‘ = exp

1N

1 1

(icj

- $j)

+jik

(3.68)

which, when substituted into (3.9), gives a real u as

l N i a f* k j + - - In -. 2j,1 2ax f

u= --

1

(3.69)

To obtain the N-soliton solution in the long-wave limit of the Nperiodic wave solution, we set @‘!eal

= II,

j

=

1, 2, . . . ,N ,

(3.70)

in (3.67). From the expansion for small k j and k, with finite uj and a,,

-

4kjkl (aj - a1)’

+ O(k;) = k j k l M i + O(k?),

(3.71)

and from (3.61), (3.67), (3.70), and (3.71), in the limit of k j , k, + 0, we obtain

(3.72) where 0, and M j l are given by (3.8) and (3.38), respectively. When k j = 0 (j = 1, 2, . . . ,N )

f

=

1

N

n<-1p=o,

y=O,1 j = l

m=l

(3.73)

which means that f is factorized by k j . Therefore, the leading terms of (3.72) are given by those on the order of k j of

3.1

57

Multisoliton Solutions

yielding the following expression for f in the long-wave limit

. j , k ...., m , n

I -

ie,

+ -1

M,,

M12 ie2

+ -a21

[a]:

p f j . k, ...I m , n

..

MIN

M2N

(3.75)

and it follows from (3.69) that

(3.76) in the same limit. Expression (3.76) with (3.75) corresponds to that of the N-soliton solution given by (3.9) together with (3.36)-(3.38). 3.1.2 Derivation of the N-Soliton Solution by the Theory of Linear Algebra The second method for obtaining the N-soliton solution of the BO equation is from the theory of linear algebra. It is well known that the system of linear algebraic equation is solved by Cramer’s formula, which expresses the solution in a determinant form. The systematic use of this formula makes the derivation of the N-soliton solution more transparent and clarifies the structure of the solution.* This discussion follows that of Matsuno [89].

58

3 The Benjamin-Ono Equation

Consider the system of linear algebraic equations for unknowns and v l :

Vk

(3.77) k= 1

N

1 MfkVl = - 1,

j = 1,2,..., N,

(3.78)

k= 1

where Mjk is the (j, k) element of matrix M defined in (3.37) and (3.38) and M’ a matrix with elements Mjk = M:j. (3.79) Matrices M and Mt are nonsingular for real x and t, that is, det M # 0,

(3.80)

det Mt # 0.

(3.8 1)

Relation (3.80) is a consequence of (3.10), (3.11), and (3.36), and (3.81) derives from (3.79) and (3.80). Given relations (3.80) and (3.81), (3.77) and (3.78) are solved uniquely by using Cramer’s formula as vj

N

=

1 fikjfdet M,

k= 1

uf =

N

-

C f i s f d e t M*,

k= 1

j = 1,2,..., N ,

(3.82)

j = 1 , 2 , . .., N ,

(3.83)

where f i j k denotes the cofactor of Mjk defined by M f i = f i M = I, I : unit matrix. Then it follows from (3.82) and (3.83) that N

1vf

j=1

N

=

-1

N

1fiTJdet

j=1 k = l

M*

= -

1 (lfijkfdet M

(3.84)

N

N

N

j=l

k=l

j=1

(3.85) Similarly, from (3.9) and (3.36)-(3.38), fijj/det M

N

+ 1fi2Jdet M*) j=1

(3.86)

59

3.1 Multisoliton Solutions

with u j = Ajj/det M,

j

=

1 , 2 , . . .,N ,

(3.87)

where we have used the property of differentiating a determinant with only the diagonal elements depending on x [see (3.37) and (3.38)]. As a consequence of a remarkable property of the matrix M N

(3.88) which is proved in Appendix I1 [see (App. 11.22)], we obtain the useful relation N

N

N

N

Then it follows from (3.82), (3.87), and (3.89) that

and from (3.85), (3.86), and (3.90) that

Furthermore, from (3.9) we may express u(x, t) as

(3.92) which, combined with (3.91), yields N

a

Xuj= -i-hf, j= 1 ax N

Xu!=

j= 1

a

- i -ax hf*.

(3.93) (3.94)

60

3 The Benjamin-Ono Equation

By employing relations (3.95) (3.96) which are derived from (3.10) and (3.12), we obtain from (3.93) and (3.95)

and similarly from (3.94) and (3.96) (3.98) Given these relations, we now prove that u(x, t ) given by (3.9) and (3.36)-(3.38) satisfies the BO equation (3.1). For this purpose, consider the quantity

Differentiating (3.78) with respect to x gives N

due to (3.37) and (3.38). Substituting (3.100) into (3.99), using (3.78) and the explicit expression for M j k , kfjk

where

6jk

=

M t j

=(-iej

+

l/aj)djk

+

denotes Kronecker’s delta,

j # k, and noting the identity

[2/(ak

6jk

-

aj)](1

-

djk)?

= 1 for j = k,

6jk

(3.101) =

0 for

61

3.1 Multisoliton Solutions

(3.99) can be modified to the form

=

-uj’

1

- - (ujt - u j )

2

+ ZajeJ{uf - uj) 1

(3.103)

To eliminate the third and fourth terms on the right-hand side of (3.103), multiply by a j on both sides of (3.77) and (3.78) and add the resultant equations to give

Substituting (3.104) into (3.103) yields

(kf j ) N

= - ~ ( u ~ - v k ) - 2 u = 0 , j = l , 2 ,..., N , k= 1

(3.105)

where (3.91) has been used in the last step. Relations (3.105) may be identified with the system of homogeneous linear algebraic equations whose unknowns are given by the quantities in parentheses on the right-hand side of (3.99). Hence we conclude from properties of determinants and (3.81) that i u z x - $aJ{uf - u j )

+ 2uuf = 0,

j

=

1, 2 , . . . , N . (3.106)

Summing (3.106) with respect to j yields N

i

1uf,.

j= 1

I N aj(uj’ - u j ) 2 j=1

--

1

N

+ 2u 11 uf

=

j=

0,

(3.107)

and it follows from the complex conjugate expression of (3.107) together with (3.85) that N

N

i

1u j , x - -2l 1Naj[(uJ)*- u;]

j= 1

j=1

-

2u 1 uj j= 1

=

0.

(3.108)

62

3 The Benjamin-Ono Equation

Adding (3.107) and (3.108) and using (3.91) yields

(3.109) Then we have from (3.91), (3.97), (3.98), and (3.109)

or

2u2 + Hu, =

I N

1 [ajuj - a,(uJ)*] + complex conjugate.

j= 1

(3.1 1 1 )

Substituting (3.82) and (3.83) into (3.1 1 l), we obtain

2u2 + Hu,=

-

A a

kj j

Mjkaj

4det M

j,k=l

+

+ complex conjugate

tifk)

+ complex conjugate.

(3.1 12)

Since the second term and the complex conjugate expression on the right-hand side of (3.1 12) vanish identically [see (App. 11.23)],

c N

@hjaj

j,k=l ( jf k )

-

+ Mjkaj

4det M

=

0,

(3.1 13)

(3.1 12) reduces to 21.4’ + Hu, =

1 2 A..a. + complex conjugate. det M N

j=l

(3.1 14)

It follows from (3.36)-(3.38) and properties of determinant differentiation that

a % f* Ajjaj -i z h ~ + complex =; conjugate, lm (3.115)

3.1

63

Multisoliton Solutions

which, compared with (3.1 14), yields i

a

2 u 2 + H u x = ---1n-. 2at

f* j-

(3.116)

Finally, differentiating both sides of (3.1 16) with respect to x and using the definition of u (3.9), we obtain the equation 4uu,

+ Hu,, = - u t ,

(3.1 17)

which is the BO equation (3.1). Thus, we have completed the proof that expression (3.9) together with (3.36)-(3.38) is an exact N-soliton solution of the BO equation derived from the theory of linear algebra. Three interesting relations follow from this solution. The first is an expression of u(x, t) in terms of uj and (j= 1, 2, . . . , N), which are solutions of the system of linear algebraic equations (3.77). Multiply v j on both sides of (3.77), sum with respect to j from 1 to N, and use (3.91) to obtain

UT

+ complex conjugate,

(3.1 18)

where the identity (3.1 19)

has been used. Expression (3.118) is similar to the KdV N-soliton solution in which u is expressed in terms of the squared eigenfunctions C431.

It follows by differentiating (3.77) s times with respect to x, multiplying by aSvj/axs, and summing over j = 1, . . . , N that

(3.120)

Integrating (3.120) with respect to x from - 0 0 to 00 and using the boundary condition uj + 0 as 1x1 -,00, we obtain the second relation

64

3 The Benjamin-Ono Equation

The third relation concerns an initial condition which evolves into pure N solitons as time goes to infinity [92]. As shown in (3.91), an Nsoliton solution is represented in terms of uj (j= 1,2,. . . , N), which are solutions of (3.77). Therefore, it is natural to start with (3.77) to find an initial condition that evolves into pure N solitons. It follows from (3.77) with t = 0 that

Assume the initial form of oJ{x, 0) to be uJ{x, 0) = (ix and set

+ l)-',

xjo=O,

j = 1, 2,. . . , N,

j = l , 2 ,..., N,

(3.123a) (3.1 23b)

in (3.122). Substituting (3.123a) and (3.123b) into (3.122) yields the system of N nonlinear equations for the amplitudes a j :

1N

k=l (k+j)

1 =1(1 a j - ah 2

t),

j = 1,2,..., N.

(3.124)

These equations characterize the zeros of the Laguerre polynomial LN of order N, as shown in Section 3.3 [see (3.297)]. If we identify a j with thejth zero of LN, an initial shape of u evolves completely into pure N solitons with the amplitudes equal to the zeros of LN corresponding to (3.123a) and (3.123b).0 Using (3.123a) and (3.91), the initial condition corresponding to (3.123a) and (3.123b) is represented by a simple Lorentzian profile l N

1

u(x, 0) = [UJ{X, 0) 2 j=1

+ uj*(x, O)]

=

N x2 1 '

~

+

(3.125)

3.1.3 Derivation of the N-Soliton Solution by the Pole Expansion Method

The third method for obtaining the N-soliton solution of the BO equation is the pole expansion method. This method originated in a study of the time evolution of positions of the poles of special solutions This statement will also be verified by means of a quite different method in Section 3.3.

3.1

65

Multisoliton Solutions

of the KdV equation; the original concept was described by Kruskal [93].Thickstun [94] applied it to the KdV soliton solutions and clarified the mechanism of soliton interaction. Airault el al. [95] developed the pole expansions of rational and elliptic solutions of the KdV and Boussinesq equations and found a relationship between the motion of the poles of these solutions and the time evolution of certain types of one-dimensional many-body problems. The pole expansion method was then studied extensively and was shown to be applicable to a wide class of nonlinear evolution equations [96,97]. The pole expansion method was also employed to obtain the Nsoliton [90,91,98] and the N-periodic wave [99] solutions of the BO equation. We now proceed with a simple derivation of the N-soliton solution of the BO equation as developed by Case [98]. Asdemonstrated in Section 3.1.1, ifthe N-soliton solution is assumed to be in form (3.9) together with (3.10), then the time evolution of the nth pole x, is governed by a nonlinear equation (3.39). Differentiating (3.39) with respect to r and using (3.39) again to eliminate is, a,*,and in!we obtain the second-order equation

(s # n)

=-2-

a 1 ”

axns.j=l (s #j )

1 (xs-xxj)

2,

n = 1, 2,. .., N. (3.126)

This system of equations is equivalent to an N-body Hamiltonian system interacting with a potential proportional to the inverse square of the distance. It is important to note that (3.126) can be rewritten in the Lax form [lo01 (3.127) aL/at = BL - LB = [B,LI, where L and B are the N x N matrices whose elements are given by

which may be easily verified by direct calculation.

66

3 The Benjamin-Ono Equation

If we now define the N x N matrix K [ x ( t ) ] with elements Kj,

then it follows that [98] K[x(t)] = U{K[x(to)]

=

(3.130)

djlxl(t),

+ ( t - to)L[x(to),a ( t o ) ] } u - l , (3.131)

where the N x N matrix U is given by a solution of the evolution equation aU/at

=

U ( t o )= I, I : unit matrix,

BU,

(3.132)

and x(to) and i ( t o ) are vectors with components x(t0) = [xl(tO), xZ(tO), . . . a(tO)

= [al(tO),

a.2(t0),

7

XN(tO)l,

. . > aN(tO)l, *

(3.133) (3.134)

where the conditions imposed on x j are Im x,Oo) > 0, x,(t0) # xl(to)

j = 1,2, ..., N

(3.135)

for j # 1.

(3.136)

To prove (3.131), consider the quantity (3.137)

J(t) = U - 'K[x(t)]U.

Differentiating (3.137) with respect to t gives aJ/at

=

+

U - ~ { K C ~ ( ~[ )KI, B I I U

(3.138)

by (3.132). Using the relation [95],

[ K , B] = L (3.138) reduces to aJ/at

=

K[i(t)],

(3.139)

u-~LU.

(3.140)

-

Differentiating (3.140) once more with respect to t and using (3.127), we find a2J/at2 =

u-ya/at - [B,L ] > U = 0.

(3.141)

+ cZ(t - to),

(3.142)

Therefore, J has a form J ( t ) = ~1

where c 1 and c 2 are constant matrices. Setting t c1 =

KCx(t0)l

=

to in (3.137),

(3.143)

67

3.1 Multisoliton Solutions

by (3.132). Matrix c2 is also determined from (3.140) with t c2 =

=

to as

LCx(to), i ( t 0 ) l .

(3.144)

+ ( t - to)LCx(to), i ( t O ) l ,

(3.145)

Thus, J ( t ) becomes J ( t ) = KCx(t0)l

which, combined with (3.137), yields relation (3.131). The N-soliton solution is now constructed as

n N

(X -

j= 1

xj)

by the definition of K[x(t)], K[x(t)]}, by det A B = det BA, = det U - '{XI - K[x(t)]}U, = det{xI - U-'K[x(t)]U}, by U - ' U = I, by (3.131), = det{xI - KCx(to)l - ( t - to)LCx(to), i(tO)l>, (3.146) = det{xI - Z}, = det{xI -

where Z is the N x N matrix with elements Zim

=

61nxdt0) + (t - to)L,m(to)*

(3.147)

It follows from (3.9), (3.10), and (3.146) that i d 2ax

u = - - In

i a 2 ax

/I,fl /jIln (x - xT)

= - - ln[det(xI

]

(x - xj)

- Z*)/det(xI

-

Z)],

(3.148)

which is an explicit expression of the N-soliton solution. Expression (3.148) may be reduced to that of the N-soliton solution derived by the bilinear transformation method. Assume an asymptotic form of x j ( t )for large t as xj(t)

N

ajt

+ i/aj + xoj,

t

-+

co, j = 1, 2,. . . , N.

(3.149)

Taking the limit to -+ co,(3.147) becomes Zi, = 6,,(ajto = di,(ajt

+ i/aj + x o j ) + 6i,(t - to)aj + (1 - 6,,)2i/(ai + i/aj + xoj) + (1 - 6,,)2i/(a, - a,).

- a,) (3.150)

Substituting (3.150) into (3.148), we obtain (3.9) with (3.36)-(3.38).

68

3 The Benjamin-Ono Equation

The pole expansion method can also be applied to obtain the Nsoliton solutions of the higher-order BO equations [ l o l l and the N periodic wave solution of the BO equation [99, 1023.

3.2 Backlund Transformation and Conservation Laws of the Benjamin-Ono Equation In this section the Backlund transformation of the BO equation is formulated on the basis of the bilinear transformation method. An infinite number of conservation laws of the BO equation are then constructed from the Biicklund transformation of the BO equation and the structure of conserved quantity is clarified. 3.2.1 Backlund Transformation We first write the BO equation in the convenient form u,

+ 4uu, + PHu,,

=

0,

(3.15 1)

where P (>0) is a parameter characterizing the magnitude of the dispersion. Introducing the dependent variable transformation i

a ax

u = -P-ln

2

f' -, j-

(3.152)

where f is given by (3.46) and f ' by (3.47), (3.151) is transformed into the bilinear equation i l l , f ' . f = /3Dl f ' .f .

(3.153)

Let u be another solution of (3.151), that is, i a u=-b-ln-, 2 ax

iD, g' g = pD;g' .g.

g' g

(3.154) (3.155)

69

3.2 Backlund Transformation and Conservation Laws

The relation connecting the two solutions u and u, the Backlund transformation of the BO equation, is given in terms of the bilinear variables as [lo31 (iD, - 2iLDx - PO: - p)f . g = 0, (iDr - 2i1Dx (POx

(3.156)

- p)f‘.g’ = 0,

(3.157)

+ U)f

(3.158)

.g‘ = ivf’g,

where 1,p, and v are arbitrary constants. Iff and f ’ satisfy (3.153) then g and g’ also satisfy (3.155), provided that Eqs. (3.156)-(3.158) hold for f,f’,g, and 9’. We show this by considering the quantity

P = [(ill,

-

P D 2 ) f ’ . flg’g - flf[(iD,

-

BD:)g‘.g].

(3.159)

Using formula (App. 1.6.1), P is converted to

P

= (iDtf’ .g’)fg

- f’g’(iDr f ‘ 9 ) - P(DZ f’. f)g’g

+ Pflf(D2g’

*

9).

(3.160) Substituting (3.156) and (3.157) into (3.160) and using formulas (App. 1.6.1) and (App. I.6.3), P becomes

This last expression vanishes identically owing to (3.158) and (App. 1.1.5), that is,

P

=

0.

(3.162)

Therefore, (3.155) follows from (3.162) and (3.153), and the proof is complete. The superposition formula and the commutability relation may be derived similarly on the basis of (3.156)-(3.158), which has been detailed by Nakamura [103].

70

3 The Benjamin-Ono Equation

To transform (3.156)-(3.158) into a form written in the original variables, we introduce the potential functions ii and V through the relations u = ii,, (3.163) (3.164)

u = V,, or i i = - iP l n - - f' , 2 f

(3.165) (3.166)

from (3.152) and (3.154), respectively. Note also the relations that are derived using (3.50):

and

-1( I 2 Dividing (3.158) by fg' yields

-

is f: iH)u = ---. 2 f'

(3.168)

(3.169) Substituting (3.163)-(3.168) into (3.169), we obtain the space part of the Backlund transformation written in original variables as (ii

+ V),

=

-1

+ v exp[-2i(ii

- V)/fl]

-

iH(ii - V),.

(3.170)

Introducing a function w by ii - V = -iPw/2

(3.171)

and E by ,l = v =

(3.170) becomes* -($/2)P-

w,

-2/E,

+ (1 - e-W)/E = u,

Equation (3.173) will be used extensively in this text.

(3.172) (3.173)

71

3.2 Backlund Transformation and Conservation Laws

where P - is an operator defined by

P-

=

31 - iH).

(3.174)

The time part of the Backlund transformation, (3.156) and (3.157), is rewritten as

( i i

-

( a4

i--2zA-

2 i A k ) Inif

-

Pax21nfg a2 -B a2

ln-;-P-lnf'g'-P

. x:)

-

22:)

(In 5 $) -

In

= 0,

-p=O.

ax2

Subtracting (3.175) from (3.176) yields

(i:

-p

-

P$

(In 5 +

In

(3.175) (3.176)

5)

Introducing (3.165)-(3.168) into (3.177), we obtain the time part of the Backlund transformation written in original variables as (U - V ) ,

=

2A(U

- V ) , + 2i(P - ij),H(U

Eliminating the term (U

(U - V),

=

- V), -

iP(U

+ fi), by using (3.170) gives + 2i(U - V),H(U - V),

+ V),,.

2A(U - V), - 2v(U - a), exp[-2i(U - V ) / P ] - PH(U - ij),.

(3.178)

(3.179)

Finally, by introducing (3.171) and (3.172) into (3.179) it follows that w, = -PHw,,

4

- -(1 E

- e-")w,

+ Pw,Hw,.

(3.180)

It is interesting to note the relation u,

+ 4uu, + PHu,,

=

w,

+ BHw,, + -4( 1 E

- e-")w,

-

pw,Hw,

1

,

(3.181)

72

3 The Benjamin-Ono Equation

which follows from u in (3.173) and the properties of the H operator (see Appendix 111). Therefore, (3.173) and (3.180) imply (3.151). If we expand w formally in powers of E as w=

c w,E”, W

(3.182)

n= 1

then it can be seen that (3.151) and (3.181) also imply (3.180). The forms (3.173) and (3.180) correspond to those of the Backlund transformation of the KdV equation, (2.230) and (2.233).

3.2.2

Conservation Laws

It follows by integrating (3.180) from - co to 00 and using a property of the H operator, (App. III.18), together with the boundary condition w+Oaslxl-+cothat w dx

=

0,

(3.183)

which means that the function w is a conserved density. Substituting (3.182) into (3.183), we obtain

* dIn ,=o, I-& dt

(3.184)

where

I, =

J-

03

dx.

(3.185)

Relation (3.184) must hold for arbitrary E, therefore dIJdt

=

0,

n

=

1 , 2,... .

(3.186)

The I, defined in (3.185) is the nth conserved quantity of the BO equation. To derive the explicit functional form of w,, we introduce (3.182) into (3.173) and compare the E“ term on both sides of (3.173).

73

3.2 Backlund Transformation and Conservation Laws

The first few expressions of I, constructed from these w, are given as

Il = J

(3.187)

udx, -m

(3.188) (3.189) (3.190)

n 2 3. (3.191)

As seen from (3.173) with j? = 0, w = -ln(l - m ) =

1 un O0

n=l

-&",

n

(3.192)

implying that the term that does not contain B has the form u"/n in the expression of w,. It may be informative to discuss another construction of the Backlund transformation described by Bock and Kruskal [1043. They considered the associated linear equation 41

+ 4% + B H q x x = 0

(3.193)

and observed that the conserved density for the BO equation (3.194)

q'o' = u, q(l)

= u2

q(2)

= u3

+ iBHu,, + B($uHu, + $Huu,)

(3.195) - +~'uxx,

(3.196)

74

3 The Benjamin-Ono Equation

is satisfied by (3.193). They then assumed the existence of an infinite series

= EblU

+ EZb,(UZ + @Hu,) + E 3 b 3 [ U 3 + P($uHu, + ZHuu,) (3.197)

where 6, (j= 1,2, . . .) are unknown constants. Solving(3.197) inversely for u under the condition that this inverse contains no derivative of q higher than the first, they found an expression

which is the analogue of the Miura transformation [lo51 of the KdV equation. By substituting (3.198) into (3.193), it is confirmed that q is a conserved density, that is, q dx = 0.

dt

(3.199)

Introducing expansion (3.197) into (3.198) and comparing the E" term on both sides of (3.198), the qn(n = 1,2, .. .)are determined successively, the first few corresponding to (3.194)-(3.196). We now reconsider the Backlund transformations (3.173) and (3.180). All information concerning the conservation laws is included in (3.173) and therefore the study of (3.173) may help to clarify their structure. However, (3.173) is highly nonlinear, owing to the term e - w , and hence intractable in the present form. To overcome this, we differentiate (3.173) with respect to x to obtain iP 2

- -P-

w,,

+ -E1 e-ww, = u,.

Eliminating the term e - w by (3.173) yields U,

+

UW,

+ iP2 P - w,, + -

(3.200)

75

3.2 Backlund Transformation and Conservation Laws

Introducing (3.182) into (3.201) and comparing the E” term on both sides of (3.201), we obtain a recurrence formula for w, as [lo61 w1

=

(3.202)

u,

(3.203) This formula is a starting point for the following discussion. The first few expressions of w, derived from (3.202) and (3.203) are w2

U’ =-

w3

=-

Wq

u4 =-

2

+ -iP2P - U x r

u3 + -iP( U P 3 2 4

(3.204)

+ P - uu,)

u,

P’

- - P - u,,,

4

iP +( P - u’u, + UP- uu, + U ’ P 2

- p’ [P(.P4

u,),

(3.205)

u,)

+ P-(uu,), + U P - u,, + (3.206)

which reduce to the conserved quantities (3.188)-(3.190) after integrating with respect to x. Since (3.203) includes a derivative term w , + ~ , , , it is not clear whether (3.203) is integrable. To see this, it is convenient to introduce Cj (j2 1) by (3.207)

Cl = w1,

c . =wJ. - - piP- w . J

s= 1

s= 1

J-i,x,

j 2 2.

(3.208)

76

3 The Benjamin-Ono Equation

which is linear in C j (j = 1,2, . . . ,n explicitly as

n - s l -s2 -

... - s j -

1

X

+ l), and its solution is obtained

I

w s , w s , ~ - ~ w , , w n - s , - s , sj+l,

s,= 1

n 2 1.

(3.210) Terms w,+ are constructed recursively from (3.207), (3.208), and (3.210). From (3.207) and (3.208) it may be noted that 03

W

I, = /-,w,, dx

=

/-,C,, dx,

n

=

1,2,. . . .

(3.211)

The important relations which hold between conserved quantities follow from the recurrence formula (3.203). Some notation must be introduced before deriving these relations. The functional derivative 6IJ6u is defmed in the relation

(3.212) The derivative D/Du is

where the action of the operator a/au,,, on Hf,f being an arbitrary function of u, u x , . . . ,is defmed by

aHf p u n ,

=

H afpu,,,

n = 0, 1,2,. . . .

