Homotopic Approximate Solutions for the Perturbed CKdV Equation with Variable Coefficients

This work concerns how to find the double periodic form of approximate solutions of the perturbed combined KdV (CKdV) equation with variable coefficients by using the homotopic mapping method. The obtained solutions may degenerate into the approximate solutions of hyperbolic function form and the approximate solutions of trigonometric function form in the limit cases. Moreover, the first order approximate solutions and the second order approximate solutions of the variable coefficients CKdV equation in perturbation εu n are also induced.


Introduction
To solve the nonlinear partial differential equation (NPDE) has been an attractive research topic for mathematicians and physicists. Nonlinear evolution equations with variable coefficients can describe the physical phenomenon more accurately, and it is of great significance to study how to find the solutions of nonlinear evolution equations with variable coefficients.
In recent years, many researchers have developed various approaches for attaining the exact solutions and approximate of NPDE, such as the inverse scattering method [1], homogeneous balance method [2], elliptic function method [3], and perturbation method [4]. A recently reported analytic approximate method, the homotopic mapping method proposed in [5], has been applied to solve many nonlinear problems in engineering and technology effectively, like the nonlinear vibration of [6], boundary layer flow of [7], and so on [8][9][10][11][12]. However, the above works only studied the soliton approximate solutions of equations with constant coefficients. In this work, we applied the homotopic mapping method to the variable coefficients perturbed CKdV equation and obtained the approximate solution of the Jacobi elliptic function form.

Model and Homotopy Mapping
In this work, we focus on the perturbed CKdV equation with variable coefficients where ( ), ( ), ( ) is any function about , is a disturbance term, and is the sufficiently smooth function. This equation is widely used in the field of plasma physics [13], fluid mechanics [14], and quantum field theory [15]. It is fascinating to observe that when ( ), ( ), ( ) is constant, = 0, (1) becomes the well-known combined KdV equation, the equation in plasma physics which describes the acoustic wave propagation of a small-amplitude ion without Landau decay. It can be used as a model equation in fluid mechanics; related research can be referred to in [16,17]. When = ( ), (1) becomes the forced combined KdV equation, and the exact solutions of various forms are given in [18,19], such as solitary wave solutions, trigonometric function solutions, and Jacobian elliptic function solutions. When ( ) = 0, [20] studied the elliptic function solution of composite form. The Scientific World Journal Next, we study the approximate solution of (1). In order to simplify (1), set where 1 , 3 are any constants. By setting 2 ( ) = ( ) 1 / ( ), (1) can be expressed as where * ( , , , , ) For the sake of convenience, let = , = , ( , , , , ) = * ( , , , , ), and (3) can be written as the study on the solution of (1) is translated into the solution of (5). In order to get the solution of (5), we lead in homotopy mapping. where is parameter, V is auxiliary function, and (V) + (V) = 0.
One can easily prove that ( , 1) = 0 and (5) is the same, so the solution ( , ) of (5) is the solution of ( , ) = 0 when under the condition → 1. be the solution of ( , ) = 0; by [22] we can know this series is uniformly convergent in the ∈ [0, 1]. Thus, it yields that

Conclusion
This work studies the perturbed CKdV equation with variable coefficients by using the homotopic mapping method, and two degree approximate solution of the Jacobi elliptic function form are obtained, which can degenerate to solitary wave approximate solution and trigonometric function approximate solution in the limit cases. Furthermore, the approximate solution of the perturbed CKdV is also obtained. Our results show that the homotopic mapping method is applicable to the variable soliton equations. How to apply this method to high degree and high dimension system remains to be further studied.