AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 327682 10.1155/2012/327682 327682 Research Article Soliton Solutions for the Wick-Type Stochastic KP Equation Guo Y. F. 1, 2 Ling L. M. 2 Li D. L. 1 Zhu Peicheng 1 Department of Information and Computation of Science Guangxi University of Technology Liuzhou, Guangxi 545006 China gxut.edu.cn 2 Institute of Applied Physics and Computational Mathematics P.O. Box 8009 Beijing 100088 China iapcm.ac.cn 2012 13 12 2012 2012 10 09 2012 28 11 2012 2012 Copyright © 2012 Y. F. Guo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The Wick-type stochastic KP equation is researched. The stochastic single-soliton solutions and stochastic multisoliton solutions are shown by using the Hermite transform and Darboux transformation.

1. Introduction

In recent decades, there has been an increasing interest in taking random effects into account in modeling, analyzing, simulating, and predicting complex phenomena, which have been widely recognized in geophysical and climate dynamics, materials science, chemistry biology, and other areas, see [1, 2]. If the problem is considered in random environment, the stochastic partial differential equations (SPDEs) are appropriate mathematical models for complex systems under random influences or noise. So far, we know that the random wave is an important subject of stochastic partial differential equations.

In 1970, while studying the stability of the KdV soliton-like solutions with small transverse perturbations, Kadomtsev and Petviashvili  arrived at the two-dimensional version of the KdV equation:(1.1)utx=(uxxx+6uux)x+3α2uyy, which is known as Kadomtsev-Petviashvili (KP) equation. The KP equation appears in physical applications in two different forms with α=1 and α=i, usually referred to as the KP-I and the KP-II equations. The number of physical applications for the KP equation is even larger than the number of physical applications for the KdV equation. It is well known that homogeneous balance method [4, 5] has been widely applied to derive the nonlinear transformations and exact solutions (especially the solitary waves) and Darboux transformation , as well as the similar reductions of nonlinear PDEs in mathematical physics. These subjects have been researched by many authors.

For SPDEs, in , Holden et al. gave white noise functional approach to research stochastic partial differential equations in Wick versions, in which the random effects are taken into account. In this paper, we will use their theory and method to investigate the stochastic soliton solutions of Wick-type stochastic KP equation, which can be obtained in the influence of the random factors.

The Wick-type stochastic KP equation in white noise environment is considered as the following form: (1.2)Utx=(f(t)Uxxx+6g(t)UUx)x+3α2f(t)Uyy+W(t)R(U,Ux,Uxx,Uxxxx,Uyy), which is the perturbation of the KP equation with variable coefficients: (1.3)utx=(f(t)uxxx+6g(t)uux)x+3α2f(t)uyy, by random force W(t)R(U,Ux,Uxx,Uxxxx,Uyy), where is the Wick product on the Hida distribution space (S(d))* which is defined in Section 2, f(t) and g(t) are functions of t, W(t) is Gaussian white noise, that is, W(t)=B˙(t) and B(t) is a Brownian motion, R(u,ux,uxx,uxxxx,uyy)=βuxxxx+6γux2+6γuuxx+3α2βuyy is a function of u,ux,uxx,uxxxx,uyy for some constants β,γ, and R is the Wick version of the function R.

This paper is organized as follows. In Section 2, the work function spaces are given. In Section 3, we present the single-soliton solutions of stochastic KP equation (1.2). Section 4 is devoted to investigate the multisoliton solutions of stochastic KP equation (1.2).

2. SPDEs Driven by White Noise

Let (S(d)) and (S(d))* be the Hida test function and the Hida distribution space on d, respectively. The collection ξn=e(-x2/2)hn(2x)/(π(n-1)!)1/2,n1 constitutes an orthogonal basis for L2(), where hn(x) is the d-order Hermite polynomials. The family of tensor products ξα=ξα1,,αd=ξα1ξα1(αd) forms an orthogonal basis for L2(d), where α=(α1,,αd) is d-dimensional multi-indices with α1,,αd. The multi-indices α=(α1,,αd) are defined as elements of the space 𝒥=(0)c of all sequences α=(α1,α2,) with elements αi0 and with compact support, that is, with only finite many αi0. For α=(α1,α2,), we define (2.1)Hα(ω)=i=1hαi(ω,ηi),ω(S(d))*.

