Soliton Solutions for the Wick-Type Stochastic KP Equation

Abstract

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:𝑢_𝑡𝑥=𝑢_𝑥𝑥𝑥+6𝑢𝑢_𝑥𝑥+3𝛼²𝑢_𝑦𝑦, (1.1) 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 [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: 𝑈_𝑡𝑥=𝑓(𝑡)◊𝑈_𝑥𝑥𝑥+6𝑔(𝑡)◊𝑈◊𝑈_𝑥^◊_𝑥+3𝛼²𝑓(𝑡)◊𝑈_𝑦𝑦+𝑊(𝑡)◊𝑅^◊𝑈,𝑈_𝑥,𝑈_𝑥𝑥,𝑈_{𝑥𝑥𝑥𝑥},𝑈_𝑦𝑦, (1.2) which is the perturbation of the KP equation with variable coefficients: 𝑢_𝑡𝑥=𝑓(𝑡)𝑢_𝑥𝑥𝑥+6𝑔(𝑡)𝑢𝑢_𝑥𝑥+3𝛼²𝑓(𝑡)𝑢_𝑦𝑦, (1.3) by random force 𝑊(𝑡)◊𝑅^◊(𝑈,𝑈_𝑥,𝑈_𝑥𝑥,𝑈_{𝑥𝑥𝑥𝑥},𝑈_𝑦𝑦), where ◊ is the Wick product on the Hida distribution space (𝑆(ℝ^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)=𝐵̇(𝑡) and 𝐵(𝑡) is a Brownian motion, 𝑅(𝑢,𝑢_𝑥,𝑢_𝑥𝑥,𝑢_{𝑥𝑥𝑥𝑥},𝑢_𝑦𝑦)=𝛽𝑢_{𝑥𝑥𝑥𝑥}+6𝛾𝑢²_𝑥+6𝛾𝑢𝑢_𝑥𝑥+3𝛼²𝛽𝑢_𝑦𝑦 is a function of 𝑢,𝑢_𝑥,𝑢_𝑥𝑥,𝑢_{𝑥𝑥𝑥𝑥},𝑢_𝑦𝑦 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(ℝ^𝑑)) and (𝑆(ℝ^𝑑))^∗ be the Hida test function and the Hida distribution space on ℝ^𝑑, respectively. The collection 𝜉^𝑛=𝑒^{(−𝑥2/2)ℎ𝑛(√2𝑥)/(𝜋(𝑛−1)!)1/2},𝑛≥1 constitutes an orthogonal basis for 𝐿²(ℝ), where ℎ_𝑛(𝑥) is the 𝑑-order Hermite polynomials. The family of tensor products 𝜉_𝛼=𝜉_{𝛼1,…,𝛼𝑑}=𝜉_𝛼1⊗⋯⊗𝜉_𝛼1(𝛼∈ℕ^𝑑) forms an orthogonal basis for 𝐿²(ℝ^𝑑), where 𝛼=(𝛼₁,…,𝛼_𝑑) is 𝑑-dimensional multi-indices with 𝛼₁,…,𝛼_𝑑∈ℕ. The multi-indices 𝛼=(𝛼₁,…,𝛼_𝑑) are defined as elements of the space 𝒥=(ℕ^ℕ₀)_𝑐 of all sequences 𝛼=(𝛼₁,𝛼₂,…) with elements 𝛼_𝑖∈ℕ₀ and with compact support, that is, with only finite many 𝛼_𝑖≠0. For 𝛼=(𝛼₁,𝛼₂,…), we define 𝐻_𝛼(𝜔)=^∞_𝑖=1ℎ_𝛼𝑖𝜔,𝜂_𝑖,𝜔∈𝑆ℝ^𝑑∗. (2.1)

