Exact Traveling Wave Solutions for Wick-Type Stochastic Schamel KdV Equation

F-expansionmethod is proposed to seek exact solutions of nonlinear partial differential equations. Bymeans of Hermite transform, inverse Hermite transform, and white noise analysis, the variable coefficients andWick-type stochastic Schamel KdV equations are completely described. Abundant exact traveling wave solutions for variable coefficients Schamel KdV equations are given. These solutions include exact stochastic Jacobi elliptic functions, trigonometric functions, and hyperbolic functions solutions.

In this paper we use F-expansion method for finding new periodic wave solutions of nonlinear evolution equations in mathematical physics, and we obtain some new periodic wave solutions for Schamel KdV equations.This method is more powerful and will be used in further works to establish more entirely new solutions for other kinds of nonlinear partial differential equations arising in mathematical physics.The effort in finding exact solutions to nonlinear equations is important for the understanding of most nonlinear physical phenomena, for instance, the nonlinear wave phenomena observed in the fluid dynamics, plasma, and optical fibers [1,2].Many effective methods have been presented such as homotopy analysis method [22], variational iteration method [23,24], tanh-function method [25][26][27], homotopy perturbation method [28][29][30], tanh-coth method [26,31,32], Exp-function method [33][34][35][36][37][38], Jacobi elliptic function expansion method [39][40][41][42], and F-expansion method [43][44][45][46].The main objective of this paper is using F-expansion method to construct the exact traveling wave solutions for Wick-type stochastic Schamel KdV equations via the Wicktype product, Hermite transform, and white noise analysis.If (1) is considered in a random environment, we can get stochastic Schamel KdV equations.In order to give the exact solutions of stochastic Schamel KdV equations, we only consider this problem in white noise environment.

Description of the F-Expansion Method
In order to simultaneously obtain more periodic wave solutions expressed by various Jacobi elliptic functions to nonlinear wave equations, we introduce an F-expansion method which can be thought of as a succinctly overall generalization of Jacobi elliptic function expansion.We briefly show what Fexpansion method is and how to use it to obtain various periodic wave solutions to nonlinear wave equations.Suppose a nonlinear wave equation for (, ) is given by where  = (,) is an unknown function and Ψ 1 is a polynomial in  and its various partial derivatives in which the highest order derivatives and nonlinear terms are involved.In the following we give the main steps of a deformation F-expansion method.
Step Hence, under the transformation in (4), then, (3) can be transformed into ordinary differential equation (ODE) as follows: Step 2. Suppose that () can be expressed by a finite power series of () of the form where  0 ,  1 , . . .,   are constants to be determined later, while   () in ( 6) satisfies and hence holds for (): where , , and  are constants.
Step 3. The positive integer  can be determined by considering the homogeneous balance between the highest derivative term and the nonlinear terms appearing in (5).Therefore, we can get the value of  in (6).
Setting each coefficient of this polynomial to be zero yields a set of algebraic equations for  0 ,  1 , . . .,   and .
Step 5. Solving the algebraic equations with the aid of Maple we have  0 ,  1 , . . .,   and  can be expressed by , , and .Substituting these results into F-expansion (6), then a general form of traveling wave solution of (3) can be obtained.
Step 6.Since the general solutions of ( 6) have been well known for us, choose properly , , and  in ODE (7) such that the corresponding solution () of it is one of Jacobi elliptic functions (see Appendices A, B, and C) [43][44][45].

White Noise Functional Solutions of (2)
In this section, we employ the results of Section 3 by using Hermite transform to obtain exact white noise functional solutions for Wick-type stochastic Schamel KdV equations (2).The properties of exponential and trigonometric functions yield the fact that there exists a bounded open set H ⊂ R + × R,  < ∞,  > 0 such that the solution (, , ) of ( 9) and all its partial derivatives which are involved in ( 9) are uniformly bounded for (, , ) ∈ H×  (), continuous with respect to (, ) ∈ H for all  ∈   (), and analytic with respect to  ∈   (), for all (, ) ∈ H. From Theorem 4.1.1 in [47], there exists (, , ) ∈ (S) −1 such that (, , ) = Ũ(, )() for all (, , ) ∈ H ×   () and (, ) solves ( 2) in (S) −1 .Hence, by applying the inverse Hermite transform to the results of Section 3, we get exact white noise functional solutions of (2) as follows.
(i) Exact stochastic Jacobi elliptic functions solutions: with (ii) Exact stochastic trigonometric solutions: with (iii) Exact stochastic hyperbolic solutions: with We observe that, for different forms of  1 ,  2 , and  3 , we can get different types of exact stochastic functional solutions of (2) from ( 34)- (38).

Example
It is well known that Wick version of function is usually difficult to evaluate.So, in this section, we give non-Wick version of solutions of (2).Let   = Ḃ  be the Gaussian white noise, where   is the Brownian motion.We have the Hermite transform [47]: Suppose that where  1 ,  2 , and  3 are arbitrary constants and () is integrable or bounded measurable function on R + .Therefore, for  1 () 2 () 3 () ̸ = 0, exact white noise functional solutions of (2) are as follows: with with Physics [  ()] 2 =  4 () +  2 () + ,    ()

Summary and Discussion
We have discussed the solutions of SPDEs driven by Gaussian white noise.There is a unitary mapping between the Gaussian white noise space and the Poisson white noise space.This connection was given by Benth and Gjerde [48].From [47, section 4.9] and by the aid of the connection, we can derive some stochastic exact soliton solutions, which are Poisson white noise functions in (2).In this paper, using Hermite transformation, white noise theory, and F-expansion method, we study the white noise functional solutions for Wick-type stochastic Schamel KdV equations.This paper shows that F-expansion method is sufficient to solve the stochastic nonlinear equations in mathematical physics.The method which we have proposed in this paper is standard, direct, and computerized method, which allows us to do complicated and tedious algebraic calculation.

C.
The ODE and Jacobi elliptic functions: for relation between values of , , and  and corresponding () in ODE, see Table 1.