stochastic soliton like solution - kdv equation

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The stochastic soliton-like solutionsof stochastic KdV equations

Yong Chen a,b,c,*, Qi Wang b,d, Biao Li b,d

a Department of Mathematics, Ningbo University, Ningbo 315211, Chinab Key Laboratory of Mathematics Mechanization, Chinese Academy of Sciences, Beijing 100080, China

c Department of Physics, Shanghai Jiao-Tong University, Shanghai 200030, Chinad Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, China

Accepted 21 June 2004

Communicated by Prof. M. Wadati

Abstract

By means of a generalized method and symbolic computation, we consider a stochastic KdV equation Ut+

f(t)UUx+ g(t)Uxxx= W(t)R(t, U, Ux, Uxxx). We construct new and more general formal solutions. At the same

time, we recover all the solutions found by Xie [Phys. Lett. A 310 (2003) 161]. The solutions obtained include the non-

travelling wave and coefficient functions stochastic soliton-like solutions, singular stochastic soliton-like solutions,

stochastic triangular functions solutions.

2004 Elsevier Ltd. All rights reserved.

1. Introduction

As is well known, the motion of long, unidirectional, weakly nonlinear water waves on a channel can be described by

the Kortewegde Vries (KdV) equation. When the surface of the fluid is submitted to a nonconstant pressure, or when

the bottom of the layer is not flat, a forcing term has to be added to the equation. This term is given by the gradient

of the exterior pressure or of the function whose graph defines the bottom. We are interested in the case when the forc-

ing term is random, which is a very natural approach if it is assumed that the exterior pressure is generated by a tur-

bulent velocity field for instance. We also assume that this random force is of white noise type[15]. In 1983, for the first

time M. Wadati answers an interesting question, how does external noise affect the motion of solitons?. In [1],theKdV equation under Gaussian noise is studied and it is showed that a soliton under Gaussian noise satisfies a diffusion

equation in transformed coordinates; the deformation of the soliton during the propagation is explicitly obtained; the

phenomenon is designated as the diffusion of soliton. In 1990, a nonlinear partial differential equation which describes

wave propagations in random media is presented by Wadati[2]. The stochastic equation (17) in[2]is useful for study of

similar problem in hydrodynamics and plasmas physics. Recently, the stochastic KdV equation arises when modelling

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

doi:10.1016/j.chaos.2004.06.049

* Corresponding author. Address: Department of Mathematics, Ningbo University, Ningbo 315211, China.

E-mail address: [email protected](Y. Chen).

Chaos, Solitons and Fractals 23 (2005) 14651473

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the propagation of weakly nonlinear waves in a noisy plasma [26]. The remarkable achievement (see[16]and refer-

ences for detail) of the study of stochastic partial differential equation have been obtained, for example, de Bouard and

Debussche[6] use function space similar to those introduced by Bourgain to prove well posedness results for the KdV

equation inL2-function and obtain the global existence ofL2(R) solution when the covariance operator of the noise is

HilbertSchmidt in L2(R). Holden et al. [7]gave while noise functional approach to research stochastic partial differ-

ential equations in Wick version. More recently, based on the theory in [7], using Hermite transform and the homog-

enous balance method, Xie studied Wick-type stochastic KdV equation and obtained stochastic soliton solution of thisequation[8].

In this paper, we would like to further extended the method presented by Fan [911] and recently improved by

Chen et al. [1214], to find stochastic soliton-like solutions of a Wick-type stochastic KdV equation as the following

form[8]:

UtftU}Ux gtUxxx Wt}R}t;U;Ux;Uxxx 1:1

which is the perturbation of the KdV equation with variable coefficients

utftuux gtuxxx 0 1:2

by random force W(t)R(R, U, Ux, Uxxx), where f(t) and g(t) are functions of t, W(t) is Gaussian white noise, i.e.,

Wt _Bt and B(t) is a Brownian motion, R(u, ux, uxxx) = auux buxxx is a functional ofu, ux and uxxx for someconstantsa,b and R is the Wick version of the functionalR. For more detail about the exchange between Wick-type

stochastic equation and common partial differential equation, the reader is advised to see the remarkable achievement

by Holden et al. [7] and the second section of Ref. [8] by Xie. As a result, we construct new and more general formal

solutions for Eq.(1), which include the nontravelling wave and coefficient function s stochastic soliton-like solutions,

singular stochastic soliton-like solutions, stochastic triangular functions solutions.