If 𝑛∈ℕ is fixed, let (𝑆)^𝑛₁ consist of those 𝑥=∑_𝛼𝑐_𝛼𝐻_𝛼∈⊕^𝑛_𝑘=1𝐿²(𝜇) with 𝑐_𝛼∈ℝ^𝑛 such that ‖𝑥‖²_1,𝑘=∑_𝛼𝑐²_𝛼(𝛼!)²(2ℕ)^𝑘𝛼<∞ for all 𝑘∈ℕ with 𝑐²_𝛼=|𝑐_𝛼|²=∑^𝑛_𝑘=1(𝑐_𝛼^(𝑘))² if 𝑐_𝛼=(𝑐_𝛼⁽¹⁾,…,𝑐_𝛼^(𝑛))∈ℝ^𝑛, where 𝜇 is the white noise measure on (𝑆^∗(ℝ),ℬ(𝑆^∗(ℝ))),𝛼!=∏^∞_𝑘=1𝛼_𝑘! and (2ℕ)^𝛼=∏_𝑗(2𝑗)^𝛼𝑗 for 𝛼=(𝛼₁,𝛼₂,…)∈𝒥. The space (𝑆)^𝑛₋₁ can be regarded as the dual of (𝑆)^𝑛₁. (𝑆)^𝑛₋₁ consisting of all formal expansion 𝑋=∑_𝛼𝑏_𝛼𝐻_𝛼 with 𝑏_𝛼∈ℝ^𝑛 such that ‖𝑋‖_−1,−𝑞=∑_𝛼𝑏²_𝛼(2ℕ)^−𝑞𝛼<∞ for some 𝑞∈ℕ, by the action ⟨𝑋,𝑥⟩=∑_𝛼(𝑏_𝛼,𝑐_𝛼)𝛼! and (𝑏_𝛼,𝑐_𝛼) is the usual inner product in ℝ^𝑛.

𝑋◊𝑌=∑_𝛼,𝛽(𝑎_𝛼,𝑏_𝛽)𝐻_𝛼+𝛽 is called the Wick product of 𝑋 and 𝑌, for 𝑋=∑_𝛼𝑎_𝛼𝐻_𝛼,𝑌=∑_𝛼𝑏_𝛼𝐻_𝛼∈(𝑆)^𝑛₋₁ with 𝑎_𝛼,𝑏_𝛼∈ℝ^𝑛. We can prove that the spaces (𝑆(ℝ^𝑑)),(𝑆(ℝ^𝑑))^∗(𝑆)^𝑛₁, and (𝑆)^𝑛₋₁ are closed under Wick products.

For 𝑋=∑_𝛼𝑎_𝛼𝐻_𝛼∈(𝑆)^𝑛₋₁ with 𝑎_𝛼∈ℝ^𝑛, ℋ(𝑋) or 𝑋 is defined as the Hermite transform of 𝑋 by ℋ(𝑋)(𝑧)=𝑋(𝑧)=∑_𝛼𝑎_𝛼𝑧^𝛼∈ℂ^𝑛 (when convergent), where 𝑧=(𝑧₁,𝑧₂,…)∈ℂ^ℕ (the set of all sequences of complex numbers) and 𝑧^𝛼=𝑧^𝛼1₁𝑧^𝛼2₂⋯𝑧^𝛼𝑛_𝑛⋯ for 𝛼=(𝛼₁,𝛼₂,…)∈𝒥. For 𝑋,𝑌∈(𝑆)^𝑁₋₁, by this definition we have 𝑋◊𝑌(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 ℂ^𝑁 defined by (𝑧¹₁,…,𝑧¹_𝑛)⋅(𝑧²₁,…,𝑧²_𝑛)=∑^𝑛_𝑘=1𝑧¹_𝑘𝑧²_𝑘, where 𝑧^𝑖_𝑘∈ℂ. Let 𝑋=∑_𝛼𝑎_𝛼𝐻_𝛼∈(𝑆)^𝑛₋₁. Then the vector 𝑐₀=𝑋(0)∈ℝ^𝑛 is called the generalized expectation of 𝑋 denoted by 𝔼(𝑋). Suppose that 𝑓∶𝑉→ℂ^𝑛 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 ℝ^𝑛. Then the Wick version 𝑓^◊(𝑋)=ℋ⁻¹(𝑓∘𝑋)∈(𝑆)^𝑛₋₁.

