Chaos, Solitons and Fractals 23 (2005) 175–181

www.elsevier.com/locate/chaos

A modified B€acklund transformation andmulti-soliton solution for the Boussinesq equation

Yi Zhang a,b,*, Deng-yuan Chen b

a Department of Mathematics, Zhejiang Normal University, Jinhua 321004, PR Chinab Department of Mathematics, Shanghai University, Shanghai 200436, PR China

Accepted 5 April 2004

Communicated by Prof. M. Wadati

Abstract

It is shown that a modified B€acklund transformation is presented by the dependent variable transformation. Starting

from it, a new representation of N-soliton solution and a class of novel multi-soliton solution of the Boussinesq

equation have been derived. We also find the novel soliton solution may be deduced by limiting.

� 2004 Elsevier Ltd. All rights reserved.

1. Introduction

In the soliton theory, a remarkable aspect is that the nonlinear evolution equations governing them are exactly

solvable. Especially, it has been shown by analytical investigations that there is the existence of N -soliton solution

describing multiple collision solutions is a common algebraic feature of these nonlinear evolution equations. There

exists several direct methods to construct N-soliton solution, among them the bilinear method [1]. The advantage of this

method is that the N -soliton solution is represented by a purely algebraic procedure. It is significant to search for

bilinear N-soliton solution representation of a given nonlinear evolution equations in the investigation of its integra-

bility [2–7]. Not only has the Hirota bilinear method provide a powerful tool in searching for soliton solution, but it has

also a close relationship to the B€acklund transformations (BTs). By applying well chosen exchange formulas, bilinear

BTs can been derived for a variety of soliton equations [8–12]. These bilinear BTs consist of a set of parameter

dependent bilinear equations. Starting from the bilinear BTs, the multi-soliton solution of the nonlinear evolution

equations may be generated and the structure of the solution be clarified.

In this paper we present a new form BTs in bilinear which has the much more desirable property of transforming

between N -soliton and N þ 1-soliton solution of the Boussinesq equation. The mathematics used here is only an ele-

mentary theory of linear partial differential equation. We also present a process to show that the new representation of

N -soliton solution and the representation of the bilinear equations are the same for recovering the solutions of the

Boussinesq equation. The inverse scattering scheme for the Boussinesq equation has a third order eigenvalue equation,

which is discovered by Zakharov [13].

The paper is organized as follows. In Section 2, we will present a modified bilinear BTs for the Boussinesq equation.

In Section 3, based on the modified bilinear BTs, a new representation of N -soliton solution for the Boussinesq equation

* Corresponding author.

E-mail address: zy2836@163.com (Y. Zhang).

0960-0779/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.chaos.2004.04.006

176 Y. Zhang, D. Chen / Chaos, Solitons and Fractals 23 (2005) 175–181

is obtained in agreement with Hirota’s standard expression. In Section 4, some exact novel solutions of equation are

found by modified bilinear BTs. Finally Section 5 summarizes the results.

2. A modified form of BT for the Boussinesq equation

The Boussinesq equation reads [14]

utt � uxx � ð3u2Þxx � uxxxx ¼ 0; ð2:1Þ

which describes motions of long waves in one-dimensional arises in a nonlinear lattices and in shallow water under

gravity. Through the dependent variable transformation

u ¼ 2ðln f Þxx; ð2:2Þ

Eq. (2.1) becomes

ðD2t � D2

x � D4xÞf � f ¼ 0; ð2:3Þ

where defining

Dmx D

nt f � g ¼ ð@x � @x0 Þmð@t � @t0 Þnf ðx; tÞgðx0; t0Þ j x0¼x;t0¼t:

Before we proceed, it is instructive to recall the following facts. The Boussinesq equation is also a continuum

approximation to equation for one-dimensional nonlinear lattices. Toda and Wadati proposed a one-dimensional

nonlinear lattice with neighboring particles interacting through an exponential potential function, which admits exact

analytic solution [15]; Wadati also presented the B€acklund transformation too [16]. By continuing, the multi-soliton

solution for Eq. (2.1) can be constructed; Hirota and Satsuma later obtained intuitively general N-soliton solution [9].