If n is fixed, let (S)1n consist of those x=αcαHαk=1nL2(μ) with cαn such that x1,k2=αcα2(α!)2(2)kα< for all k with cα2=|cα|2=k=1n(cα(k))2 if cα=(cα(1),,cα(n))n, where μ is the white noise measure on (S*(),(S*())),α!=k=1αk! and (2)α=j(2j)αj for α=(α1,α2,)𝒥. The space (S)-1n can be regarded as the dual of (S)1n. (S)-1n consisting of all formal expansion X=αbαHα with bαn such that X-1,-q=αbα2(2)-qα< for some q, by the action X,x=α(bα,cα)α! and (bα,cα) is the usual inner product in n.

X Y = α , β ( a α , b β ) H α + β is called the Wick product of X and Y, for X=αaαHα,Y=αbαHα(S)-1n with aα,bαn. We can prove that the spaces (S(d)),(S(d))*(S)1n, and (S)-1n are closed under Wick products.

For X=αaαHα(S)-1n with aαn, (X) or X~ is defined as the Hermite transform of X by (X)(z)=X~(z)=αaαzαn (when convergent), where z=(z1,z2,) (the set of all sequences of complex numbers) and zα=z1α1z2α2znαn for α=(α1,α2,)𝒥. For X,Y(S)-1N, by this definition we have XY~(z)=X~(z)·Y~(z) for all z such that X~(z) and Y~(z) exist. The product on the right-hand side of the above formula is the complex bilinear product between two elements of N defined by (z11,,zn1)·(z12,,zn2)=k=1nzk1zk2, where zki. Let X=αaαHα(S)-1n. Then the vector c0=X~(0)n is called the generalized expectation of X denoted by 𝔼(X). Suppose that f:Vn is an analytic function, where V is a neighborhood of 𝔼(X). Assume that the Taylor series of f around 𝔼(X) has coefficients in n. Then the Wick version f(X)=-1(fX~)(S)-1n.

Suppose that modeling considerations lead us to consider the SPDE expressed formally as A(t,x,t,x,U,ω)=0, where A is some given function, U=U(t,x,ω) is the unknown generalized stochastic process, and the operators t=/t,x=(/x1,,/xd) when x=(x1,,cd)d. If we interpret all products as wick products and all functions as their Wick versions, we have (2.2)A(t,x,t,x,U,ω)=0. Taking the Hermite transform of (2.2), the Wick product is turned into ordinary products (between complex numbers), and the equation takes the form (2.3)A~(t,x,t,x,U~,z1,z2,)=0, where U~=(U) is the Hermite transform of U and z1,z2, are complex numbers. Suppose that we can find a solution u=u(t,x,z) of (2.3) for each z=(z1,z2,)𝕂q(r) for some q,r, where 𝕂q(r)=z=(z1,z2,) and α0|zα|2(2)qα<r2. Then under certain conditions, we can take the inverse Hermite transform U=-1u(S)-1 and thereby obtain a solution U of the original Wick equation (2.2). We have the following theorem, which was proved by Holden et al. in .

Theorem 2.1.

Suppose that u(t,x,z) is a solution (in the usual strong, pointwise sense) of (2.3) for (t,x) in some bounded open set G×d and z𝕂q(r) for some q,r. Moreover, suppose that u(t,x,z) and all its partial derivatives, which are involved in (2.3), are bounded for (t,x,z)G×𝕂q(r), continuous with respect to (t,x)G for all z𝕂q(r), and analytic with respect to z𝕂q(r) for all (t,x)G. Then there exists U(t,x)(S)-1 such that u(t,x,z)=(U~(t,x))(z) for all (t,x,z)G×𝕂q(r) and U(t,x) solves (in the strong sense in (S)-1) (2.2) in (S)-1.

3. Single-Soliton Solution of Stochastic KP Equation

In this section, we investigate the single-soliton solutions of the Wick-type stochastic KP equation (1.2). Using the similar idea of the Darboux transformation about the determinant nonlinear partial differential equations, we can obtain the soliton solutions of (1.2), which can be seen in the following theorem.

Theorem 3.1.

For the Wick-type stochastic KP equation (1.2) in white noise environment, one has the single-soliton solution U(S)-1 for KP-I: (3.1)U=λ22k( sech (Φ¯2))2,whenα=1 and for KP-II: (3.2)U=2a2k sech 2(Φ¯1(t,x,y)),whenα=i, where Φ¯(t,x,y)=λx+λ2y+4λ30tf(s)ds+4λ3βB(t)-2λ3βt2 and (3.3)Φ¯1(t,x,y)=ax-2aby+4(a3-3ab2)0tf(s)ds+4β(a3-3ab2)(B(t)-12t2).

