Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2012, Article ID 327682,9 pages doi:10.1155/2012/327682

Research Article

Soliton Solutions for the Wick-Type Stochastic KP Equation

Y. F. Guo,1'2 L. M. Ling,2 and D. L. Li1

1 Department of Information and Computation of Science, Guangxi University of Technology, Liuzhou, Guangxi 545006, China

2 Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, China Correspondence should be addressed to Y. F. Guo, guoyanfeng@yahoo.com.cn

Received 10 September 2012; Accepted 28 November 2012 Academic Editor: Peicheng Zhu

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 [3] arrived at the two-dimensional version of the KdV equation:

Utx = (Uxxx + 6uux )x + 3a2Uyy, (1.1)

which is known as Kadomtsev-Petviashvili (KP) equation. The KP equation appears in physical applications in two different forms with a = 1 and a = 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 [6], as well as the similar reductions of nonlinear PDEs in mathematical physics. These subjects have been researched by many authors.

For SPDEs, in [7], 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:

Utx = f mUxxx + 6gmwux)% + 3a2/muyy + W (t)0R<( U,UX,UXX,U

which is the perturbation of the KP equation with variable coefficients:

utx = f (t)uxxx + 6g (t)uux)x + 3a2/(t)uyy, (1.3)

by random force W(t)^R<(U,Ux,Uxx,Uxxxx,Uyy), where ♦ is the Wick product on the Hida distribution space (S(Rd))* which is defined in Section 2, /(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) = puxxxx + + 6juuxx + 3a2 ¡uyy is a function of u, ux, uxx, uxxxx, uyy for some constants ¡5, y, 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(Rd)) and (S(Rd))* be the Hida test function and the Hida distribution space on Rd, respectively. The collection ln = e(-x2/2)hn(^/2x)/(n(n-1)!)1/2, n > 1 constitutes an orthogonal basis for L2(R), where hn(x) is the d-order Hermite polynomials. The family of tensor products ¿,a = ^a1,...,ad = &a1 ® ••• ® £a1 (a e Nd) forms an orthogonal basis for L2 (Rd), where a = (a1,.. .,ad) is d-dimensional multi-indices with ai,...,ad e N. The multi-indices a = (a\,...,ad) are defined as elements of the space J = (N^)c of all sequences a = (a1,a2,...) with elements ai e No and with compact support, that is, with only finite many ai = 0. For a = (a1, a2,...), we define

Ha(w) = J7hai({v,nt}), w e (s(Rd))*. (2.1)

If n e N is fixed, let (S)1 consist of those x = £aCaHa e e^L2^) with Ca e Rn such that ||x||1/fc = %ac2a(a!)2(2N)fca < » for all k e N with ca = |Ca|2 = I,nk=1(cia))2 if Ca = (c(a),...,c{ai)) e Rn, where fi is the white noise measure on (S*(R),B(S*(R))), a! = U.k=1ak! and (2N)a = :(2j)a' for a = (a1 ,a2,...) e J. The space (S)n1 can be regarded as the dual of

(S)n. (S)"1 consisting of all formal expansion X = ^a baHa with ba e Rn such that \\X\\_1-q

b2a(2N)_qa < go for some q e N, by the action (X,x) = ^a(ba,ca)a! and (ba,ca) is the usual inner product in Rn.

XQY = Xap(aa, bp)Ha+p is called the Wick product of X and Y, for X = ^a aaHa, Y = habaHa e (S)^ with aa,ba e Rn. We can prove that the spaces (S(Rd)), (S(Rd))*(S)", and (S)n1 are closed under Wick products.

For X = ^a aaHa e (S)nj with aa e Rn, H(X) or X is defined as the Hermite transform of X by H(X)(z) = XX(z) = ha aaza e Cn (when convergent), where z = (z1, Z2,...) e CN (the set of all sequences of complex numbers) and za = za1 za2 •••z■■■ for a = (a^a2,...) e J. For X,Y e (S)N1, by this definition we have X^Y(z) = XX(z) ■ Y(z) for all z such that XX(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 CN defined by (z1, ...,z") ■ (z2, ...,z") = X n=1 z\z2k, where zik e C. Let X = ha aaHa e (S)n1. Then the vector C0 = XX(0) e Rn is called the generalized expectation of X denoted by E(X). Suppose that f : V ^ Cn 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 Rn .Then the Wick version f°(X) = H-1(f o XX) e (S)-1.

