Annals of Physics 323 (2008) 3059–3064

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Annals of Physics

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The exact solution and integrable properties to thevariable-coefficient modified Korteweg–de Vries equation

Yi Zhang *, Jibin Li, Yi-Neng LvDepartment of Mathematics, Zhejiang Normal University, Guest Road 288, Jinhua 321004, PR China

a r t i c l e i n f o

Article history:Received 25 March 2008Accepted 21 April 2008Available online 3 May 2008

Keywords:Solitonvc-mKdV equationBäcklund transformationLax pairsBidirectional wave interaction

0003-4916/$ - see front matter � 2008 Elsevier Indoi:10.1016/j.aop.2008.04.012

* Corresponding author. Fax: +86 579 82298188E-mail address: zy2836@163.com (Y. Zhang).

a b s t r a c t

In this paper, a variable-coefficient modified Korteweg–de Vries(vc-mKdV) equation is investigated. With the help of symboliccomputation, the N-soliton solution is derived through the Hirotamethod. Then the bilinear Bäcklund transformations and Lax pairsare presented. At last, we show some interactions of solitary waves.

� 2008 Elsevier Inc. All rights reserved.

1. Introduction

It is known that to find exact solutions of the nonlinear evolution equations (NLEEs) is always oneof the central themes in mathematics and physics. In the past few decades, there are noticeable pro-gress in this field. And various methods have been developed, such as the inverse scattering transfor-mation (IST) [1], the Bäcklund transformation (BT) [2], Darboux transformation (DT) [3,4], Hirota’sbilinear method [5,6], Wronskian technique [5,7], and so on.

However, to our knowledge, most of aforementioned methods are related to the constant-coeffi-cient models. Recently, there has been a growing interest in studying variable-coefficient NLEEs[8,9]. When the inhomogeneities of media and nonuniformity of boundaries are taken into accountin various real physical situations, the variable-coefficient NLEEs often can provide more powerfuland realistic models than their constant-coefficient counterparts in describing a large variety of realphenomena, as seen, e.g., in the coastal waters of oceans [10], superconductors [11], space and labo-ratory plasmas [12] and optical-fiber communications [13].

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3060 Y. Zhang et al. / Annals of Physics 323 (2008) 3059–3064

It is found that the variable-coefficient KdV (vc-KdV) equation, which is from Bose–Einstein con-densates and fluid dynamics, can be used to describe the water waves propagating in a channel withan uneven bottom and deformed walls and the trapped quasi-one-dimensional Bose–Einstein conden-sates with repulsive atom–atom interactions. In Ref. [14], Tian and Gao put their focus on a variablecoefficient higher-order nonlinear Schrödinger (vcHNLS) model which can be used to describe thefemtosecond pulse propagation, applicable to, e.g., the design of ultrafast signal-routing and disper-sion-managed fiber-transmission systems, and new transformation was proposed from vcHNLS modelto its constant-coefficient counterpart. The mKdV-typed equation, on the other hand, has been discov-ered recently, e.g., to model the dust-ion-acoustic waves in such cosmic environments as those in thesupernova shells and Saturn’s F-ring. In this paper, we will consider the variable-coefficient modifiedKorteweg–de Vries (vc-mKdV) equation, which can be read as

ut þ gðtÞuxxx þ f ðtÞu2ux þ lðtÞuþ qðtÞux ¼ 0; ð1:1Þ

where gðtÞ, f ðtÞ, lðtÞ and qðtÞ are all analytic functions.Eq. (1.1) has attracted considerable attention in many different physical fields including ocean

dynamics, fluid mechanics and plasma physics [15,16]. For example, when lðtÞ ¼ 0, it can be used toinvestigate the dynamics hidden in the plasma sheath transition layer and inner sheath layer.