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,…,𝑐_𝑑)∈ℝ^𝑑. If we interpret all products as wick products and all functions as their Wick versions, we have 𝐴^◊𝑡,𝑥,𝜕_𝑡,∇_𝑥,𝑈,𝜔=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 𝐴𝑡,𝑥,𝜕_𝑡,∇_𝑥,𝑈,𝑧₁,𝑧₂,…=0, (2.3) where 𝑈=ℋ(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,…)∈𝕂_𝑞(𝑟) for some 𝑞,𝑟, where 𝕂_𝑞(𝑟)=𝑧=(𝑧₁,𝑧₂,…)∈ℂ^ℕ and ∑_𝛼≠0|𝑧^𝛼|²(2ℕ)^𝑞𝛼<𝑟². Then under certain conditions, we can take the inverse Hermite transform 𝑈=ℋ⁻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 [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⊂ℝ×ℝ^𝑑 and 𝑧∈𝕂_𝑞(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×𝕂_𝑞(𝑟), continuous with respect to (𝑡,𝑥)∈𝐺 for all 𝑧∈𝕂_𝑞(𝑟), and analytic with respect to 𝑧∈𝕂_𝑞(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×𝕂_𝑞(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]∈(𝑆)₋₁ for KP-I: 𝑈[1]=𝜆²2𝑘sechΦ2²,𝑤ℎ𝑒𝑛𝛼=1 (3.1) and for KP-II: 𝑈[1]=2𝑎²𝑘sech²Φ₁𝑡,𝑥,𝑦,𝑤ℎ𝑒𝑛𝛼=𝑖, (3.2) where Φ(𝑡,𝑥,𝑦)=𝜆𝑥+𝜆²𝑦+4𝜆³∫^𝑡₀𝑓(𝑠)𝑑𝑠+4𝜆³𝛽𝐵(𝑡)−2𝜆³𝛽𝑡² and Φ₁𝑡,𝑥,𝑦=𝑎𝑥−2𝑎𝑏𝑦+4𝑎³−3𝑎𝑏^2𝑡₀𝑓(𝑠)𝑑𝑠+4𝛽𝑎³−3𝑎𝑏²𝐵(𝑡)−12𝑡². (3.3)

Proof. Taking the Hermite transform of (1.2), the equation (1.2) can be changed into 𝑈_𝑡𝑥=𝑓(𝑡)+𝛽𝑊(𝑡,𝑧)𝑈_{𝑥𝑥𝑥𝑥}+6𝑔(𝑡)+𝛾𝑊(𝑡,𝑧)𝑈𝑈_𝑥𝑥+3𝛼²𝑓(𝑡)+𝛽𝑊(𝑡,𝑧)𝑈_𝑦𝑦, (3.4) where 𝑈 is the Hermite transform of 𝑈; the Hermite transform of 𝑊(𝑡) is defined by 𝑊(𝑡,𝑧)=∑^∞_𝑘=1𝜂_𝑘(𝑡)𝑧_𝑘 where 𝑧=(𝑧₁,𝑧₂,…)∈(ℂ^ℕ)_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 𝑢_𝑡𝑥=𝑓(𝑡)+𝛽𝑊(𝑡,𝑧)𝑢_𝑥𝑥𝑥+6𝑢𝑢_𝑥𝑥+3𝛼²𝑓(𝑡)+𝛽𝑊(𝑡,𝑧)𝑢_𝑦𝑦. (3.5) Let 𝐹(𝑡,𝑧)=𝑓(𝑡)+𝛽𝑊(𝑡,𝑧); then (3.5) can be changed into 