Let f be a solution of Eq. (2.3) and g another one of the same equation. The BTs of Eq. (2.1) which relate f and g in

bilinear forms are given by [17]

ðDt þ aD2xÞf � g ¼ 0; ð2:4Þ

ðaDtDx þ Dx þ D3xÞf � g ¼ kfg; ð2:5Þ

where k is an arbitrary parameter and a2 ¼ �3.

In order to obtain a modified BT in bilinear form, we make

f ! enf ; ð2:6Þ

g ! egg; ð2:7Þ

where n ¼ x1t þ k1xþ nð0Þ; g ¼ r1t þ h1xþ gð0Þ; nð0Þ; gð0Þ are arbitrary constants.

By expanding f ; g in the following way

f ¼ 1þ ef ð1Þ þ e2f ð2Þ þ e3f ð3Þ þ � � � ; ð2:8Þ

g ¼ 1þ egð1Þ þ e2gð2Þ þ e3gð3Þ þ � � � ; ð2:9Þ

then Eqs. (2.6) and (2.7) are transformed into

½Dt þ ðx1 � r1Þ�f � g þ a½Dx þ ðk1 � h1Þ�2f � g ¼ 0; ð2:10Þ

faDx þ ðk1 � h1Þ½Dt þ ðx1 � r1Þ� þ ½Dx þ ðk1 � h1Þ�3gf � g ¼ kfg ð2:11Þ

respectively, where using the following bilinear operator identity

Dmt D

nxe

nf � egg ¼ enþgðDt þ x1 � r1ÞmðDx þ k1 � h1Þnf � g; ð2:12Þ

equating the constant terms, we have

x1 þ ak21 ¼ 0; ð2:13Þ

ak1x1 þ k31 � k ¼ 0: ð2:14Þ

Y. Zhang, D. Chen / Chaos, Solitons and Fractals 23 (2005) 175–181 177

Thus we we arrive at the modified BTs of Boussinesq equation as follows

ðDt þ aD2x þ 2akDxÞf � g ¼ 0; ð2:15Þ

½akDt þ aDtDx þ ð1þ 6k2ÞDx þ 3kD2x þ D3

x �f � g ¼ 0; ð2:16Þ

where k is an arbitrary parameter. Then, from the bilinear BTs (2.15) and (2.16), we can obtain the exact multi-soliton

solutions for the Boussinesq equation.

3. A new representation of N-soliton solution for the Boussinesq equation

In this section, we start from the bilinear BTs (2.15) and (2.16) and derive a new representation of N -soliton solution

for the Boussinesq equation.

In order to obtain solution of Eqs. (2.15) and (2.16), we substitute (2.8) and (2.9) into (2.15) and (2.16) and equating

coefficients of �, yields a system of linear equation to be solved.

f ð1Þt þ af ð1Þ

xx þ 2akf ð1Þx � gð1Þt þ agð1Þxx � 2akgð1Þx ¼ 0; ð3:1aÞ

af ð1Þtx þ akf ð1Þ

t þ ð1þ 6k2Þf ð1Þx þ 3kf ð1Þ

xx þ f ð1Þxxx þ agð1Þtx � akgð1Þt � ð1þ 6k2Þgð1Þx þ 3kgð1Þxx � gð1Þxxx ¼ 0; ð3:1bÞ

f ð2Þt þ af ð2Þ

xx þ 2akf ð2Þx � gð2Þt þ agð2Þxx � 2akgð2Þx ¼ �ðDt þ aD2

x þ 2akDxÞf ð1Þ � gð1Þ; ð3:1cÞ

af ð2Þtx þ akf ð2Þ

t þ ð1þ 6k2Þf ð2Þx þ 3kf ð2Þ

xx þ f ð2Þxxx þ agð2Þtx � akgð2Þt � ð1þ 6k2Þgð2Þx þ 3kgð2Þxx � gð2Þxxx

¼ �½akDt þ aDtDx þ ð1þ 6k2ÞDx þ 3kD2x þ D3

x �f ð1Þ � gð1Þ ð3:1dÞ

and so on.