The rest of this paper is organized as follows. In Section 2, we establish a generalized method. In Section 3, we apply

the generalized method to a stochastic KdV equation and obtain some exact analytical solutions for this model. A short

summary and discussion are given in final.

2. Summary of the generalized method

In the following we would like to outline the main steps of our general method:

Step 1. For a given nonlinear partial differential equation (NPDE) system with some physical fields ui(x,y, t)

(i= 1, . . . , n) in three variables x,y, t,

Fiui; uit; uix; uiy; uitt; uixt; uiyt; uixx; uiyy; uixy; . . . 0: 2:1

We express the solutions of the NPDE by the new more general ansatz

uin ai0 Xmij1

aij/j bij/j1

ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiX4q0

hq/q

vuut

80; h4 0: 2:5

b. Kink shaped soliton-like solutions.

/ ffiffiffiffiffiffiffiffiffiffiffiffi

h22h4

r tanh

ffiffiffiffiffiffiffiffiffi h2

2

r n

!; h0 h

22

4h4; h1 h3 0; h2 0: 2:6

c. Soliton-like solutions.

/ h2 sech2 1

2

ffiffiffiffiffih2

p n

2ffiffiffiffiffiffiffiffiffih2h4

p tanh 1

2

ffiffiffiffiffih2

p n

h3 ; h0 h1 0; h2 >0: 2:7(ii) Jacobi and Weierstrass doubly-like periodic solutions.

/ ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi

h2m2h42m2 1

s cn

ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffih2

2m2 1

r n

!; h4 0; h0 h

22m

21 m2h42m2 12

; 2:8

/ ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffim2h42 m2

s dn

ffiffiffiffiffiffiffiffiffiffiffiffi ffiffih2

2 m2r

n

!; h4 0; h0 h

221 m2

h42 m22; 2:9

/ ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi

h2m2h4m2 1

s sn

ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffi h2m2 1

r n

!; h4 >0; h2 0; 2:11

whereg2 4h1

h3 and g3 4h0

h3 are called invariants of Weierstrass elliptic function. The Jacobi elliptic functions aredoubly periodic and possess properties of triangular functions:

sn2n cn2n 1; dn2n 1 m2sn2n;snn0 cnndnn; cnn0 snn; dnn0 m2snncnn:

When m ! 1, the Jacobi functions degenerate to the hyperbolic functions, i.e.snn ! tanh n; cnn ! sechn;

whenm ! 0, the Jacobi functions degenerate to the triangular functions, i.e.snn ! sin n; cnn ! cos n:

The more detailed notations for the Weierstrass and Jacobi elliptic functions can be found in Refs. [19,20].

Y. Chen et al. / Chaos, Solitons and Fractals 23 (2005) 14651473 1467
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Remark 1

1. Generalization: The method proposed here is more general than the method [911] by Fan, the method [1214]

improved by Chen et al. and the method [15] improved by E. Yomba. Firstly, compared with the method [911]

and the improved method [1214] the restriction on n(x,y, t) as merely a linear function x,y, t and the restriction

on the coefficients ai0, aij, bij (i= 1,2, . . . ; j= 1,2, . . . , mi) as constants are removed. Secondly, compared with the

improved method [15] by E. Yomba, the Eq. (2.3) that the new variable /= /(n) satisfies is more general. More

importantly, we add terms bij/j1

ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiP4q0hq/

qq

in new ansatz (2.2), so more types of solutions would be expected

for some equations.