Suppose that modeling considerations lead us to consider the SPDE expressed formally as A(t,x,dt, Vx,U,w) = 0, where A is some given function, U = U(t,x,w) is the unknown generalized stochastic process, and the operators dt = d/dt, Vx = (d/dx1, ...,d/dxd) when x = (x1r ...,cd) e Rd .If we interpret all products as wick products and all functions as their Wick versions, we have

A()(t,x,öt, Vx,U,w) = 0. (2.2)

Taking the Hermite transform of (2.2), the Wick product is turned into ordinary products (between complex numbers), and the equation takes the form

A( t,x,dt, Vx,Ü,z1,z2,..^ = 0, (2.3)

where U = H(U) is the Hermite transform of U and z1r z2,... are complex numbers. Suppose that we can find a solution u = u(t,x,z) of (2.3) for each z = (z1r z2,...) e Kq(r) for some q,r, where Kq (r) = z = (z1 ,z2,...) e CN and ^ a / 0 |za|2(2N)qa < r2. Then under certain conditions, we can take the inverse Hermite transform U = H-1u e (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 [7].

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 c R x Rd and z e Kq(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) e G x Kq(r), continuous with respect to (t,x) e G for all z e Kq(r), and analytic with respect to z e Kq(r) for all (t,x) e G. Then there exists U(t,x) e (S)_1 such that u(t,x,z) = (U(t,x))(z) for all (t, x,z) e G x 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[1] e (S)_1 for KP-I:

U[1] = sech( , when a = 1 (3.1)

2k V V 2 and for KP-II:

U[1] = 2|- sech2((t,x,y)^, when a = i, (3.2)

where 0(t, x, y) = Xx + l2y + 413 f0 f (s)ds + 4X3ßB(t) - 2X3ßf and

®i(t,x,y) = ax - 2aby + 4(a3 - 3ab2) J" f (s)ds + 4ß(a3 - 3ab2) (ß(t) - 2t^j. (3.3)

Proof. Taking the Hermite transform of (1.2), the equation (1.2) can be changed into Ufx = f (t) + pW(t, z)] Uxxxx + 6 [g(t) + yW(t, z)] (uUx)x + 3a2 f (t) + pW(t, z)] Uyy,

where U is the Hermite transform of U; the Hermite transform of W (t) is defined by W(t, z) = Jhk=1 (t)zk where z = (z1, z2,...) e (CN)c is parameter.

Suppose that g (t) +yW (t, z) = k[/(t)+pW (t, z)] .Let u = kU. From (3.4), we can obtain

utx = /(t) + pw(t, z)] (uxxx + 6uux)x + 3a2 /(t) + pW(t, z)] uvv. (3.5)

Let F(t, z) = /(t) + pW(t, z); then (3.5) can be changed into

utx = F(t, z)(uxxx + 6uux)x + 3a2 F(t, z)uyy. (3.6)

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):

Sy = a^tyxx + a~luS,

, ^ (3.7)

St = 4F(t, z)Sxxx + 6F(t, z)u$x + 3F(t, z) (avy + ux)$.

Then we can obtain the Wick-type Lax pair of (1.2):

fay = aTlfyxx + a^uOfa,

fat = 4(f (t) + pW (t))Wxxx + 6(f (t) + pW (t))OuOfax (3.8)

+ 3 (f (t) + pW (t)) ♦ (avy + ux) Ofa.

Let fa1 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 fa[1] = fax _ (fa1xOfa1^(~1^)Ofa, where fa is an arbitrary solution of (3.8), then fa[1] satisfies the following equations:

fay [1] = a-^xx[1] + a_1u[1]Ofa[1],

fat [1] = 4(f (t)+ pW (t))Ofaxxx[1] + 6(f (t)+ pW (t))Ou[1]fax[1] (3.9)

+ 3(f (t) + pW (t))0(avy [1] + ux [1])Ofa[1],

where u[1] = u + 2(fa1xOfa10(_1))J, v[1] = v + 2(fa1xOfa10(_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

fay = a 1faxx,

(3.10)

fat = 4( f (t) + jW (t)) Ofaxxx-If a = 1, (3.10) have the exponential function solution

fai( t,x,y,z) = exp°{ ^(t,x,y,z)} + 1, (3.11)

^ = Ax + A2y + 4A3 0 f (s)ds + j№(t)^, (3.12)

and A 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:

U[1] = 2 (fa1x ♦fa10(-1) )0fa = A2(sech°( f))2, (3.13)

f(t,x,y) = Ax + A2y + 4A3 f f (s)ds + 4A3pB(t).