𝑢_𝑡𝑥=𝐹(𝑡,𝑧)𝑢_𝑥𝑥𝑥+6𝑢𝑢_𝑥𝑥+3𝛼²𝐹(𝑡,𝑧)𝑢_𝑦𝑦. (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): 𝜙_𝑦=𝛼⁻¹𝜙_𝑥𝑥+𝛼⁻¹𝜙𝑢𝜙,_𝑡=4𝐹(𝑡,𝑧)𝜙_𝑥𝑥𝑥+6𝐹(𝑡,𝑧)𝑢𝜙_𝑥+3𝐹(𝑡,𝑧)𝛼𝑣_𝑦+𝑢_𝑥𝜙. (3.7) Then we can obtain the Wick-type Lax pair of (1.2): 𝜙_𝑦=𝛼⁻¹𝜙_𝑥𝑥+𝛼⁻¹𝑢◊𝜙,𝜙_𝑡=4𝑓(𝑡)+𝛽𝑊(𝑡)◊𝜙_𝑥𝑥𝑥+6𝑓(𝑡)+𝛽𝑊(𝑡)◊𝑢◊𝜙_𝑥+3𝑓(𝑡)+𝛽𝑊(𝑡)◊𝛼𝑣_𝑦+𝑢_𝑥◊𝜙. (3.8)
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[1]=𝜙_𝑥−(𝜙_1𝑥◊𝜙₁^◊(−1))◊𝜙, where 𝜙 is an arbitrary solution of (3.8), then 𝜙[1] satisfies the following equations: 𝜙_𝑦[1]=𝛼⁻¹𝜙_𝑥𝑥[1]+𝛼⁻¹𝑢[1]◊𝜙[1],𝜙_𝑡[1]=4𝑓(𝑡)+𝛽𝑊(𝑡)◊𝜙_𝑥𝑥𝑥[1]+6𝑓(𝑡)+𝛽𝑊(𝑡)◊𝑢[1]𝜙_𝑥[1]+3𝑓(𝑡)+𝛽𝑊(𝑡)◊𝛼𝑣_𝑦[1]+𝑢_𝑥[1]◊𝜙[1], (3.9) where 𝑢[1]=𝑢+2(𝜙_1𝑥◊𝜙₁^◊(−1))^◊_𝑥, 𝑣[1]=𝑣+2(𝜙_1𝑥◊𝜙₁^◊(−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 𝜙_𝑦=𝛼⁻¹𝜙_𝑥𝑥,𝜙_𝑡=4𝑓(𝑡)+𝛽𝑊(𝑡)◊𝜙_𝑥𝑥𝑥. (3.10)
If 𝛼=1, (3.10) have the exponential function solution 𝜙₁𝑡,𝑥,𝑦,𝑧=exp^◊𝜑𝑡,𝑥,𝑦,𝑧+1, (3.11) where 𝜑=𝜆𝑥+𝜆²𝑦+4𝜆^3𝑡₀𝑓(𝑠)𝑑𝑠+𝛽𝐵(𝑡), (3.12) 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: 𝑈[1]=2𝑘𝜙_1𝑥◊𝜙₁^◊(−1)𝜆◊𝜙=²2𝑘sech^◊Φ2², (3.13) where Φ𝑡,𝑥,𝑦=𝜆𝑥+𝜆²𝑦+4𝜆^3𝑡₀𝑓(𝑠)𝑑𝑠+4𝜆³𝛽𝐵(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 𝑈[1]=𝜆²2𝑘sechΦ2², (3.15) where Φ𝑡,𝑥,𝑦=𝜆𝑥+𝜆²𝑦+4𝜆^3𝑡₀𝑓(𝑠)𝑑𝑠+4𝜆³𝛽𝐵(𝑡)−2𝜆³𝛽𝑡². (3.16) In particular, when 𝑓(𝑠)=1 we can obtain the solution of (2.2), respectively, as follows: 𝑈[1]=𝜆²2𝑘sech²12𝜆𝑥+𝜆²𝑦+4𝜆³𝑡+4𝜆³𝛽𝐵(𝑡)−2𝜆³𝛽𝑡². (3.17) If 𝛼=𝑖, (3.10) have the exponential function solution 𝜙₁𝑡,𝑥,𝑦,𝑧=exp^◊𝜑₁𝑡,𝑥,𝑦,𝑧+exp^◊−𝜑₁𝑡,𝑥,𝑦,𝑧, (3.18) where 𝜑₁𝑡,𝑥,𝑦,𝑧=𝜆𝑥+𝑖𝜆²𝑦+4𝜆^3𝑡₀𝑓(𝑠)𝑑𝑠+𝛽𝐵(𝑡), (3.19)𝜑₁ is the conjugation of 𝜑₁ 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: 𝑈[1]=2𝑘𝜙_1𝑥◊𝜙₁^◊(−1)◊𝜙=2𝑎²𝑘sech^◊Φ₁𝑡,𝑥,𝑦², (3.20) where Φ₁𝑡,𝑥,𝑦=𝑎𝑥−2𝑎𝑏𝑦+4𝑎³−3𝑎𝑏^2𝑡₀𝑓(𝑠)𝑑𝑠+4𝑎³−3𝑎𝑏²𝛽𝐵(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]=2𝑎²𝑘sech²Φ₁𝑡,𝑥,𝑦, (3.22) where Φ₁𝑡,𝑥,𝑦=𝑎𝑥−2𝑎𝑏𝑦+4𝑎³−3𝑎𝑏^2𝑡₀𝑓(𝑠)𝑑𝑠+4𝛽𝑎³−3𝑎𝑏²𝐵(𝑡)−12𝑡². (3.23)