2. Feasibility: For the generalization of the ansatz, naturally more complicated computation is expected than ever

before. Even if the availability of computer symbolic systems like Maple or Mathematica allows us to perform

the complicated and tedious algebraic calculation and differentiation on a computer, in general it is very difficult,

sometime impossible, to solve the set of over-determined partial differential equations in (step 3). As the calculation

goes on, in order to drastically simplify the work or make the work feasible, we often choose special function forms

for ai0, aij, bij(i= 1,2, . . . ; j= 1,2, . . . , mi) and n, on a trial-and-error basis.

3. Further extendable: In fact, We naturally present a more general a ansatz, which reads,

uin ai0 Xmij1

aij/j bij/j fij/j1 ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiXr

q0hq/

qvuut kij ffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiPr

q0hq/

qsuj

8>>>>>>>:9>>>>=>>>>;; 2:12

where ai0, aij, bij,fij, kij (i= 1,2, . . . ; j= 1,2, . . . , mi) and n are differentiable function to be determined later. When

ai0, aij, bij,fij, kij(i= 1,2, . . . ; j= 1,2, . . . , mi) are constants and n is linear function with respect to x,y and t in the above

ansatz, we have studied in Refs. [1618]. Therefore, for some nonlinear equations, more types of solutions would be

expected.

3. The stochastic soliton solutions of stochastic KdV equations

In this section, we will give exact solutions of Eq.(1.1)by the generalized method. Taking the Hermite transform of(1.1), we get the equationfUtt;x;z ht;z eUt;x;z eUxt;x;z st;z eUxxxt;x;z 0; 3:1where ht;z ft a eWt;z, st;z gt b eWt;z, the Hermite transformation of W(t) is defined byeWt;z P1k1gktzk when z= (z1,z2, . . . ,)2 (CN)c is parameter. We first solve Eq.(3.1).

According to the generalized method, we suppose that the solutions of(3.1)are the form

eUt;x;z a0 a1/n b1 ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffih0 h1/n h2/2n h3/3n h4/4nq a2/2n b2/nffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffih0 h1/n h2/2n h3/3n h4/4n

q ; 3:2

wherea0= a0(t, z), a1= a1(t, z), a2= a2(t, z), b1= b1(t, z), b2= b2(t, z),n= xp(t, z) + q(t, z) and /(n) satisfy(2.3).

Then substituting(3.2) and (2.3)into(3.1), collecting coefficients of monomials of/(n), ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiP4q0hq/qq and x of theresulting systems numerator (Notice that h(t, z),s(t, z),a0(t, z),a1(t, z),a2(t, z),b1(t, z),b2(t, z),p(t, z),q(t, z) are all inde-

pendent ofx), then setting each coefficients to zero, we obtain the following over-determined PDEs system.

8ht;zpt;za0t;zb2t;zh2 60h4st;zpt;z3b1t;zh1 10ht;zpt;za1t;zb2t;zh1 144h4st;zpt;z3b2t;zh0 6h3b1t;z o

otqt;z 8ht;zpt;zb1t;za1t;zh2 32st;zpt;z3b2t;zh22

10ht;zpt;zb1t;za2t;zh1 30st;zpt;z3b1t;zh2h3 8b2t;z ootqt;z

h2

12ht;zpt;za2t;zb2t;zh0 4 oota2t;z 84st;zpt;z3b2t;zh1h3 6h3ht;zpt;za0t;zb1t;z 0;

3:3

1468 Y. Chen et al. / Chaos, Solitons and Fractals 23 (2005) 14651473
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12h4b2t;z ootpt;z 0; 3:4