(3.214)

By using a property of the H operator [see (App. III.4)]

fHg

=

-gHf

+ HCfS - ~ ~ f ) ( H s ) l ,

(3.215)

+ HGn,

(3.21 6)

we may derive a relation 6IJ6u = DwJDu

where fi is a certain function of u, u,, . . ., Hu, Hu,,. . . . We now prove the first relation awj/& = (j- l ) ~ ~ - j~ 2, 2,

(3.217)

77

3.2 Backlund Transformation and Conservation Laws

which is verified by mathematical induction. If we assume (3.217) up to j = n, then it follows by differentiating (3.203) with respect to u that

1

(3.2 18)

where we have used a commutation relation (3.219)

where f is an arbitrary function of u, u x , . . . , and (3.217) is used in passing to the second equality of (3.218). Substituting the relations

s= 1

s= 1

n- 1

(n - s - l)W,.P- wn-s.-

s= 1

n-2

=

C (n - s - 1)ws,xP- W n - s -

1

s= 1

(3.221)

into (3.218), we obtain

78

3 The Benjamin-Ono Equation

Integrating (3.222) with respect to x and using the boundary condition w, + 0 as 1x1 + co,we have aw,+l/au = nw,,

(3.223)

which implies that relation (3.217) holds for j = n proof. The second relation to be proved is

+ 1, completing the

which is a consequence of the important fact that the functional derivative of a conserved quantity is a conserved density. To verify this statement, we use (3.213), (3.216), and (3.217) to obtain

=

(n - 1 ) ~ ~+-

z m

s=

as

awn

(- 1)” 7ax au,, 1

+ HE,.

(3.225)

Integrating (3.225) with respect to x yields (3.224), by noting the definition of I,, (3.185), and formula (App. 111.2),

1m

HE, dx = 0.

(3.226)

m

The third relation is

+ A] = 1s!(n(n--s l)! Kn-,[~]AS, - l)! n- 1

K[u

,=(J

n = 1, 2 , . . .,

(3.227)

where

K,[u]

=

61,,/6~

(3.228)

and A is an arbitrary constant. This relation follows by repeated use of (3.224).# Finally, we shall comment on the inverse scattering transform of the BO equation. The derivation of the inverse scattering transform can be Recent topics concerning the conservation laws of the BO equation appear in the references [107-1091. In [107], an infinite number of conservation laws were constructed from Lie algebra.Theconservedquantitieswhich depend explicitly on time were presented in [l08, 1091 as an application of this theory.

79

3.3 Asymptotic Solutions

performed formally from the Backlund transformation once the Backlund transformation has been constructed. (The procedure is the same as that for the KdV equation demonstrated in Section 2.4.) We define wave functions t,b and I// by

*

Slf, *’ = s’lf’

(3.229)

=

(3.230)

and introduce these into (3.175), (3.176), and (3.169). Taking relations (3.167) and (3.168) into account, we obtain the inverse scattering transform of the BO equation in the form [lo31

+ p az/axz + 2iu, - ~ H U+, p)$ (i alat - 2il alax + p az/ax2 - 21.4, - ~ H U+, p)$’ (ip a/ax + 2~ + A)+‘ - V+ (i alat - 2il a/ax

= 0,

(3.231)

= 0,

(3.232)

= 0.

(3.233)

One can verify that eliminating $ and I,$‘ from (3.231)-(3.233) reproduces the original BO equation (3.151). At the present time, however, the standard technique of the inverse scattering method [3,4] cannot be applied directly to the system of equations (2.231)-(2.233), and we shall not pursue this problem further.

3.3 Asymptotic Solutions of the Benjamin-Ono Equation The methods developed in the preceding sections are convenient tools for obtaining special solutions of the BO equation. The N-soliton and N-periodic wave solutions are very important and are characteristic of integrable nonlinear evolution equations. From the mathematical point of view, however, the initial value problem must be studied for full understanding of the general nature of solutions. In this section we shall develop an approximate method for solving an initial value problem of the BO equation and investigate the asymptotic behavior of solutions in the zero dispersion limit. An explicit example of an initial condition which evolves into pure N solitons is also presented, and it is shown that the amplitudes of solitons are then closely related to the zeros of the Laguerre polynomial of order N! The method presented in this section was developed by Matsuno [110-1121.

80

3 The Benjamin-Ono Equation

3.3.1 Method of Exact Solution We now consider the initial condition

t = 41,

u(x, 0 ) = Uo4(<),

’0,

4(&

(3.234) =

0,

(3.235)

where uo and 1 are representative values of u(x, 0) and x , respectively, and 4 is a dimensionlessfunction characterizing the profile of the initial disturbance. Tentatively assume that the initial condition given by (3.234) and (3.235)evolves into pure N solitons for large values of time. The amplitude of each soliton may be determined from the system of equations derived from an infinite number of conservation laws of the BO equation. To show this, consider first the two-soliton case. For large t , the solution is approximated by

a superposition of two solitons with amplitudes a, and a,. To determine a, and a,, estimate the conserved quantities I, and I, given by (3.188) and (3.189) at both t = 0 and t + co. Keeping (3.236) in mind, we obtain

4

(a1

=P

16 (a:

-

+ a2) = I,[u(x, O)] = + a:) = I ~ [ u ( x O)] , =

J-

W

w,(x, 0 ) dx,

(3.237)

w ~ ( x0) , dx.

(3.238)

1, 2 , . . . ,

(3.239)

1,2, .. .,

(3.240)

n = 1,2,. . . ,

(3.241)

W

j

n

0

= =

(3.242)

= UOllB,

(3.237) and (3.238) can be rewritten as Yl

y:

+ Y2

+ y:

s1.

(3.243)

= s2.

(3.244)

=

81

3.3 Asymptotic Solutions

Values y , and y, are determined from the roots of an algebraic equation of order two

- sly

y2

+ 3s: - s2) = 0

(3.245)

as y, =

&, - JW),

y, = Ks1

(3.246)

+ J-1,

(3.247)

where we have assumed y1 < y,. To yield real and positive yl and y,, it is necessary that the following conditions are satisfied:

(3.248) (3.249) This discussion may be extended to the general N-soliton case. In this case, however, we must evaluate the nth conserved quantity I , for pure N-soliton solution. For this purpose, we employ the Backlund transformation of the BO equation (3.173)and (3.180). We first evaluate I , for the one-soliton solution (3.3). Note that the relation wtCu,(x, 01 = -aw,Cu,(x,

01

(3.250)

holds since u,(x, t) is a function of x - at. Substituting (3.250) into (3.180),integrating from - cc to x, and using the boundary condition w + 0 as 1x1 + 00, we obtain

w

+ pHw,

4 - -(1 E

- eCw) -

Integrating (3.251)once from - 00 to

-

p

00

p

w,Hw,dy

=

0. (3.251)

yields

s_mmdx J;mWyHwyd y

=

0.

(3.252)

Substituting the relation

(3.253)

82

3 The Benjamin-Ono Equation

which derives from (3.173) and (3.3), into (3.252) yields

However, since the second term on the right-hand side of(3.254) vanishes due to formula (App. 111.25), (3.254) becomes

It follows from (3.255), (3.182), and (3.185) that In[us(x,t ) ] = nfl(a/4)”- l,

n

=

I, 2, . . . .

(3.256)

For pure N solitons C(3.9) with (3.36)-(3.38)], (3.256) can be replaced by N

I,[u(x, t ) ] = np

1(aj/4)”-l,

n

=

j= 1

1, 2,. . .,

(3.257)

since, as will be shown in Chapter 4, u(x, t) is expressed as a superposition of N separated solitons in the limit of large values oft. Using the fact that I, is a conserved quantity I,[u(x, t)] = I,[u(x, O)],

n = 1, 2,. . . ,

(3.258)

from (3.257), (3.258), and (3.185) we find that

c (aj/4)n-l N

np

m

O)] dx,

= /-mw,,[u(x,

j= 1

n = 1 , 2 , . .. .

(3.259)

The first N + 1 expressions of (3.259) may be used to determine a,, a 2 , . . . , aN. The equations given by (3.259) can be rewritten in terms of y j and s,, defined by (3.239) and (3.241), as N

xfl-l=s,,

j= 1

n = l , 2 ,..., N + 1 .

(3.260)

For n = 1, Eq.(3.260) gives the total number N of solitons that arise from a given initial condition

N = s l = - t7 Jm dJ(t)d5.

+

X

-m

(3.261)

For 2 1. n I N 1, the equations defined by (3.260) give N equationsfor N unknown functions y,, y,, .. .,y , from which the amplitudes

83

3.3 Asymptotic Solutions

of each soliton can be determined. To show this, it is convenient to introduce the elementary symmetric functions of y,, y 2 , . . .,yN as Y,

+ Y 2 + ... + y,,

01

=

*2

= Yly2

(3.262)

+ y 2 y 3 + ." + Y N - l y N ,

ON-1

=

yly2"'yN-1

*N

=

ylY2".yN-

(3.263)

+ ". + y 2 y 3 " ' y N ,

(3.264) (3.265)

The unknown quantities y,, y 2 , . . . ,y, are given by the roots of the single algebraic equation of order N [l 1 11

y" - oIY-'

+ C T Z ~ N - +~

* * *

+ (-1),0,

= 0.

(3.266)

The coefficients o,, 02,. . . ,a, are uniquely determined by a recursion formula

(-l)"rn*,,, starting with

+ ( - l ) m - 1 ~ 2 ~ m+- l... +

=

The first few terms a,,,are given as

0,

= s2.

01

= s2,

6 2

= K-S3

(3.268)

+ Si),

= g-sg

+

(3.269)

+ +$),

03 = 4 ( ~ 4- b 2 ~ 3

04

m

+

s,cl s,+, = 0, 2, 3, ..., N , (3.267)

S , , , - ~ O ~-

+

3 4 ~ 2 ~ +s: 4

(3.270) - S;S~

+ 4~:).

(3.271)

The structure of the BO equation may be clarified by studying the properties of the algebraic equation (3.266). 3.3.2 Asymptotic Solution in Zero Dispersion Limit We shall now investigate the asymptotic behavior of solutions when the effect of dispersion is very small, that is, fl N 0. In this case, the number of solitons arising from an initial condition d(() increases indefinitely, as seen from (3.261). Therefore, we can introduce the number density function F(a) of solitons where F(a) da gives the

84

3 The Benjamin-Ono Equation

number d N of solitons with amplitudes within the interval (a, a as

+ da)

(3.272)

d N = F(a) da.

Equation (3.259) is then approximated by the integral equations

=gl"g"(<)d<, n

-m

B+O,

n = l , 2 ,..., (3.273)

where we have used (3.191). The solution of (3.273) can be found to be [llO, 11110 F(a) =

CJ

4nu0

~

I

(3.274)

dt,

4,,,,&(<) >

where the integration interval is restricted by the condition 4u04(t) > a.

(3.275)

It can be seen from (3.274) that the amplitudes of solitons do not exceed four times the maximum of the initial perturbation (3.234) F(a) = 0

for a > max 4u(x, 0).

(3.276)

We now give explicit examples of F(a) for two different initial conditions. (i) Rectangular well 1

F(u) =

G

for for

o/4nu0

-3 I < < I , < > f, < < -3,

(3.277a) (3.277b)

for a < 4U0, for a L 4u0,

(3.278a) (3.278b)

1

N = a/n; The proof of (3.274) is given in Appendix IV.

(3.279)

85

3.3 Asymptotic Solutions

(ii) Lorentzian profile

4(<)= l/(? + 11, F(u) = N

rb

o/2auo)(4u0/a - 1)’’’

(3.280)

-=

for a 4u0, (3.281a) for a 2 4u0, (3.281b) (3.282)

= CT.

The space distribution of solitons for large values of time is also obtained using (3.274). Since the individual soliton with amplitude a moves with velocity a and is found at x = at when t -, CO, we have from (3.275) 0 < x / t < max 4u(x, 0).

a = x/t,

(3.283)

Denoting the number of solitons within the space interval (x, x as k(x, t ) dx, we obtain k(x, t) dx

+ dx)

F(a) da,

(3.284)

4 x 9 t ) = (l/t)F(x/t),

(3.285)

=

which, combined with (3.283), yields where F is given by (3.274). So far we have been concerned only with the initial condition defined in (3.234). However, these results can be generalized for the initial condition including the negative region where u(x, 0) < 0. In this case, after the lapse of a large amount of time, the initial disturbance is assumed to decay into separated N solitons with the amplitudes aj (j= 1, 2, . . . ,N) and the tail part. (The tail part is a sort of dispersive wave.) The value of the conserved quantity may be assumed to be the sum of the I, for the solitons and the I, for the tail. Under these assumptions, the equations given by (2.259) can be replaced by N 1(aj/4)”-’ + I,- = J-mwn(x, 0) dx, m

aB

j= 1

n = 1,2,. . .,

(3.286)

where & denotes the value of I,[u(x, t)] for the tail. For small B, (3.286) is approximated by

np JOm(a/4)”-lF(a)d a

+

=

l/n

J-

m m

u”(x, 0) dx,

n = 1 , 2 , ... .

(3.287)

86

3 The Benjamin-Ono Equation

Introducing F(a) from (3.274) yields

l/n that is

u(x, 0 ) > 0

-

u"(x, 0) dx-+

I,,= l/n

J

u(x. 0 ) < 0

r,,= l/n

J-

m

u"(x, 0)dx,

B

U"(X, O)dx,

(3.288)

OD

+

0,

(3.289)

which implies that, for small /I, the tail arises from the negative region of the initial disturbance u(x, 0). The total number N of solitons arising from +(<)is given in the limit of B-+ 0 as N = JoWF(a)da =

h,,,

>O

4 5 ) d5.

(3.290)

Hence, the number of solitons depends on the positive region of the initial disturbance, and for small B (or large c) at least one soliton arises if the initial disturbance satisfies the condition

(3.291) as can be seen from (3.290). 3.3.3 An Initial Condition Evolving into Pure N Solitons The initial conditions we have considered so far have been rather general. We shall now present an initial condition which evolves into pure N solitons after the lapse of a large amount of time. It has the form u(x, 0) =

N WB)'

+ 1'

N: positive integer.

(3.292)

The amplitude of each soliton corresponding to (3.292) is identified with the zeros of the Laguerre polynomial of order N. To prove these statements, let us first define the Laguerre polynomial of order N by N

u.Y) =

Z(-V( N -Nr )!! ( r ! ) 2J f .

r=O

(3.293)

87

3.3 Asymptotic Solutions

The LN(y) satisfies the following second-order ordinary differential equation (1 - y)Lk NLN = 0, (3.294)

+

+

where the prime denotes differentiation with respect to y. Let y j (j= 1 , 2 , . . . , N ) be N zeros of L N ( y ) .Then the y j have the properties j = 1,2,..., N ,

L&j) = 0, yj>o,

c ( y j - yk)-'

for j # k , j , k = 1 , 2 ,..., N , (3.296)

yj#yk

N

k= 1 W+j)

(3.295 )

j = 1,2, ..., N .

= $(l - y;'),

(3.297)

To show (3.297)*,put LN(y)in the form

(3.298) It follows by direct calculation that N

(3.299) N

N

2 (y -

Li = 2LN

j= 1

yj)-'

1

k=l (k+i)

(yj

- y&)-l.

(3.300)

Substituting (3.299) and (3.300) into (3.294) and rearranging, we obtain N

1 ( y - yj)-'

j= 1

[

N

&-1 (k

2y,{yj - y k ) - l

+ 1 - yj

*n

1

= 0.

(3.301)

Relation (3.301) holds for arbitrary y, and therefore the quantity in the parentheses must be zero identically, which implies (3.297). We now define the matrix A whose elements are given by Ajk

=

8jk(Z

+ y y l ) + 2(1 -

6jk)bj

- yk)-',

For a discussion of properties (3.295) and (3.296) see [113].

(3.302)

88

3 The Benjamin-Ono Equation

where z is an arbitrary complex number. As will be shown in Appendix I1 [see (App. 11.411, the following remarkable identity holds: det A(z) = (z

+ l)N.

(3.303)

Given these conditions, we employ the N-soliton solution of the BO equation C(3.9) with (3.36)-(3.38)]. Putting aJ. = y j , xoj=O,

j = l , 2 ,..., N ,

(3.304a)

j = l , 2 ,..., N.

(3.304b)

in this solution, then at t = 0, the solution becomes [112]

(3.305) where we have used (3.303) in passing to the third line of (3.305). The asymptotic form of u(x, t ) corresponding to (3.305) for large values of time is expressed as N

u(x9 t)I%

(yj//3)’(X

Yi

- yjt)2

+ 1’ ,

(3.306)

since the initial condition (3.305) evolves as t + 03 into the pure N solitons with amplitudes equal to the zeros y j (j= 1 , 2 , . . . ,N) of the Laguerre polynomial of order N. The initial condition (3.292) has the same form as (3.125), which has already been derived from the linear algebraic equation. We shall now evaluate the nth conserved quantity I, (n = 1,2, . . .) of the BO equation for the initial condition (3.292). For this purpose, (3.257), (3.258), and (3.304) are used to obtain

pr- $ 1fl-’. N

Z,[u(x, O)] = I,[u(x, t)] = np

=

j= 1

j= 1

(3.307)

89

3.3 Asymptotic Solutions

Therefore, it is sufficient to calculate the quantity N

(3.308)

j= 1

=[

the solution of which is given by the well-known Euler formula as

Pn

olPn-1

- 02p,,-,

+ ... + (-

1)”-20,,-1p1 + (-l)”-’nom for n = 1,2, ..., N, (3.309) (-1)”-’CNPn-N for n > N. (3.310)

- o2Pn-2

+ .’.+

Here o l , o,, . . . ,oNareelementary symmetric functions ofy,, y,, . . . ,yN defined in (3.262)-(3.265). In this case, by comparing (3.293) and (3.298), it follows that j = 1, 2,. . . ,N.

oj = [N!/(N - j ) ! ] ’ / j ! ,

(3.31 1)

The p,, (n = 1,2, . . .) are calculated by introducing (3.31 1) into (3.309) and (3.3 10). The first few expressions of p,, are given as

P1

=

(3.3 12)

NZ,

p z = N2(2N - l),

+ 2), p4 = NZ(14N3 - 29N2 + 22N - 6), p s = N2(42N4 - 130N3 + 165N2 - lOON + 24). p3 =

N2(5NZ- 6N

(3.3 13) (3.314) (3.315) (3.316)

In particular, for N = 1, Pn

so that In{1/[(X//3)2

= 1,

(3.317)

+ l]} = z/3/4”-l.

(3.318)

Finally, we may note a close relationship between the distribution function F(a) of solitons and that of the zeros of the Laguerre polynomial of order N in the limit of large N. Since the amplitudes of the solitons are given by the zeros of LN for the initial condition (3.292), we obtain, by substituting (3.292) into (3.274) and setting a = y, in the limit of N + co

F(y) = 2 {I!

n)(4N/y

-

l)’”,

y < 4N, y 2 4N.

(3.319) (3.320)

90

3 The Benjamin-Ono Equation

Here F ( y ) is normalized such that the total number of zeros corresponds to N: Pa,

J

Fb)dy

=

(3.321)

N.

0

The F b ) d y gives the number of zeros within the interval ( y , y + dy). Using (3.319) and (3.320) the p , defined in (3.308) are evaluated in the limit of N + 00 as N

xfl j= 1

Pn=

-

(2n)! N n + 1 , n!(n l)!

+

n = 1,2, ...,

(3.322)

which correspond to the first term on the right-hand side of (3.312)(3.316).

3.4 Stability of the Benjamin-Ono Solitons In this section the linear stability problem of the BO solitons is briefly discussed using the results obtained in Section 3.3. Let us consider the initial condition u(x, 0) which evolves completely into pure N solitons after the lapse of a large amount of time. The amplitude of each soliton is then determined by the system of equations (3.259) as N

a,

zj? (aj/4)” = /-a,w,+ j= 1

O)] d x

=

I,+

1,

n = 1,2, . . . .

(3.323) Consider the increments 6aj of the amplitude of each soliton when the initial disturbance u(x, 0) is varied infinitesimally, by 6u(x, 0), for example. It follows from (3.323) that

nj?n

--

4”

i=l

a;-’ 6aj = ~31,,+~, n

=

1,2, . . . ,N,

(3.324)

91

3.5 The Linearized Benjamin-Ono Equation

wheredl,, denotes the variation ofl, (n = 1,2, . . .,N)corresponding to 6u(x, 0). The system of equations (3.324) is linear with respect to 6aj, and the solutions are obtained using Cramer’s formula as

6Uj

1

1

46l2/n/?

a,

a2

8 61,/n/?

aN-l

aN-l

:

=

X

a,

*

*

a$-

I

...

Here the 1

1

...

1

a,

a2

...

aN

ai-1

1 aN

4N

a2

a’-1 1

a

...

.

,

-

j

=

n

1,2, ..., N .

(aj - ak).

(3.325)

(3.326)

1s k<jsN

a$-

It may be seen from (3.325) and (3.326) that a small variation in the initial disturbance results in a small change in the amplitude of each soliton since aj # ak forj # k. This means that the N-soliton solution is stable against the small disturbance.

3.5 The Linearized Benjamin-Ono Equation and Its Solution In this section the linearized BO equation is analyzed using the Fourier transform method. The initial value problem of the linearized BO equation is solved exactly, and the asymptotic behavior of the solution is investigated for large values of time. The self-similar solution of the BO equation is also discussed briefly.

92

3 The Benjamin-Ono Equation

Consider the linearized version of the BO equation in the form u,

+ Hu,, = 0.

(3.327)

If u(x, t) is represented in the form of the Fourier transform as u(x, t)

1m

=

u(k) exp{i[kx

-

w(k)t]}dk,

(3.328)

m

where k and w(k)are the wave number and frequency, respectively, then substituting (3.328) into (3.327) yields the dispersion relation (3.329)

w(k)= - k l k J , where use has been made of a formula m

eikx

P J - m xdx= i n- .

Ikl k

(3.330)

The unknown function u(k) is determined from the initial value u(x, 0) as u(k) =

1

m

211

-m

u(x, O)e-"" dx.

(3.331)

Substituting (3.331) and (3.329) into (3.328), we obtain a general solution of (3.327) as exp{i[k(x

- y)

+ k Ik It]}u(y, 0) dk dy (3.332)

where

C(x) = J1cos(nt2/2) dt, ~(x= )

1

sin(nt2/2) dt.

(3.334) (3.335)

93

3.5 The Linearized Benjamin-Ono Equation

The function K(x) defined in (3.333) has the following asymptotic expansions:

+ l)! ! 22"+ ZX4n + 3 '

(- 1)"(4n

K(x) N

(cos(x2)

+ sin(x2) -

$ -

x+ (-1)"(4n

n=O

22n+2

+ l)!!

X4 n + 3

+0O,

(3.336)

,

x

+.

- oo.

(3.337) Now assume that the initial condition vanishes rapidly when 1x1 + oo. Expanding K[(x - y)/2&] in powers of y as

with K'"'(x) = d" K(x)/dx",

(3.339)

and inserting (3.338) into (3.332), we find

The asymptotic expressions for K(")follow from (3.336) and (3.337) as

{x

+

K(")(x)N fiC(-1)"/8][(n 2 ) ! / ~ " + ~ ] , x + +a (3.342) x -,- 00, (3.343) 2 W ) " cos(x2 - n/4 + n7r/2), and the Fourier transform of u(x, 0) is represented, by using (3.331) and (3.341), as 1 v(k)=C Pn-.(-ik)" (3.344) 2nn,o n! The asymptotic expression of u(x, t) is derived by employing the above results as follows: When x/& -+ - 00, we find from (3.340) and (3.343) that u(x, t) N 2-

Re{u(x/2t) exp[i(x2/4t - n/4)]>.

(3.345)

94

3 The Benjamin-Ono Equation

When x/&

-,

+ 00, it follows from (3.340) and (3.342) that

u(x, t ) N (l/n&)[2/(~/&)~]P~,

Po # 0.

(3.346)

For lxl/& 6 1 and large values oft, the solution is approximated by the first nonvanishing term in (3.340) to yield

u(x, t ) = (~/,/G)K(x/~&)P,,

P, z 0.

(3.347)

As an explicit example of the initial condition, consider Dirac’s delta function u(x, 0) = 6(x).

(3.348)

The solution for t > 0 is given by (3.332) as

(3.349)

r(’/

and asymptotic expressions of (3.349) are found from (3.336) and (3.337) as

u(x, t )

for x/&+ for x/&

~)c~/(x/&)~I (1/&) cos(x2/4t - n/4) J -

+m, -, - m.

(3.350) (3.351)

Finally, we shall comment on the similarity solution of the BO equation. It may easily be confirmed that the BO equation (3.1) is invariant under the similarity transformations u=

d/&,

(3.352)

x = XI/&,

(3.353)

t

(3.354)

= Etl.

Therefore, u has a solution in the form =

U/&)f(x/&).

(3.355)

Substituting (3.355) into the BO equation (3.1) yields

-$(
(3.356)

where the prime denotes‘differentiation with respect to the similarity variable 5;

< = x/&.

(3.357)

If the boundary condition

f(+oo)

=0

(3.358)

3.5 The Linearized Benjamin-Ono Equation

95

is imposed, (3.356) reduces, after integration by <,to

-& + 2f’ + Hf’= 0.