In particular, when 𝑓(𝑠)=1 we can obtain the solution of (2.2) as follows: 𝑈[1]=2𝑎²𝑘sech²𝑎𝑥−2𝑎𝑏𝑦+4𝑎³−3𝑎𝑏²𝛽𝑡−2𝑡²+𝛽𝐵(𝑡). (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 have𝜙[2]=𝜕−𝜙𝜕𝑥^2𝑥[1]𝜙₂[1]𝜕−𝜙𝜕𝑥𝜙^1𝑥₁𝜙, (4.1) where 𝜙₂[1] 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 𝜙₂[1]=𝜙_2𝑥−𝜙_1𝑥𝜙₁𝜙₂,𝑢[2]𝜕=𝑢+2²𝜕𝑥²ln𝑊𝜙₁,𝜙₂. (4.2)(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 𝜙₁,𝜙₂,…,𝜙_𝑁 be different and independent solutions of (3.8). We define the Wronski determinant 𝑊 of functions 𝑓₁,…,𝑓_𝑚 as𝑊𝑓₁,…,𝑓_𝑚=det𝐴,𝐴_𝑖𝑗=𝑑^𝑖−1𝑓_𝑗𝑑𝑥^𝑖−1,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]∈(𝑆)₋₁ satisfying 𝑈[𝑁]=2𝑘𝜕²𝜕𝑥²ln^◊𝑊^◊𝜙₁,…,𝜙_𝑁. (4.5)

Proof. From [6], it is easy to see that the function 𝜙[𝑁]=𝑊𝜙₁,…,𝜙_𝑁,𝜙𝑊𝜙₁,…,𝜙_𝑁 (4.6) satisfies the following equations: 𝜙_𝑦[𝑁]=𝛼⁻¹𝜙_𝑥𝑥[𝑁]+𝛼⁻¹𝑢[𝑁]𝜙[𝑁],𝜙_𝑡[𝑁]=4𝐹(𝑡,𝑧)𝜙_𝑥𝑥𝑥[𝑁]+6𝐹(𝑡,𝑧)𝑢[𝑁]𝜙_𝑥[𝑁]+3𝐹(𝑡,𝑧)𝛼𝑣_𝑦[𝑁]+𝑢_𝑥[𝑁]𝜙[𝑁], (4.7) where 𝑢[𝑁]=𝑢+2(𝜕²/𝜕𝑥²)ln𝑊(𝜙₁,…,𝜙_𝑁) and 𝑣[𝑁]=𝑣+2(𝜕/𝜕𝑥)ln𝑊(𝜙₁,…,𝜙_𝑁).
Then we have the Wick-type form 𝜙[𝑁]=𝑊^◊𝜙₁,…,𝜙_𝑁,𝜙𝑊^◊𝜙₁,…,𝜙_𝑁 (4.8) satisfying the following equations: 𝜙_𝑦[𝑁]=𝛼⁻¹𝜙_𝑥𝑥[𝑁]+𝛼⁻¹𝑢[𝑁]◊𝜙[𝑁],𝜙_𝑡[𝑁]=4𝑓(𝑡)+𝑊(𝑡)◊𝜙_𝑥𝑥𝑥[𝑁]+6𝑓(𝑡)+𝑊(𝑡)◊𝑢[𝑁]◊𝜙_𝑥[𝑁]+3𝑓(𝑡)+𝑊(𝑡)◊𝛼𝑣_𝑦[𝑁]+𝑢_𝑥[𝑁]◊𝜙[𝑁], (4.9) where 𝑢[𝑁]=𝑢+2(𝜕²/𝜕𝑥²)ln^◊𝑊^◊(𝜙₁,…,𝜙_𝑁).
In particular, taking 𝑢=0, 𝑣=0, we can obtain the 𝑁-soliton solution of (1.2): 𝑈[𝑁]=2𝑘𝜕²𝜕𝑥²ln^◊𝑊^◊𝜙₁,…,𝜙_𝑁. (4.10) When 𝛼=1 and 𝛼=𝑖, 𝜙₁,…,𝜙_𝑁 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 [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).