Proof.

Taking the Hermite transform of (1.2), the equation (1.2) can be changed into (3.4)U~tx=[f(t)+βW~(t,z)]U~xxxx+6[g(t)+γW~(t,z)](U~U~x)x+3α2[f(t)+βW~(t,z)]U~yy, where U~ is the Hermite transform of U; the Hermite transform of W(t) is defined by W~(t,z)=k=1ηk(t)zk where z=(z1,z2,)()c is parameter.

Suppose that g(t)+γW~(t,z)=k[f(t)+βW~(t,z)]. Let u=kU~. From (3.4), we can obtain (3.5)utx=[f(t)+βW~(t,z)](uxxx+6uux)x+3α2[f(t)+βW~(t,z)]uyy. Let F(t,z)=f(t)+βW~(t,z); then (3.5) can be changed into (3.6)utx=F(t,z)(uxxx+6uux)x+3α2F(t,z)uyy. Now we consider the soliton solutions of (3.6) using Darboux transform. It is more convenient to consider the compatibility condition of the following linear system of partial differential equations, that is, Lax pair of (3.6): (3.7)ϕy=α-1ϕxx+α-1uϕ,ϕt=4F(t,z)ϕxxx+6F(t,z)uϕx+3F(t,z)(αvy+ux)ϕ. Then we can obtain the Wick-type Lax pair of (1.2): (3.8)ϕy=α-1ϕxx+α-1uϕ,ϕt=4(f(t)+βW(t))ϕxxx+6(f(t)+βW(t))uϕx+3(f(t)+βW(t))(αvy+ux)ϕ.

Let ϕ1 be a given solution of (3.8). Using the idea of the Darboux transformation about the determinant nonlinear partial differential equations, by direct computation, it is easy to know that if supposing that ϕ=ϕx-(ϕ1xϕ1(-1))ϕ, where ϕ is an arbitrary solution of (3.8), then ϕ satisfies the following equations: (3.9)ϕy=α-1ϕxx+α-1uϕ,ϕt=4(f(t)+βW(t))ϕxxx+6(f(t)+βW(t))uϕx+3(f(t)+βW(t))(αvy+ux)ϕ, where u=u+2(ϕ1xϕ1(-1))x, v=v+2(ϕ1xϕ1(-1)).

Since (3.6) is nonlinear, it is difficult to solve it in general. In particular, taking u=0 and v=0, then from (3.8), we have (3.10)ϕy=α-1ϕxx,ϕt=4(f(t)+βW(t))ϕxxx.

If α=1, (3.10) have the exponential function solution (3.11)ϕ1(t,x,y,z)=exp{φ(t,x,y,z)}+1, where (3.12)φ=λx+λ2y+4λ3(0tf(s)ds+βB(t)), and λ is an arbitrary real parameter. Then we can obtain the single-soliton solution of (3.6). By (3.11) and (3.12) there exists a stochastic single-solitary solution of (1.2) as following: (3.13)U=2k(ϕ1xϕ1(-1))ϕ=λ22k(sech(Φ2))2, where (3.14)Φ(t,x,y)=λx+λ2y+4λ30tf(s)ds+4λ3βB(t). Since exp{B(t)}=exp{B(t)-(1/2)t2} (see Lemma  2.6.16 in ), (1.2) has the single-soliton solution (3.15)U=λ22k(sech(Φ¯2))2, where (3.16)Φ¯(t,x,y)=λx+λ2y+4λ30tf(s)ds+4λ3βB(t)-2λ3βt2. In particular, when f(s)=1 we can obtain the solution of (2.2), respectively, as follows: (3.17)U=λ22ksech2(12(λx+λ2y+4λ3t+4λ3βB(t)-2λ3βt2)). If α=i, (3.10) have the exponential function solution (3.18)ϕ1(t,x,y,z)=exp{φ1(t,x,y,z)}+exp{-φ¯1(t,x,y,z)}, where (3.19)φ1(t,x,y,z)=λx+iλ2y+4λ3(0tf(s)ds+βB(t)),φ¯1 is the conjugation of φ¯1 and λ is an arbitrary complex parameter. Let λ=a+ib, according to (3.9), from (3.18) and (3.19) there exists a stochastic single-solitary solution of (1.2) as follows: (3.20)U=2k(ϕ1xϕ1(-1))ϕ=2a2k(sech(Φ1(t,x,y)))2, where (3.21)Φ1(t,x,y)=ax-2aby+4(a3-3ab2)0tf(s)ds+4(a3-3ab2)βB(t). Same as the former case, since exp{B(t)}=exp{B(t)-(1/2)t2}, (1.2) has the single-soliton solution (3.22)U=2a2ksech2(Φ¯1(t,x,y)), where (3.23)Φ¯1(t,x,y)=ax-2aby+4(a3-3ab2)0tf(s)ds+4β(a3-3ab2)(B(t)-12t2).