For a generalized variable-coefficient NLEEs, it is not completely integrable unless the variablecoefficients satisfy some specific constraint conditions. It has been shown that in Refs. [17] theconstraint conditions on the coefficient functions for some variable-coefficient NLEEs to be mappedto the completely integrable constant-coefficient counterparts are precisely the same as those forsuch equation to possess Painlev�e properties. Eq. (1.1) can pass the Painlev�e test when f ðtÞ, gðtÞ,lðtÞ satisfy

f ðtÞ ¼ 6C2

0

gðtÞe2R

lðtÞ dt ; ð1:2Þ

where C0–0 is an arbitrary constant. That is to say, under the condition (1.2), Eq. (1.1) becomesintegrable.

The organization of the paper is as follows. In Section 2, under the Painlev�e-integrable condition,we obtain the bilinear forms and N-soliton solution of Eq. (1.1). We present the bilinear BTs andthe Lax pairs for the vc-mKdV equation in Section 3. In Section 4, we will show the interaction of mul-tiple solitons. Finally, the conclusions and discussions will be given in Section 5.

2. Bilinear forms and N-soliton solution of vc-mKdV equation

In this section, we will apply the Hirota method to construct N-soliton solution for the vc-mKdVequation. Based on this method, we first transform the vc-mKdV equation into bilinear forms, thenconstruct N-soliton solution for it.

Under the transformation

u ¼ IC0e�R

lðtÞ dt o

oxln

F�

FðI ¼

ffiffiffiffiffiffiffi�1p

Þ; ð2:1Þ

where F� is the conjugation of F. Eq. (1.1) can be transformed into the following bilinear forms

D2x F � F� ¼ 0; ð2:2aÞ½Dt þ gðtÞD3

x þ qðtÞDx�F � F� ¼ 0; ð2:2bÞ

where Dt and Dx are the Hirota bilinear operators introduced by

Dmx Dn

t a � b ¼ ð@x � @x0 Þmð@t � @t0 Þnaðx; tÞbðx0; t0Þjx0¼x;t0¼t :

Provide that F can be expanded as perturbation series

Fðx; tÞ ¼ 1þXN

j¼1

Fjðx; tÞej; ð2:3Þ

Y. Zhang et al. / Annals of Physics 323 (2008) 3059–3064 3061

where e is an arbitrary parameter. Substituting Eq. (2.3) into Eqs. (2.2) and equating coefficients of e,then through the standard process of Hirota method, we can derive the N-soliton solution of Eq. (1.1).For instance, the one-soliton solution can be expressed as

u ¼ C0e�R

lðtÞ dtk1sechn1; ð2:4Þ

with

n1 ¼ k1x� k31

ZgðtÞdt � k1

ZqðtÞdt þ nð0Þ1 :

Two-soliton solution can be written as follows

u ¼ IC0e�R

lðtÞ dt o

oxln

1� Ien1 � Ien2 � en1þn2þA12

1þ Ien1 þ Ien2 � en1þn2þA12

� �; ð2:5Þ

where

nj ¼ kjx� k3j

ZgðtÞdt � kj

ZqðtÞdt þ nð0Þj ðj ¼ 1;2Þ; eA12 ¼ k1 � k2

k1 þ k2

� �2

:

In a similar procedure, we can get the N-soliton solution, which can be shown as

u ¼ IC0e�R

lðtÞ dt o

oxln

Pl¼0;1 exp

PNj¼1ljðnj � I p

2Þ þPN

16j<lljllAjl

h iP

l¼0;1 expPN

j¼1ljðnj þ I p2Þ þ

PN16j<lljllAjl

h i8<:

9=;; ð2:6Þ

where

nj ¼ kjx� k3j

ZgðtÞdt � kj

ZqðtÞdt þ nð0Þj ; eAil ¼ ki � kl

ki þ kl

� �2

; j ¼ 1; . . . ;N; 1 6 i < l 6 N:

The firstP

l¼0;1 is taken over all possible combinations of l ¼ 0;1,PN

16j<l means a summation over allpossible pairs ðj; lÞ chosen from the set f1;2; . . . ;Ng under the condition j < l.