2 o

otpt;z

2h0b2t;z b1t;zh1 0; 3:5

4ht;zpt;zb1t;zb2t;zh0 4ht;zpt;za0t;za1t;z 4st;zpt;z3a1t;zh2 2ht;zpt;zb1t;z2h1 12st;zpt;z3a2t;zh1 4 o

otb1t;z 4a1t;z o

otqt;z 0; 3:6

180h4st;zpt;z3b2t;zh1 8h4ht;zpt;za0t;zb1t;z 12ht;zpt;zb1t;za2t;zh2 14ht;zpt;za2t;zb2t;zh1 10h3ht;zpt;za0t;zb2t;z 10h3ht;zpt;zb1t;za1t;z 80h4st;zpt;z3b1t;zh2 8h4b1t;z o

otqt;z 130st;zpt;z3b2t;zh2h3

12ht;zpt;za1t;zb2t;zh2 10h3b2t;z ootqt;z 30st;zpt;z3b1t;zh23 0; 3:7

2

o

otpt;z 4h2b2t;z 3h3b1t;z 0; 3:82

o

otpt;z

2b1t;zh2 3h1b2t;z 0; 3:9

30st;zpt;z3b2t;zh1h2 4ht;zpt;za0t;zb1t;zh2 48st;zpt;z3b1t;zh0h4 60st;zpt;z3b2t;zh3h0 6ht;zpt;za0t;zb2t;zh1 18st;zpt;z3b1t;zh1h3 8ht;zpt;zb2t;za1t;zh0 8ht;zpt;zb1t;za2t;zh0 4 o

ota1t;z 4st;zpt;z3b1t;zh22

4b1t;z oot

qt;z

h2 6ht;zpt;zb1t;za1t;zh1 6b2t;zh1 ootqt;z 0; 3:10

4a1t;z ootpt;z 0; 3:11

8pt;z12h4a2t;zst;zpt;z2 ht;za2t;z2 ht;zb1t;z2h4 8pt;z2h3ht;zb1t;zb2t;z ht;zb2t;z2h2 0; 3:12

6pt;z4pt;z2h4st;za1t;z 10pt;z2h3st;za2t;z 6pt;zht;zb2t;z2h1 2ht;za1t;za2t;z 6pt;z2ht;zb1t;zb2t;zh2 h3ht;zb1t;z2 0; 3:13

12ht;zpt;zb2t;z2h4 0; 3:14

20h4pt;zb2t;z12pt;z2

h4st;z a2t;zht;z 0; 3:1510ht;zpt;zb2t;zh3b2t;z 2h4b1t;z 0; 3:16

8a2t;z ootpt;z 0; 3:17

4o

ota0t;z 4ht;zpt;za1t;zb1t;zh0 12st;zpt;z3b1t;zh3h0 3st;zpt;z3b2t;zh21

4ht;zpt;za0t;zb2t;zh0 2b1t;z ootqt;z

h1 2ht;zpt;za0t;zb1t;zh1

2st;zpt;z3b1t;zh1h2 4b2t;z ootqt;z

h0 16st;zpt;z3b2t;zh2h0 0; 3:18



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4ht;zpt;zb2t;z2h0 8ht;zpt;zb1t;zb2t;zh1 4ht;zpt;za1t;z2 8a2t;z oot

qt;z

8ht;zpt;za0t;za2t;z 12h3st;zpt;z3a1t;z 4 ootb2t;z 32st;zpt;z3a2t;zh2

4h

t;z

p

t;z

b1

t;z

2h2

0;

3:19

2

o

otpt;z

4h4b1t;z 5h3b2t;z 0; 3:20

14h3ht;zpt;zb1t;za2t;z 16ht;zpt;za2t;zb2t;zh2 12h4ht;zpt;zb1t;za1t;z 12h4ht;zpt;za0t;zb2t;z 105st;zpt;z3b2t;zh23 120h4st;zpt;z3b1t;zh3

12h4b2t;z ootqt;z 240h4st;zpt;z3b2t;zh2 14h3ht;zpt;za1t;zb2t;z 0; 3:21

2pt;z48p2t;zst;zb1t;zh24 168p2t;zh4st;zb2t;zh3 2pt;zht;z8h4b1t;za2t;z a1t;zb2t;z 9h3a2t;zb2t;z 0; 3:22

Solving the system of PDEs. (3.3)(3.22)by Maple, we obtain the following two solutions.

p F1z; h h; a1 F2z; a2 2F2z h4h3

; b2 0;

a0 F3zq 3F1zRht;zdth3F3z F1z

Rht;zdth2F2z 3F5zh3

3h3; s 1

3

ht;zF2zh3F

21z

;

b1 2ffiffiffiffiffih4

p F2zh3

; 3:23

whereF1(z), . . ., F3(z) and h(t, z) are all arbitrary functions.