(3.359)

It should be noted that the linearized version of (3.359), that is,

-&1 + Hf’= 0,

(3.360)

f(<)= K(5/24’3>,

(3.361)

has a solution where K is given by (3.333). The analytical solution of (3.359) has not yet been obtained, therefore the numerical method may be applied to it.

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Interaction of the Benjamin-Ono Solitons

In this chapter we study the properties of the N-soliton solution of the BO equation. The asymptotic behavior of the solution is derived for large value of time, and it is shown that no phase shift appears as a result of soliton collisions.0 The very complex interaction process between two solitons is then investigated in detail. The motion of poles corresponding to the two-soliton solution is drawn in a complex plane, and the nature of the interaction is clarified.11 This is in contrast to collisions that take place between KdV solitons. 'The main part of this chapter follows the discussion of Matsuno [114]. 97

98

Interaction of the Benjamin-Ono Solitons

4

4.1 Asymptotic Behaviors of the N-Soliton Solution Consider the N-soliton solution of the BO equation in the form expressed by (3.9) and (3.36)-(3.38), and expand the determinant with respect to the variables Oj (j= 1, 2, . . . , N ) as N

f N =

=

1

C i n Q ~ - n ( S 1 , S 2 ,...,Sn)es,es,...e,n n = 1 NC,,

" 1 iNe1e2. .. O N + i N - ' 1 -el&

. . .en-,en+ . . . O N

n=l a n

+ .. .,

(4.1)

with

n = l , 2 ,..., N - 1 , Q N

=

[fN]&

Qo

=

1,

=

e2 = ... = eN

--

(4.2) (4.3)

0,

(4.4)

CNCn

where the notation means summation over all possible combinations of n elements taken from 1,2, . . .,N . To investigate the behavior of (4.1), we order the velocities of each soliton as

0 < a l c a2 c ... c a N ,

(4.5)

transform to a moving frame with a velocity a,, and then take the limits t + f 00. First, consider the limit t + - CO. It follows from (3.8) and (4.1) that

e1,e2,..., e n + 1, e n + 2 7

..

*

7

=

-00,

(4.6)

8,

=

finite,

(4.7)

ON

=

+ CO.

(4.8)

Substituting (4.6)-(4.8) into (4.1), we obtain the asymptotic behavior of as

fN

f

- 1.N e1e2. . .en-,enen+. . . O N + iN-1 (l/a,,)e,e,. . .en-,en+ . . .

+ . .. .

(4.9)

4.1 Asymptotic Behaviors of the N-Soliton Solution

99

Thus, we find from (3.9) and (4.9) that

-fa In{[ 2 ax

(-i)N8102 ... e,,-le,e,,+l

-

-

an

(anen)’

+ 1’

.-.ON

t + -co,

(4.10)

which is identical to the one-soliton solution (3.3) of amplitude a,, moving to the right at constant velocity a,,. The asymptotic behavior for t + + m is derived similarly. The result is the same as (4.10)’ that is, U N

a” (anen)’

+ 1’

t + +co.

(4.1 1)

Therefore, the solution is found to evolve asymptotically as t + co into N localized solitons moving with constant velocities a , , a z , . . . ,aN in the original reference frame as (4.12) It can be seen from (4.12) that no phase shift appears as the result of collisions of solitions, in contrast to collisions that take place between the KdV solitons [see (2.64)]. This is an interesting feature of the system of solutions expressed by (3.9) and (3.36)-(3.38). In Fig. 4.1 (a)-(d) and 4.2 (a)-(d), the u(x, t ) for N = 2, 3 are plotted as a function of x for various values oft.

-50

-40

-30

-20

0

-10

10

20

30

40

50

Position

Fig. 4.1 Plot of u(x, t ) for N = 2, a , = 2.0, a2 = 4.5, xoI = xo2 = 0, and (a) - 10.0, (b) t = -0.05, (c) t = 0,(d) t = 10.0.

100

t =

54 -

3 2 -

I

Position

Fig. 4.2 Plot ofu(x, t)for N = 3,a, = 1.0,a2 = 3.0,a3 = 4.5,xol and (a) t = - 10.0, (b) t = - 1.0, (c) t = O,(d)t = 10.0.

101

= XOZ =

x03

= 0,

102

4

Interaction of the Benjamin-Ono Solitons

We shall now derive the asymptotic form of u(x, t ) when 1x1 + co while keeping t finite. The asymptotic form of fN for this limit can be obtained using expansion (4.1) as

(4.13) Thus, we find

a

i f; u(x, t ) = - - In 2ax fN

=

1 1 c -3 + an

n= 1

(4.14)

0(~-4),

X

which implies that u(x,t) has a long tail, unlike the asymptotic behavior of the KdV N-soliton solution, which decays exponentially. 4.2 Interaction of Two Solitons As shown in Section 4.1, the BO solitons are remarkable in that they cause no phase shift after interactions. However, during the interaction, the processes are very complex. Therefore, it is necessary to find a representation of solitons that brings to light more details of the interaction process. For this purpose, the interaction of two solitons is studied in this section. The motion of two solitons may be characterized by that of the poles x,(t) (n = 1,2) in a complex plane [see (3.9) and (3.10)].To show this, we rewrite the N-solition solution (3.9) with (3.10) in the form N

- 1 x - x,*(t) n=l

Im xn(t) =,= c [ x - Re x,(t)l2 + [Im xn(t)I2‘ N

(4.15)

103

4.2 Interaction of Two Solitons

The term u(x, t ) has 2N poles xl(t), x2(t), . . . ,xN(t), x:(t), x:(t), . . . ,xb(t) in the complex x plane. The trajectories of x,(t) (n = 1,2,. . . ,N ) can be obtained directly from (3.10) and (3.36)-(3.38) by solving an algebraic equation of order N . Note that Re x, gives the center position and (Im x,)- the amplitude of the nth soliton. For the two-soliton case, poles x1(t) and x2(t) are found from (3.10) and (3.32) by solving the algebraic equation of order two (4.16) The solutions of (4.16) are obtained as

(4.19)

1

+ a2

+ 4 (al - a2)4 T2}1'2)1'2 3

(4.20)

(a1a2I2

T = t + xo1 - x02

(4.21)

a1 - a2

In the following discussion, phase constants xol and xO2are assumed to be zero without loss of generality, so that

T = t.

(4.22)

104

Interaction of the Benjamin-Ono Solitons

4

In the limit of t + fco,a(t) and b(t) behave like a(t) = (a,

- a2)t + o(t-'),

b(t) = (a2 - al)/(ala2)

(4.23)

+ O(t-2).

(4.24)

Substituting (4.23) and (4.24) into (4.17) and (4.18), we find, in the same limit, that xl(t) = alt

+ a1 + o(t-'),

(4.25)

x 2 ( t ) = a2 t

+ -i + o(t-').

(4.26)

1

-

a2

The above results may also be derived directly from (4.12) and (4.15) as

1

x - (ant + xOn+ ia;') N

1

-,=1 x - (a,,t + x,,,, - ia;

1

') '

t

+

fco.

(4.27)

If we regard solitons as stable particles located at the poles, we may introduce the time t, at which two solitons collide. It is natural to define it such that the distance between two poles reduces to a minimum. In the present case, the distance I(t) between two poles at time t is found from (4.17) and (4.18) as l(t) = Ixl(t) - x2(t)I = [a2(t)

It follows from (4.19) and (4.22) that l(t) =

a2)4 t 2

+ b2(t)] p a ( ! a i 2 ) 2 b2(t) ~

11.7

2

[z

+ b2(t)]1/2. - a2)2 a1a2

ltl]"2,

(4.28)

(4.29)

by the inequality a

+ b 2 2fi,

a, b 2 0.

(4.30)

105

4.2 Interaction of Two Solitons

-1

L

0

2

8

6

4

10

12

s

Fig. 4.3 Plot of a:D as a function of s ( = a 2 / a , ) . (From reference [114], by permission of The Institute of Physics, England.)

It may be seen that l(t) reduces to a minimum when t = 0. Therefore, it is especially important to investigate the behavior of poles in the situation near t = 0. The nature of the interaction of poles xl(t) and x 2 ( t ) is characterized by the behavior of the function b(t) as seen from (4.17)-(4.19). The behavior of b(t) depends critically on the quantity

(4.3 1)

where s=

a2/al.

(4.32)

It will be shown later that D defined in (4.31) is merely the square of the minimum distance between two poles. Note that a:D decreases monotonically from 00 to - 1 as s increases from 1 to co,as shown in Fig. 4.3. We shall now study the behavior of xl(t) and x 2 ( t ) for small t under the conditions D>O, 1<s<3+2$, (4.33) D=O,

~=3+2$,

(4.34)

D < 0,

s >3

+ 2J2.

(4.35)

106

4 Interaction of the Benjamin-Ono Solitons

The discussion following is concerned only with the situation near t = 0 since we are interested in the details of the process of the collision of solitons. Case A. When D > 0, condition (4.33), the behaviors of a@),b(t), xl(t), and xz(t) for small t are =

+ W)1,

(-Jot/ltI)C1

+ O(t2)I, = -+(Jot/ItI) + i(a, + az)/(a,az)+ O(t), = %Jot/ItI) + i(a1 + az)/(a1a2) + ow.

b(t) = (It I/Jo)Cl Xl(0 XZ(t)

(4.36) (4.37) (4.38) (4.39)

It may be seen from these expressions that Re xl(t) and Re x 2 ( t ) are discontinuous at f = 0. The minimum distance between two poles is given by l(t,)

=

l(0) =

Jo.

(4.40)

Figure 4.4 shows a plot of a two-soliton solution for various values of time, with parameters given by a , = 1.0,

(4.41)

az = 5.6,

(4.42)

xo,

D

= xo2 = =

0,

0.0814,

I(t,) = I(0) = 0.285.

(4.43) (4.44) (4.45)

Figure 4.5 represents the motion of two poles for -0.3 I t I 0.3. In Fig. 4.6, Re x 1 and Re x2 are plotted as functions oft. We can see from these figures that for D > 0 two poles interchange their velocities at the instant of the collision (t = 0), which reflects the discontinuities of Re x, and Re x2 at t = 0 [see (4.38) and (4.39)]. Case B. When D = 0, condition (4.34), a(t) =

-(2t/lt1”2)[1

+ O(lt13/2)],

+ O(lt13’2)], x,(t) = - t / l t l ’ / 2 + i(0.586/a1 + lt1’/2) + O(t), x 2 ( t ) = - t / l t 1 ” 2 + i(0.586/a1- It(’/’) + O(t). b(t) = 21tl’‘z[1

(4.46) (4.47) (4.48) (4.49)

107

4.2 Interaction of Two Solitons

t=

-2.0

-1.6

-1.2

-0.8

-0.4

-0.025

0.0

0.4

0.8

1.2

1.6

2.0

Position

Fig. 4.4 Plot of a two-soliton solution with a l = 1.0, a2 = 5.6, xoI = xo2 = 0 for various values of time t . (From reference [114], by permission of The Institute of Physics, England.)

-al

0.8

P 0

r

0

r

0.6

2

t

.-2

-f

0.4

-2.0

-1.6

-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

1.6

2.0

Real Part of Pole

Fig. 4.5 Plots of the motions of two poles for -0.3 It 4 0.3. The arrows indicate the direction of motion of the poles and the dots show position of the poles at t = 0. The open arrow +- denotes motion of xl(t); the solid arrow --c denotes motions of x 2 ( t ) . (From reference [114], by permission of The Intitute of Physics, England.)

108

4

-2.0

-1.6

-1.2

-0.8

-0.4

Interaction of the Benjamin-Ono Solitons

0.0

0.4

0.8

1.2

1.6

2.0

Real Part of Pole

Fig. 4.6 Plot of Re x , ( t ) (+) and Re x2(t) (-) as functions of time. (From reference [ 1141, by permission of The Institute of Physics, England.)

t = -0.025

-2.0

-1.6

-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

1.6

2.0

Position

Fig. 4.7 Plot is the same as Fig. 4.4 except that a2 = 3 [ 1141, by permission of The Institute of Physics, England.)

+2fi.

(From reference

4.2 Interaction of Two Solitons

109

1.0 r

0.8

-

e L

0.6

-

.-2m

0.4

-

0.2

-

01 0

n r

0

2

-E

0.0

.

1

Fig. 4.10 Plot is the same as Fig. 4.4 except that u2 = 6.0. (From reference [I 141, by permission of The Institute of Physics, England.)

0.3

-

0.2

-

-0.4 -2.0

-1.6

-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

1.6

2.0

Real Part of Pole

Fig. 4.9 Plot is the same as Fig. 4.6 except that a2 = 3 + 2 f i . (From reference

[114], by permission of The Institute of Physics, England.)

110

4 Interaction of the Benjamin-Ono Solitons

Figures 4.7-4.9 show plots corresponding to those of Case A, with parameters given by a , = 1.0, (4.50)

a2 = 3

+ 2fi

xo1 = xo2

D I(?,)

=

=

5.83,

0,

(4.52)

= 0, =

(4.5 1) (4.53)

I(0) = 0.

(4.54)

We can see from (4.48) and (4.49) that two poles unite at t = 0,that is, x,(O) = x2(0).This is marked as a dot in Fig. 4.8. Case C. When D < 0, condition (4.35),

(4.58)

I(?,)

=

I(0) =

J-0.

(4.59)

Figures 4.10-4.12 show plots corresponding to those of Case A, with parameters given by a , = 1.0,

(4.60)

a2 = 6.0,

(4.61)

XO, = xo2 =

D

=

0,

-0.0544,

I(tc) = I(0) = 0.233.

(4.62) (4.63) (4.64)

111

4.2 Interaction of Two Solitons

j:L

A

5.0

t = -0.025

4.0 -.."I

9._ 3.0 -

t=

-0.3

t=

-0. -0.05

4.8

2

Q

2.0 1 .o

0.0 -2.0 -1.6

-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

1.6

2.0

Position

Fig. 4.10 Plot is the same as Fig. 4.4 except that u2 = 6.0. (From reference [I 141, by permission of The Institute of Physics, England.)

al L 0

r

0.8

-

0.6 -

2 t

E 0.4 ._ m

-E

0.2-

0.0L -2.0 -1.6

-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

1.6

1

2.0

Real Part of Pole

Fig. 4.11 Plot is the same as Fig. 4.5 except that u2 = 6.0. (From reference [I 141, by permission of The Institute of Physics, England.)

112

4 Interaction of the Benjamin-Ono Solitons

-0.4

I

-2.0

-1.6

-1.2

-0.8

-0.4

I

0.0

0.4

0.8

1.2

1.6

1

2.0

Real Part of Pole

Fig. 4.12 Plot is the same as Fig. 4.6 except that a2 = 6.0. (From reference [I 141, by permission of The Institute of Physics, England.

From these figures we find that the trajectories of two poles are continuous and have no cusps for all times, a clear difference between Cases A and C. Note also that d Re x,(t)/dt, which represents the velocity of the smaller soliton, becomes zero at two times (from Fig. 4.12 these times are seen to be at t 21 +0.03) and has a minimum value at t = 0 given by mind Re x , ( t ) / d t

=

d Re x l ( t ) / d t l , = o

= (a1

+4

2

- (a, -

~2)~/(2ala2J-o), (4.65)

where we have used (4.57).This minimum value is - 0.54 in the present example. O n the other hand, d Re x,(t)/dt, which represents the velocity of the larger soliton, does not become zero and has a maximum value at t = 0 given by max d Re x 2 ( t ) / d t = d Re ~ ~ ( t ) / d t l , = ~ = (al aJ2 (a1 - U#/(~~,U~J-D),

+

+

(4.66)

113

4.2 Interaction of Two Solitons

where we have used (4.58). In the present example this maximum value is 12.4. Although the processes of the interaction of two poles are very differentfor the three cases A-C, the profiles of the two-soliton solution obtained are similar to one another, that is, they have only one peak at the instant of the collision (see Figs. 4.4, 4.7, and 4.10). The profile of the two-soliton solution at t = 0 is found from (3.9) and (3.32) with xol = xoz = Oas u(x, 0) =

a,az(a,

+ az)x2 + (a1 + az)C(a, + az)/(a1

{ a l a z x z - [(al

- a2)l2

+ a2)/(a1- a z ) ] z } 2+ (al + aZ)’xz’

(4.67)

and the x derivative of u(x, 0) is given by u,(x, 0) = -2(a1

+ a2)x((ala2)3x4+ 2(alaz)2[(al + az)/(al - az)12x2

{3alazC(al + - az)i4 - (a1 + az)zC(al + a z m 1 - %)IZ>) x ( { a l a z x Z- [(al az)/(al - a2)]z>2 (al -

+

+

+ az)zxz})-? (4.68)

Note also that u,,(O, 0) = -2a:[(s

- l)4/(s + 1)3](s2 - 5s

+ 1).

(4.69)

It follows from (4.68) and (4.69) that u(x, 0) has different profiles depending on the value of a parameter s = az/al. For the condition 1 < s < (5

+ *)/2

= 4.79,

(4.70)

the maximum value of u(x, 0) is given by max u(x, 0) = u( +xo, 0) (s - 1)2 s+l

= a, ___ X

S

(-s’

+ 6s - 1)1/2[2~1/2 - (-sZ + 6s - l)’”]’ (4.71)

where xo =--1 1 s + 1 [(-sz

a1Jis-l

+s6s - 1 )‘Iz- l l l ” .

(4.72)

114

4 Interaction of the Benjamin-Ono Solitons

For the condition s > (5

+ J21)/2,

(4.73)

+ 1).

max u(x, 0) = u(0,O) = al(s - 1)’/(s

(4.74)

It is interesting that for 1 < s < 3 + 2$, the positions of maximum values of u(x, 0) become f x , [for 1 < s < (5 f i ) / 2 ] and 0 [for (5 + m ) / 2 < s < 3 + 2J21. These positions do not coincide with the positions corresponding to maximum values of two solitons, which are given by

+

Jo 2

(s - 1)2

Rex,(-())=-=--Re x2(-0)

=

s+l[l-T] a,&s-l

- -. fi

1’2

, (4.75) (4.76)

2

It may be seen from (4.72) and (4.75) that x, < Rex,(-0)

for s > 1.

(4.77)

However, the process of the interaction of two solitons for 1 < s < (5 m ) / 2 is not substantially different from that for 1 < s < 3 + 2$, as was shown in Case A. These circumstances are depicted in Figs. 4.13-4.15, where plots corresponding to those of Case A are shown, with parameters given by

+

a, = 1.0,

(4.78)

a2 = 4.6,

(4.79)

x,, = xo2 = 0,

(4.80)

D = 0.622,

(4.8 1)

l(t,)

=

I(0) = 0.789.

(4.82)

It follows from (4.70)-(4.72), (4.75), and (4.76) that X,

=

0.215,

max u(x, 0) = U( fx,, 0) Re xl(-0)

=

(4.83) =

2.33,

-Re ~ ~ ( - 0=)0.394.

(4.84) (4.85)

115

4.2 Interaction of Two Solitons

4.0 5.0: = -0.05

\

-2.0

-1.6

-1.2

-0.8

t=-0.025

-0.4

0.0

t=o

0.4

0.8

1.2

1.6

2.0

Position

Fig. 4.13 Plot is the same as Fig. 4.4 except that a2 = 4.6. (From reference [ 1141, by permission of The Institute of Physics, England.)

0.0 I -2.0

-1.6

-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

1.6

2.0

Real Part of Pole

Fig. 4.14 Plot is the same as Fig. 4.5 except that a2 = 4.6. (From reference [I 141, by permission of The Institute of Physics, England.)

116

4 Interaction of the Benjamin-Ono Solitons 0.4

0.3 0.2

0.1

-0.4 Real Part of Pole

Fig. 4.15 Plot is the same as Fig. 4.6 except that az = 4.6. (From reference [114], by permission of The Institute of Physics, England.)

The nature of the interaction of two solitons can be divided into two classes, depending on the initial amplitudes of two solitons, as follows: (i) For a, < u2 < (3 + 2 4 ) a , , as time goes from - co to 00, the amplitude of the larger soliton decreases from a2 to a, while the amplitude of the smaller one increases from a , to a 2 . The two solitons interchange their velocities at the instant of collision without passing through each other (see Figs. 4.5, 4.6,4.14, and 4.15). (ii) For a, > (3 + 2fi)a,, the larger soliton first absorbs the smaller one and then emits the smaller one backward. This situation is clear from the fact that the velocity of the smaller soliton becomes negative for some range of time (see Fig. 4.12, where this range is -0.03 It s 0.03). In this case the two solitons pass through each other.

The Benjamin-Ono- Related Equations

In this chapter the BO-related equations are analyzed using the bilinear transformation method developed in Chapter 2. The equations treated here are the higher-order BO equations [19-211, the higherorder KdV equations [22], the finite-depth fluid equation [27] and its higher-order equations [21, 291, and the higher-order modified KdV equations [30]. A systematic method for bilinearizing these equations is developed, and the solutions for these equations are presented. Finally, the Bicklund transformations of the higher-order KdV equations are constructed on the basis of the bilinear transformation method together with the inverse scattering transforms of the same equations [31]. 117

118

The Benjamin-Ono-Related Equations

5

5.1 Higher-Order Benjamin-Ono Equations 5 . I.1 Bilinearization

The Lax hierarchy of the BO equation is given by U, =

-(a/aX)(sr,&)

= -a K j a x ,

n

=

3,4,5, . . . ,

(5.1)

with

K , = 61,/6u,

n

3,4, 5, . . . ,

(5.2) where I , is the nth conserved quantity of the BO equation and 6/6u denotes the functional derivative defined by relation (3.212). As shown in Section 3.2, the BO equation has an infinite number of conserved quantities [see (3.187)-(3.191)] and from these quantities the K , defined in (5.2) are calculated as =

K 3 = Hu, + u’,

K4 =

+ ~ H U U+, $uHu,+ u3,

-uZ,

K , = - H u ~ , - 3UU2, - 2 ( ~ , )+ ~ (Hu,)’ + H u H u ~ ,+ ~ H u ~ u , + 2UHUU, + 2u’Hu, + u4, (5.5)

+ 2Huu3, + 2u,Hu2, + 2uHu3, + 2u2,Hu,) + $[- 13~(~,)’- 1 0 ~ ~ +~ 3u(Hu,)’ 2, + ~Huu,Hu, + 2Hu2Hu2, + 4(Hu,)(Huu,) + 2uHuHu2, + 2HuH(ux)2 + 2HuHuu,,] + 3 U 3 H U X + H U 3 U , + U Z H U U , + uHuZu,)

K , = u4,

- ~4 H u, u2,

+ us,

(5.6)

where

u,, For n

=

=

anu/axn,

n = 0, 1,2, . . . .

(5.7)

3, (5.1) and (5.3) yield U, =

- Hu,,- ~ u u , ,

(5.8)

which is the BO equation. In this expression the coefficient of the nonlinear term has been taken to be two instead of four. For n 2 4, the equations given by (5.1) are called the higher-order BO equations. Let us now define the rank of the polynomial K, as the sum of the number of factors u and the number of a/&, as is consistent with the scaling properties of the BO equation, where the Hilbert

119

5.1 Higher-Order Benjamin-Ono Equations

transform operator H does not affect the rank, since it has zero net power of x. Then the K n defined in (5.2) is a polynomial of rank n - 1. In this section we shall develop a systematic method for bilinearizing the higher-order BO equations and present their N-soliton and N periodic wave solutions. We first introduce the dependent variable transformation u

=

i 8 ln(f'/f)/ax,

(5.9)

with (5.10) (5.1 1)

I m x n > 0,

n = 1,2,..., N,

(5.12)

Imx:,
n = 1,2,..., N',

(5.13)

where xn and x: are complex functions o f t and N and N' are finite or infinite positive integers. Substituting (5.9) into (5.1) and integrating by x yields

(iDtf'.f)/fIf= -Kn,

(5.14)

where the integration constant is assumed to be zero. To transform the nth order equation of (5.1) into the bilinear equation, we impose the following n - 3 subsidiary conditions for u : uTj= -aKj+,/dx,

j

=

1,2,

..., n

- 3,

(5.15)

where z j (j= 1,2,. . . ,n - 3) are independent auxiliary variables. Equation (5.15) is transformed into the form

(iD,,f'.f)/flf= -Kj+,,

j = 1,2,..., n - 3,

(5.16)

by introducing (5.9) into (5.15). Under conditions (5.16), the nth order equation of (5.1) is assumed to be transformed into the bilinear equation P O , 211 n-3

iD,f'.f=a(")D:-'f' .f+

1 p y ) D , , D : - j - 2 f ' . f,

j= 1

(5.17)

120

5 The Benjamin-Ono-Related Equations

where a(”)and fly) (j= 1,2, . . . ,n - 3) are unknown coefficients. The procedure to determine these coefficients is as follows: First, differentiating formula (App. 1.3) exp[(ED,

+ 6 0 , ) f ‘ .f ] = exp

with respect to 6 and setting 6 = 0 gives

=

expC(:(eD,)f’ .f]at cosh

a{

[ a X ]; E

- In -

+ sinh

[

E

- lnf’f

x:

11

.