3. Bäcklund transformations and Lax pairs of vc-mKdV equation

In this section, we will derive the bilinear Bäcklund transformations and Lax pairs of Eq. (1.1).Let F1ðx; tÞ and F2ðx; tÞ be two distinct solutions of the Eqs. (2.2), which are

D2x F1 � F�1 ¼ 0; ð3:1aÞ½Dt þ gðtÞD3

x þ qðtÞDx�F1 � F�1 ¼ 0; ð3:1bÞ

D2x F2 � F�2 ¼ 0; ð3:2aÞ½Dt þ gðtÞD3

x þ qðtÞDx�F2 � F�2 ¼ 0: ð3:2bÞ

Consider the equations

F1F�1ðD2x F2 � F�2Þ � F2F�2ðD

2x F1 � F�1Þ ¼ 0; ð3:3aÞ

F1F�1½Dt þ gðtÞD3x þ qðtÞDx�F2 � F�2 � F2F�2½Dt þ gðtÞD3

x þ qðtÞDx�F1 � F�1 ¼ 0: ð3:3bÞ

It is obvious that if F1ðx; tÞ satisfies Eqs. (2.2), F2ðx; tÞ also satisfies the same equation. Therefore, Eqs.(3.3) can be regarded as a relation between the pair of solutions F1ðx; tÞ and F2ðx; tÞ, namely, the bilin-ear Bäcklund transformations of Eq. (1.1).

In the following, we will write Eq. (3.3) in more conventional forms. Making use of some bilinearoperator identities [18], Eqs. (3.3) turn into

Dx½ðDxF2 �F�1Þ �F�2F1þF2F�1 � ðDxF1 �F�2Þ�¼0; ð3:4Þ

F1F2f½DtþgðtÞD3x þqðtÞDx�F�1 �F

�2g�F�1F�2f½DtþgðtÞD3

x þqðtÞDx�F1 �F2gþ3DxðDxF1 �F�2Þ � ðDxF2 �F�1Þ¼0: ð3:5Þ

3062 Y. Zhang et al. / Annals of Physics 323 (2008) 3059–3064

If we let

DxF1 � F�2 ¼ IkF2F�1 ð3:6Þ

in Eq. (3.4), then Eq. (3.5) can be rewritten as

F1F2f½DtþgðtÞðD3x �3k2DxÞþqðtÞDx�F�1 �F

�2g�F�1F�2f½DtþgðtÞðD3

x �3k2DxÞþqðtÞDx�F1 �F2g¼0:

ð3:7Þ

Obviously, Eq. (3.7) is satisfied provided that the following equations hold:

½Dt þ gðtÞðD3x � 3k2DxÞ þ qðtÞDx�F1 � F2 ¼ 0: ð3:8Þ

Then Eqs. (3.6) and (3.8) constitute another forms of BT for the vc-mKdV equation.Next, we will follow the Ablowitz–Kaup–Newell–Segur (AKNS) approach to construct the Lax pairs

of Eq. (1.1).The linear eigenvalue problems for Eq. (1.1) can be expressed as

Ux ¼ UU; Ut ¼ VU; ð3:9Þ

where U ¼ ð/1;/2ÞT, T denotes the transpose of the matrix, while U and V are shown as follows

U ¼k e

RlðtÞ dtu

� 1C2

0eR

lðtÞ dtu k

0@

1A; V ¼

B C

D �B

� �;

where

B ¼ �4k3gðtÞ � qðtÞ � 2C2

0

e2R

lðtÞ dtgðtÞu2;

C ¼ �eR

lðtÞ dt 4k2gðtÞuþ 2kgðtÞux þ gðtÞuxx þ2C2

0

e2R

lðtÞ dtgðtÞu3 þ qðtÞu" #

;

D ¼ eR

lðtÞ dt

C20

4k2gðtÞu� 2kgðtÞux þ gðtÞuxx þ2C2

0

e2R

lðtÞ dtgðtÞu3 þ qðtÞu" #

;

and k is a parameter independent of x and t.It is easy to prove that the compatibility condition Ut � Vx þ ½U;V � ¼ 0 gives rise to Eq. (1.1), which

can show that Eqs. (3.9) are the Lax pairs of vc-mKdV equation. It is noted that the above-obtained BTsand Lax pairs can assure the complete integrability of Eq. (1.1).