We start with gð1Þ ¼ 1, corresponding to the zero solution u ¼ 0 of Eq. (2.1), we have from Eqs. (3.1a) and (3.1b)

4f ð1Þxx þ 12kf ð1Þ

x þ ð1þ 12k2Þf ð1Þ ¼ 0: ð3:2Þ

We show that f ð1Þ can be chosen the following

f ð1Þ ¼ en1 ; ð3:3aÞ

where

n1 ¼ ðh1 þ k1Þx� aðh21 � k21Þt þ nð0Þ1 ð3:3bÞ

and

h1 þ k1 ¼ �4ðh31 þ k31Þ ð3:3cÞ

for

k ¼ �k1: ð3:3dÞ

It is readily verified that the linear dispersion relation of (2.3)

X2 ¼ K2 þ K4 ð3:4aÞ

is satisfied by K ¼ h1 þ k1;X ¼ �aðh21 � k21Þ and the phase shift is expressed by

eAmj ¼ ðhm � hjÞðkm � kjÞðhm þ hjÞðkm þ kjÞ

: ð3:4bÞ

Now we take gð1Þ ¼ en1 . It is readily shown that f ð1Þ can be chosen by

f ð1Þ ¼ a1en1 þ a2en2 ; ð3:5aÞ

where

ni ¼ ðhi þ kiÞx� aðh2i � k2i Þt þ nð0Þi ; i ¼ 1; 2; ð3:5bÞ

178 Y. Zhang, D. Chen / Chaos, Solitons and Fractals 23 (2005) 175–181

a1 ¼h1 þ k2k2 � k1

; a2 ¼ � h2 þ k1k2 � k1

ð3:5cÞ

for

k ¼ �k2: ð3:5dÞ

Substitution of (3.5) into Eqs. (3.1c) and (3.1d) and solving the resulting equation, we get

f ð2Þ ¼ a12en1þn2 ; ð3:6aÞ

where

a12 ¼h1 � h2k2 � k1

; ð3:6bÞ

while all the terms higher than �2 in (3.1) can be taken to be zero and we deduce

f ¼ f2 ¼ 1þ a1en1 þ a2en2 þ a12en1þn2 : ð3:7Þ

If we choose k ¼ �k3 and gð1Þ; gð2Þ given by Eqs. (3.5) and (3.6) respectively, then Eqs. (3.1a) and (3.1b) are satisfied by

f ð1Þ ¼ b1en1 þ b2e

n2 þ b3en3 ; ð3:8aÞ

where

b1 ¼ðh1 þ k2Þðh1 þ k3Þðk1 � k2Þðk1 � k3Þ

; b2 ¼ðh2 þ k1Þðh2 þ k3Þðk2 � k1Þðk2 � k3Þ

; b3 ¼ðh3 þ k1Þðh3 þ k2Þðk3 � k1Þðk3 � k2Þ

: ð3:8bÞ

When we substitute f ð1Þ into Eqs. (3.1c) and (3.1d), then

f ð2Þ ¼ b12en1þn2 þ b13en1þn3 þ b23en2þn3 ; ð3:9aÞ

where

b12 ¼ �ðh1 � h2Þðh1 þ k3Þðh2 þ k3Þðk1 � k2Þðk1 � k3Þðk2 � k3Þ

;

b13 ¼ðh1 � h3Þðh1 þ k2Þðh3 þ k2Þðk1 � k2Þðk1 � k3Þðk2 � k3Þ

;

b23 ¼ �ðh2 � h3Þðh2 þ k1Þðh3 þ k1Þðk1 � k2Þðk1 � k3Þðk2 � k3Þ

:

ð3:9bÞ

As before, inserting Eqs. (3.8), (3.9) and (3.1), we find

f ð3Þ ¼ Ben1þn2þn3 ; ð3:10aÞ

where

B ¼ �ðh1 � h2Þðh1 � h3Þðh2 � h3Þðk1 � k2Þðk1 � k3Þðk2 � k3Þ

: ð3:10bÞ

Thus we obtain a three-soliton solution

f ¼ f3 ¼ 1þ f ð1Þ þ f ð2Þ þ f ð3Þ; ð3:11Þ

where f ð1Þ; f ð2Þ; f ð3Þ is mentioned above.