(5.19)

Substituting (5.9) and a relation

a

f’

Hu=iH--h--= ax f

a

--lnf’f ax

(5.20)

into (5.19) with t = r j , using (5.16), and then equating the tZn-’terms on both sides of (5.19), we obtain n- 1

n- 1

(5.21)

from the 6’” terms we obtain

121

5.1 Higher-Order Benjamin-Ono Equations

where A , is defined by n = 1,2, ...,

A,=(D:f'.f)/flf;

( K ~ += ~amKj+2/aXm, ) ~ ~ m

=

0,i , 2 , ...,

(5.23) (5.24)

and ,,C, is the binomial coefficient ncm

=

n! (n - m)!m!'

(5.25)

To express A,, in terms of u, u,, . . ., identity (5.19) is used. Setting t = x in (5.19), substituting (5.9) and (5.20) into the resultant equation, and then equating the eZn- terms on both sides of (5.19), we obtain the recursion formula for Azn: n- 1

A2n = l/i

C

1C2mAz(n-m)- 1UZm.x

20-

m=O

m=O

and from the 2''terms, the corresponding formula for A z n + : n

A2n+ 1 = 1/i

C m=O

2nC2mAZ(n-m)UZmx

-2

n- 1

-

m=O

Z n c 2 m + 1 ~ 2 ( n 1)1 ~ ~ ( 2 n 1l) + x.

(5.27)

These recursion formulas are solved successively starting with A,

=

1,

(5.28)

A,

=

-iu.

(5.29)

The expressions for A 2 , . . .,A , are given as A2 = -Hu,- u 2 ,

(5.30)

A , = -i(u2, - 3uHu, - u3),

(5.31)

A4 = - H u ~ , - 4UU2,

+ ~(Hu,)' + ~ u ~ H +u ,u4,

A , = -i[u4, - 5uHu3, - 10u2,Hu, - 10uzuzx i5u(~u,)2 1 0 u 3 ~ ~ ,

+

+

+

(5.32) (5.33)

122

5 The Benjamin-Ono-Related Equations

Finally, introducing (5.14), (5.26), and (5.27) into (5.17) we obtain for odd n ( = 2 ~+ 1) s- j- 1

+

-Kzs+l = a(Zs+l)Azs x

j= 1

AZ(s-j-m)-

1(Kzj+Z)Zmx

1

s - j- 1

+ 1

2(s-,3-

m=O

+

1 C z m + 1 A 2 ( s - j- l ) H ( K z j + 2 ) ( 2 m + 1)x

1

s- j- 1

1

z(s- j ) C Z m + 1 A z ( s - j - m ) -

m=O

1 H ( K 2 j + 1)(2m+ 1 ) x

(5.34)

and for even n ( = 2s) -K

2s

- a(ZS)A 2s-

1

+ s2pG’[i j= 1 +

+ +

c

s- j- 1 2(s-j-1)C2mAZ(s-m-j-1)(Kzj+2)2mx

m=O

1

s-j-2

1

m=O

s-1

z(s-j:

1 ) C 2 m + I A z ( s - m - j- 1 ) - I H ( K 2 j + 2 ) ( 2 m +

s-j-1

1~1”j”lI[i

j= 1

1

m=O

z(s-1)-

1CZmAz(s-j-m)-

s - j- 1

1

m=O

2(s-,3-

1)x

1C2m+ lAz(s-m-j-

1 ( K Z j + 1)2mx

1

1 ) H ( K 2 j + 1)(Zm+ 1)x

*

(5.35)

If we substitute (5.26), (5.27), and K , into (5.34) and (5.39, then both sides of (5.34) and (5.35) are represented by polynomials of u, ux, . . . with rank 2s and 2s - 1, respectively. Equating the coefficients of H q Z s - l ) x , . . . ,uZson both sides of (5.34) and those of u ( z s - 2 ) x ,...,u 2 s - 1 on both sides of (5.39, we can derive the simultaneous linear algebraic equations for unknown coefficients a(”) and f l)(j= 1,2, . . . ,n - 3), from which these coefficientsare determined. Thus, we have established a method for bilinearizing the higher-order BO equations.

5.1

123

Higher-Order Benjamin-Ono Equations

We now demonstrate the procedure for bilinearization by explicit examples. We shall take the higher-order BO equations with n = 4,5,6. (i) n = 4 It follows from (5.35) with s = 2 that

+

-K4 = a(4)A3 @;O[iAiK3

+ AoH(K3),l.

(5.36)

Substitution of (5.3), (5.4), (5.29), and (5.1 1) into (5.36) yields ~ 2 ,-

$Huu, - &Hu, - u3 ia(4)- /3'$))uzx + 2/?i4)Huu, + (3ia(4)+ @~))uHu,+ (ia(4)+ f14))u3.

= (-

(5.37)

Equating the coefficients of u z x ,Huu,, uHu,, and u3 on both sides of (5.37), we obtain -ia'4' - 81 (4) = 1,

(5.38)

-3,

(5.39)

3ia(4)+ 8\41 = -$,

(5.40)

28';~)=

ia(4) + 8';') = - 1,

(5.41)

from which a(4)and 8';')are determined as a(") = i/4

(5.42)

B';"

(5.43)

(ii) n = 5 It follows from (5.34) with s

=

=

-2.

2 that

- K 5 = a(5)A4+ /3(:)[iA2K3+ iAO(K3)Zx /?i5'[iA1K4 + AoH(K4),].

+

+ 2A,H(K,),I (5.44)

Substituting (5.3)-(5.9, (5.28)-(5.30), and (5.32) into (5.44) and equating the coefficients of Hu3,, uuzx, . . ., u4 on both sides of (5.44), we have formally 10 equations but essentially three independent equations for three unknown coefficients a(5),fl\5), and 8'25): -a(5) + i/j\5) - 8\5) = 1,

(5.45)

- $fl'25) = $,

(5.46)

2iP:)

38\51 = -2.

(5.47)

124

5

The Benjamin-Ono-Related Equations

These equations have been obtained by comparing the coefficients of Hu3,, (u,)’, and Hu’u,, respectively. Thus, a@),8t5), and pi5)are determined as = 11 27

(5.48)

= -$j,

(5.49)

= -2

(5.50)

(.J5)

pi”

3.

(iii) n = 6 It follows from (5.34) with s = 3 that -K6

+

+

= a ( 6 ) ~ 5/ J \ 6 ) [ i ~ 3 ~ 3 3i~,(~,),,

3AzH(KdX AoH(K3)zxI

+ Bi6’CiA2K4 + iAO(Kd2, + 2A IH(K4LI

+ ps6)[iA1K5 + A 0 H ( K 5 ) , ] .

(5.5 1)

Substituting (5.3)-(5.6), (5.28)-(5.31), and (5.33) into (5.51) and equating the coefficients of u4,, H u x u Z x , .. .,u5 on both sides of (5.51), we have formally 20 equations but essentially four independent equations for four unknown coefficients a(@, pi6)’,and fi‘)’:

at6),

-ia(6)

+ fl$@ = - 1,

- /I1 (6) -

(5.52)

These equations have been obtained by comparing the coefficients of u4,, H u x u z x , u,Hu2,, and HuZHu2,. Thus, a(@,fl‘f), and pi6) are determined as c1(6)

= 91 6b .

(5.56)

Pi6’ = 3%

(5.57)

8\61 =

(5.59)

- 38 -

5.1

Higher-Order Benjamin-Ono Equations

125

5.1.2 Solutions

Once the bilinearization has been performed, the solutions are constructed following the procedure developed in Chapter 2. The N periodic wave solutions of the nth BO equation (n = 4,5,6) may be expressed compactly in the form

a f* 1 k , + i-ln--, ax f N

u= -

m= 1

(5.60) (5.61) (5.62)

with

(5.63) (5.64) (5.65) (5.66)

(5.68) where k,, +):, a,): and 8:) (m = 1,2,. . . ,N ; n = 4,5,6) are arbitrary constants, * denotes complex conjugate, indicates the summation over all possible combinations ofp, = 0, 1, p2 = 0,1,. . . ,pN = 0,1, and l i y ) mmeans the summation over all possible combinations of N elements under the condition 1 < m. The N-soliton solution may be obtained from (5.60)-(5.68) by setting 8:)=n, m = l , 2,..., N , n = 4 , 5 , 6 , (5.69)

126

5 The Benjamin-Ono-Related Equations

and taking the long-wave limit k, + 0 (m = 1,2,. . . , N), keeping finite. The result is expressed as

)c:

a

u = i-ln-, ax

f* f

(5.70)

f = det M,

(5.71)

where M is an N x N matrix whose elements are given by

with c):

for m = I,

(5.72)

for m # I,

(5.73)

and 62) (n = 4, 5 6 ) being arbitraty constants. If we set

(5.74) then the functional form of the solution is the same for all n, the only difference being the velocity (n - 1)(~!3‘“~/2”-~ of the solitons.

5.2 Higher-Order Kortewegde Vries Equations 5.2.1 Bilinearization Consider the higher-order KdV equations

3 , 4 , 5 , . . . , (5.75) where I,,is the nth conserved quantity of the KdV equation. The first few expressions of K, are given as (5.76) K 3 = u l X + 3u2, U, =

-(a/ax)(sz,/su) = -aK,,/ax,

n

=

(5.77) + 10uu2x -t 5(ux)’ + 10u3, = + 14UU4, -t 28u,u3, + 2 1 ( U ~ , ) -k~ 70U2U2, + ~ O U ( U , +) ~ 35u4, (5.78) K6 = us, + 18UU6, -k 54U,U5x + 114U2,u4, + 69(~3,)’ -k 126u2u4, + 4 6 2 ( ~ , ) ~ u +~ , 504uu,u3, + 3 7 8 u ( ~ ~ ,+) ~420u3u2, + 630u2(u,)2 -k 126~’. (5.79)

K4 = u~~

K5

U6,

127

5.2 Higher-Order Korteweg-de Vries Equations

These expressions may be generated from the recursion formula C43,541 aK,, l/ax

=

a 3j a ~x 3 + 4u a K j a x + 2u, K ,

(5.80)

starting with K , = 3. In (5.76)-(5.79), the terms K, are normalized to make the coefficient of the highest derivative term equal to one since numerical factors that multiply K , can be absorbed into the time variable in (5.75). For n = 3, Eq. (5.75) with (5.76) is the KdV equation U, = - ~ 3 , -

~uu,.

(5.81)

The procedure for bilinearizing the higher-order KdV equations is the same as that for the higher-order BO equations developed in Section 5.1. To bilinearize the nth order equation (5.75), impose n - 3 subsidiary conditions for u:

u,, = - d K j + , / a x ,

j = 1, 2, ..., n - 3,

(5.82)

where T~ (j= 1,2, . . . , n - 3) are independent auxiliary variables. Introducing the dependent variable transformation

u = 2 a 2 Infiax2,

(5.83)

into (5.82) and integrating by x yields (DKjD,f.f)/f2 = - K j + z ,

j = 1,2,. . . , n - 3.

(5.84)

Under conditions (5.82) [or (5.84)], the nth order KdV equation may be transformed into the bilinear equation [22] n-3

n = 3,4, 5,. . ., (5.85)

where a(") and fly) (j= 1,2, . . . , n - 3) are unknown coefficients. Consider now an identity of the bilinear operators8 exp(&D,)a.b = [exp(c a/& (~/b)][co~h(~D,)b. b], 4cOsh(eD,)b. b = :[exp(ED,)

+ exp(-cD,)]b.

b.

(5.86)

128

5

The Benjamin-Ono-Related Equations

which is derived from formulas (App. 1.4) and (App. 1.5). Setting a =hi, b = f and expanding the E ~ ' " + ' terms, we obtain 1 1 D:'"+lDrj f .f 2 (2m l)!

-

+

1 = c(2s + 1)!(2m m

s=o

-

(

I"/)ID:("'-')f. 2 ~ ) ! aTja x Z s + l

f ] (5.87)

by using formula (App. 1.1.6). Here we note the relation a2s+2

lnf

aTjaX2s+l

1 8 2 s - 1 au 2 aX2S-1aTj

2

ax2s

'

(5.88)

which is derived from (5.82) and (5.83). Substitution of (5.88) into (5.87) yields

From (5.75) and (5.83) it also follows that

Substituting (5.89) and (5.90) into (5.85), we obtain the desired relation

(5.9 1) which is used to determine a(") and by). To express (0;'"'.f)/f2 in terms of u, u x , . . . ,formula (App. 1.5) is employed together with (5.83) to give cosh(eD,) f .f= exp

(5.92)

5.2 Higher-Order Kortewegde Vries Equations

Comparing the formula

E””

129

terms on both sides of (5.92), we obtain a useful

C(,,,)is over all sets of nonnegative integers sI

where the summation satisfying the condition

C Is, = m. I

(5.94)

The first five expressions for (0:’”’.f)/f’ are given in Appendix I [see (App. IS.I)-(App. 1.5.5)]. If we define the rank of the polynomial ua0u: . . . up; as I

that is, by the sum of the number of factors u and half the number of d/ax, then both sides of (5.91) are represented by the polynomials of u, u,, . . . with rank n - 1. Equating the coefficients of u ~ - ~ ( u , ) .’ ., . , u ~ ( , , - ~on ) , both sides of (5.91), we can derive simultaneous linear algebraic equations for unknown coefficientsa(”)and /?y)(j= 1,2,. . .,n - 3). This procedure is now performed explicitly in the case of the higherorder KdV equations with n = 4,5,6. (i) n = 4 In this case (5.91) becomes

Substituting (5.76), (5.77), (App. 1.5.1), and (App. 1.5.3) into (5.95) yields u4,

+ iouuz, +

+ 102 + 15u3) + /?\4)[3(uz,+ 3u’)u + u4, + 6uu2, + 6(u,)’] = ( - a(4) + B\4))u4, + (- 1 5 ~ r ‘ ~ +) 9/3\4))uu2x + 6/?\4)(u,)z + (- 15a(4)+ 9@’;0)u3. +

= - Q ( ~ ) ( U ~ , 15uu,

(5.96)

130

5 The Benjamin-Ono-Related Equations

Comparing the coefficients of u4,, uu2,, (u,)’, and u3 on both sides of (5.96), we obtain -a(4) + pi4) = 13 (5.97)

- 15d4’ + 9pi4’ = 10, 6pi4’ = 5,

(5.98) (5.99)

from which d4)and pi4)are determined to be 4(4)

= -16 9

pp’ = 2.

(5.100)

(5.101)

(ii) n = 5 In this case (5.91) becomes

(5.102) Substituting (5.76)-(5.78), (App. 1.5.1), (App. 1.5.2), and (App. 1.5.4) we obtain, by straightforward algebra, u6x

+ 14UU4, + 28u,u3, + 21(u2,)’ + 70U2U2x+ ~OU(U,)’+ 35u4 = ( - d 5 )+ pi5)+ pi5’)u6, + (-28d5’ + 16pi” + 13/3i5))uu4, + (24fl:) + 30~~5))u,u3, + ( -35d5) + 23p\” + 20p$5))(~2,)2 + (- 2 1 0 8 ) + 908i5)+ 6O/3‘25))u2u2,+ (60p\” + 75/3‘25))u(u,)’ + (- 1 0 5 ~ (+~458i5) ) + 30pi5))u4. (5.103)

Equating the coefficients of u6,, uu4,, . . ., u4 on both sides of (5.103), we obtain linear algebraic equations for a(’), pi5),and pi5’:

-fp + p y + pi3 = 1, -2801‘~)+ 168‘:)

+ 13pi5)= 14,

+ 30/$” = 28, -35a(” + 23pi5)+ 2O8i5)= 21, -210d5) + 90p\’) + 60/?‘,5)= 70, 608\” + 758i5)= 70, - 105d5)+ 45/3i5)+ 30/3i5) = 35. 248‘:’

(5.104) (5.105) (5.106) (5.107) (5.108) (5.109) (5.1 10)

5.2 Higher-Order Kortewegde Vries Equations

131

It may easily be seen that only two of seven equations are independent. Thus, the unknown coefficients d5),fit5),and Pi5)are determined as a(5’

= 51 - sc1, 1

(5.111) (5.1 12)

fi‘25’

=

c1,

(5.1 13)

where c1 is an arbitrary constant. If we do not introduce the auxiliary variable z2, then this constant can be assumed to be zero. (iii) n = 6 In this case (5.91) becomes

(5.1 14) Substituting (5.76)-(5.79) and (App. 1.5.1)-(App. 1.5.5) and equating the coefficients of u S x , uu6x, u x u 5 x , U Z ~ U (u3J’, ~ ~ , U ’ U ~ ~ , U U ~ U u(uZx)’, ~ ~ , u 3 u Z x~, ’ ( u , ) ’ , and u5 on both sides of (5.1 14), we have the 12 equations:

+ fit6)+ fii6)+ Pi6) + fib6) = 1, -45d6’ + 27fi‘p’ + 20fii6’ + 17fib6’= 18, 36fi‘p’ + 50fi16)+ 56fib6’ = 54, -210d6’ + 132fi‘p’ + l15fii6’ + 112fiS6’= 114, 60fi‘p’ + 70/3i6’ + 70fib6)= 69, - 6 3 0 ~ ~ ‘+~ 252fi‘p) ) + 145fii6)+ 112fi6)= 126, 2lOfiCp) + 385/?i6’ + 490fi6’ = 462, 504fit6’ + 540fii6’ + 504fib6’ = 504, - 1575a(@+ 693fi‘p)+ 430fii6) + 343fib6)= 378, + lO50fii6)+ 500/3‘,6) + 350fi6) = 420, -3150~r‘~) 630fi‘p’ + 675/3i6’ + 630fi6’ = 630, -945~4~’ + 315fi(i6’ + 150fii6’+ lO5fib6’= 126. -a(6)

(5.115) (5.116) (5.117) (5.118) (5.119) (5.120) (5.121) (5.122) (5.123) (5.124) (5.125) (5.126)

132

5 The Benjamin-Ono-Related Equations

It may be seen that only three of these equations are independent. Thus, and pi6)as we can determine a(6),pi6),,):3/' (5.127) (5.128) (5.129) (5.130) where c2 is an arbitrary constant. If we do not introduce the auxiliary variable z3,then c2 can be assumed to be zero. 5.2.2 Solutions

The N-soliton solutions of the higher-order KdV equations are constructed following the same procedure developed in Chapter 2 for the KdV equation itself(see Section 2.2), and these may be expressed in a compact form as u

=

2 a2

(5.131)

Infiax2,

with

1 = 1,2,..., N, n = 4,5,6 ,..., (5.133)

exp A E = (al - U , , , ) ~ / ( U ,

+ a,)',

I , m = l , 2,..., N, n = 4 , 5 , 6 ,..., (5.134)

where al and Sl")(I = 1,2,. . . ,N, n = 4,5,6,. . .) are arbitrary constants. If we set

Sl"" = Sl"' +

n-3

1 a:jzj

j= 1

(5.135)

in (5.133), then the structure of the N-soliton solution (5.131) is the same for all n except the velocity a:("-') of solitons. The situation is similar to that of the higher-order BO equations; this is an interesting property of the Lax hierarchy of nonlinear evolution equations.

133

5.3 The Finite-Depth Fluid Equation

5.3 The Finite-Depth Fluid Equation and Its Higher-Order Equations 5.3.1 Bilinearization

Consider the finite-depth fluid equation [23-261 u,

with Gu(x, t ) =

P

+ 2uu, + Gu,,

Jyrn

I

{coth[$

(x' - x)] - sgn(x' - x) ~ ( x 't, ) dx',

where sgn(x' - x)

(5.136)

= 0,

=

il 0

-1

for x < x', for x' = x, for x > x',

(5.137) (5.138a) (5.138b) (5.138~)

and 1-' is a positive parameter characterizing the depth of fluid. Physically, Eq. (5.136) describes long waves in a stratified fluid of finite depth. Mathematically, it reduces to the BO equation u,

+ 2uu, + Hu,, = 0

(5.139)

in the deep-water limit 1 + 0 and to the KdV equation 242:

+

UUf

+ +fff= 0

(5.140)

in the shallow-water limit 1 + co,where new variables 1: and 2 defined by (5.141) (5.142) have been introduced in (5.140). Therefore, the finite-depth fluid equation may possess characteristics common to both the BO and KdV equations. First, we shall bilinearize (5.136). Let us evaluate the complex integral

134

5 The Benjamin-Ono-Related Equations

2i h

"

0

-R

c

R

Fig. 5.1 The integral path C. See text for discussion and definition of variables.

along the integral path C shown in Fig. 5.1. It is assumed that the complex functionf(z) is such thatf(z - i/l) has no zero inside C. It follows from the well-known Cauchy's residue theorem that I is modified in the limit of R + co and r + 0 to P

J:a

p x ' [ia

coth

-

x)]

ax

= - "lnj(x

ax

-i

)f(x

]

In f ( x ' - i/A) dx' f ( x ' + i/l)

i)

+ + c,

(5.144)

where constant c is determined by the behavior off(z) at infinity [65]. We now introduce into (5.136) the dependent variable transformation (5.145)

where f + ( x , t ) = f(x - i/A70,

(5.146)

f-(x, t) = f ( x

(5.147)

+ i/A, t).

Then using the relation Gu, = -

a2

ax

lnf+ f-

+ lu,

(5.148)

135

5.3 The Finite-Depth Fluid Equation

which stems from (5.144), and a formula m

J-

m

sgn(x‘ - x)u,.,. dx’ = -2

S(x’ - x)u,. dx‘ = -2u,,

(5.149)

m

we obtain the bilinear equation [27, 651 (iD,

+ iADx - Df)f+ .f- = 0,

(5.150)

where the integration constant has been assumed to be zero. Note that, by employing formula (App. 1.2), (5.150) can be rewritten in the form (iD, + iAD, - Df) exp( - iA- ‘Dx)f-f

=

0.

(5.151)

For real u satisfying the boundary condition u + 0 as 1x1 + co,fmay be written in the form [27] f(x, t)

n N

=

n= 1

(1

+ exp A[A(Im x,)x

0 < AImx, < TC,

n

=

-

Re x,]},

1,2,..., N ,

(5.152) (5.153)

f + =f?,

(5.154)

where x, (n = 1,2, . . . , N) are complex functions of t. The function f(x - i/A, t) has no zero inside C because of condition (5.153). It is now straightforward to show (5.148) directly by using (5.145), (5.152), and a formula ~ 0 t h [ ( d / 2 ) ( ~-’ x)] dx’ = cosh(Ayx’) + cos(y)

-

’

2A- csc(y) sinh(Ayx) cosh(Ayx) cos(y) ’

+

0 < y < a. (5.155)

The reduction of the bilinear equation (5.150) or (5.151) to the bilinearized BO and KdV equations is done as follows [27]: (i) The BO limit (A + 0) We introduce the quantities (5.156)

Re x, = a Re f,, Im x, = (n/1)(l

-

1Im 2,).

(5.1 57)

136

5 The Benjamin-Ono-Related Equations

In the leading order of l (5.152) becomes N

f- N (- nA)N fl (x n=

- 2,)

1

= (-

(5.158)

nA)Nf

and (5.150) reduces to (iD, - Df)f*

.f= 0,

(5.159)

which is the bilinear form (3.17) of the BO equation. We note here that the condition ImX, > 0, n = 1 , 2,..., N , (5.160) which is an assumption imposed onfin deriving (3.17), holds because of (5.153) and (5.157). (ii) The KdV limit (A -+ 00) We now introduce the quantities Rex, = A-’ Re X,,

(5.161)

Im x, = 1-3/2 Im 2,

(5.162)

together with (5.141) and (5.142). Then (5.152) becomes N

f ( x , t) =

fl [l + exp(X Im 2, - Re X,)]

n= 1

and (5.151) reduces to D,(Df

=f

+ f D , ” ) f . f = 0.

(5.163)

(5.164)

In deriving (5.164), we have used the expansion

(5.165) and formula (App. 1.1.3). Equation (5.164) is the bilinearized KdV equation. In discussing the periodic wave solution of (5.136), it is appropriate to introduce the dependent variable transformation

(5.166) instead of (5.145). Then (5.136) is transformed into the bilinear equation [iD,

+ i(A + 2u0)D,

-

Df

+ c ] f + .f- = 0,

(5.167)

137

5.3 The Finite-Depth Fluid Equation

where uo is a constant and c an integration constant depending generally on time. Furthermore, (5.167) can be rewritten in the form {i[D,

+ (A + 2uo)D,]

sinh(iA- ID,)

+ (02 - c) cosh(iA-'D,))f.f=

0

(5.168) using formulas (App. 1.2) and (App. 1.1.2). This expression will be used in Section 5.3.3.