In particular, when f(s)=1 we can obtain the solution of (2.2) as follows: (3.24)U=2a2ksech2(ax-2aby+4(a3-3ab2)(t-β2t2+βB(t))).

4. Multisoliton Solutions of Stochastic KP Equation

At the same time, the multisoliton solutions of stochastic KP equation can be also considered. It is evident that the Darboux transformation can be applied to (3.9) again. This operation can be repeated arbitrarily. For the second step of this procedure we have(4.1)ϕ=(x-ϕ2xϕ2)(x-ϕ1xϕ1)ϕ, where ϕ2 is the fixed solution of (3.9), which is generated by some fixed solution ϕ2 of (3.8) and independent of ϕ1. We know that (4.2)ϕ2=ϕ2x-ϕ1xϕ1ϕ2,(4.3)u=u+22x2lnW(ϕ1,ϕ2). By using N-times Darboux transformation, the formula (4.3) can be generalized to obtain the solutions of the initial equations (3.8) without any use of the solutions related to the intermediate iterations of the process.

Let ϕ1,ϕ2,,ϕN be different and independent solutions of (3.8). We define the Wronski determinant W of functions f1,,fm as(4.4)W(f1,,fm)=detA,Aij=di-1fjdxi-1,i,j=1,2,,m.

Theorem 4.1.

For the Wick-type stochastic KP equation (1.2) in white noise environment, one has the N-soliton solution U[N](S)-1 satisfying (4.5)U[N]=2k2x2lnW(ϕ1,,ϕN).

Proof.

From , it is easy to see that the function (4.6)ϕ[N]=W(ϕ1,,ϕN,ϕ)W(ϕ1,,ϕN) satisfies the following equations: (4.7)ϕy[N]=α-1ϕxx[N]+α-1u[N]ϕ[N],ϕt[N]=4F(t,z)ϕxxx[N]+6F(t,z)u[N]ϕx[N]+3F(t,z)(αvy[N]+ux[N])ϕ[N], where u[N]=u+2(2/x2)lnW(ϕ1,,ϕN) and v[N]=v+2(/x)lnW(ϕ1,,ϕN).

Then we have the Wick-type form (4.8)ϕ[N]=W(ϕ1,,ϕN,ϕ)W(ϕ1,,ϕN) satisfying the following equations: (4.9)ϕy[N]=α-1ϕxx[N]+α-1u[N]ϕ[N],ϕt[N]=4(f(t)+W(t))ϕxxx[N]+6(f(t)+W(t))u[N]ϕx[N]+3(f(t)+W(t))(αvy[N]+ux[N])ϕ[N], where u[N]=u+2(2/x2)lnW(ϕ1,,ϕN).

In particular, taking u=0, v=0, we can obtain the N-soliton solution of (1.2): (4.10)U[N]=2k2x2lnW(ϕ1,,ϕN). When α=1 and α=i, ϕ1,,ϕN are represented by the corresponding forms (3.11) and (3.18), where λ,a,b take the different constants.

Remark 4.2.

However, in generally, in the view of the modeling point, one can consider the situations where the noise has a different nature. It turns out that there is a close mathematical connection between SPDEs driven by Gaussian and Poissonian noise at least for Wick-type equations. It is well known that there is a unitary map to the solution of the corresponding Gaussian SPDE, see . Hence, if the coefficient f(t) is perturbed by Poissonian white noise in (1.2), the stochastic single-soliton solution and stochastic multisoliton solutions also can be obtained by the same discussion.

Acknowledgments

This paper is supported by National Natural Science Foundation of China (no. 11061003) and Foundation of Ph.D. of Guangxi University of Technology (no. 03081587).

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