(3.14)

Since exp°{B(t)} = exp{B(t) _ (1/2)t2} (see Lemma 2.6.16 in [7]), (1.2) has the single-soliton solution

f))2, (315)

0(t,x,y) = Xx + X2y + 4X3 I" f (s)ds + 4X3pB(t) _ 2X3pt2. (3.16)

In particular, when f (s) = 1 we can obtain the solution of (2.2), respectively, as follows:

U[1] = 22sech2Q (Xx + X2y + 4X3t + 4X3pB(t) _ 2X3pt2^. (3.17)

If a = i, (3.10) have the exponential function solution

fa1 ( t, x, y, z) = exp° { ^ ( t, x, y,z)} + exp° {( t, x, y,z)}, (3.18)

(t, x, y, z) = Xx + iX2y + 4X^| f (s)ds + pB(t)J, (3.19)

^ is the conjugation of ^ and X is an arbitrary complex parameter. Let X = a+ib, according to (3.9), from (3.18) and (3.19) there exists a stochastic single-solitary solution of (1.2) as follows:

U[1] = k (fa1xOfa10(_1^Ofa = (sech°(®1(t,x,y)))\ (3.20) where

®1 (t,x,y) = ax _ 2aby + 4(a3 _ 3ab2) J f (s)ds + 4(a3 _ 3ab2)pB(t). (3.21)

Same as the former case, since exp°(B(t)} = exp{B(t) _ (1 /2)t2}, (1.2) has the single-soliton solution

U[1] = 2a2sech^®1 (t,x,y^, (3.22)

®1 (t, x, y) = ax _ 2aby + 4 (a3 _ 3ab2) J f (s)ds + 4p(a3 _ 3ab2) ^B(t) _ 11^. (3.23)

In particular, when f (s) = 1 we can obtain the solution of (2.2) as follows:

U[1] = ^k-sech2 ^ax - 2aby + 4(a3 - 3ab2) (t - ^t2 + . (3.24)

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

= / A _ fo* [1] W A _ jx \

*[2]H s - A* - j*)*' (41)

where *2[1] is the fixed solution of (3.9), which is generated by some fixed solution *2 of (3.8) and independent of *1. We know that

<M1] = <2x - ^fc, (4.2)

u[2] = u + 2 — ln W«i,<2). (4.3)

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 <1 ,<2,...,<N be different and independent solutions of (3.8). We define the Wronski determinant W of functions f1,...,fm as

di-1 f ■

W(fi.....fm) = det A, Aj = dxrf, i,j = 1,2.....m. (4.4)

Theorem 4.1. For the Wick-type stochastic KP equation (1.2) in white noise environment, one has the N-soliton solution U[N] e (S)-1 satisfying

U[N] = k dx2ln°W <......<n). (4.5)

Proof. From [6], it is easy to see that the function

] = W«1.....<N<

< W«1.....<n)

satisfies the following equations:

fay [N] = a-1faxx[N] + a-1u[N]fa[N],

fat[N] = 4F (t,z)faxxx[N]+ 6F(t,z)u[N]fax [N ] (4.7)

+ 3F(t, z) (avy [N] + ux[N])fa[N],

where u [N] = u + 2(d2/dx2) ln W (fa1,...fa) and v [N ] = v + 2(d/dx) ln W (fa1,...,faN). Then we have the Wick-type form

fa[N] = W 0(fa.....faN,fa) (4.8)

fa W^fa1.....faN)

satisfying the following equations:

fay [N] = a-1faxx[N] + a_1u[N]Ofa[N],

fat[N ] = 4(f (t) + W (t))Ofaxxx[N] + 6(f (t) + W (t))Ou[N]Ofax [N] (4.9)

+ 3(f (t) + W (t))O(avy [N ] + ux [N])Ofa[N],

where u[N] = u + 2(d2/dx2)ln°W°(fa1,...fa).

In particular, taking u = 0, v = 0, we can obtain the N-soliton solution of (1.2):

U[N] = k dx2ln°W0(fa1.....faN). (4.10)

When a = 1 and a = i, fa1,...,faN are represented by the corresponding forms (3.11) and (3.18), where X, 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 [7]. 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).

References

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