Exact Solutions for the Wick-Type Stochastic Schamel-Korteweg-de Vries Equation

We consider the Wick-type stochastic Schamel-Korteweg-de Vries equation with variable coefficients in this paper. With the aid of symbolic computation and Hermite transformation, by employing the -expansion method, we derive the new exact travelling wave solutions, which include hyperbolic and trigonometric solutions for the considered equations.


Introduction
The partial differential equations arise in many physical fields like the condense matter physics, fluid mechanics, plasma physics, optics, and so on, which exhibit a rich variety of nonlinear phenomena.It is known that to find exact solutions of the partial differential equations is always one of the central themes in mathematics and physics.
where  = (, ) denotes the unknown function of the space variable  and time  and the parameters , , and  are constants which refer to the activation trapping, the convection, and the dispersion coefficients, respectively.Equation (1) arises in number of scientific models, such as plasma physics and optical fibre.This equation describes the nonlinear interaction of ion-acoustic waves when electron trapping is present and also it governs the electrostatic potential for a certain electron distribution in velocity space.In addition, a generalized KdV equation is a special case of (1) which has been studied in a variety of mathematical physics contexts.Equation (1) incorporates the well-known KdV equation when  = 0 and the Schamel equation when  = 0 [5].
In [6], Kangalgil used the extended (  /)-expansion method to obtain some hyperbolic function solutions and trigonometric functions with free parameters of (1).In [7], by using the sine-cosine method and the extended tanh method, Yang and Tang obtained the soliton-like solutions, the kink solutions, and the plural solutions of (1).
When the inhomogeneities of media and nonuniformity of boundaries are taken into account in various real physical situations, the variable coefficient partial differential equations often can provide more powerful and realistic models than their constant coefficient counterparts in describing a large variety of real phenomena.Recently, the importance of taking random effects into account in modeling, analyzing, simulating, and predicting complex phenomena has been widely recognized in geophysical and climate dynamics, materials science, chemistry, biology, and other fields [8,9].Stochastic partial differential equations are appropriate mathematical models for complex systems under random influence in fluids.
The Wick-type stochastic Schamel-Korteweg-de Vries equation with variable coefficients is one of the most important stochastic partial differential equations and it has many applications.In [10], Holden et al. gave white noise functional approach to research stochastic partial differential equations in Wick versions.In [11], Li et al. introduced a new direct method called the (  /, 1/)-expansion method to look

Some Basic Concepts and Main Steps
Assume that S(R  ) and (S(R  )) * are the Hida test function space and the Hida disturbance space on R  , respectively.And let ℎ  () be the -order Hermite polynomials.Put We have that the collection {  } ≥1 constitutes an orthogonal basis of  2 (R).
We can prove that the spaces S(R  ), S(R  ) * , (S) 1 , and (S) −1 are closed under Wick products.
For  = ∑      ∈ (S)  −1 with   ∈ R  , the Hermite transform of  is denoted as follows: where  = ( 1 ,  2 , . ..) ∈ C  (the set of all sequences of complex numbers), For ,  ∈ (S)  −1 , we can find for all  such that X() and Ỹ() 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 Let  = ∑      , ∈ (S)  −1 .Then the vector  0 = X(0) ∈ R  is called the generalized expectation of  and is denoted by ().Suppose that  :  → C  is an analytic function, where  is a neighborhood of ().Assume that the Taylor series of  around () has coefficients in R  , and we can find the Wick version We define the Wick exponential of  ∈ (S)  −1 as follows: We can find that the Wick exponential has the same algebraic properties as the usual exponential with the use of the Hermite transformation.For example, exp ♢ { + } = exp ♢ {}♢ exp ♢ {}.Suppose that modeling considerations lead us to consider an SPDE expressed formally as (, ,   , ∇  , , ) = 0, where  is some given function and  = (, , ) is unknown (generalized) stochastic process, where the operators Firstly, we interpret all products as Wick products and all functions as their Wick versions indicate this as  ♢ (, ,   , ∇  , , ) = 0. (16) Secondly, we take the Hermite transformation of ( 16).This turns Wick products into ordinary products and the equation takes the following form: where Ũ = H() is the Hermite transformation of  and  1 ,  2 , . . .are complex number.Suppose we find a solution  = (, , ) of the equation Ã(, ,   , ∇  , , ) = 0 for each  = ( 1 ,  2 , . ..) ∈ K  (), where Then, under certain conditions, we can take the inverse Hermite transformation  = H −1  ∈ () −1 and obtain a solution  of the original Wick equation ( 16).We have the following theorem, which was proved by Holden et al. [10].

Description of the (𝐺 󸀠 /𝐺,1/𝐺)-Expansion Method
In this section, we describe the main steps of the (  /, 1/)expansion method for finding travelling wave solutions of nonlinear evolution equations.As preparations, consider the second order linear ordinary differential equation and we let for simplicity here and after.Using ( 19) and (20) yields From the three cases of general solutions of the linear ordinary differential equation ( 19), we have the following.
Case 1.When  < 0, the general solution of the linear ordinary differential equation ( 19) is and we have where  1 and  2 are two arbitrary constants and Case 2. When  > 0, the general solution of the linear ordinary differential equation ( 19) is and we have where  1 and  2 are two arbitrary constants and Case 3. When  = 0, the general solution of the linear ordinary differential equation ( 19) is and we have where  1 and  2 are two arbitrary constants.

Advances in Mathematical Physics
Now we consider a nonlinear evolution equation, say in two independent variables  and ,  (,   ,   ,   ,   ,   , . ..) = 0. (28) In general, the left-hand side of (28) is a polynomial in  and its various partial derivatives.The main steps of the (  /, 1/)-expansion method are as follows.
Step 3. Determine the positive integer  in (30) by using the homogeneous balance between the highest order derivatives and the nonlinear terms appearing in the ordinary differential equation (29).More precisely, we define the degree of () as [()] = , which gives rise to the degree of other expressions as follows: Therefore, we can get the value of  in (30).In some nonlinear equations the balance number  is not a positive integer.In this case, we make the following transformations [13]: (a) When  = /, where / is a fraction in the lowest terms, we let then substituting (32) into (29) to get a new equation in the new function V() with a positive integer balance number.(b) When  is a negative number, we let then substituting (33) into (29) to get a new equation in the new function V() with a positive integer balance number.

Exact Solutions of Stochastic Schamel-Korteweg-de Vries Equation
In this section, we will give exact solutions of (2).
Case 1 (hyperbolic function solution).If  < 0, substituting (46) into (44) and using ( 21) and ( 23), the left-hand side of (44) becomes a polynomial in  and .Setting the coefficients of this polynomial to be zero, we yield a system of algebraic equations in  1 ,  1 , , and  as follows: Solving the above algebraic equations, we get the following solutions: when 3 1 /( 2 (7 − 48) 2 − 3 2 ) > 0, then From these solutions, we obtain the hyperbolic function solutions of (34). where , and  1 and  2 are two arbitrary constants.