5.3.2 Soliton Solution Now we shall seek the soliton solution of (5.136) that is real and finite for all x and t and satisfies the boundary condition u + 0 as 1x1 -+ 00. In. this case it is convenient to employ the bilinearized equation (5.150). For the one-soliton case, it may be confirmed by direct substitution that f- =f*+= 1

+ f+@+iy

(5.169)

satisfies (5.150). Here

8=x a

=

- at -

A(l

-y

6,

(5.170)

cot y),

0 < y < n,

(5.171)

where y and 6 are constants and a is the soliton velocity. Transforming to the original variables by means of (5.145), we find U =

Ay sin y cosh[Ay(x - at - S)]

+ cos 7'

(5.172)

The BO one-soliton solution is obtained from (5.172) by introducing a new constant V through the relation

y

=

n(l

-

(5.173)

A/V)

and taking the deep-water limit A -+ 0 to yield u=

2v VZ(X - Vt - 6)Z

+ 1'

(5.174)

138

5 The Benjamin-Ono-Related Equations

where we have used the expansions for small A: sin y

=

sin(nA/V) N nA/V,

cos y

=

-cos(n1/V)

(5.175)

= - 1 + &nA/V)2,

+ n(l

-

1/V) cot(nA/V)]

cosh[Ay(x - at - S ) ]

1:

1

a = A[1

‘v

(5.176) (5.177)

V,

+ ~ ( ~ T A ) ~-( Vx t - ~ 5 ) ~ . (5.178)

On the other hand, the KdV one-soliton solution follows from (5.172) by introducing new independent variables t and f by (5.141) and (5.142), new constants V and 6 by the relations = 1-1/2v,

(5.179)

6 = pq,

(5.180)

y

and then taking the shallow-water limit A u

=

V2/{cosh[V(f

+

co to yield

-

(f - V 3 2 - b)], with the use of the expansions for large I : = --sech2[: V2

sin y

=

sin(1-

cos y

=

cos(A- 1’2V)

a

=

A[1 - A-’I2V cot(1- 1/2V)] N v2/3.

V,

(5.182)

1 - 7/2/21,

(5.183)

V ) N 15

(5.181)

(5.184)

The N-soliton solution is obtained similarly using the technique developed in Chapter 2, and it may be expressed in the form [27]

j=

r=0,1

with

0, = x

- ant -

a, = A(l 8 1 -

“

1 exp 1 pn(1ynen + n=l

S,,

- y, cot y,),

= (ar - a d 2 (a1 - a d 2

Qn)

+

(N)

1

1 plprnnlrn

I
n = l , 2 ,..., N ,

3

(5.185)

(5.186)

0 < y,, < n, n = 1, 2 , . . . , N , (5.187)

+ L2(7l + + Yrn)”

where y, and 6, (n = 1, 2, . . . ,N ) are real constants.

(5.188)

139

5.3 The Finite-Depth Fluid Equation

The expressions of the BO and KdV N-soliton solutions are derived from (5.185) by taking the limits A + 0 and A + m, respectively. (i) The BO limit (A + 0) Introduce constants V , by n = 1 , 2,..., N. (5.189) n(l - A/V,), Following the procedure developed for the periodic wave solution of the BO equation in Section 3.1, it may be shown, in the limit of A + 0, that

yn

=

f N (- niA)Ndet M, with the aid of the expansion for small II =

eAIm

1 - 4(7rA)’/(& - V,)’

(5.190)

+ O(A4),

(5.191)

where M is an N x N matrix whose elements are given by Mnm =

{

i(x - V,t - 6,)

+ Vn-’

for n

=

rn,

(5.192)

for n # rn. (5.193) Y(V, - Vm) Except for the nonessential numerical factor ( - niA)N,expression (5.190) corresponds to the BO N-soliton solution (3.36)-(3.38). (ii) The KdV limit (A + co) Introduce constants V, and 8, by =

A-

1/2

V,, 6, = A- 1/28,, fl

n = l , 2 ,..., N,

(5.194)

n = l , 2 ,..., N

(5.195)

and new variables 1 and f by (5.141) and (5.142), respectively. Then it follows from (5.185)-(5.188) that

with

A”,,

=

(V,

-

V,)’/(V,

+ V,)’,

I, rn

=

1,2, . . . , N,

(5.197)

which is essentially the same as the N-soliton solution of the KdV equation (2.33) and (2.34). Thus, we have demonstrated that the present N-soliton solution (5.185) with (5.186)-(5.188) reduces to both the BO and the KdV N-soliton solutions in appropriate limits.

140

5

The Benjamin-Ono-Related Equations

We shall now investigate the asymptotic form of the N-soliton solution of the finite-depth fluid equation for large values of time. For this purpose, we order the parameters as 71

(5.198)

> Y Z > ' * * > YN,

so that in view of (5.187) aj are ordered as a l > a2 > ... > aN.

(5.199)

Transforming to the reference frame which moves with velocity a, (n = 1,2,. . .,N), we find, in the limit o f t + - 00, U N

AYn

cosh{Ay,,[x - ant - 6,

In the limit of t

+

sin Y n

+ (Ay,,)-

c,,,Z1l A,,,,,]}

+ cos y,,' (5.200)

+ 00 we have

(5.201) Therefore, the present solution is found to evolve asymptotically as t + f00 into N localized solitons with constant velocities al, a 2 , .. . ,aN in the original reference frame. It may be seen from (5.200) and (5.201) that, as t -+ + co, the trajectory of the nth soliton is shifted by the quantity A, n- 1

A,

=

5

1A,,,,, - m = n +

(m= 1

A,,,,)/Ay,,,

1

n

=

1 , 2 , . . . , N, (5.202)

relative to the trajectory as t -+ - 00. In the BO limit (A -+ 0), (5.202) reduces, by using (5.189) and (5.191), to

=O,

n = l , 2 ,..., N.

(5.203)

Thus, no phase shift results from the collision of solitons, which corresponds to the asymptotic expression (4.12).

141

5.3 The Finite-Depth Fluid Equation

In the KdV limit ( A + oo), from (5.188), (5.194), and (5.202) we obtain lim

A+ m

An = lim A1”A,, d+m

which corresponds perfectly to formula (2.64) for the KdV equation.

5.3.3 Periodic Wave Solution In this section we shall present the periodic wave solution of the finite-depth fluid equation (5.136) by employing an exact method developed in Section 2.2 [28]. It is appropriate in this case to use the bilinear equation given by (5.168), that is,

with

+

F(D,, D,) = i[D, (A + 2u,)D,] sinh(iA- ‘D,) (0: - C) cosh(iA- ‘0,).

+

(5.206)

From the procedure developed in Section 2.2, the expression

f = 9(q; z)

1 exp(nq + n2z), 00

=

z < 0,

(5.207)

n=-m

q

=

i(kx

+ ot + q o )

(5.208)

is an exact one-periodic wave solution of (5.205), provided that the conditions

F(rn) = =

m

1 ~ [ ( 2 n- rn)ik, (2n - rn)io]exp{[n2 + (n - r n ) ’ ] ~ } n=-m 0,

rn = 0, 1,

(5.209)

142

5

The Benjamin-Ono-Related Equations

are satisfied. Here k, o,and qo are wave number, frequency, and phase constant, respectively, and T is a parameter related to the amplitude of the original wave. Introducing the notations A,

= Ao(kl- ',T) =

A,

= A,(kl-',

c cosh(2nklm

n = -a

') exp(2n2z),

03

1 coshC(2n - l)kl-']

T) =

n=-m

exp{[n'

(5.210)

+ (n - l)']~}, (5.21 1)

the equations given by (5.209) may be rewritten as

F(0) = F(1) =

+ + (2Uo + L)]kAo,k + C A o = 0 + [0 + (2Uo + l)]kAl,k C A I = 0,

-k212Ao,kk

[O

-k21'A,,kk

where Aj,k Aj,kk

= aAj/ak, =

a2Aj/ak2,

(5.212) (5.213)

j=O,1

(5.2 14)

j = 0, 1.

(5.215)

It follows from (5.212) and (5.213) that AO,kkAl

o = k'l

a

-

AOAl,kk

-

AOAl,k

-

+

( 2 ~ 0 l)k

DkAo * A1 - ( 2 ~ 0+ L)k, (5.216) ak where Dk denotes a bilinear operator with a variable k. Using (5.210) and (5.21 l), we obtain the series expansion of DkAo.A , as =

DkA,. A , =

k'l

- In

W

1-

n,n'=

w

{Dk cosh(2nkl- '). coshC(2n' - 1)kl-J)

x exp{[2n2

+ n" + (n' - l)']~}

' I{(2n + 2n' - 1) sinhC(2n - 2n' 2fln,n,=-m

--

+ l)kl-']

+ (2n - 2n' + 1) sinhC(2n + 2n' - 1)kl- '1) x exp{[2n2 + nr2 + (n' - l)']~}

=

l - ' {-2 sinh(kl-')e' + 2[3 sinh(kl-') + sinh(3kl-')]e3' - 6 sinh(3kl-')e5' + . . .}.

(5.2 17)

143

5.3 The Finite-Depth Fluid Equation

Substituting (5.217) into (5.216) yields o = k2 -

- 2 cosh(kA- ')er

+ 6[cosh(kA- ') + cosh(3kA- 1)]e3r

-2

+ 2[3 sinh(kA- ') + sinh(3kA- ')]e3r

sinh(kA- ')er

- 18 cosh(3kA- 1)e5r+ . . . -6

+

( 2 ~ 0 A)k.

+

sinh(3kA- ')e5r . . . (5.218)

Expression (5.207), with (5.208) and (5.218), constitutes the oneperiodic wave solution of (5.205). Now consider the two limiting cases of A + 0 and A + co. (i) The BO limit (A + 0) We now introduce a new amplitude parameter by z = Z' - kA-', T' < 0.

(5.219)

Substituting (5.219) into (5.207) and (5.218), we find, in the limit of

A -b 0,

limfT

=

1-0

limf(x & iA-', t ) = 1 + exp(Tq

1-0

+ r'),

(5.220)

(5.221) which, in (5.166), produce the one-periodic wave solution (3.57) of the BO equation written in the original variable. (ii) The KdV limit (A + 00) Rescaling wave number and frequency by

k

(5.222)

=

(5.223) and space and time coordinates by (5.142) and (5.141), respectively, and substituting these into (5.207) and (5.218), we find in the limit of A + co

T= lim f = C 03

1-m

n=-m

exp(nq + A),

+ 631 + qo), 1 1 - 30q2 + 81q4 + ... k3, o = -2uoR + 3 1 - 6q2 + 9q4 + . . q=

(5.224) (5.225)

i(k2

q

= el,

(5.226)

144

5 The Benjamin-Ono-Related Equations

Nonperiodic

Periodic

h+O

Nonperiodic Finite-Depth Fluid

h

h+O

L--m

*

Periodic Finite-Depth Fluid

+m

Nonperiodic KdV

L-m 4

Periodic KdV

Fig. 5.2 Interrelationshipamong six types of solutions. Here L denotes period and I the (depth of fluid)-'.

which is essentially the one-periodic wave solution of the KdV equation presented in Section 2.2. This argument may be extended to the twoperiodic wave solution, which is detailed in [28]. The interrelationship among solutions of the KdV, the finite-depth fluid, and the BO equations, for both periodic and nonperiodic cases, is schematically represented in Fig. 5.2. 5.3.4 Higher-Order Equations

The Lax hierarchy of the finite-depth fluid equation (5.136) is expressed as

145

5.3 The Finite-Depth Fluid Equation

where I, is the nth conserved quantity of the finite-depth fluid equation, the first few of which are given as [115] m

(5.228)

dx,

I,

=

I,

=

f_,tu2 dx,

I3

=

[ (h3+ juGu,)dx,

m

(5.229)

m

J-, [&4“ + J-, {bU”+ m

I4 =

m

=

(5.230)

J-m

&’Gu,

+ &u,)’ + $(Gu,)~+ &Gu,]

+’GU,

- &u,Gu2

dx, (5.231)

+ ~ u ( u , )+~&(Gu,)’

+ ~ u ~ G ~-u ~, u, G u , , + ~ u G ~ u , , , + 1[&2Gu, + &(u,)’ + ~ G U , ) ~+]~L2uGu,}dx.

(5.232)

For n = 3, (5.227) and (5.230) yield the finite-depth fluid equation (5.136). For n 2 4, Eqs. (5.227) are called higher-order finite-depth fluid equations. For n = 4, we find from (5.227) and (5.231) the first higher-order equation as

In this section we shall outline a method for bilinearizing (5.227). As an example, the first higher-order equation will be bilinearized and its N-soliton solution will be presented. The procedure for the bilinearization is the same as that for the higherorder BO and KdVequations. First, impose n - 3 subsidiary conditions for u : utj = -aKj+,/ax,

j

=

1,2, ..., n - 3,

(5.234)

with z j (j= 1, 2, . . . , n - 3) being independent auxiliary variables. Equations (5.234) may be transformed into the form (iD,,f+ .f-)/f+

f- = - K j + , ,

j = 1 , 2 , . . ., n - 3, (5.235)

146

5 The Benjamin-Ono-Related Equations

by the dependent variable transformation (5.145). Under conditions (5.235), the nth-order equation of (5.227) may be bilinearized as iD, f +.f- =

n- 2

1 ay)LjD;-j-'f+

j=O n-3

.f-

n-i-3

(5.236) where ay), By), and $) are unknown constants. The procedure to determine these unknown constants, using (5.19) and (5.148), is the same as that for the higher-order BO equations. Instead of treating the general case, we shall illustrate the procedure in the case of n = 4. From the E term of (5.19) with f ' = f +,f = f- , and t = 71, we have

a2 D,f+.fa f+ a7, ax l n f + f - + f+f- ar, In -.f-

D,,Dxf+ .f- =-

f+f-

-

(5.237)

Substituting (5.148), (5.234), and (5.235) into (5.237) yields

f+f-

'f-

= -Cur, = u3

+ 1 /:2, dx' + uK3= GK3,, - AK3 + UK,

+ G2u,, + 2Guu, + uGu, - Lu2 - LGu,.

It follows from the E', E', and and t = x, that

E'

(5.238)

terms of (5.19), with f ' = f + ,f = f - ,

(5.239)

off+.f- a In f + + a2 In f +f- D,f+ .f- f+ff +f- ax f- ax2 ~

=x :( -1n-

;)2

+-lnf+ 2: :

(5.240)

f-,

Ef+.f- = -In a3 f++ 2 D,f+ .f- a2 lnf+ ff+f-

ax3 fDEf+*f-

+

a3

f+

f+f-

a

=-ln-+3-ln---lnf+ ax3 f- ax

f + f - ax2 a f+ - In ax ff + a2 ff- ax2

+ (5.241)

147

5.3 The Finite-Depth Fluid Equation

Substituting (5.145) and (5.148) into these expressions, we obtain

D,f+*f-- -iu, f+foff+.f- = -Gu, + LU - u’, f+fD:f+ .f- = -i(u,, - 3uGu, + 3Lu2 - u3). f+fThe expressions for 0: f +. f-/f+f- ( n (5.29)-(5.31) by the formal replacement

=

(5.242) (5.243) (5.244)

1, 2, 3) are obtained from

Hu,+ Gu, - Lu.

(5.245)

Noting the relation

i &f+.f- = - K , ,

n = 3 , 4,...,

f+f-

(5.246)

which is derived from (5.227) and (5.145), Eq. (5.236) with n becomes -K

- ~ ( 4E ) f +.f+ Lu‘j4)

4 -

0

+

f+fB‘j4) Dz,Dxf+*ff+ f-

f+f-

=

4

f+f(5.247)

Substituting (5.238), (5.242)-( 5.244), and K4 = u3

+ SUGU,+ SGUU,- &, + :G2~,, + ~ A G u , (5.248)

into (5.247), we have -(u3

+ ~ u G u +, $Guu, =

- ;u,, + ~ G ’ u , , + $LGu,) - Gu, - iab4)(~,,- ~ u G u ,+ 311.4’ - u3) + + L’a$4’( - iu)

+ B‘j4)(u3+ G’u,, + 2Guu, + uGu, - Lu’

=

- u’)

AGu,) 2B‘j4)Guu, - iab4)uxx -

+ + + + + + + A( - 3iab4) - a‘;‘) - j?’f))u’ + L’(U\~’ - iui4))u.

(iab4) j?’t))u3 (3iub4’ /?‘j4))uGu, /?(14)GZ~,, A( -a\4) - /3\4))G~,

+ AU

(5.249)

148

5 The Benjamin-Ono-Related Equations

Comparing the coefficients of u3, uGu,,. . . ,u on both sides of (5.249), we obtain the linear algebraic equations for a?) (j= 0, 1,2) and /?';I as iab4) + B(t)= - 1,

(5.250)

- 2, 3

(5.251)

-3,

(5.252)

3i44) + /?I") =

28';" = -ia'4'0

=1 4,

(5.253)

B';"

=

fit)

= -34 9

(5.255)

= 0,

(5.256)

1 - i.44) = 0 a(4)

(5.257)

-

-3ia(4) 0 - a1 (4) -

-$,

(5.254)

from which a?) and fl\4)are determined to be

(5.258) (5.259) (5.260) (5.261) Thus, the first higher-order finite-depth fluid equation has been bilinearized as i D , f + . f - = ($ill:

+ SAD:

- iid2D,

- $D,,D,)f+

- f - (5.262)

with a subsidiary condition

iDrlf+ . f - = (0:- idD,)f+ o f - . Using (5.263), (5.262) can be rewritten as 1291 i D , f + . f - = [$D: - $D,,(D, - 2id)]f+ . f - .

(5.263) (5.264)

The higher-order equations for n 2 5 can be bilinearized similarly, but we shall not discuss them here. Instead, we shall show that the nthorder equation (5.236) reduces to the nth-order BO and KdV equations. (i) The BO limit (d + 0) In the deep-water limit 1 + 0, (5.236) reduces to

149

5.3 The Finite-Depth Fluid Equation

where f is defined by (5.158). Obviously, (5.265) corresponds to the bilinear form of the nth order BO equation (5.17). (ii) The KdV limit (A + co) In the shallow-water limit A + co, it is appropriate to introduce scalings (5.141) and (5.142) together with

tj = A ' / 2 ~ j ,

j = 1 , 2 , . . . , n - 3.

(5.266)

It follows by noting formulas (App. 1.1.3) and (App. 1.2) that

iDtf+ .f- = iD, exp( - iA- 'Dx)f.f = iA"2Di exp( - iA- 'l2Di)f.f (5.267)

n-3

n-i-3

(5.269) Substituting (5.267)-(5.269) and comparing the Ao term on both sides of (5.236), we obtain, in the limit of A + 00,

(5.270)

150

5 The Benjamin-Ono-Related Equations

If we set (5.271) n-j-3

by’ = By’ 1

s=o

( - i)” - j +

(n -j

-

+ s - l)!’

j = 1 , 2 , . . ., n

- 3, (5.272)

Eq. (5.270) has the same form as the bilinear form of the nth-order KdV equation (5.85). Once the bilinear equation is obtained, the solutions are constructed following the procedure developed in Chapter 2. For the first higher-order finite-depth fluid equation in the bilinear form, (5.262) with (5.263), the N-soliton solution may be expressed in the form [29]

f = p =1o , 1 ~ with

“

X PC Pn(Aynen n=l

9,=x-lin-anzl -hn,

an = A(1 - yncot yn),

-

an = ban 3 2

+ $an - &AyJ2,

1

(N)

+ i y n ) + 11 PIpLmAlm <m

7

n = 1,2,..., N, 0 < y n < n, n

=

(5.273)

(5.274)

1, 2, ..., N,

(5.275) (5.276) (5.277)

where y n and 6, are real constants. It may be shown that (5.273) reduces to the N-soliton solution of the first higher-order BO equation in the deep-water limit (A -, co) and to that of the first higher-order KdV equation in the shallow-water limit (l -, 0) [29].

5.4 Higher-Order Modified Korteweg-de Vries Equations 5.4.1

Bilinearization

The Lax hierarchy of the modified KdV equation is given by U, =

-(d/ax)(61,,/Js~)

= -13K,/dx,

n

=

3,4,. . . . , (5.278)

where I , is the nth conserved quantity of the modified KdV equation.

151

5.4 Higher-Order Modified Korteweg-de Vries Equations

The first few K, derived from the functional derivative of I, are

(5.279) + 2u3, (5.280) K4 = u4, + lOu(~,)~ + 1Ou2u2, + 6u5, K 5 = u6x 4- 14U2U4x+ 5 6 ~ ~ -I~ 42u(u2,)2 ~ 3 , + ~O(U,)~U~, (5.28 1) + 70u4u2, + 1 4 0 ~ ~ ( u +, ) ~20u7, ~ 2~ 258, ~ ~ 2 ~ ~210(U,)2U4, 4, K6 = Us, f 18U2U6, + 1 0 8 ~ ~ + + 1 2 6 ~ ~ +~ 2138~(~3,)’ , + 7 5 6 ~ ~ ~ +2 ~1 ~0 30, 8 ~ ~ + 182(u2J3 + 7 5 6 ~ ~ ( u ~+, 3) 1~ 0 8 ~ ~ ( u , ) ~+u ~42ou6u2, , (5.282) + 798u(uJ4 + 1 2 6 0 ~ ~ ( u ,+) ~70u9.

K 3 = u2,

For n = 3, (5.278) and (5.279) yield the modified KdV equation U, = - ~ 3 , -

6~~~4,.

(5.283)

For n 2 4, (5.278) gives the higher-order versions of the modified KdV equation. To bilinearize (5.283), we introduce the dependent variable transformat ion

.a ax

u = z-ln-.

f’

f

(5.284)

It may then be seen, by noting formulas (App. 1.3.1)-(App. I.3.3), that the coupled bilinear equations [55]

D , f ‘ * f = -D:f‘.f,

(5.285)

D,”f’ .f = 0

(5.286)

are equivalent to (5.283). For the purpose of bilinearizing the nth-order modified KdV equation, we first impose n - 3 subsidiary conditions for u : urj = - a K j + , / d x ,

j = 1,2,..., n - 3,

(5.287)

where ti ( j = 1,2, . . . ,n - 3) are independent auxiliary variables. Equations (5.278) and (5.287) reduce, by the dependent variable transformation (5.284), to

(iD,f’.f)/flf= -Kn, (iD,,f’-f)/j”f= -Kj+2,

n = 3,4, ...,

(5.288)

j = 1,2, ..., n - 3,

(5.289)

~ ~ ~ 3 ~

152

5 The Benjamin-0110-Related Equations

respectively. Under conditions (5.287), the nth-order modified KdV equation may be transformed into coupled bilinear equations for f ‘ and .f [30]:

D,f’.f=

#‘)D”’-3f’-f+

n- 3

1 /$”)D,Di(n-j-21f’.f,

n 2 4,

j= 1

(5.290) (5.291)

D f f ’ * f = 0,

where a(”)and /$”) (j= 1,2, . . . ,n - 3) are unknown coefficients. Now comparing the & ’ t n - j P 2 ) terms on both sides of identity (5.19) with t = z j , we obtain

a2(n-j-m-2)

x

ax’(.- j - m - 2 )

1’

f‘

-

f

n-j-3

a2(n-j-m-3)+1 aX2(n-j-rn-3)+1

lnflf.

(5.292)

Equations (5.291), (5.284), and the definition of the D operator yield

(5.293) from which we obtain the relation

(5.294) It follows from (5.287) and (5.294) that

_ a _a hf’f=2 /;,uur, dx‘ a z j ax

=

-2

J;:

dx’ (5.295)

5.4

Higher-Order Modified Kortewegde Vries Equations

153

and from the translation invariance of 1, that

(5.296) which implies that (K,),u is a perfect derivative, that is,

Substituting (5.289), (5.295), and (5.297) into (5.292) and dividing both sides by f If, we obtain

(5.298) where we have set A, = (D: f’. f)/fIf,

n

=

0, 1 , . . ..

(5.299)

The expressions for J , defined by (5.297) are derived, using (5.279)(5.281), as 53

=

UUZ,

- &.4,)2

+ 3u4,

(5.300)

To express A, in terms of u, u,, . . ., substitute (5.284) and (5.294) into identity (5.19) with t = x and compare the cZn-’terms and the zZn

154

5

The Benjamin-Ono-Related Equations

terms on both sides of the resultant equation. Then we find the recursion formulas for A,, and A,,,+ as

(5.303)

m=O i

n- 1

n

(5.304) These formulas are solved successively starting with A , A = - iu. Several explicit forms of An are given as

=

1 and

A , = 1,

(5.305)

A , = -iu,

(5.306)

A2

=0

by (5.291),

+ 2u3) = -iK3 by (5.279), A, = -2uuzx + 2(u,)2 - 2u4, A5 = -i[u4, + 1 0 ~ ~ + ~ l2o U , ( ~ , ) ~+ 6uS] A3

= -i(u2,

= -i K ,

A6 = -4uu4, A7

A8

=

-i[u6,

=

-iK5

=

by (5.280),

(5.307) (5.308) (5.309) (5.310)

+ 8 ~ ~ -~4(u2,), 3 , - 4ou3uzx- 16u6, (5.311) + 14U2u4, + 5 6 ~ ~ +~ 42U(u~,)~ ~ 3 , + 70(U,)2U2x

+ 7Ou4u2, + 140U3(u,)2 + 20u7] by (5.281),

(5.312)

- 6 6 ~+ ~12u,u5, ~ ~ - 26u2,u4, - 1 1 2 ~ ~ ~ 4420U2(U2,)2 , - 28Ou(~,)~u,, - 616u5u2, - 280u4(u,)’ + 14O(uJ4 + ~ O ( U ~-, )132u8, ~ (5.313)

+ 18u2u6, + 108uu,us, + 1 8 6 + 252(uJ2u,, ~ ~ ~ + 84u4u4, + l 8 O u ( ~ ~ ,+) ~6 7 2 + 1~ 3 ~4 4~ ~ + 224(~2,)~+ 588u3(u2J3 + 252~2(ux)2u2,+ 168u6u2, + l26Ou(~,)~+ 1512u5(uJ2 + 28u9]. (5.314)

A , = -i[u8,

~ ~ ~ ~~ ~~ ~ ~~

155

5.4 Higher-Order Modified Korteweg-de Vries Equations

Note that AZn-3 =

(5.315)

-iKn,

for n = 3, 4, 5, but this relation does not hold generally for n 2 6. Introducing (5.288) and (5.298) into (5.290) we obtain K n= - - & ' ) A Z n - 3

+

c fl)[ c

n-3

n-j-2

j= 1

m=O

2(n-j-2)CZmA2m~Kj+2~Z(n-j-m-2)x

n- j - 3

+ 2i 1 m=O

Z(n-j-Z)C2mAzm+l(Jj+2)2(n-j-m-3)x

3

9

n 2 4. (5.316)

Equating the coefficients of u Z n v 3u, ~u Z x~, .. . ,-u ~~ ( ~on- both ~ ) sides ~ of (5.316), we can derive simultaneous linear algebraic equations for unknown coefficients a(") and fiy) (j= 1,2, . . .,n - 3). We shall now apply the method of bilinearization, as developed so far, to the higher-order modified KdV equations (5.278) with n = 4, 5, 6 and present the explicit N-soliton solutions for these equations. (i) n = 4 It follows from (5.316) with n = 4 that

Substituting (5.279), (5.280), (5.300), (5.305)-(5.307), and (5.310) into (5.317) yields u4x

+ lOu(uJ2 + 10uzu2, + 6u5 =

-

fii49u4,

10(~(4)lO(01'~)- fi\4))u2u2x- 6 ( ~ r (~ )/?(:))u'.

