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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 249043, 17 pages
http://dx.doi.org/10.1155/2013/249043
Research Article

The Study of the Solution to a Generalized KdV-mKdV Equation

1School of Finance, Chongqing Technology and Business University, Chongqing 400067, China
2School of Finance, Southwestern University of Finance and Economics, Chengdu 611130, China

Received 26 April 2013; Accepted 2 June 2013

Academic Editor: Shaoyong Lai

Copyright © 2013 Xiumei Lv et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A mathematical technique based on an auxiliary equation and the symbolic computation system Matlab is employed to investigate a generalized KdV-mKdV equation which possesses high-order nonlinear terms. Some new solutions including the Jacobi elliptic function solutions, the degenerated soliton-like solutions, and the triangle function solutions to the equation are obtained.

1. Introduction

Many powerful techniques have been established during the past decades for the study of the nonlinear dispersive partial differential equations [19]. The inverse scattering method, the Backlund transformation, the Darboux transformation, the Painleve’ analysis, the pseudospectral method, the finite differences method, and the sine-cosine ansatz are used to acquire solitary wave solutions and compactons solutions for some nonlinear equations. Wadati [1] employed the potential function to handle the KdV equation, and the resulting equation was solved iteratively by making use of a formal series for the potential function. Wadati [2] developed the trace method to handle the KP equation. The method, developed by Malfliet and Hereman [3], is heavily used in the literature to deal with nonlinear partial differential equations. Fan and Zhang [4] introduced a useful extended method that combined the standard method with the Riccati equation. The extensive method was effectively used by many researchers to investigate exact solutions for a lot of nonlinear models (see [10, 11]).

With the development of the symbolic computation system, the direct methods for constructing travelling wave solutions to differential equations become feasible. With the help of Mathematica, Sirendaoreji and Jiong [12, 13] used the auxiliary equation method to investigate KdV and mKdV equations, Boussinesq equations, sine-Gordon equations, and the nonlinear Klein-Gordon equations, respectively. The Jacobi elliptic function expansion method is confirmed as a powerful technique to solve some nonlinear differential equations (see [14]).

Zhang et al. [15] used a sub-ODE technique to investigate the exact solution of the following nonlinear dispersive KdV-mKdV equation: where , and are constants. The bell type solitary wave solution, the kink type solitary wave solution, the algebraic solitary wave solution, and the sinusoidal travelling wave solution of (1) with exponent were expressed explicitly in [15]. In fact, (1) is turned into a KdV equation if , and an mKdV equation if .

In this paper, we further develop the work in [15] for the study of (1). By using a mathematical technique different from those in previous works [110], we obtain the exact travelling wave solutions including Jacobi elliptic function solutions, degenerated soliton solutions, and triangle function solutions for (1) with exponent . Many of the solutions obtained are different from those presented in Zhang et al.’s work [15].

2. Brief Description of the Approach

To illustrate the basic concept of the auxiliary differential method, we consider that the nonlinear partial differential equation has the form Using the transformation and , (2) turns into the following nonlinear ordinary different equation:

We seek for the solutions of (3) in the form where are constants which will be determined later. The parameter is a positive integer and can be determined by balancing the highest-order derivative terms and the highest power nonlinear terms in (3). The highest degree of can be calculated by

We assume that represents the solutions of the following auxiliary differential equation where are real constants.

Substituting (4) and (6) into (3) and equating the coefficients of all powers of and to be zero in the resulting equation, several algebraic equations will be obtained. Then, solving these algebraic equations by the symbolic computation system Matlab and combining (4) and the solutions of the auxiliary equation (6), we can get the exact solutions for (2).

3. Exact Travelling Wave Solutions to (1)

Firstly, setting yields which turns (1) into

To find the traveling solutions for (1), we use the wave variable , where and . The wave variable transforms (8) into the following ordinary differential equation: From (9), we have . Therefore, we choose the ansatz where may be determined by which possesses several types of solutions listed in Table 1 (see [16]).

tab1
Table 1

In Table 1, functions , , and , are Jacobian elliptic functions with modulus , which have the properties ,,, , , , , and . Setting yields ,, and . When , it derives that , , and .

Substituting (10) and (11) into (9) and letting each coefficient of be zero, we obtain

Solving (12) with the help of Matlab, we acquire the following solutions: where the exponent , and .

3.1. The Jacobi Elliptic Function Solutions to (1) in the Case of

In order to make that wave number a real-valued number, we have to choose constants and modulus ( depends on ) to satisfy some restrictions. However, in this section, we allow to take values in complex number domain. From the expression of , and in (13) and the solutions listed in Table 1, we derive the following Jacobi elliptic function solutions for (1) in the case :

Remark 1. From the properties of Jacobi elliptic functions, we know that when the module or , sn, cn, and dn degenerate into triangular or hyperbolic functions, from which we can obtain soliton and triangular function solutions of (1) in the case . In other word, when , we can deduce Jacobi elliptic functions solutions, soliton, and triangular function solutions for (1).

Remark 2. When , , , and , turns into which was obtained by Yomba [17].

Remark 3. Supposing , ( and are arbitrary constants). When in , we obtain where . Solution (19) is in full agreement with that presented in [17] by Yomba.

3.2. The Jacobi Elliptic Function Solutions to (1) in the Case of

Similarly as in Section 3.1, we allow to take values in complex number field. Making use of (14) and the solutions listed in Table 1, we obtain the following Jacobi elliptic function solutions for (1) in the case :

When , we can also obtain Jacobi elliptic functions solutions, soliton, and triangular function solutions for (1).

3.3. The Jacobi Elliptic Function Solutions to (1) in the Case of??

From the expressions of (15), we get the following jacobi elliptic function solutions to (1):