The procedure described above of solving soliton solution for the Boussinesq equation suggest to us a form of

possible N -soliton solution

f ¼ fN ¼Xlj¼0;1

expXNj¼1

lj nj

"(þ

XNm¼1;m6¼j

ð1� lmÞAjm

#þ

XN16 j6m

ljlmðBjm þ piÞ); ð3:12aÞ

Y. Zhang, D. Chen / Chaos, Solitons and Fractals 23 (2005) 175–181 179

where

eAjm ¼ hj þ kmkm � kj

; eBjm ¼ hj � hmkj � km

ð3:12bÞ

and the first sum is taken over all possible combination of lj ¼ 0; 1.It is easily shown that the new representation of N-soliton solution (3.12) is equivalent to the Hirota’s N-soliton

solution formulas [9].

4. The novel multi-soliton solution of the Boussinesq equation

In the previous section, we presented a modified BTs in bilinear and drive a new representation of the N-soliton

solution for the Boussinesq equation. Due to the inherent nature of the linear differential equation with constant

coefficients, one may utilize arbitrary constant k. It is shown that the novel multi-soliton solutions can be deduced from

BTs. In what follows we explain this procedure.

To obtain novel multi-soliton solutions it is more convenient to deal with arbitrary constant k. For this purpose letus again consider Eq. (3.4), but k is satisfied with

k ¼ �k1: ð4:1Þ

From Eqs. (3.1a) and (3.1b), we have

f ð1Þ ¼ g1en1 ; ð4:2aÞ

where g1 ¼ a1t þ b1xþ c1; c1 being arbitrary parameter, corresponding coefficients as follows

a1 ¼�2aðh1 þ k1Þð1þ 12h1k1Þ

8h1 � 4k1; b1 ¼

�3ðh1 � k1Þðh1 þ k1Þ2h1 � k1

: ð4:2bÞ

Substituting (4.2) into Eqs. (3.1a) and (3.1b) gives

f ð2Þ ¼ ce2n1 ; ð4:3aÞ

where

c ¼ 1þ 12k211þ 12h21

: ð4:3bÞ

Finally, we find

f ðiÞ ¼ 0; iP 3: ð4:4Þ

Therefore the series (2.9) truncates and becomes

f ¼ 1þ g1en1 þ ce2n1 : ð4:5Þ

Now, we take

gð1Þ ! g1en1 ; gð2Þ ! ce2n1 ; gðiÞ ¼ 0 ðiP 3Þ; ð4:6Þ

where g1; c are mentioned above.

In a similar procedure, one can establish the form of multi-soliton solution as

f ¼ 1þ f ð1Þ þ f ð2Þ þ f ð3Þ; ð4:7aÞ

where

f ð1Þ ¼ ða1g21 þ a2g1 þ a3t þ a4Þen1 ; f ð2Þ ¼ ðb1g21 þ b2g1 þ b3t þ b4Þe2n1 ; f ð3Þ ¼ c1e3n1 ð4:7bÞ

and

180 Y. Zhang, D. Chen / Chaos, Solitons and Fractals 23 (2005) 175–181

a1 ¼ � aðh1 þ k1Þ2

a1 þ 2ah1b1

;

a2 ¼ f�aðh1 þ 2k1Þa1 þ ð1þ 12h1k1 þ 18k21Þb1 þ ah1½�a1 þ 2aðh1 þ 2k1Þb1�gða1 þ 2ah1b1Þ