Thus from(3.2) and (3.23),we obtain two families of exact solutions of Eq.(3.1)as follows. For simplicity, we omit

polynomial-like, rational-like, triangular-like periodic solutions in this paper.Family 1. When h0= h1= h4= 0, h2> 0,

eU1t;x;z F3z F2zh2 sech2

ffiffiffiffih2

p2 n

h3

; 3:24

wheren= xF1(z) + q(t, z), q(t, z), h(t, z) and s(t, z) are determined by(3.23).

Family 2. When h0= h1= 0, h2> 0

eU2x; t;z F3z

F2zh2 sech2ffiffiffiffi

h2

p2 n

2ffiffiffiffiffiffiffiffiffih2h4p

tanh ffiffiffiffih2

p2 n h3

2F2zh4h22 sech4

ffiffiffiffih2

p2 n

h3

2ffiffiffiffiffiffiffiffiffih2h4p

tanh ffiffiffiffih2

p2 n h32

2

h52

2

ffiffiffiffiffih4

ph3

F2zsech4

ffiffiffiffih2

p2 n

tanh

ffiffiffiffih2

p2 n

2ffiffiffiffiffiffiffiffiffih2h4p tanh ffiffiffiffih2p2 n h32

sech6ffiffiffiffi

h2

p2 n

ffiffiffiffiffiffiffiffiffih2h4

p

2ffiffiffiffiffiffiffiffiffih2h4

p tanh

ffiffiffiffih2

p2 n

h3

30BBB@

1CCCA; 3:25wheren= xF1(z) + q(t, z), q(t, z), h(t, z) and /(t, z) determined by(3.23).

By(3.24) and (3.25), the definition ofeW, Theorem 2.1 in Ref. [7] and exp}fBtg expfBt 12t2g (see Lemma

2.6.16 in[6]), we have the following stochastic solitary solutions:

Family 1.

U1t;x F3 4 F2h2exp}

ffiffiffiffiffih2

p xF1 Qt

exp}ffiffiffiffiffih2p xF1 Qt 1}2h3 ; 3:26

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where

Qt 3F1h3F3 F1h2F2Rftdt aBt 1

2t2 3F4h3

3h3;

F1, F2, F3 and F4 are all arbitrary constants.

Family 2.

U2x; t F3 4exp}ffiffiffiffiffih2p nF2h2

2ffiffiffiffiffiffiffiffiffih2h4p exp}ffiffiffiffiffih2p n 2ffiffiffiffiffiffiffiffiffih2h4p h3exp}ffiffiffiffiffih2p n h3}exp}ffiffiffiffiffih2p n 1 32exp

}2ffiffiffiffiffih2p nh22h4F22ffiffiffiffiffiffiffiffiffih2h4p exp}ffiffiffiffiffih2p n 2ffiffiffiffiffiffiffiffiffih2h4p h3exp}ffiffiffiffiffih2p n h3}2}exp}ffiffiffiffiffih2p n 1}2h3

32exp}ffiffiffiffiffih2p n 1}exp}2ffiffiffiffiffih2p nF2ffiffiffiffiffih4p h522

2ffiffiffiffiffiffiffiffiffih2h4p exp}ffiffiffiffiffih2p n 2ffiffiffiffiffiffiffiffiffih2h4p h3exp}ffiffiffiffiffih2p n h3}2}exp}ffiffiffiffiffih2p n 1}3h3 128exp

}3ffiffiffiffiffih2p nF2h4h322ffiffiffiffiffiffiffiffiffih2h4p exp}ffiffiffiffiffih2p n 2ffiffiffiffiffiffiffiffiffih2h4p h3exp}ffiffiffiffiffih2p n h3}3}exp}ffiffiffiffiffih2p n 1}3h3 ; 3:27

where n = xF1+ Q(t),

Qt 3F1h3F3 F1h2F2Rftdt aBt 1

2t2 3F4h3

3h3;

F1, F2, F3 and F4 are all arbitrary constants.