+(4)

-

-

(5.318)

Comparing the coefficients of ugX,u(uJZ, uzuZx,and u5 on both sides of (5.318), we obtain only one independent equation for two unknown coefficients a(4)and j?:), -(a(4)

- /p)= 1,

(5.319)

from which d4)and fip)are determined as (-J4)

= - 1 + c(4) 1 ,

pi41 =

44),

(5.320) (5.321)

156

5 The Benjamin-Ono-Related Equations

where c(;L)is an arbitrary constant. If we do not introduce the auxiliary variable zl, c‘p) can be taken to be zero. (ii) n = 5 With n = 5, it follows from (5.316) that K5 =

-ia‘5’

2

I-. 3- j

+ 1 815’ 1 2 ( 3 - j ) C Z m ~ 2 m ( K j + 2 ) 2 ( 3 - j - m ) x j=1

2-j

+ 2i 1 2 ( 3 - , , C z m + , A , m + 1 ( J j + 2 ) 2 ( 2 - j - m ) x m=O

].

(5.322)

Substituting (5.279)-(5.281), (5.300), (5.301), (5.305)-(5.309), and (5.312) into (5.322) and comparing the coefficients of u6x,U ’ U ~ ~. .,.,u7 on both sides of (5.322) yields eight equations, but essentially one independent equation for three unknown coefficients a(5),fl\5), and given by -a(5)

+ fl\5) +

=

1,

(5.32 3)

which has been obtained by comparing the coefficient of us,. Thus, a(5),@f),and fi5) are determined as a(5) =

- 1 + 4 5 ) + 43,

(5.324)

fly) = c(5)1 9 fly’ = c p ,

(5.325) (5.326)

where 4’) and ci5)are arbitrary constants. If we set ci5)= need not introduce the auxiliary variables z1 and z2.

=

0 we 9

(iii) n = 6 With n = 6, it follows from (5.136) that K6

3

[:g

+ 1816’

--

j= 1

3-j

2(4-,,C~mA~m(Kj+,)~(,-j-m)x

+ 2im1 2(4-j)C2m+1A2m+1(Jj+2)~(3-j-m)x =O

].

(5.327)

Substituting (5.279)-(5.281), (5.300)-(5.303), (5.305)-(5.3 lo), and (5.313) into (5.327) and comparing the coefficients of usx,t4zu6x,.. .,u9 on both sides of (5.327) yields formally 16 equations, but essentially

157

5.4 Higher-Order Modified Korteweg-de Vries Equations

two independent equations for four unknown coefficients a(@, p'p),

p',"), and pi6)given by

-a(6)

-28a(@

+ p'p) + p'p) + /rp) =

1,

+ 76p'p) + 68pi6)+ 7Op'p) = 70.

(5.328) (5.329)

Equations (5.328) and (5.329) have been obtained by comparing the coefficients of us, and u9, respectively. Thus, a(6),S:"), pi6),and pi6)are determined as a(6)

= -18

+ i c y ) + icy),

(5.330) (5.331)

(5.333)

where ci6)and ci6)are arbitrary constants. If we choose c \ ~ = ' cL6) = 0, we need not introduce the auxiliary variables T~ and T ~ Note . that, if we = ( 5 ) = ( 5 ) = 0 in (5.320), (5.321), and (5.324)-(5.326), then set c(4) 1 c2 c3 for n = 4 , 5 the higher-order modified KdV equations can be transformed, without introducing the auxiliary variables T~ and T ~ into , the coupled bilinear equations '

D,f'.f = -Df"-3f'.fr

(5.3 34)

off'.f = 0.

(5.335)

These bilinear equations include the bilinear equations (5.285) and (5.286) of the original modified KdV equation as a special case of n = 3. However, for n 2 6, we cannot generally transform the higher-order modified KdV equations into such simple forms as (5.334) and (5.335). In the case of the higher-order KdV equations, even the first higherorder equation cannot be transformed into the bilinear equation without introducing an auxiliary variable, as shown in Section 5.2. This is an interesting difference between the higher- order KdV equations and the higher-order modified KdV equations. Since the modified KdV equation is related to the KdV equation by the Backlund transformation [1051, it seems worthwhile to study the relation between these, higher- order equations from the viewpoint of the Backlund transformation, which will be shown in the next section.

158

5

The Benjamin-Ono-Related Equations

Solutions

5.4.2

The N-soliton solutions of the nth-order modified KdV equation with n = 3,4,5,6 are expressed compactly in the forms

with

[

8,= aI eAlm

=

-

-

n- 3

1a?hj -

j= 1

(aI - a,>2/(aI+ a,)2,

1

,

1 = 1,2,. . ., N,

(5.338)

1 , m = 1,2 ,..., N , (5.339)

where aIand d1 are arbitrary constants. If we set

6, = 6, +

n- 3

1a:jzj,

(5.340)

j= 1

the form of the solution is the same for all equations, the only difference being the velocity a;("-') of solitons. This is a common property of the solutions of the Lax hierarchy of nonlinear evolution equations.

5.5 Backlund Transformations and Inverse Scattering Transforms of Higher-Order Korteweg-de Vries Equations 5.5.1 Backlund Transformations

In this section we shall derive the Backlund transformation of the Lax hierarchy of the KdV equation, which is written as u, = -i?Kn(u)/i3x,

n

=

3,4,. . .,

(5.341)

5.5

159

Transformations of Higher-Order Korteweg-de Vries Equations

on the basis of the bilinear transformation method [31]. As shown in Section 5.2, Eqs. (5.341) are transformed into bilinear equations through the dependent variable transformation a2

u = 2lnf. ax2

(5.342)

Denote another solution of (5.341) as a2

u’ = 2-lnf’

(5.343)

ax2

and assume that f ’also satisfies the spatial part of the Backlund transformation 0;f ’ . f = 1.x (5.344) where 1 is an arbitrary parameter. Furthermore, introduce the quantit ies w, = u, (5.345) (5.346)

w; = u’,

a

v=i-ln-. ax

f‘ f

(5.347)

Using (5.342), (5.343), and (5.345)-(5.347), (5.344) is rewritten as (w:

+ w,) - uz = 1,

(5.348)

which leads to the relation

w; + w, = 2v2 + 21.

(5.349)

On the other hand, it follows from (5.342), (5.343), and (5.345)-(5.347) that

which, combined with (5.349), yields

w:

=

-iv,

w, = iv,

+ v z + 1, + v2 + 1.

(5.351) (5.352)

To derive the time part of the Backlund transformation, substitute

(5.345) and (5.346) into (5.341), integrate with respect to x , and use the

160

5 The Benjamin-Ono-Related Equations

boundary condition, K ( u ) + 0 as 1x1 + co. Then we obtain the relation W;

-

-[K,(w:)

W, =

-

K,,(w,.)].

(5.353)

On the other hand, it follows from (5.342), (5.343), (5.349, and (5.346) that

w ;- w, = 2 -

a2

atax

f' a ln-=2--. jax

o,fy fy

(5.354)

Equating (5.353) and (5.354) and substituting (5.351) and (5.352) into the resultant equation, we obtain 2--a ax

Rf'*f- - [K,( -iu, f'f

+ u2 + A) - K,(iu, + u2 + A)].

(5.355)

We now introduce the relation n- 1

K,( fiu,

+ u2 + A ) = 1 C,,,K,,-,,,( f iu, + u2)Am,

(5.356)

m=O

where Cn,m =

(2n m!(n

-

2)!(n

- m - l)!

- 1)!(2n - 2m - 2)!.

(5.357)

This relation is derived by the formula [54]

61,

-dx =

2(2n

-

3)In-1,

n

=

2, 3 , . . . ,

(5.358)

as follows: Repeated use of (5.358) and the definition of K, (= dI,,/du) gives

(5.359) On the other hand, since K,(u) is expressed in the form

K,(u) = ~ 2 ( , , - 2 ) , + ...

+ CU"-~,

c # 0,

(5.360)

5.5 Transformations of Higher-Order Korteweg-de Vries Equations

161

we obtain from (5.359) K,(u

+ A ) = K,(u) + 1 5 m. m=l

(5.361) Moreover, we note the result from Adler and Moser [116]

where R, is derived from the functional derivative of the nth conserved quantity I", of the modified KdV equation

Rn(u)= STJdU,

(5.363)

explicit forms of which have been given in (5.279)-(5.282). It should be noted that

Rn(- u) =

- Rn(u).

(5.364)

Introducing (5.356) into (5.355) and using (5.362), we obtain, after integration with respect to x, Dt f' * f/fY = i

n-2

C Cn,m AmRn- m(u),

m=O

(5.365)

where the relation K, = constant has been used and the integration constant has been taken to be zero. It is important to observe that (5.365)reduces to the nth order modified KdV equation in the limit of A-0 U, =

-aRn(u)/ax,

=

3,4,. . . .

(5.366)

162

5 The Benjamin-Ono-Related Equations

The Backlund transformation (5.349) and (5.365) is rewritten in terms of the original variables w and w' as [117] w;

+ w, = -&w'

-

w)2

+25

(5.367)

m=O

which follow from (5.349), (5.365), (5.342), (5.343), and (5.345)-(5.347). Equations (5.367) and (5.368) with n = 3 correspond to the Backlund transformation of the KdV equation (2.163) and (2.166), first developed by Wahlquist and Estabrook [SO]. To express the right-hand side of (5.365) in bilinear form, we must bilinearize K,-,(u) under condition (5.344), following the procedure developed for the higher-order modified KdV equations (see Section 5.4). As shown in Section 5.4, the nthorder modified KdV equation (5.366) is transformed into the coupled bilinear equations D,f'.f=

dn)D:"-3f'-f+

n- 3

1 fl)DrjD:("-i-2)f'.f, (5.369)

j= 1

(5.370)

D y -f= 0, under the n - 3 subsidiary conditions

orsf'.f= & + Z ) ~ f s + I f r . f +

s- 1

1

p ? + Z ) ~ ~ , ~ f ( ~ - j ) f.f, '

j= 1

s = 1,2, .. ., n - 3,

(5.371) through the dependent variable transformation (5.347). Substitution of (5.366) and (5.347) into (5.369) gives

(5.372)

We now define the quantities A,(I)

=

Aj,n(A) =

(D:f' .f)/f% (DrjXf'*f)/f%

A2 =

A

by (5.344),

(5.373) (5.374) (5.375) (5.376)

5.5 Transformations of Higher-Order Korteweg-de Vries Equations

163

Using these definitions, (5.372) may be rewritten as

-

n- 3

K,(u)

=

1

- ~ C ~ ' " ) A ~ , ,- ~ i( O ) j?$")Aj,z(n-j-z)(0). (5.377) j= 1

The recursion formulas corresponding to (5.303) and (5.304) are obtained by noting the relation a2

-lnf'f= ax2

A2(1) -

=

1 + uz,

(5.378)

which is a consequence of (5.344) and (5.347) by

m=O

In the limit of 1 -,0, (5.379) and (5.380) reduce to (5.303) and (5.304), respectively. To express An@)in terms of An(0),we expand A,@) in powers of 1 as W21

An@) =

j=O

ii$")1jAn-2J(0),

n = 0, 1,2,. . . ,

(5.381)

where ii$") are coefficients that will be determined later, and we may set a,-,-O, -(Zn) -

n = l , 2 ,...,

(5.382)

since A 2 ( 0 ) = 0 by (5.344). In (5.381) the notation [n/2] means the maximum integer that does not exceed 4 2 . Then it is straightforward to derive the recursion formulas for a$")as iip) = (n/n - 2j)ii$"- ') 2!,z") = (2n - l)~!,Z!;2).

for 0 I j I [n/2] - 1,

(5.383) (5.384)

164

The Benjamin-Ono-Related Equations

5

These relations follow by substituting (5.38 1) into (5.379) and (5.380) and comparing the coefficients of A j on both sides of the resultant equations. The solution of (5.383) and (5.384) may be obtained as Zj") = n!/2j(n - 2j)!j!,

(5.385) O Ij 5 [n/2], with an initial condition = 1, which is derived from the relation A 2 ( 4 = L. Substituting (5.385) into (5.381) and solving these equations with respect to A,(O), we obtain (5.386) with

(5.387) Now Eq. (5.292) is rewritten in terms of (5.373)-(5.376) as

However

a

an

BJ.. n(A) = --lnf'f= a T j ax.

a an-2 aTja X n - 2

___

2

(0

a + L) = aTj -u2 = BjJ0), an-2

-

axn-2

(5.389) and A"j,

=

A"j, n(O),

(5.390)

by (5.375), the definition of Substituting (5.386), (5.389), and (5.390) into (5.388) yields, after some algebra,

n

=

1 LSa:2n)Aj,2(n-s,(A).

s=o

(5.391)

5.5 Transformationsof Higher-Order Korteweg-de Vries Equations

165

Introducing (5.386) and (5.391) into (5.377), we obtain

Finally, substituting (5.392) into (5.365) and using the definition of An and A , n , we obtain n-4

D~~0;'"

/

n-m-2

f .f)

- m - j - s- 2 )

I

- C,,n-sAn-3D: f ' .f - (Cn,n-2- 3Cn,n-3)An-2Dxf'.$ (5.393) The subsidiary conditions corresponding to (5.371) are expressed as

D,, f '.f

=

right-hand side of (5.393) with n replaced by 1 + 2, 1 = 1, 2,. .., n - 3. (5.394)

Equations (5.344), (5.393), and (5.394) constitute the Backlund transformation of the nth-order KdV equation expressed in bilinear form. For n = 4, Eqs. (5.393) and (5.394) give D, f ' .f = d4)D; f ' .f - 10(d4)+ 1)AD: f ' .f

+ B\4'(D,,D;f'

D,,f'.f=

-D:f'-f-

*f - LDrIf'

of

1,

+ 15d4)A2DXf ' .f

(5.395)

3AD,f'*f,

(5.396)

where L Y ( ~ )and pi4)are given by (5.320) and (5.321), respectively. Finally, we shall derive the conserved quantity of the nth-order KdV equation. For this purpose, expand w' - w as w' - w = 2q

m

+ 1 f'ln")q-m, m= 1

q

= A 112,

(5.397)

166

5 The Benjamin-Ono-Related Equations

and substitute (5.397) into (5.367). The result is expressed in-the form of the recursion formula as

fl“’

(5.398)

= -u,

f y = 51 u x ,

(5.399)

Substituting (5.397) into (5.368), integrating with respect to x, and using the boundary condition, we obtain

which implies that a l

1;) = J-mfB)dx,

rn

=

1,2,.

..,

(5.402)

are the conserved quantities of the nth-order KdV equation. We see that the functional form of the conserved quantities 12) is the same for all n, since the spatial part of the Backlund transformation (5.367) does not depend on n. This is a remarkable property of the Lax hierarchy of the KdV equation. 5.5.2 Inverse Scattering Transforms As discussed in Section 2.5, the inverse scattering transform can be derived from the Backlund transformation. To show this explicitly in the case of the higher-order KdV equations, define the wave function by =f’/f, (5.403)

*

*

and note the identity of the bilinear operator exp(sDx

+ hD,,)f’.f

(5.404)

167

5.5 Transformations of Higher-Order Korteweg-de Vries Equations

which is derived from (App. 1.3). Substituting (5.403) into (5.404) yields exp(sD,

+ 6Drj)f ’.f f’

(5.405) On the other hand, from the subsidiary condition for u [sec (5.82)], urj = -8Kj+,/ax,

j = I, 2 , , . , , n - 3,

(5.406)

and (5.342), we have, after integration with respect to x, a2

aTjax l n f =

2-

(5.407)

-Kj+’.

Introducing (5.342) and (5.407) into (5.405) and comparing the order 6 terms on both sides of (5.405), we obtain

(5.408) It follows from (5.405) with 6

=

0 and (5.342) that

If we use (5.408) and (5.409) expanded in powers of E, we can express the right-hand side of (5.393) in terms of . . . , +rj, . . . , u, u, . . . . It is important to note that the resultant expressions are linear with respect to +. Eliminating + r j in (5.393) by using the subsidiary conditions (5.394), we can express (5.393), which describes the time evolution of the inverse scattering transform, in terms of I), . . ., u, u,, . . . .

+, +,,

+,,

168

5

The explicit expressions for n ILt

= -$sX

- 1oU$3,

=

- 5(U2,

The Benjamin-Ono-Related Equations

4, 5 are given as

+ 3u2 + 3A2)$,,

n = 4, (5.410)

7(3u - A ) $ S x - 70(2u2 - 2Au + A’)$,, - ~cu,, iouuzX + io(2U3 - 3 ~ +~ 3 ~22 ~ ) 1 + , - 3 5 c - u ~ ~ (u - A)2]ux*, t~ = 5. (5.411)

= -$7,

-

+

+

The spatial evolution of the inverse scattering transform may be obtained by noting the relation

(5.412) w‘ - w = 2*.&, which derives from (5.342), (5.343), (5.349, (5.346), and (5.403), and substituting (5.412) into (5.367) to yield *2x

+ u*

=

(5.413)

A*.

Thus, we have completed the derivation of the inverse scattering transform from the Backlund transformation written in bilinear form. We note that the time evolution of the inverse scattering transform is expressed in a more compact form as n- 2 *t

=

i

1

m=O

Cn,mAmEn-m(i$xI$)

+ cJI,

(5.414)

where c is an integration constant. Equations (5.347), (5.368), and (5.403) have been used in deriving (5.414). It may be seen that Eqs. (5.413) and (5.414) with n = 3 correspond to the inverse scattering transform of the KdV equation first derived by Gardner et al. [l].

Topics Related to the Benjamin-Ono Equation

This chapter is concerned with recent topics related to the BO equation. The equations considered here are the modified BO equation, the derivative nonlinear Schrodinger equation, and the perturbed BO equation. In Section 6.1 the modified BO equation [32] is constructed from the Backlund transformation of the BO equation, and the Nsoliton and N-periodic wave solutions are presented. The derivative nonlinear Schrodinger equation, briefly discussed in Section 6.2, derives from the nonlinear self-modulation problem of the BO equation; it describes the long-time behavior of the complex amplitude of the basic wave train [34]. Finally, in Section 6.3 the effect of small dissipation on the BO equation is considered. Owing to dissipation, the amplitude of BO solitons decays slowly in time, and a system of 169

170

6 Topics Related to the Benjamin-Ono Equation

equations describing the long-time behaviors of amplitudes and phases of BO solitons are derived by employing the multiple time-scale expansion.

6.1 The Modified Benjamin-Ono Equation

The modified BO equation is generated from the Backlund transformation of the BO equation and may be written as [32]

w, + Hw,,

+ 2y(l - ePW)w,- w,Hw,

= 0.

(6.1)

Equation (6.1) has been obtained from (3.180) by setting 2/E = Y,

(6.2)

p = 1.

(6.3)

It is clear from the construction of the Backlund transformation of the BO equation that (6.1) is transformed into the system of bilinear equations (iD, (iD,

+ 2yD,

+ 2yD,

D: - p)f.g = 0,

(6.4)

DE - p ) f ’ * 9‘ = 0,

(6.5)

-

-

(D, - 7i)f.g’ = -7if’g by introducing the dependent variable transformation

w

=

1n(glf/gf’),

(6.6) (6.7)

whereJf’g, and g‘ have the forms (6.8) (6.9) (6.10) (6.1 1)

6. I

171

The Modified Benjamin-Ono Equation

with the conditions Im xJ(t) > 0,

j = 1, 2 , . . . , N ,

(6.12)

Im xJ(t) < 0,

j = 1, 2, . . . , N’,

(6.13)

Im yj(t)> 0,

j = 1, 2 , . . . , M ,

(6.14)

Im y;(t) < 0,

j = 1, 2, . . . , M‘,

(6.15)

where x,(t), xj(t), yJ(t), and y;(t) are complex functions o f t ,and N , N‘,

M , and M‘ are positive finite or infinite integers. Equations (6.4)-(6.6) correspond to Eqs. (3.156)-(3.158), respectively. Transformation (6.7) follows from (3.171), (3.165), and (3.166). It is an easy exercise to

reconstruct (6.1) from (6.4)-(6.6) and (6.7). We shall now construct the N-soliton and N-periodic wave solutions of the coupled bilinear equations (6.4)-(6.6). For N = 1, it is confirmed by direct substitution that the following pair of solutions are satisfied by (6.4)-(6.6):

f = 1 + exp(i51 - 41 + f ’ = 1 + exp(i51 - 41 9 = 1 + exp(i51 + 41 + $A

(6.18)

+ exp(i5, + $ J ~- $l).

(6.19)

g’ = 1

$11,

(6.16)

$11,

(6.17)

Here (6.20) (6.21) (6.22) with k , , 41,xol, and <‘p) being real constants. To satisfy (6.12)-(6.15), the following conditions must be imposed on k , , 41,and $1 : ($1 - 41Ml

’0,

(6.23)

+ 41Ykl

< 0.

(6.24)

($1

172

6 Topics Related to the Benjamin-Ono Equation

The present solution may be expressed in terms of the original variable in the form w = In =

cos (1 cos t,

+ cash(+, - 41)

+ cash(+, + 41)

sinh +1 sinh 4, cos cosh +1 cosh 41’

- 2 arctanh

(, +

(6.25)

considering (6.7) and (6.16)-(6.19). We next consider the one-soliton solution of (6.4)-(6.6), which is simply derived from the one-periodic wave solution by taking the long-wave limit k, -,0. To show this, we set

<‘P’ = n in (6.20) and take the limit k, (6.20)-(6.22) that

+ 0,

41 = k,/(a,

-

(6.26)

keeping a , finite. It follows from

(6.27)

2Y),

+1 N kl/%

+ (k:/2)(~ - a,t - ~ 0 1 ) ~

(6.28)

cos 5 1

N

-1

sinh+,

kl/(al

sinh

+,

NN

kl/al,

(6.31)

cash

4,

1

(6.32)

(6.30)

- 2y),

+ +[kl/(al - 27)12, cosh +, 1 + gk,/a,)2. N

(6.29)

(6.33)

N

Substituting these expressions into (6.25), we obtain, in the limit of -, 0, the one-soliton solution

k,

w = In =

(x (x

alt - xOl)’ - U l t - XOl)’ -

- 2 arctanh

+ [Val - Ma, - 2y)12 + [l/u, + l/(u, - 2 M

(x - a,t

2/a,(a, - 2Y)

- xO1)’+ [l/ai

+ l/(ul

-

2y)12’

(6.34)

Note that (6.34) behaves asymptotically for large x as

w

2:

-[4/al(ul

-

2~)]x-~,

(6.35)

which has the same asymptotic form as that of the BO soliton solution [see (4.14)].

173

6.1 The Modified Benjamin-Ono Equation

For general N, the solutions are constructed following the procedure developed in Section 2.2. The N-periodic wave solution may be expressed as [32]

f = 1 1exp p=o.

Ir, 1

pjjirj

- 4j

(N)

+ $ j ) + j l
IN

(N)

f‘ = 1 1exp 11pj(irj - 4j - $ j ) + j 1< pk j p k A j k p=o,

where = kJjx - a j t -

xoj)

a j = k j coth 4 j + 2y,

+ t!’),

exP(2$j) = (aj + kj)/(aj - k j ) , exp A j , =

(aj - a,)’ - ( k j - k,)’ (aj

- a,)’

- (kj

+ k,)”

3

9

1 7

(6.36) (6.37)

j = 1, 2 , . . ., N,

(6.40)

j = 1 , 2,..., N,

(6.41)

j = 1,2, ..., N,

(6.42)

j , 1 = 1, 2 , . . . , N,

(6.43)

j = 1 , 2,..., N,

(6.44) (6.45)

j = 1,2, ..., N,

with k j , 4 j ,x O j ,and t)O)( j = 1,2, . . .,N) being real constants. The present N-periodic wave solution reduces to the N-soliton solution in the long-wave limit. The procedure to derive the N-soliton solution is the same as that developed for the BO N-soliton solution (see Section 3.1), that is, we set

r)O)=.n,

j = l , 2 ,..., N,

(6.46)

and take the long-wave limit k j + O (j= 1, 2, ..., N), keeping a j (j = 1,2, . . . ,N) finite. The result is expressed in terms of the bilinear variables as w = ln(r’/#f’).