4. Interaction of multiple solitons

In this section, we will show the interaction of solitary waves via a list of pictures.It is well-known that some equations, such as KdV equation, can only be used to describe the uni-

directional water wave interaction [19], some equations admit the bidirectional wave interaction,including head-on and overtaking collisions, such as Boussinesq system and AKNS system [20,21].We know that the constant-coefficient mKdV equation cannot be used to describe the bidirectionalsoliton interaction, but the vc-mKdV equation admits it. And we will show it as below.

Fig. 1 describes the head-on collision of two solitary waves along opposite directions of propaga-tion for Eq. (2.5). In contrast, Fig. 2 depicts the overtaking collision of two solitary waves along samedirections of propagation via Eq. (2.5), which can show the large-amplitude solitary wave with fastervelocity overtakes the small-amplitude one, after collision, the shorter is left behind. Fig. 3 shows thehead-on collision of two left-going solitary waves and one right-going solitary wave for Eq. (2.6),which can be regarded as the combination of Figs. 1 and 2.

Through the above pictures, we can know that the vc-mKdV equation possesses bidirectional waveinteractions including head-on and overtaking collisions, which is unlike the mKdV equation exhibit-ing the unidirectional interactions only. As a matter of fact, for many other variable-coefficient NLEEs,they also admit bidirectional wave interactions while their constant-coefficient counterparts cannot

0

x100500-50-100

3

2.5

2

1.5

1

0.5

Fig. 2. The overtaking collision of two solitary waves via Eq. (2.5), where k1 ¼ 1:25, k2 ¼ 2:5, gðtÞ ¼ 1, qðtÞ ¼ �0:75, lðtÞ ¼ 0,cðtÞ ¼ 2, n11 ¼ 0, n21 ¼ 1, n31 ¼ 0; t ¼ �8; 0;8.

2

x

1.5

1

100

0.5

0500-50-100

x6040

5

20

4

3

0

2

1

-200

-40-60

Fig. 1. The head-on collision of one left-going solitary wave and one right-going solitary wave for Eq. (2.5), where k1 ¼ 1,k2 ¼ 0:5, gðtÞ ¼ 1, qðtÞ ¼ �0:75, lðtÞ ¼ 0, cðtÞ ¼ 2, n11 ¼ 0, n21 ¼ 1; t ¼ �150; 0;150:

0

x100500-50-100

3

2.5

2

1.5

1

0.5

Fig. 3. The head-on collision of two left-going solitary waves and one right-going solitary wave via expression (2.6) with N ¼ 3,k1 ¼ 1, k2 ¼ 0:5, k3 ¼ 1:5, gðtÞ ¼ 0:53, qðtÞ ¼ �0:05, lðtÞ ¼ 0, cðtÞ ¼ 2, n11 ¼ 0, n21 ¼ 1, n31 ¼ 0; t ¼ �50; 0;50.

Y. Zhang et al. / Annals of Physics 323 (2008) 3059–3064 3063

3064 Y. Zhang et al. / Annals of Physics 323 (2008) 3059–3064

be used to describe this property, because comparing with the constant-coefficient NLEEs, the vari-able-coefficient NLEEs have more parameters in their solutions.

5. Discussions and conclusions

In this paper, under the constraint condition (1.2), the exact solution of vc-mKdV equation are de-rived by means of the Hirota method. Besides giving the solution, we present many properties of theaforementioned equation, such as the Bäcklund transformations and Lax pairs, which can be regardedas a sufficient condition to be integrable. At last, we show that the vc-mKdV equation possesses thebidirectional wave interactions by means of a few pictures. Furthermore, we can also get the Wrons-kian solution via Wronskian technique, and we will show it in another paper. For the constant-coef-ficient mKdV equation, many other types of solutions and properties have been obtained by manyauthors under different approaches. So are there any other forms of exact solutions and propertiesto the vc-mKdV equation? We hope to investigate in the future.

Acknowledgments

The authors express their sincere thanks to Prof. Yi-Shen Li for his enthusiastic guidance. This workis supported by the National Natural Science Foundation of China (No. 10671179 and No. 10771196).

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