Abstract

The Exp-function method combined with -expansion method is employed to investigate the equation with -dependent coefficients. The solitary wave solutions and periodic wave solutions to the equation are constructed analytically under certain circumstances. The results presented in this paper improve the previous results.

1. Introduction

The research work of nonlinear evolution equations in applied mathematics and theoretical physics has been going on for the past forty years. It is known that many physical phenomena are often described by nonlinear evolution equations. Searching for exact traveling wave solutions of nonlinear evolution equations plays an important role in the study of these nonlinear physical phenomena, for example, the wave phenomena observed in fluid dynamics, elastic media, optical fibers, and so forth. By dint of some new methods for obtaining exact solutions of nonlinear evolution equations, many new results have been published in this area for a long time. Here, it is worth to mention that the two methods, the Exp-function method [1–4] and -expansion method [5–8], can be combined to form one method [9–15].

In this paper, by using Exp-function method combined with -expansion method, we will study the generalized equation having -dependent coefficients [16]. Consider where and are functions of the time variable related to the linear decay or growth of the wave. The functions and are the time-dependent nonlinear and dispersion coefficients, respectively, with , , and being integers. Generally, (1) is not integrable.

Specially, when and with being a constant, (1) degenerates to the following generalized KdV equation with variable coefficient: which has been solved by using the Jacobi elliptic function expansion method and derived some new soliton-like solutions in [17]. Yu and Tian [18] studied the variable coefficient KdV equation (2) and obtained some new soliton-like solutions including nonsymmetrical kink solutions, compacton solutions, solitary pattern solutions, triangular function solution, and Jacobi and Weierstrass elliptic function solutions using the auxiliary equation method.

In 2009, by using solitary wave ansatz in the form of and functions, respectively, Triki and Wazwaz [16] obtained exact bright and dark soliton solutions for (1) in the cases and . Besides, we have recently derived some exact solutions of (1) by using Exp-function method combined with -expansion method in the cases and [19].

In this paper, we further extend the works made in [16, 19] by investigating (1). Using Exp-function method combined with -expansion method, we establish new solitary wave solutions and periodic solutions of (1) in the cases and , which is different from those presented in the previous works [16, 19]. It is shown that the variable coefficients , , and and the exponents , , and are the main factors to cause the qualitative change in the physical structures of the solutions.

2. Description of the Method

In this section, we review the combining of the Exp-function method with -expansion method [14, 15] at first.

Given a nonlinear partial differential equation, for instance, in two variables, as follows: where is in general a nonlinear function of its variables, we firstly use the Exp-function method to obtain new exact solutions of the following Riccati equation: where and are arbitrary constants, then using the Riccati equation (4) as auxiliary equation and its exact solutions, we obtain exact solutions of the nonlinear partial differential equation (3).

Seeking for the exact solutions of (4), we introduce a complex variable , defined by where is a constant to be determined later, is an arbitrary constant, and Riccati equation (4) converts to where prime denotes the derivative with respect to .

According to the Exp-function method, we assume that the solution of (6) can be expressed in the following form: where , , , and are positive integers which are given by the homogeneous balance principle, , are unknown constants to be determined. To determine the values of and , we usually balance the linear term of the highest-order in (6) with the highest-order nonlinear term. Similarly, we can determine and by balancing the linear term of the lowest-order in (6) with the lowest-order nonlinear term we obtain , . For simplicity, we set and ; then (7) becomes Substituting (8) into (6), equating to zero the coefficients of all powers of () yields a set of algebraic equations for , , , , , , and . Solving the system of algebraic equations by using Maple, we obtain the new exact solution of (4), which is read as follows: where , and are free parameters.

Consider where , and are free parameters.

By choosing properly values of , , , , we find many kinds of hyperbolic function solutions and triangular periodic solutions of (4), which are listed as follows.(i)When , , , , and solution (9) becomes (ii)When , , , , and solution (9) becomes (iii)When , , , , , and solution (10) becomes (iv)When , , , , , and solution (10) becomes (v)When , , , , , and solution (10) becomes (vi)When , , , , , and solution (10) becomes

For simplicity, in the rest of the paper, we consider .

3. Application to the Equation with -Dependent Coefficients

Balancing the order of the nonlinear term with the term in (1), we obtain so that To get a closed-form solution, it is natural to use the transformation and when , (1) becomes

This means that all the evolution terms that satisfy the condition contribute to the soliton formation.

In order to obtain new exact travelling wave solutions for (20), we use where and are functions of to be determined later, and substituting the (21) into (20), we obtain

Now, we assume that the solution of (22) can be expressed in the following form: where is a positive integer that is given by the homogeneous balance principle, and is a solution of (4). Balancing term with term in (22) gives . Therefore, we obtain

Substituting (24) into (22) and using the Riccati equation (4), collecting the coefficients of , we have Because the expresses to these coefficients of in (25) are too lengthiness, so we omit them; setting the coefficients to zero yields a set of algebraic equations as follows: Solving the algebraic equations obtained above, we can have the following three sets of solutions.

Case 1. Consider where , and are arbitrary nonzero constants.

Case 2. Consider where , and are arbitrary nonzero constants.

Case 3. Consider where , and are arbitrary nonzero constants.

Thus from (24), (27), (28), and (29) we obtain families of exact solutions to (22) as follows: where is a solution of (4).

Substituting new solutions (9) and (10) of Riccati equation into solutions (30), using the transformation (19), we have the following several families of solutions to (1).

Family 1. Consider where ,
If we set , , and in (31), we obtain
Setting , , and in (31), we get

Family 2. Consider where , , and , .
If we set , , , and in (35), we obtain Setting , , , and in (35), we get Setting , , , and in (35), we have Setting , , , and in (35), we have