- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 327682, 9 pages
Soliton Solutions for the Wick-Type Stochastic KP Equation
1Department of Information and Computation of Science, Guangxi University of Technology, Liuzhou, Guangxi 545006, China
2Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, China
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.
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: which is known as Kadomtsev-Petviashvili (KP) equation. The KP equation appears in physical applications in two different forms with and , 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: which is the perturbation of the KP equation with variable coefficients: by random force , where is the Wick product on the Hida distribution space which is defined in Section 2, and are functions of , is Gaussian white noise, that is, and is a Brownian motion, is a function of for some constants , and is the Wick version of the function .
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 and be the Hida test function and the Hida distribution space on , respectively. The collection constitutes an orthogonal basis for , where is the -order Hermite polynomials. The family of tensor products forms an orthogonal basis for , where is -dimensional multi-indices with . The multi-indices are defined as elements of the space of all sequences with elements and with compact support, that is, with only finite many . For , we define
If is fixed, let consist of those with such that for all with if , where is the white noise measure on and for . The space can be regarded as the dual of . consisting of all formal expansion with such that for some , by the action and is the usual inner product in .
is called the Wick product of and , for with . We can prove that the spaces , and are closed under Wick products.
For with , or is defined as the Hermite transform of by (when convergent), where (the set of all sequences of complex numbers) and for . For , by this definition we have for all such that and exist. The product on the right-hand side of the above formula is the complex bilinear product between two elements of defined by where . Let . Then the vector is called the generalized expectation of denoted by . Suppose that is an analytic function, where is a neighborhood of . Assume that the Taylor series of around has coefficients in . Then the Wick version .
Suppose that modeling considerations lead us to consider the SPDE expressed formally as , where is some given function, is the unknown generalized stochastic process, and the operators when . If we interpret all products as wick products and all functions as their Wick versions, we have Taking the Hermite transform of (2.2), the Wick product is turned into ordinary products (between complex numbers), and the equation takes the form where is the Hermite transform of and are complex numbers. Suppose that we can find a solution of (2.3) for each for some , where and . Then under certain conditions, we can take the inverse Hermite transform and thereby obtain a solution 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 is a solution (in the usual strong, pointwise sense) of (2.3) for in some bounded open set and for some . Moreover, suppose that and all its partial derivatives, which are involved in (2.3), are bounded for , continuous with respect to for all , and analytic with respect to for all . Then there exists such that for all and solves (in the strong sense in ) (2.2) in .
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 for KP-I: and for KP-II: where and
Proof. Taking the Hermite transform of (1.2), the equation (1.2) can be changed into
where is the Hermite transform of ; the Hermite transform of is defined by where is parameter.
Suppose that . Let . From (3.4), we can obtain Let ; then (3.5) can be changed into 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): Then we can obtain the Wick-type Lax pair of (1.2):
Let 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 , where is an arbitrary solution of (3.8), then satisfies the following equations: where , .
Since (3.6) is nonlinear, it is difficult to solve it in general. In particular, taking and , then from (3.8), we have
If , (3.10) have the exponential function solution where and 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: where Since (see Lemma 2.6.16 in ), (1.2) has the single-soliton solution where In particular, when we can obtain the solution of (2.2), respectively, as follows: If , (3.10) have the exponential function solution where is the conjugation of and is an arbitrary complex parameter. Let , according to (3.9), from (3.18) and (3.19) there exists a stochastic single-solitary solution of (1.2) as follows: where Same as the former case, since , (1.2) has the single-soliton solution where
In particular, when we can obtain the solution of (2.2) as follows:
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 where is the fixed solution of (3.9), which is generated by some fixed solution of (3.8) and independent of . We know that By using -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 be different and independent solutions of (3.8). We define the Wronski determinant of functions as
Theorem 4.1. For the Wick-type stochastic KP equation (1.2) in white noise environment, one has the -soliton solution satisfying
Proof. From , it is easy to see that the function
satisfies the following equations:
where and .
Then we have the Wick-type form satisfying the following equations: where .
In particular, taking , , we can obtain the -soliton solution of (1.2): When and , are represented by the corresponding forms (3.11) and (3.18), where 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 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.
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).
- E. Weinan, X. Li, and E. Vanden-Eijnden, “Some recent progress in multiscale modeling,” in Multiscale Modelling and Simulation, vol. 39 of Lecture Notes in Computational Science and Engineering, pp. 3–21, Springer, Berlin, Germany, 2004.
- P. Imkeller and A. H. Monahan, “Conceptual stochastic climate models,” Stochastics and Dynamics, vol. 2, no. 3, pp. 311–326, 2002.
- B. B. Kadomtsev and V. I. Petviashvili, “Stability of combined waves in weakly dispersing media,” Doklady Akademii Nauk SSSR, vol. 192, no. 4, pp. 753–756, 1970.
- R. Hirota, The Direct Method in Soliton Theory, Cambridge University Press, Cambridge, UK, 2004.
- Z. Dai, Y. Huang, X. Sun, D. Li, and Z. Hu, “Exact singular and non-singular solitary-wave solutions for Kadomtsev-Petviashvili equation with p-power of nonlinearity,” Chaos, Solitons and Fractals, vol. 40, no. 2, pp. 946–951, 2009.
- V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons, Springer, Berlin, Germany, 1991.
- H. Holden, B. Øksendal, J. Ubøe, and T. Zhang, Stochastic Partial Differential Equations: A Modeling, White Noise Functional Approach, Birkhäuser, Basel, Switzerland, 1996.