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

Abstract

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.

1. 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.

In [1], Lee and Sakthivel used the exp-function method to obtain some exact travelling waves solutions for the following Schamel-Korteweg-de Vries equation [2–4].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 and the Schamel equation when [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 -expansion method to look for travelling wave solutions of nonlinear evolution equations. The determination of exact solutions to the variable coefficients partial differential equations is a complicated problem that challenges researchers greatly. We will use their theory and the -expansion method to give exact solutions of the following Wick-type stochastic Schamel-Korteweg-de Vries equation with variable coefficients:where is the Wick production on the Hida distribution space , namely, is the white noise functional space which is defined in [12], and , and are white noise functionals.

This paper is organized as follows. In Section 2, we briefly describe some basic concepts on Wick-type and main steps of finding solutions. In Section 3, we describe the -expansion method. In Section 4, we use white noise analysis and Hermite transformation to obtain a number of Wick versions of hyperbolic, trigonometric, and rational solutions. The conclusions are given in the final section.

2. Some Basic Concepts and Main Steps

Assume that and are the Hida test function space and the Hida disturbance space on , respectively. And let be the -order Hermite polynomials. PutWe have that the collection constitutes an orthogonal basis of .

If we denote as -dimensional multi-indices with , we have that the family of tensor products forms an orthogonal basis for Let be the th multi-index number in some fixed ordering of all -dimensional multi-indices We assume that this ordering has the following property:We defineand denote multi-indices as elements of the space of all sequences with elements and with compact support, that is, with only finitely many We denote .

Fixing , let consist of thosewhere with , and , where is the white noise measure on ,

The space consists of all formal expansionswhere The family of seminorms gives rise to a topology on , and we can regard as the dual of by the actionwhere is the usual inner product in .

For with , is called the Wick product of and .

We can prove that the spaces , and are closed under Wick products.

For with , the Hermite transform of is denoted as follows:where (the set of all sequences of complex numbers), when .

For , we can findfor 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

Let Then the vector is called the generalized expectation of and is denoted by Suppose that is an analytic function, where is a neighborhood of Assume that the Taylor series of around has coefficients in , and we can find the Wick version .

We define the Wick exponential of 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, .

Suppose that modeling considerations lead us to consider an SPDE expressed formally as , 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

Secondly, we take the Hermite transformation of (16). This turns Wick products into ordinary products and the equation takes the following form:where is the Hermite transformation of and are complex number. Suppose we find a solution of the equation for each , where

Then, under certain conditions, we can take the inverse Hermite transformation and obtain a solution of the original Wick equation (16). We have the following theorem, which was proved by Holden et al. [10].

Theorem 1. Suppose is a solution (in the usual strong, pointwise sense) of (17) for in some bounded open set , and for all , for some Moreover, suppose that and all its partial derivatives, which are involved in (17), 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 ) (16) in .

3. Description of the -Expansion Method

In this section, we describe the main steps of the -expansion method for finding travelling wave solutions of nonlinear evolution equations. As preparations, consider the second order linear ordinary differential equationand we letfor simplicity here and after. Using (19) and (20) yieldsFrom the three cases of general solutions of the linear ordinary differential equation (19), we have the following.

Case 1. When , the general solution of the linear ordinary differential equation (19) isand we havewhere and are two arbitrary constants and .

Case 2. When , the general solution of the linear ordinary differential equation (19) isand we havewhere and are two arbitrary constants and .

Case 3. When , the general solution of the linear ordinary differential equation (19) isand we havewhere and are two arbitrary constants.

Now we consider a nonlinear evolution equation, say in two independent variables and ,In general, the left-hand side of (28) is a polynomial in and its various partial derivatives. The main steps of the -expansion method are as follows.

Step 1. By coordinates transformation and with , (28) can be reduced to an ordinary differential equation on with

Step 2. Suppose that the solution of the ordinary differential equation (29) can be expressed by a polynomial in and aswhere satisfies the second-order linear ordinary differential equation (19), , and are constants to be determined later, and .

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 letthen substituting (32) into (29) to get a new equation in the new function with a positive integer balance number.
(b) When is a negative number, we letthen substituting (33) into (29) to get a new equation in the new function with a positive integer balance number.

Step 4. Substituting (30) into (29), using (21) and (23) (here Case is taken as an example), the left-hand side of (29) can be converted into a polynomial in and , in which the degree of is not larger than one. Equating each coefficient of the polynomial to zero yields a system of algebraic equations about and Then we solve the algebraic equations with the aid of Maple or Mathematica. Substituting the values of , and obtained into (30), one can obtain the travelling wave solutions expressed by the hyperbolic functions of (29).