þ 2aðh1 þ k1Þ2b1ðaa1 þ 6h1b1Þ=ðb1ð1þ 12h21ÞÞ;a3 ¼ �2ðh1 þ k1Þf�3ah1ðh1 þ k1Þa1b2

1 þ aa1½�a21 þ ð2þ 3h21 þ 39h1k1Þb21�

þ 3b1½�3h1ð1þ 4h21Þb21 þ k1ð2a21 þ b2

1 � 36h21b21Þ�g=ðb1ð1þ 12h21Þða1 þ 2ab1ÞÞ;

b1 ¼ � a1ða1 � 2ak1b1Þ2aðh1 þ k1Þ2

; b3 ¼a3ða1 � 2ak1b1Þ2aðh1 þ k1Þ2

;

b2 ¼a½a3ðh1 þ k1Þ2 � 4a1h1a1b1� � a1½a21 þ 6ð�h21 þ 2h1k1 þ k21Þb

21�

6ðh1 þ k1Þ4;

b4 ¼1

12aðh1 þ k1Þ6fa1a31 þ 2aa1ð4h1 þ k1Þa21b1 þ 6½a4ðh1 þ k1Þ4a1 þ 2b1ððh1 þ k1Þ3 þ a1ð�4h21 þ k21Þa1b1Þ�

� 12að3cðh1 þ k1Þ6 þ b1½a4k1ðh1 þ k1Þ4 þ b1ð�2a2ðh1 þ k1Þ4 � 2a1h1ð�h21 þ h1k1 þ k21Þb1ÞÞ�g;

c1 ¼�a3cþ ðb4 � a2cÞa1 � 2a½a4cðh1 þ k1Þ2 þ b1ð�b4h1 � a2ch1 � 2a2ck1 � b2b1 þ a1cb1Þ�

6aðh1 þ k1Þ2:

ð4:7cÞ

We now consider the another choice for the k in (4.6) and set

k ¼ �k2: ð4:8Þ

Applying similar technique, the another of the novel multi-soliton solution for Eqs. (2.15) and (2.16) is derived as

f ¼ 1þ f ð1Þ þ f ð2Þ þ f ð3Þ; ð4:9aÞ

where

f ð1Þ ¼ ðA1g1 þ A2Þen1 þ en2 ;

f ð2Þ ¼ ðB1g1 þ B2Þen1þn2 þ B3e2n1 ;

f ð3Þ ¼ C1e2n1þn2 ;

f ðiÞ ¼ 0; iP 4

ð4:9bÞ

and

A1 ¼h1 þ k2k2 � k1

;

A2 ¼ � 2ðh1 þ k1Þ2ð12k2h1 � 12k1h2 � 1Þð1þ 12h21Þðk1 � k2Þ2

;

B1 ¼h2 � h1h2 þ k1

; B2 ¼a1 þ 2ah1b1 � 2ah2b1 � 2ak1b1

2aðh2 þ k1Þ2;

B3 ¼ �ð4acðh1 þ k1Þ2 þ 2acðh21 � k21Þ þ 4acðh1 þ k1Þk2 � A2a1 þ 2aA2k2b1 � 2aA1b21Þ

=ð2ah21 þ 8ah1k1 þ 6ak21 � 4ah1k2 � 4ak1k2Þ;

C1 ¼B2a1 þ 2a½�cðh1 þ k1Þð3h1 � 2h2 þ k1Þ þ b1ðB2h2 þ B1b1Þ�

2aðh1 þ k1Þðh1 þ 2h2 þ 3k1Þ:

ð4:9cÞ

In a similar manner, more novel multi-soliton solution analogous (4.5) and (4.7) can be studied too, but the details

will not be pursued further.

5. Conclusions and discussions

By the dependent variable transformations, a modified bilinear BTs can be obtained. Starting from the modified

bilinear BTs, a new representation of N -soliton solution and a class of novel multi-soliton solutions of the Boussinesq

Y. Zhang, D. Chen / Chaos, Solitons and Fractals 23 (2005) 175–181 181

equation have been derived. By taking the limit k2 ! k1, we find that (4.5) is just derived in (3.4), (4.9a) can be similarly

obtained through (4.7a). It means that the novel soliton solution may be deduced by limiting. It would be reasonable in

theory to continue to find novel solutions but more involved as N increases. We expect that this would be solved in the

future.

Acknowledgements

We would like to express our thanks to the referees for their valuable suggestions and timely help. This project is

supported by the National Science Foundation of China and the Special Funds for Major Specialities of Zhejiang

Province.

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