(6.47)

174

6 Topics Related to the Benjamin-Ono Equation

Here

f = det M, f' = det M',

(6.48)

gj = det L,

(6.50)

det L',

(6.51)

=

(6.49)

where M , M', L, and L' are N x N matrices whose elements are given

(6.52) (6.53) (6.54) (6.55) (6.56) The asymptotic forms of the present N-soliton solution for large values oft are readily obtained from (6.47)-(6.56) as

which implies that no phase shift occurs on collision of solitons. The situation is the same as that for the BO N-soliton solution discussed in Chapter 4. From the Backlund transformation of the modified BO equation, a new nonlinear evolution equation, which may be called the second modified BO equation, can be generated, and this process may continue infinitely. The concept of the chain of the Backlund transformation for the KdV equation first appeared in [53]. As is well known, the modified KdV equation is generated from the Backlund transformation of the KdV equation [ 1051. Similarly, the second modified KdV equation is constructed from the Backlund transformation of the modified KdV equation [ 5 3 ] . Furthermore, the modified finitedepth fluid equation [ 8 11 and the modified Sine-Gordon equation [ 8 2 ] are also generated from the Backlund transformations of the

175

6.2 The Derivative Nonlinear Schrodinger Equation

BO Equation

Finite-Depth Fluid Equation

BT

Modified BO Equation

L

Modified Finite-Depth Fluid Equation

BT

2...2 BT

..-

--).

BT

nth Modified 80 Equation

-

nth Modified Finite-Depth Fluid Equation

Fig. 6.1 New nonlinear evolution equations generated from the chain of Backlund transformations of the finite-depth fluid, the BO, and the KdV equations. Here BT denotes the Backlund transformation and 6 the depth of fluid.

finite-depth fluid equation and the Sine-Gordon equation, respectively. In Fig. 6.1, the new nonlinear evolution equations generated by the chain of the Baacklund transformations of the finite-depth fluid, the BO, and the KdV equations are depicted along with their interrelationships.

6.2 The Derivative Nonlinear Schrodinger Equation In this section we shall consider the nonlinear evolution equation iu,

+ uxx= iu(lu12)x,

(6.58)

where u is the complex amplitude of the basic wave train. This equation governs the nonlinear self-modulation of weak periodic wave solutions of the BO equation [34]. It should be noted that Eq. (6.58) is different from the derivative nonlinear Schrodinger equation ia,

+ uxx = i(aIal2),,

(6.59)

which is known to be completely integrable [ltS], and also different from the derivative nonlinear Schrodinger equation as expressed in (2.259).

176

6 Topics Related to the Benjamin-Ono Equation

We now seek a solution of (6.58) in the form [34] a(x, t ) = A(x)e-’R‘, ~ ( x =) F(x)ei6@),

(6.60) (6.61)

where F ( x ) and $(x) are real functions and 0 is a constant. Substituting (6.60) and (6.61) into (6.58) yields a system of ordinary differential equations

b4- 4(Q - C1)u’ + 4 2 - 4; = + c1u-l,

(duldx)’ = d+/dx

V(U), (6.62)

(6.63)

where u = F2

(6.64)

and c1 and c2 are integration constants. To obtain a bounded solution of (6.62), we decompose V(u) in the form V(u) = (u - u)(u - B)(u - y)(u - S),

with the condition

6 2 0 s y I p Iu.

(6.65) (6.66)

An oscillating solution may then be expressed in terms of the Jacobian elliptic function (cn) as

(6.67) where and

(6.69) A solitary wave solution is obtained from (6.67)-(6.69) by setting c1 = p, for example, as U =

2B(B

+ Y) - (28 + r)(B - Y) sech’ 5 2(8 + Y) + (B - Y) sech’ 5 (6.70)

6.2 The Derivative Nonlinear Schrodinger Equation

177

where use has been made of the relations

k = 1, cn(& 1) = sech 5,

a

+ /3 + y + 6 = 0.

(6.72) (6.73) (6.74)

Since (6.58) is invariant under the transformations

ii(x7Z)

=

a(x, t ) exp[i(-Ax/2

x = x - At - xo,

+ A2t/4 + &)I.

t = t - to, where A, xo, t o , and 4o are arbitrary real constants, a

(6.75) (6.76) (6.77)

solitary wave solution with a propagation velocity A may be constructed from the present stationary solution (6.70) by employing (6.75)-(6.77). Whether (6.58) possesses multisoliton solutions is currently an unsolved problem. However, the result of the numerical solution involving a head-on collision of two solitary waves [34] strongly suggests that (6.58) is completely integrable, and it may therefore have an N-soliton solution, an infinite number of conservation laws, the Backlund transformation, etc., which are common properties of completely integrable nonlinear evolution equations. Finally, we note that (6.58) exhibits at least three conservation laws, their explicit forms being written as

(6.78)

(6.80) where a. is a boundary value of a, that is, a. = a( f 00, t).

(6.81)

178

6 Topics Related to the Benjamin-Ono Equation

6.3 The Perturbed Benjamin-Ono Equation Throughout this text, we have been concerned only with nonlinear evolution equations that include the effects of nonlinearity and dispersion ; these may be appropriate model equations that describe physical systems. However, there exist many situations in which another important effect,that of dissipation, cannot be neglected. In these cases, the equations must be modified to include the effect of dissipation. In this final section we shall consider the system that is described by the BO equation with a small dissipation effect [119]. The model equation may be written in the form u,

+ 4uu, + Hu,,

= ERCU],

(6.82)

where R is a functional of u representing the effect of dissipation and E is a small positive parameter. Equation (6.82) may be called the perturbed BO equation. When E = 0, (6.82) reduces to the BO equation. Although the exact method for solving (6.82) is not known, we can treat it by a perturbation method.*To apply this method to (6.82), introduce the multiple time scales tJ. = ~ j t , j = 0, 1,2,. . . , (6.83) and expand u into an asymptotic series m

u =

1

(6.84)

&jUj,

j=O

where u j is assumed to be a function of x, t o , t,, t 2 , . . . , u j = uj(x, t o , t l , t 2 , . . .),

j = 0, 1, 2,. . . .

(6.85)

As a consequence of (6.83), the time derivative is replaced by m

alat = p a / a t j . j=O

(6.86)

Substituting (6.84)-(6.86) into (6.82) and comparing the ~ j ( = j 1,2,. . .) terms on both sides of (6.82), we obtain a system of equations, the first two of which are

The method used here is the multiple time-scale expansion [120, 1211.

179

6.3 The Perturbed Benjamin-Ono Equation

Note that the equations that derive from the ~j terms are linear with respect to uj for j 2 1, while only the lowest-order equation (6.87) is nonlinear. To illustrate the method, take the one-soliton solution of (6.87) as the lowest-order one uo

= a/(z2

+ l),

(6.89)

0,

(6.90)

with z = a(x -

(6.9 1) t = ato + t o ( t l ,t 2 , . . .>, tO
E,

LCfI

=Lo+ 4(u0f), + HfXx= 0.

(6.92)

Function g defined by (6.93) satisfies the equation L*b]

gto

+ 4Uog, + Hgxx = 0.

(6.94)

Operator L* is an adjoint to L. Multiplying (6.88) by g and integrating with respect to x from - co to 00, we obtain (JYrn9U, dx)to

=

J;m91RCUol

- U o , t , ) dx,

(6.95)

with the aid of (6.94). If g depends on x and t only in the combination x - at,, then the right-hand side of (6.95) is not made to depend on to by introducing a new integration variable x - at, instead of x. In this situation, (6.95) is integrated as (6.96)

180

6 Topics Related to the Benjamin-Ono Equation

which implies that u , diverges for large t o . To eliminate this undesirable behavior, the following nonsecularity condition may be imposed : (6.97) Functionsf and g may easily be found as follows: Differentiate (6.87) with respect to the independent parameters p j included in u o . In the one-soliton case, we can set p1 = a and p z = t. Then (auo/aPj)t, + 4(u0 aUo/aPj)x + H(auo/apj)xx = 0,

(6.98)

which means that ~ ~ a u ~ /= a 0. P~l

(6.99)

Therefore (6.100)

fj = auo/aPj and

(6.101) For the one-soliton case, we find from (6.89), (6.100), and (6.101) that f1

=

g1 =

auo

1 - 2z =

J-m x

(6.102)

(z2 + 1 ) Z ’

au,

aadx’

1 2 a z Z 1’

= --

+

(6.103) (6.104) (6.105)

The right-hand side of (6.88) becomes

(6.106)

181

6.3 The Perturbed Benjamin-Ono Equation

by the definition off, andf2. The nonsecularity conditions (6.97) J - m m g j ~ ~ ~ UU oo, t~, ) -d x = 0,

now yield equations for a and

j =1

, ~

(6.107)

5 as (6.108) (6.109)

These results may be rewritten in terms of the original variable t as

(6.110)

by noting d t p t = atlato

+

&

aylat,

+

0(&2)

(6.112)

and (6.91). This result can be extended to the case where the lowest-order solution uo consists of N solitons, and the independent parameters included in uo are N amplitudes a j (j= 1,2, . . ., N) and N phases ti (j = 1,2,. . . ,N). If we set these parameters as p . = a. J

J ,

P N + j = t j ,

j=1,2,...,N,

(6.113)

j = 1,2,-..,N7

(6.114)

then the 2N independent solutions of (6.94) are constructed as dx‘,

r5,

j = 1, 2 ,. . . ,2N.

(6.115)

The nonsecularity conditions (6.97) are then replaced by 2N simultaneous equations for 2N unknown quantities apj/atl (j= 1,2, . . . ,2N) 2N

s=l

7 dPS at

g j ) = { R [ u o ] ,g j } ,

aPs

j

=

1,2,. . . ,2N, (6.116)

182

6 Topics Related to the Benjamin-Ono Equation

where (6.117) from which we can determine the t l dependences of the parameters pi 0’ = 1 , 2 , . . . ,2N). As explicit examples, we shall consider two forms of R given as

R[u]

=

-u,

(6.118)

R[u]

=

u,,.

(6.119)

For the one-soliton case, we obtain from (6.110) and (6.111) the following results: (i) R[u] = -u a

=

(ii) R[u]

(6.120)

= a,e-’&‘,

to+ (a0/2&)(1- e-’&‘).

(6.121)

+ 2&a;t)-”’,

(6.122)

= u,,

a = a,(l

t = 5 0 + (l/cao)[(l + 2&a;t)”z - I], (6.123) to are initial amplitude and initial phase of a soliton,

where a. and respectively. Obviously, these results reduce to the unperturbed cases when E = 0. Once the t l dependences of parameters have been determined, one can proceed to the first-order equation (6.88), which describes the time evolution of the nonsoliton part. Being a linear equation for u l , it may be more tractable than the original nonlinear equation (6.82). However, the details will not be discussed here. Finally, we shall comment on the conservation laws. Because of the effect of dissipation, the terms I , given by (3.187)-(3.191) are no longer conserved and depend explicitly on time. The time evolutions of I , can be expressed as dt

’In R[u] dx,

n

=

1,2,...

(6.124)

For the one-soliton case, the approximate solution of (6.82) u = uo

+ &U]

(6.125)

183

6.3 The Perturbed Benjamin-Ono Equation

satisfies (6.124) up to the order E. To show this, note that I,

=

n = 1,2, .. .,

n(a/4)”-’,

(6.126)

for the one-soliton solution (6.89) [see (3.256)]. From (6.126), we obtain 61,/6~= (~/4)(n- 1 ) ( ~ / 4 ) ’ -6~ ~ / 6 ~ , n = 2,3,. . . . (6.127) On the other hand, it follows from (6.126) with n m

$ /-mu2 dx

=

=

2 that

(n/4)a,

(6.128)

which, by taking the functional derivative, yields u = (n/4)

(6.129)

6a/6u.

Combining (6.127) with (6.129), we obtain the useful relations 61,/6u = 1,

61,/6u

=

(n

(6.130) -

I)(u/~)”-~ =u4(n - l)(a/4)”-’[1/(z2

+ I)],

n 2 2.

(6.131) Then dl,--

dt

c m

Ej

j=o

a

-

atj

I&,

+

a +-

=

at,

61,

+m: J

EUI)

Im 6u

1

(4% at, 6u

u=uo

U,

dx]

u=u0

) u ~ ]d x

+ 0(c2)

+ 0(e2),

(6.132)

in view of the conservation laws aI,(u,)/dt,

=

0,

n

=

1, 2,. . . .

(6.133)

184

6 Topics Related to the Benjamin-Ono Equation

Substituting (6.88) into (6.132) and integrating by parts, we obtain

(6.134) However, the second term on the right-hand side of (6.134) vanishes because of (6.131), that is, 61,/6u

Lo cc uo,

(6.135)

and (6.87), the lowest-order equation for uo. Thus,

I

61, dln R[uo] dx = E 6u dt u=uo On the other hand, it follows from (6.124) that

f_, OD

61, -

R[u] d x = E

61, J-, 6u

1

u=uo

+ O(E~).

R[u0] d x

+ 0(c2).

(6.136)

(6.137)

Therefore, we see from (6.136) and (6.137) that relation (6.124) is satisfied by (6.125) up to the order E. The multiple time-scale expansion method developed here can be applied to a wide class of nonlinear dispersiveand dissipative equations. The interested reader may refer to [122, 123, 1243.

Appendix

I

Formulas of the Bilinear Operatorso

The definition of the bilinear operators is given by 1. The following formulas are easily derived from the definition of the bilinear operators : D y a . 1 = ama/dx", (App.I.1.1)

-

DFa .b

=

( - 1)"'DFb a,

DFa.a

=

0

for oddm,

D,D,a . 1 = D,D, 1 .u

=

D , a . b = 0 - a a 6, D:m+ ' a , .b

= +Dim'

-

a2a/axat,

' D , a 6.

(App.I.1.2) (App.I.l.3) (App.I.l.4) (App.I.1.5) (App.I.l.6)

The formulas given in this Appendix are found mainly in References [ S ] , [lo], and ~131. 185

186

Appendix I

2. The following formula is a fundamental property of the bilinear operator: exp(ED,)a(x) .b(x)

=

a(x

+ E ) ~ ( x- E).

(APP.I.2)

This formula is proved as follows: exP(EDx)a(x) . b(x) x’=x

by definition

by exchanging the order of sum by replacing n by n

+s

If we set a ( x ) = exp(p,x) and b(x) = exp(p, x) in (App.I.2) and compare the P term on both sides of (App.I.2), we obtain the formula D: exP(P,x).exP(Pzx)

=

(PI

-

P Z Y exP(P1 + PZb. (APP.I.2.1)

Let F ( D , , D,) be a polynomial of D, and D,. Then it follows from (App.I.1.1) and (App.1.2.1) that

which is very useful in deriving the N-soliton solution. 3. The following formulas are useful in transforming bilinear equations into the original forms of nonlinear equations:

187

Formulas of the Bilinear Operators

where a = a(x, t) and b = b(x, t). The left-hand side of (App.I.3) is a(x + E, t + 6)b(x - E, t - 6 ) by (App.I.2). On the other hand, the right-hand side of (App.I.3) reduces to

+

+

+

+

exp{+ In[a(x E, t 6)/b(x E, t 611 - 3 ln[a(x - E, t - 6)/b(x - E , t - S)] + 4ln[a(x E, t 6)b(x E, t 611 + 3 ln[a(x - E, t - 6)b(x - ,c, t - S)]} = exp{ln[a(x E, t 6)b(x - E, t - S)]) = a(x + E, t + 6)b(x - E, t - 6),

+

+

+

+

+

+

where we have used Taylor's formula exp(e d/dx

+ 6 d/dt)a(x, r ) = a(x + E, t + 6).

Therefore, (App.I.3) has been proved. = ln(a/b) and p = In(ab). It follows from (App.I.3) with Let 6 = 0, expanding both sides in powers of E, and comparing the E" terms, that

. b)/ab

=

(App.1.3.1)

4 x 9

(0%. b)/ab = Pxx

+

(App.I.3.2)

4. The following formula is a special case of (App.I.3): exp(sD,)a. b = {exp[2 cosh(s a/ax) In b]}[exp(e d/ax)(a/b)], (APP.I.4) which follows by setting 6 exp[exp(s d@x) ln(a/b)] 5.

=

0 and noting the relation

+

+

exp{ln[a(x E)/b(x E)]} = a(x + E)/b(x + E ) = exp(s a/dx)(a/b).

=

If we set a = b = f i n (App.I.4) and use (App.I.1.3), we have cosh(ED,)f.f

=

exp[2 cosh(s a/ax) In f].

(App.I.5)

188

Appendix I

Let u = 2(ln f ) x x and u,, = a"u/ax" (n = 0, 1,2,. . .). Comparing the E" terms on both sides of (App.I.5) yields the formulas

(~x2f-f)/fZ = u,

(App.I.5.1)

(D:f*f)/f2

= uzx

(Dlf-f)/f'

= ~4~

(D:f*f)/fz

+ 3u2,

(App.I.5.2)

+ I 5uuzx+ 15u3, (App.I.5.3) + 210UzUzx+ 1 0 5 ~ ~ 9 35(U~,)~ = usx + 28uu4,

(oi0f'f)/f2 = us, + 45uu6,

f 21h,,u4,

+ 630u2u4, + 315Ou3u,,

+ 1575u(u2x)z + 945u5.

(App.I.5.4) (App.I.5.5)

6. The following formula is very important in discussing the Backlund transformation in bilinear formalism : exp(DdCexp(D,)a .b l . Cexp(D,)c . dl = exp K D , - D3){exp[&D, D3) * {exPC%D2 0 3 ) - D,]c * b},

+

+

+ D,]a. d} (APP.I.6)

where Dj

=

+

j = 1, 2, 3,

E ~ D , 6,D,,

with E~ and d j being arbitrary constants. Using (App.I.2), (App.I.6) is proved as follows: The left-hand side of (App.I.6) becomes exp(D,)Ca(x + 6 2 , t + 6,)b(x - E 2 , t - 6211 . Cc(x e3, t 6,)d(x - 4 , t - S,)] = a(x E, E,, t 6, 6,)b(x E , - E,, t 6 , - 6,) x c(x - E , E3, t - 6 , 63)d(x - E , - E 3 , t - 6 , - 6 3 ) .

+ + + + + + + +

+

+

On the other hand, the right-hand side of (App.I.6) reduces to

+

exPC%Dz - 03)ICdx + E i + (62 t 61 + (6, + 63)/2) x d(x - E , - ( E ~ ~ ~ ) /t2-, 6, - (6, 6,)/2)] . Cc(x - el (8, + ~ ~ ) /t2-, 6, ( 6 , 6,)/2) X b(x E l - (Ez E3)/2, t 6 , - ( 6 , + 63)/2)] = a(x E~ E , , t + d1 6,)d(x - E , - E ~ t , - 6 , - 6 , ) x c(x - E , E , , t - 6, 6,)b(x E , - E,, t 6 , - 6,),

+

+ +

+

+

+

+

+

+

+

+

+

+

+ +

+

189

Formulas of the Bilinear Operators

which implies (App.I.6). The following formulas are derived from (App.I.6) by appropriate choice of ej and d j : (D,u. b)cd - ~ b ( D , c* d ) = (D,u * c)bd - ~ c ( D , b* d ) = D , u ~ bc, .

D,[(D,u * b) * cd =

(App.I.6.1)

+ a b . (D,C

*

d)]

(D,D,~*d)cb - ~d(D,D,c.b) + (D,~*d)(D,c*b) - ( D , u .~ ) ( D , c*

(App.I.6.2)

b),

(0:~ b)cd . - ~ b ( D ; cd) .

=

D,[(D,u. d ) * cb

+ ad.( D , c . b)],

(App.I.6.3)

( D ~ .ub)cd - ~ b ( D : c. d ) = iD:ad. cb @,[(D:u

+ + ~ ( D , u d) .(D,c *

(D'$

*

b)

*

d ) .cb

+ ad.( D ~ Cb)], *

- UU(D:C* C) ~D,(D:u C) * cu + ~D,(D,ZU.C) . ( D , c . = ~D:(D,u . C) .UC. *

U)CC

-

=

(App.I.6.4)

U)

(App.I.6.5)

7. The following formula is also useful in transforming the Backlund transformation written in bilinear operators into a form written in original variables:

+

[exp(GD,)u . b]cd ub[exp(6D,)c. d l = [exp(GD,)u .dlcb ud[exp(6DX)c. b] - exp(6DX/2)[2 sinh(6DX/2)u c] [2 sinh(6DX/2)b

+

-

(APP.I.7) which may be verified by repeated use of (App.I.2). From (App.I.7), we obtain the formulas

+ ~ b ( D , c d ) = (D,u d)cb + ad(D,c. b), (D:u . d)cb + ~ d ( D : c b) = (D:u .d)cb + ad(D:c . b) - ~ ( D , u .c)(D,b d).

(D,u. b)cd

*

*

(App.I.7.1)

*

*

(App.I.7.2)

8. To prove the cornmutability relation, the following formula is utilized : Cexp(6 d/dx) exp(eD,)u. b]c - [exp(G a/dx) exp(eD,)a . clb = 2{exp[(e + 6) d/dx]u} x {exp[g-e 6) d/dx] sinh[(--E 6)D,]b ' c } . (App.I.8)

+

+

190

Appendix I

This formula is proved by using (App.I.2). It follows by comparing the terms B*E" (m,n = 0, 1,2,. . .) on both sides of (App.I.8) that ( D , u . b)c - ( D , u . c)b = -a(D,b

( D ~ .ub)c - (D$ . c)b =

- 2a,(D,

(D;u . b)c - (D;u * c)b =

- 3a,,(D,

*

(App.I.8.1)

c),

+ a(D, b c),, b . C ) + 3a,(D, b . c),

b . C)

- $a[(D;b . C )

*

+ 3(D,b. c),]

(App.I.8.2) (App.I.8.3)

( D f a * b ) , c- ( D $ ~ . c ) , b= - a x x ( D x b * ~-) a,(D,b.c), + &[(D;b .C ) 3(D, b .c),,]. (App.I.8.4)

+

Appendix

11

Properties of the Matrices M and A

We consider the N x N matrices whose elements are given by Mjk Ajk

=

+ a,:') + 2(1 + y,") + 2( 1 -

djk(i0j

= djk(z

6jk)(aj

djk)(yj

-

-

', '.

ak)-

yk)-

(App.II.1) (App.II.2)

[See (3.17), (3.38), and (3.302).] These matrices have the properties

(App.II.3) det A = ( z 191

+

(App.II.4)

192

Appendix I1

We shall first prove (App.II.3). Using the definition of Kronecker's delta and the matrix product rule, I is converted to N

I

=

1

(MS)jk(I - d j k ) j.k= 1

c N

= j1.

...,j s + I = 1

M j I j 2 M j z j... , M JrJs . . + 1( 1 - d j , j , + l )

N

=

j1.

C

....j s + I = 1

MjijzMjzj3

. . . Mj.j.+ l M j s +ljg(Mjs+1 j l ) -

1

(App.II.5) On the other hand, it follows from the definition of the matrix M that =

( M j s +l i t ) -

(iejl

+ ai, '1-

+ t-(aj,,

' d j s +1j1

I

- ajXl -

d j s +l j l ) .

(App.II.6) Substituting (App.II.6) into (App.II.5) and using the properties of Kronecker's delta, (App.II.7) d j k ( 1 - h j k ) = 0,

(I (aj

-

hjk)2 =

ak)hjk

=

I - 6Jk.

(App.II.8)

7

(App.II.9)

0,

(App.I I. 5) reduces to

where use has been made of the formula (ajs+ - u j l ) (

f:

U;;~U?;,'

m= 1

Consider the s (j1,j2,

=

a;,+ I - ayl. (App.II.1 I )

+ 1 cyclic permutations of the indices,j,.,j,, . . . ,j,%+ I

. . . , j s + 1)

. . ., j 2 ) -, . . .

-, ( j 2 , j 3 ,. . . , j l )-,( j 3 , j 4 ,

-,

(.is+ l,.i13.. . ,.is).

(App.II.12)

193

Properties of the Matrices M and A

Since the quantity M j l j 2 M j 2 j.3. . Mjs+ljl in (App.II.lO) is invariant under these permutations, we obtain

I

=

l/(s

+ 1)

jl.

N

1 ..., j.+

x [(a;,, I - a;,)

M j l j 2M j 2 j 3

1 .

I

=1

.M .

Js+

.