Step 5. Similar to Steps 3 and 4, substituting (30) into (29), using (21) and (25) (or (21) and (27)), we obtain the travelling wave solutions of (29) expressed by trigonometric functions (or expressed by rational functions).

4. Exact Solutions of Stochastic Schamel-Korteweg-de Vries Equation

In this section, we will give exact solutions of (2).

Taking the Hermite transformation of (2), we can get the equationwhere is a parameter.

For the sake of simplicity, we denote ,  ,  , and . In the following, we apply the -expansion method to construct the travelling wave solutions of (34). In order to obtain exact solutions of (34), we consider the transformationwhere and will be determined later. Substituting (35) into (34), we haveThis implies

Equation (37) means These imply that is a constant; denote it by . Suppose and , where , and are real constants. From (38), we obtainThen by (38) and (39), we get the following ordinary differential equations with constant coefficients:

Integrating (40) once, and considering the constants of integration as zero, we can findWhen we consider the transformationswhere , we can rewrite (41) as

For simplicity, (43) can be written aswhere , and .

By balancing between and in (44), we get Consequently, we assume where and are constants to be determined later satisfying .

By employing the -expansion method with the aid of symbolic computation, we derive new exact travelling wave solutions, which include hyperbolic, trigonometric, and rational solutions for (34); there are three cases to be discussed as follows.

Case 1 (hyperbolic function solution). If , 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 , and as follows:

Solving the above algebraic equations, we get the following solutions: when , thenFrom these solutions, we obtain the hyperbolic function solutions of (34).

Theorem 2. Suppose that and Then (34) admits an exact solutionwhere, and   and are two arbitrary constants.

Case 2 (trigonometric function solution). If , substituting (46) into (44) and using (21) and (25), 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 , and as follows:Solving the above algebraic equations, we get the following solutions: when , thenFrom these solutions, we obtain the hyperbolic function solutions of (34).

Theorem 3. Suppose that and Then (34) admits an exact solutionwhere, and   and are two arbitrary constants.

Case 3 (rational function solutions). If , substituting (46) into (44) and using (21) and (27), 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 , and as follows:Solving the above algebraic equations, we get the following solutions: when , thenFrom the above expression of , and , we can get a contradiction. In Case , (34) does not admit an exact solution.

In order to get exact solutions of (2), we give the following conditions: () suppose , and satisfy the conditions that there exist a bounded open set , and such that , and are uniformly bounded for all , continuous with respect to for all and analytic with respect to , for all .

Under condition () Theorem 1 implies that there exists such thatwhich solves (34) for all which solves (2). From (57), we can find that is the inverse Hermite transformation of .

Hence, by Theorems 2 and 3, we yield the solutions of (2) as follows.

Theorem 4. Suppose that and Then (2) admits an exact solutionwhere and are two arbitrary constants.

Theorem 5. Suppose that and Then (2) admits an exact solutionwhere, and   and are two arbitrary constants.

Since Wick versions of functions are usually difficult to evaluate, we will give some non-Wick versions of solutions of (2) in special cases.

Let , where is a constant. is an integral or bounded measurable function on is Wiener white noise: that is, ,   is a Brown motion. We have the Hermite transforms: , where is parameter, and is defined in the second section.

Using for nonrandom and the definitions of , and , we have

Hence, by Theorems 4 and 5, we yield the solutions of (2) in special cases as follows.

Theorem 6. Suppose that and Then (2) admits an exact solutionwhere, and   and are two arbitrary constants.

Theorem 7. Suppose that and Then (2) admits an exact solutionwhere, and   and are two arbitrary constants.

5. Conclusion

On the earlier works given in the reference list, the authors considered the exact solutions of the Schamel-Korteweg-de Vries equation. But in this paper, we use the -expansion method to study the Wick-type stochastic Schamel-Korteweg-de Vries equation. We derive some new exact travelling wave solutions, which include hyperbolic and trigonometric solutions for the considered equations. The obtained solutions with free parameters may be important to expose most complex physical phenomena or to find new phenomena. It is shown in this paper that the -expansion method, with the help of symbolic computation like Maple or Mathematica, is direct, concise, and elementary.

Compared to other methods, like the exp-function method [1], the extended -expansion method [6], the sine-cosine method, and the extended tanh method [7], the -expansion method is effective and powerful in finding exact solutions of many other nonlinear evolution equations in mathematical physics, applied mathematics, and engineering. The equations are very difficult to be solved by traditional methods.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work is supported by the NSF of China (nos. 11401223 and 11626070), the NSF of Guangdong (no. 2015A030313424), the Science and Technology Program of Guangzhou (no. 201607010005), and the Scientific Program of Guangdong Province (no. 2016A030310262).