Remark 2

(1) Due to the arbitrariness ofF1, F2, F3, F4and f(t), it is not difficult to verify that the solution 3.10,3.11obtained in

[8]can be reproduced by the solution(3.26)obtained by us. But, to our knowledge, the other solutions obtained

were not reported before.

Fig. 1. (a) denotes a soliton-like solution scenario given by U1, where F1=F2=F3=F4=h2=h3=a = 2,ft 3sech12 ttanh1

2t,bt 1

2t2. (b) and (c) denote the interaction scenario of soliton-like solutions.

Fig. 2. (a) denotes the boomerang-like soliton solution scenario given by U1, where F1=F2=F3=F4=h2=h3=a = 2, f(t) = 0,

bt 1

20 t

2

. (b) and (c) denote the interaction scenario of boomerang-like soliton solutions.

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(2) The more general soliton-like solutions obtained by the generalized method contain the some arbitrary differen-

tiable functions and some arbitrary constants, which may enable one to discuss the behavior of solutions as a func-

tion of these arbitrary differentiable functions and some arbitrary constants and this also provide enough freedom

to build up solutions that may correspond to a particular physical situation, or initial condition have some desired

features, which means a great variation in the solutions. In order to understand the significance of these soliton-

like solutions obtained in the paper, by choosing the special functions, we find some very interesting special solu-

tions(3.26), which including soliton-like solutions, snake-like soliton and Boomerang-like soliton, and choose the

solutions(3.26)to be figured. Their interaction scenario also were shown in fellow figures ( Figs. 13).

4. Summary and discussion

In summary, based on the generalized method and symbolic computation, by means of the theory of stochastic

partial differential equations in Wick version [7], we study stochastic KdV equation Ut+ f(t)UUx+ g(t)Uxxx=

W(t)R(t, U, Ux, Uxxx). Some new and more general formal solutions for Eq. (1.1) are constructed, which include

the nontravelling wave and coefficient functions stochastic soliton-like solutions, singular stochastic soliton-like solu-

tions, stochastic triangular functions solutions, at the same time, the results in[7]are recovered. By choosing the specialfunctions in the new and more general formal solution obtained, we would find some very interesting special solutions,

for instance, from solution (3.26), soliton-like soliton solutions, snake-like soliton solutions, boomerage-like soliton

solutions are fund, and their scenario and their interaction scenario are shown by the figures.

Acknowledgments

We would express our sincere thanks for Prof. Wadati for his valuable suggestion and kind help. The work was sup-

ported by the National Outstanding Youth Foundation of China (no. 19925522).

References

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[4] de Bouard A, Debussche A. J Funct Anal 1998;154:215.

[5] de Bouard A, Debussche A. Physica D 1999;134:200.

[6] de Bouard A, Debussche A. J Funct Anal 1999;169:532.

[7] Holden H, ksendal B, Ube, Zhang T. Stochastic partial differential equations: a modeling, white noise functional approach.

Birhkauser 1996.

[8] Xie YC. Phys Lett A 2003;310:161.

[9] Fan E. Chaos, Solitons & Fractals 2003;15:55966.



Fig. 3. (a) denotes the snake-like soliton solution scenario given by U1, where F1=F2=F3=F4=h2=h3=a = 2,ft 2sin4

5t 3, bt 1

2t2. (b) and (c) denote the interaction scenario of snake-like soliton solutions.

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[12] Chen Y, Wang Q, Li B. Chaos, Solitons & Fractals 2004;22:67582.

[13] Chen Y, Wang Q, J Mod Phys C, in press.

[14] Chen Y, Wang Q, Commun Theor Phys, in press.

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stochastic soliton like solution - kdv equation

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