IJI

+ (ayl - a?,) + ... + (a;, - ~ 7 ~ + ~ )=] /0,2 (App.II.13)

which proves (App.II.3). For n = 1, 2, (App.II.3) become N

1

(M")jk

=

O,

(App.II.14)

j.k=l

respectively. We now introduce the characteristic polynomial of the matrix M :

It follows from the well-known Hamilton-Cayley theorem that

From (App.II.16) with 1 = 0 cN = ( - l)Ndet M ,

(App.II.18)

which, substituted into (App.II.17), yields (- 1 ) M a = M N

+ c 1 M N - ' + ... + c N - M ,

(App.II.19)

where fi is an N x N matrix that consists of the cofactor of M [see (3.84)]. Since det M # 0, by (3.80), we obtain from (App.II.19)

194

Appendix I1

It follows from (App.II.3) and (App.11.20) that

(App.II.21) For n

= 1,

2, (App.II.21) become

c N

i.k=l

(@)jk

=

(App. 11.22)

0,

respectively. We shall now prove (App.II.4). For this purpose, we introduce the N x N matrices B and C as Bjk = ykI

(App.II.24)

1'

(App.II.25)

C = AB,

and then calculate the ( j , k) component of the matrix C to obtain

(App.II.26) where a prime appended to a sum indicates that the singular term s = j is omitted.6 Using the identity N

k- 1

N

N

(App. 11.27) The notation c' will be used throughout this Appendix.

195

Properties of the Matrices M and A

(App.II.26) becomes Cjk = yjk-lz

+ $-2

-

2

c yy-’ z y y m - 1 - 2yjk-l Z ( Y ~

k- 1

N

N

m= 1

s= 1

s= 1

-

Yj)-’

(App.II.28) Note that the yj ( j = 1, 2, . . . , N ) satisfy the system of equations [see (3.297)] N Z
-

yk)-’

-

=

yil),

j = 1, 2 , . .., N .

(App.II.29)

Substituting (App.II.29) into the last term on the right-hand side of (App.II.28), we obtain m= 1

where N

cy;,

P k

s= 1

k

= 0, 1 , 2, . . . .

(App.II.31)

Then from (App.II.30)

Where we have used the following properties of the determinant: AC11

c 1 2

* * ‘

CIN

Ac21

c22

* ’ *

C2N

.

AcNl

cN2

”.

.

. cNN

=A

c 1 1

c 1 2

”‘

C 1 N

c 2 l

c22

‘*.

C2N

cN2

. .’.

.

cNl

.

.

.

(App.II.33)

196

Appendix I1

c12

"'

CIN( C2N

. .

CNN

Repeating this procedure, we finally arrive at the formula 1

det C = (z

1 + l)N . ..

y, y,

...

yy-1

*.-

".-'

y,

... y;-1

. . ... .

1

= (z

+

det B.

On the other hand, it follows from (App.II.25) that det C

=

and from (App.II.24) that det B

=

det A B = det A det B,

n

(Jj

- Yk) #

0

lsk<jsN

by (3.296). Equating (App.II.35) and (App.II.36) yields [det A - (z

+ l)N] det B = 0,

and by (App.II.37), (App.II.38) becomes det A - (z

+ l)N = 0,

which proves (App.II.4). A technique used in obtaining (App.II.4) has an application to the algebra concerning eigenvalues of certain matrices related to the zeros of classical polynomials. The method is very interesting since it is purely algebraic. Let yj (j= 1,2,. . . ,N) be N zeros of a certain polynomial of order N , and consider the N x N matrices A @ ) and given by

C'P' = A'P'B >

(App.II.41)

197

Properties of the Matrices M and A

where B is an N x N matrix defined by (App.II.24) and p a positive integer. We first calculate the (j, k ) component of the matrix CP) as

N

N

(App.II.42) However, the identity

P- 1

-

N

k-D-

1

k-p-

1

1

(-1)'

r=O

r+p-1

cr f p - 1

(App.II.43)

J

holds with .Cr = n ! / ( n - r ) ! r ! ,

(App.II.44)

and pk is defined by (App.II.31), which may be verified by mathematical induction. Substituting (App.II.43) into (App.II.42) yields P- 1

N

r= 1

s= 1

cy = y;-1z + -1 (- l y k - l c r y yC'(Yj k-p-

+ (-

l)p

- (-

1)P

1

1

r=O

2

- ys)-(P-F)

r+p- lCrgpk-p-r-

1

k-o- 1 r=O

r+p-

1c,$-p-l

(App.II.45)

198

Appendix I1

We now consider, as an example, the Hermite polynomial of order N , H N ( y ) .In this case, y j ( j = 1,2,. . . ,N) satisfy the relations [125] H&j)

=

j = 1, 2 , . . . , N ,

0,

(App.II.46) (App.II.47)

C'(Yj - ys)-2

s= 1

=

+(N - 1) - 1 3 Y2 j

=

&[2(N

(App.II.48)

7

N

N

C ' ( y j - yJ4

s= 1

+ 2) - yf][2(N

-

1) - yf].

(App.II.50)

For p = 1, (App.II.45) becomes k- 2

C$' = Y : - ' z -

C yjPk-,,-,- + (k - 1 ) ~ ! - ~ , 1

r=O

(App.II.51)

and it follows, by employing the procedure used in deriving (App.II.35), that det C") = z" det B,

(App.II.52)

and therefore det A(')(z) = det C(')/det B

=

z",

(App.II.53)

since det B # 0 by the fact that y j # Yk f o r j # k. Formula (App.II.53) implies that the eigenvalues Iz of the matrix A(')(O), which are given by the roots of the characteristic equation det[A(')(O) - AI] = 0,

I : unit matrix,

(App.II.54)

are all zero. For p = 2, (App.II.45) and (App.II.47) yield r=O

- +(k -

i)(k - 2)y;-3.

(App.II.55)

199

Properties of the Matrices M and A

It follows from (App.II.55) that det C(') =

n[z N

-

k= 1

(k - I)] det B,

(App.II.56)

and therefore N

fl [x - ( k - l)],

det I I ' ~ ) ( Z =)

k= 1

(App.II.57)

which implies that the eigenvalues of the matrix A(2)(0)are all distinct and given by k = l , 2 , . . . , N.

-(k-l), For p that

=

(App.II.58)

4, it follows from (App.II.45) and (App.II.47)-(App.II.49)

C$) = [ Z -

sk2

+ i ( k - l)(k

- l)]yF-'

-

2)(2N - k

+ l)~jk-~

A(k - l)(k - 2)(k - 3)(k - 4)$5 k-5

f r=O

&r

+ 1)(r + 2)(r +

Then det 04) =

n[z N

-

k= 1

(App.II.59)

3)YjPk-r-S.

i ( k 2 - l)]

det B,

(App.II.60)

and det A(4)(z) =

N

fl [ z - 8 k 2 - l)],

k= 1

(App.II.6 I)

which implies that the eigenvalues of the matrix A(4)(0) are all distinct and given by

-i(k2

- I),

k

=

1, 2, ..., N.

(App.II.62)

We can also show by using (App.II.45)-(App.II.50) that, for p = 3 and p 2 5, det C@) cannot be reduced to such simple forms as (App.II.52), (App.II.56), and (App.II.60), since C$) includes the term yjk+p-2 for odd p and ~ j k + P - for ~ even p, which is higher order than yjk-' for p = 3 and p 2 5. In general, ( y j - yJP for p 2 2 is equal to a polynomial in y j of degree p or p - 2, depending on whether p is even or odd [125]. In these cases the characteristic equations must

200

Appendix I1

be solved by other methods. The eigenvalues of A(2)(0)and A(4)(0) can also be obtained by a quite different procedure using the residue theorem [1251. However, the method presented here is purely algebraic, and it clarifies the reason why the eigenvalues of the matrices A@)(O) and A(P)(0)(p 2 5 ) cannot be obtained in a simple manner. The eigenvectors may be calculated by employing Cramer’s formula, but we shall not perform this calculation here.

Appendix

111

Properties of the Hilbert Transform Operator

The Hilbert transform operator is defined as

where P denotes the principal value. The following properties are fundamental: (App.III.1)

H2f= -f,

(App.III.2)

20 1

202

Appendix 111

To show these formulas, define the Fourier transforms offand g by

J-m.f(k)e'k" dk, W

f(x)

=

g(x)

=

(A pp. I I I. 5 )

1m

mg(k)eik"dk,

(App.III.6)

and note the following formula, which is derived using a contour integral : H e i k X = i sgn(k)eikX, (App. I I I. 7) where sgn(k)

=

i

for k > 0, for k = 0, for k < 0.

I 0

-1

Then Hf

1-

m

W

=

(App.III.8a) (A pp. I I I .8b) (App.II1.8~)

mfHeik"dk = i /-,f sgn(k)eik"dk, (App.III.9)

and H2f= i

[

W

J - m

Tsgn(k)Heik" dk

W

= - J-mfeikx

dk

=

-

J - w

f[sgn(k)12eik" dk

-A

=

since [sgn(k)I2 = 1 by definition. This proves (App.III.1). From (App.III.9) and a formula dx = 27c6(k),

(App.III.10)

where 6(k) is Dirac's delta function, Hfdx

J-

00

=

=

i

m

m

dkfsgn(k) J-meikxdx m

27ci Jwmfsgn(k)d(k)dk

=

0,

since sgn(k)6(k)

=

0

(App.III.11)

203

Properties of the Hilbert Transform Operator

by (App.III.8). This proves (App.III.2). Now consider

J-

J-mdx1-mdk, J-mdk2f ( k 1 ) ~ ( k 2 ) e i k 1 X H e i k Z X m

m

f Hg dx

=

m

m

00

m

m

=

i J-mmdx JPmdk, ~-mdk2f(kl)Q(k2)ei(k1ikz)x sgn(k2)

=

2ni J-/k,

=

-2ni J-mf(kl)&-kl)

m

m

+ k 2 ) sgn(k2)

J-/k2f(k1)c7(k2)d(kl

m

and similarly

J-

m

gHfdx

=

JJm

=

2ni

(App.III.12)

m

-2ni

m

sgn(k,) d k , ,

m

m

~ ( k , ) f ( - k , )sgn(k,) d k ,

f ( k , ) g ' ( - k , ) sgn(k,) d k , , (App.III.13)

where, in passing to the second equality of (App.III.l3), the integration variable k , has been replaced by - k , and a formula sgn( - k l ) = - sgn(k,) has been used. Comparing (App.III.12) and (App.III.13), we obtain (App.III.3). Formula (App.III.4) is proved as follows: Consider m

m

Hfg

=

J-/k,

=

i S_mmdkl Jqndk2f(k,)g(k2) sgn(k,

+ k2).

(~pp.111.14)

J-mdk, J-mdk2f(k,)~(k2){eik1xHeik~x + eikzXHeiklX m

=

J-/k2f(k,)4(k2)He"L'+kz1

m

+ H[(Heiklx)(~eik~X)]} -

sgn(k,) sgn(k2) sgn(k,

+ k2)]ei(k1

+kz)x.

(APP.III.IS)

204

Appendix 111

Therefore

Hfs - {fHg + sHf+ m H f ) ( H s ) l J W

m

+ sgn(k1) sgn(k2)l

J-,dkl J-/k2f(k1)Q-(k2)"

=i

x sgn(k,

+ k 2 ) - sgn(k,) - ~ g n ( k ~ ) } e ~ ( ~(App.III.16) l+~~)~.

However, it is clear from the definition of function sgn(k) that c1

+ sgn(k1) sgn(k2)l sgn(k, + k 2 ) = sgn(k1) + sgn(k2). (App.III.17)

which, substituted into (App.III.16), gives (App.III.4). It follows as a consequence of (App.III.1) and (App.III.3) that

S_mmf Hfdx

1m

m

=

(App.III.18)

0,

J-,fs dx. W

W.f)(Hs) dx

=

(App.III.19)

We define the projection operators P, by

P, = $ ( I

+ iH),

(App.III.20)

= +(I

- iH).

(App.III.2 1)

P-

The following formulas are then verified using (App.III.l)-(App.III.3): (App.III.22)

(p*)2= p * ,

J-

m m

P, P-

=

P - P,

(P*f)(P, 9) dx

=

slam

=

(App.III.23)

0,

f(P7 P*)g dx

=

0. (App.III.24)

The following useful formula is derived using properties (App.III.1)(App.III.4): m

(App.111.25)

205

Properties of the Hilbert Transform Operator

where the integration with respect to x should be interpreted as a principal value. It follows from (App.III.4) with g =fthat f H f = + H [ f ' - (Hf)']

=

HF,

(App.III.26)

where F = +[f' - (Hf)'].

(App.III.27)

It should be noted from (App.III.1) and (App.III.3) that

J-

m

F dx = 0.

(App.III.28)

m

We denote the Fourier transform of F as F,that is, m

F(y) =

mF(k)eikydk, m

F ( k ) = 1/2n J-mFCy)e-ikY dy.

(App.III.29) (App.III.30)

From (App.III.28) and (App.III.30) we derive the relation

F(0) = 1/2n

m

FCy) dy = 0.

(App.III.31)

J-m

Then J

= =

J-:

m

lim

n-tm

=

J;mfHfdY

J:?

SlmHF(y)dy

J:? dk F(k) sgn(k) J: _eikydy lim i J:? J-mdk F(k) sgn(k)[eikx/ik+ n6(k)], lim i

n+m

m

=

n+ m

(App.III.32)

206

Appendix 111

where we have used (App.III.7) and a formula lim eiky/ik= -n6(k).

(App.ITI.33)

y--m

By considering (App.111.lo), J is estimated as

J-

m

J = lim 2nF(k)/k k-0

sgn(k)d(k) dk m

J-

+ lim 2ninF"(O) n- m

m

m

sgn(k)d(k) dk

=

0, (App.III.34)

since limk+,, F(k)/k is finite by (App.III.31) and (App.III.11). This completes the proof of (App.III.25).

Appendix

Iv

Proof of (3.274)

Define functions F(a) and g ( k ) by (AppJV. la) (App.IV.1 b)

F(a) =

F(a) =

J-

m

m

g(k)eik”dk.

(App.IV.2)

Introducing (App.IV.1) and (App.IV.2) into (3.273) yields m

(44)”- ‘F(a)da

=

207

(uiI/n)

J4,

@(<) d<. (App.IV.3)

208

Appendix IV

On the other hand, it follows from (App.IV.2) that

J-

1m

W

(44)"- 'F(a) da

=

W

-

J-,

dk g ( k ) J-mW(a/4)"- 'eikad a

W

dk g ( k ) ( 4 i ) - @ - ')

= 2n(i/4)"=

J-

an- ' / a k n -

1W

_eikad a

m

' ) ( k ) d ( k )dk

2n(i/4)"- 'g("- "(O),

(App.IV.4)

where 9'"- ' ) ( k ) = d"- 'g(k)/akn-'.

Substituting (App.IV.4) into (App.IV.3) gives

Thus

Introducing (App.IV.7) into (App.IV.2) yields

(App.IV.5)

209

Proof of (3.274)

where use has been made of the formula

with

JOm

eikxdk = nh(x; -a, a),

h(x; -a, a) =

{

1, 0,

1x1 < a, 1x1 > a.

a > 0,

(App.IV.9)

(AppJV. 10a) (AppJV.lob)

It follows from (App.IV.8)-(App.IV.10) that

F(a) =

(R

rJ/4nuo

which proves (3.274).

J

4u04

=.

d<,

for a > 0, ( ~ p p . 1 v .la) 1

(I

for a < 0, (App.IV.l lb)

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99. K. M. Case. Meromorphic solutions of the Benjamin-Ono equation. Phys. A %, 173-182 (1979). 100. J. Moser. Three integrable Hamiltonian systems connected with isospectral deformations. Ado. Math. 16, 197-220 (1975). 101. K. M. Case. Benjamin-Ono-related equations and their solutions. Proc. Nut. Acad. Sci. U.S.A. 76, 173-182 (1979). 102. K. M. Case. The Benjamin-Ono and related equations [I]. Phys. D 3, 185-192 (1981). 103. A. Nakamura. Backlund transform and conservation laws of the Benjamin-Ono equation. J. Phys. SOC.Japan 47, 1335-1340 (1979). 104. T. L. Bock and M. D. Kruskal. A two-parameter Miura transformation of the Benjamin-Ono equation. Phys. Lett. A 74, 173-176 (1979). 105. R. M. Miura. Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation. J. Math. Phys. 9, 1202-1204 (1968). 106. Y. Matsuno. Recurrence formula and conserved quantity of the Benjamin-Ono equation. J. Phys. SOC.Japan 52,2955-2958 (1983). 107. A. S. Fokas and B. Fuchssteiner. The hierarchy of the Benjamin-Ono equation. Phys. Lett. A 86,341-345 (1981). 108. H. H. Chen, Y. C. Lee, and J.-E. Lin. On a new hierarchy of symmetries for the Benjamin-Ono equation. Phys. Lett. A 91, 381-383 (1982). 109. L. J. F. Broerand H. M. M.Ten Eikelder. Constantsofthemotion fortheBenjaminOno and related equations. Phys. Left. A 92, 56-58 (1982). 110. Y.Matsuno. Number density function of Benjamin-Ono solitons. Phys. Lett. A 87, 15-17 (1981). 111. Y. Matsuno. Asymptotic properties of the Benjamin-Ono equation. J. Phys. SOC. Japan 51,667-674 (1982). 112. Y. Matsuno. Soliton and algebraic equation. J. Phys. SOC. Japan 51, 3375-3380 (1982). 113. H.Hochstadt. “The Functions of Mathematical Physics.” Wiley, New York, 1971. 114. Y. Matsuno. Interaction of the Benjamin-Ono solitons. J . Phys. A 13, 1519-1536 (1980). 115. J. Satsuma, M. J. Ablowitz, and Y. Kodama. On an internal wave equation describing a stratified fluid with finite depth. Phys. Lett. A 73,283-286 (1979). 116. M. Adler and J. Moser. On a class of polynomials connected with the Korteweg-de Vries equation, Comm.Math. Phys. 61, 1-30 (1978). 117. Y. Matsuno. The Backlund transformations of the higher-order Korteweg-de Vries equation. Phys. Lett. A 77, 100-102 (1980).

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Author Index

Dikii, L. A,, 41, 213 Dobbs, L. D., 212

A Ablowitz, M. J., 2, 211, 215, 216, 217 Acrivos, A., 212 Adler. M., 161, 216 Ahmed. S.. 217 Airault, H.. 65, 215

E Egri. R . , 212 Estabrook, F. B., 32, 162, 213

B F

Benjamin, T. B., 55. 212 Bock, T. L., 73, 216 Broer, L. J. F., 216 Bruschi, M., 217 Bullough, R. K., 2. 21 I

Fokas. A. S., 216 Fuchssteiner, B., 216

G C

Gardner, C. S., I , 41. 168, 211. 213 Gel'fand, I . M . , 41. 213 Greene. J. M., I . 41. 168, 211, 213

Calogero, F., 2. 211, 215. 217 Case. K. M., 65, 215, 216 Caudrey. P. J., 2, 211 Chen, H. H., 214, 215. 216 Chu. F. Y . F.. 213 Choodnovsky, D. V.. 215 Choodnovsky, G. V., 215

H Hirota, R.. 2. 6. I I . 33. 38, 41. 45, 212-215 Hochstadt. H., 216

D Davis, R. E., 212 Degasperis, A., 2, 21 I de Vries, G.. 5 . 213

1

Ishirnori, Y . , 214 219

Author Index

J Jeffrey, A., 213, 216 Joseph, R. I . , 212

K Kaup, D. J., 211, 216 Kakutani, T., 213 Kawahara, T., 216 Keener, J. P., 216 KO, D. R. S., 212 Kodama, Y.,216, 217 Korteweg, D. J., 5 , 213 Kotem, T., 213 Kruskal, M. D., I , 41, 65, 73, 168, 211, 213, 215, 216 Kubota, T., 212

L Lamb, G. L., 2, 21 I Lax, P. D., 45, 21 I Lee, Y. C., 214, 215, 216

M Maslowe, S. A.. 216 Matsuno, Y.,57, 79. 97. 212, 214-216 Mckean, H. P.. 65, 215 Mclaughlin, D. W., 213, 216 Miura, R. M., 1, 41, 168, 211-213, 216 Moser, J., 65, 161, 215, 216

N Nakamura, A.. 24, 30, 38, 212-216 Nayfeh, A. H., 216 Newell. A. C., 211, 216

0 Oishi, S., 17, 213, 215 Olshanetsky, M. A,, 217 Ono, H., 5 5 , 212

P Pereira, N. R., 215 Perelomov, A. M., 217

R Redekopp, L. G., 216 Rogers, C., 30, 212 Rosales, R., 17, 213

S Satsuma. J . , 33, 212-216 Sawada, K., 213 Scott, A. C., 213 Segur, H., 2, 21 1 Shabat. A. B., 21 I Shadwick, W. F., 30, 212

T Tanaka, M., 213, 216 Tanaka, S., 213 Ten Eikelder, H. M. M., 216 Thickstun, N., 65, 215 Toda, M.. 213, 214

W Wadati, M., 213 Wahlquist, H. D., 32, 162, 213 Z

Zakharov. V. E., 21 I

Subject Index

A Adjoint operator, 179 Associated linear equation, for BO equation, 73 Asymptotic behavior, 98, 99 Asymptotic solution, 79, 83

B Backlund transformations chain of, 175 of BO equation, 68, 69 of higher-order KdV equation, 158, 162 of KdV equation, 30, 31 Benjamin-Ono equation, 43, 48, 68 Benjamin-Ono-related equations, 1 17 Bilinear equation, 7 Bilinearization of BO equation, 50, 54, 68 of finite-depth fluid equation, 133 of higher-order BO equations, I18 of higher-order finite-depth fluid equations, 145 of higher-order KdV equations, 126 of higher-order modified KdV equations, I50 of KdV equation, 6 Bilinear operators, 6, 185 Binomial coefficient, 121 BO equation, see Benjamin-Ono equation

80-related equations, see Benjamin-Ono-related equations Boussinesq equation, 43 C

Characteristic polynomial, 193 Cofactor, 58, 193 Collisions of solitons, 99 Commutability relations, 34 Conservation laws of BO equation, 72 of higher-order KdV equations, 166 of KdV equation, 39 Conserved density, 39 Contour integral, 18 Cramer’s formula, 58, 91 Cylindrical KdV equation, 43

D Dependent variable transformation, 6, 7, 119, 134, 151, 170 Derivative nonlinear Schriidinger equation, 43, 175 Determinant. 52, 195 Dirac’s delta function, 22, 94, 202 Dispersion relation, 10, 21, 92 Dispersive wave, I8 Dissipation, 178 22 I

222

Subject Index

E Eigenvalue, 198. 199 Euler formula, 89

F Finite-depth fluid equation. 43, 133 Flux, 39 Fourier transform. 92. 202, 205 Fredholm first minor, 23 Functional derivative. 76

G Gel’fand-Levitan-Marchenko equation, 24 Generalized Toda equation, discrete analogue of, 45 Gram determinant, 23

H Hamilton-Cayley theorem, 193 Hermite polynomial, 198 Higher-order BO equations, 118 Higher-order finite-depth fluid equations, I45 Higher-order KdV equations, I26 Higher-order modified KdV equations, I50 Hilbert transform operator, 43, 48, 201 Hirota’s theorem, 14

I Interaction, BO solitons, 97 Inverse scattering method, 1 Inverse scattering transforms of BO equation, 79 of higher-order KdV equations, 166 of KdV equation, 42

J Jacobian elliptic function, 176

K Kadomtsev-Petviashvili equation, 44 KdV equation, 6 initial value problem of. 22 Kronecker’s delta. 60. 192

L Laguerre polynomial, 86 Lax form. 65 Lax hierarchy, 45, 118. 144. 150 Linearized BO equation, 92 Linearized KdV equation. 1 I Lorentzian profile, 48, 64. 85

M Miura transformation, 74 Model equation for shallow-water waves, 43 Modified BO equation, 170 Modified KdV equation, 38, 42 Multiple time-scale expansion, 178

N N-body Hamiltonian system, 65 Nonlinear differential-difference equation, 44 Nonlinear network equation, 44 Nonlinear partial difference equation, 44 Nonlinear wave equation, envelope soliton solutions, 42 Nonsecularity condition. 180. 181 N-periodic wave solution of BO equation, 55 of higher-order BO equations, 125 of modified BO equation, 173 N-soliton solution of BO equation, 52 of finite-depth fluid equation, 138 of higher-order BO equations, 125 of higher-order finite-depth fluid equation, 150 of higher-order KdV equations, 132

223

Subject Index of higher-order modified KdV equations, I58 of KdV equation, I I . 16 of modified BO equation, 173 Number density function, BO solitons, 83

P Periodic wave solution of finite-depth fluid equation, 141 of KdV equation, 24 Perturbed BO equation. 178 Phase shift formula for BO equation, 99, 140 for finite-depth fluid equation, 140 for KdV equation, 16, 141 Pole expansion method, 64 Poles. BO solitons. 103 Potential function, 70 Projection operator. 204

R Rank of polynomial, 118, 129 Rational solution, 4.5 Rectangular well, 84 Recursion formula for KdV equation. 40, 41. 127 for BO equation, 75 Riccati equation, 40 Ripple, 18 S Sawada-Kotera equation, 7

Self-dual nonlinear network equation. 44 Similarity solution, BO equation, 94 Similarity transformation. 94 Similarity variable, 94 Sine-Gordon equation, 42 Soliton solution, generalized, 17 Stability. BO solitons. 90 Superposition formula, 34

T Tail, 8.5 Taylor’s formula, 187 Theta function, 25, 29 Toda equation, 44 Translation invariance, 1.53 Two-dimensional KdV equation, 44 Two-dimensional nonlinear Schriidinger equation, 44 Two-dimensional Sine-Gordon equation, 44

v Vandermonde determinant, 91 Volterra system, 44

2

Zeros of Hermite polynomial, 198 of Laguerre polynomial, 64,87

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