International Journal of Differential EquationsVolume 2011 (2011), Article ID 451420, 11 pagesdoi:10.1155/2011/451420

Research Article

Extended Jacobi Elliptic Function Expansion Method to the ZK-MEW Equation

Weimin Zhang

School of Mathematics, Jiaying University, Meizhou, Guangdong 514015, China

Received 3 May 2011; Accepted 25 June 2011

Academic Editor: Wen Xiu Ma

Copyright © 2011 Weimin Zhang. 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.

The extended Jacobi elliptic function expansion method is applied for Zakharov-Kuznetsov-modified equal-width (ZK-MEW) equation. With the aid of symbolic computation, we construct some new Jacobi elliptic doubly periodic wave solutions and the corresponding solitary wave solutions and triangular functional (singly periodic) solutions.

1. Introduction

It is one of the most important tasks to seek the exact solutions of nonlinear equation in the study of the nonlinear equations. Up to now, many powerful methods have been developed such as inverse scattering transformation [1], Backlund transformation [2], Hirota bilinear method [3], homogeneous balance method [4], extended tanh-function method [5], Jacobi elliptic function expansion method [6] and Ma’s transformed rational function method [7].

Recently [8], an extended-tanh method is used to establish exact travelling wave solution of the Zakharov-Kuznetsov-modified equal-width (ZK-MEW) equation. In this paper, an extended Jacobi elliptic function expansion method is employed to construct some new exact solutions of the Zakharov-Kuznetsov-modified equal-width (ZK-MEW) equation.

As known, the Zakharov-Kuznetsov (ZK) equation are given by where is the isotropic Laplacian. The ZK equation governs the behavior of weakly nonlinear ion-acoustic waves in a plasma comprising cold ions and hot isothermal electrons in the presence of a uniform magnetic field [9]. In [9], the ZK equation is solved by the sine-cosine and the tanh-function methods. In [10], the numbers of solitary waves, periodic waves, and kink waves of the modified Zakharov-Kuznetsov equation are obtained.

The regularized long wave (RLW) equation given by appears in many physical applications and has been studied in [11]. Gardner et al. [12] solved the equal width equation by a Petrov Galerkin method using quadratic B-spline spatial finite-elements.

The modified equal width (MEW) equation given by has been discussed in [11]. The MEW equation is related to the RLW equation. This equation has solitary waves with both positive and negative amplitudes. The two-dimensional ZK-MEW equation which first appeared in [13] is given by where , are constants. In [13], some exact solutions of the ZK-MEW equation (1.5) was obtained by using the tanh and sine-cosine methods. More detailed description for ZK-MEW equation (1.5) the reader can find in paper [13]. In this paper, we will give some new solutions of Jacobi elliptic function type of ZK-MEW equation by using an extended Jacobi elliptic function method.

The remainder of the paper is organized as follows. In Section 2, we briefly describe the extended Jacobi elliptic function expansion method. In Section 3, we apply this method to ZK-MEW equation to construct exact solutions. Finally, some conclusions are given in Section 4.

2. The Extended Jacobi Elliptic Function Expansion Method

In this section, the extended Jacobi elliptic function expansion method is proposed in [14]. Consider a given nonlinear wave equation, say in two variables We make the transformation where is a constant to be determined later. Then (2.1) reduceds to a nonlinear ordinary differential equation (ODE) under(2.2) By the extended Jacobi elliptic function expansion method, introduce the following ansatz where are constants to be determined later, is an Jacobi elliptic function, namely, , is the modulus of Jacobian elliptic functions. Positive integer can be determined by balancing the highest-order linear term with the nonlinear term in (2.3). After this, substituting (2.4) into (2.3), we can obtain a system of algebraic equations for . Solving the above-mentioned equations with the Mathematica Software, then can be determined. Substituting these obtained results into (2.4), then a general form of Jacobi elliptic function solution of (2.1) can be given.

3. ZK-MEW Equation

In this section, we employ the extended Jacobi elliptic function expansion method to ZK-MEW equation that is given by (1.5). The transformation converts (1.5) into ODE Integrating (3.1) and setting the constant of integration to zero, we obtain Substituting (2.4) into (3.2) to balance with , we find . Thus, the solution admits in the form where are constants to be determined later.

Notice that

3.1. The Case of

Substituting and (3.3) into (3.2), making use of (3.4), we obtain a system of algebraic equations, for , and of the following form: Solving the system of the algebraic equations with the aid of Mathematica we can distinguish two cases, namely the following.

Case 1.

Case 2.

Substituting (3.6), (3.7) into (3.3), respectively, yield, the following solutions of ZK-MEW equation: Notice that , , and , , we can obtain solitary wave solutions and sin-wave solutions from (3.8) and (3.9), respectively,

Here we only give the graph of (3.8) (see Figure 1) and (3.10a) (see Figure 2) and the other graphs of equations are similar to discussing.

(a)

(b)

Figure 1: (a) The graph of (3.8) with at , and (b) is its plane when .

(a)

(b)

Figure 2: (a) The graph of (3.10a) with at and (b) is its plane when (the case is the same to (3.8) when ).

3.2. The Case of

The analysis proceeds of this case is as for Section 3.1. Substituting and (3.3) into (3.2), making use of (3.4), we obtain a system of algebraic equations, for and of the following form: Solving the system of the algebraic equations with the aid of Mathematica we can distinguish three cases, namely, The following.

Case 1.

Case 2.

Case 3.

Substituting (3.13), (3.14), and (3.15) into (3.3), respectively, yields the following solutions of ZK-MEW equation: Notice that , , thus we can obtain solitary wave of solutions of ZK-MEW equation from (3.16) and (3.17), respectively, Here we only give the graph of (3.17) (see Figure 3) and (3.19) (see Figure 4) and the other graphs of equations are similar to discussing.

(a)

(b)

Figure 3: (a) The graph of (3.17) with at , and (b) is its plane when .

(a)

(b)

Figure 4: (a) The solitary wave graph of (3.19) with at , and (b) is its plane when (the case is the same to (3.17) when ).

3.3. The Case of

The analysis proceeds of this case is as for Sections 3.1 and 3.2. Substituting and (3.3) into (3.2), making use of (3.4), we obtain a system of algebraic equations, for , and of the following form: Solving the system of the algebraic equations (3.20) with the aid of Mathematica, we can distinguish three cases, namely, the following.

Case 1.

Case 2.

Case 3.

Substituting (3.21), (3.22), and (3.23) into (3.3), respectively, yield the following solutions of ZK-MEW equation: Especially, when , and , , thus we can obtain solutions of ZK-MEW equation from (3.24) and(3.25)

Remark 3.1. In these solutions, (3.10a), (3.19) and (3.27) have been obtain in [8], the others solutions are new solutions for the ZK-MEW equation.

4. Conclusions

The extended Jacobi elliptic function expansion method was directly and effectively employed to find travelling wave solutions of the nonlinear ZK-MEW equation. Using the method, we found some new solutions of Jacobi elliptic function type that were not obtained by the sine-cosine method, the extended tanh-method, the mapping method, and other methods. In the limiting case of the Jacobi elliptic function (namely, modulus setting 0 or 1), we also obtained the solutions of sin-type, cos-tye, tanh-type, and sech-type. The extended Jacobi elliptic function expansion method can be applied to some other nonlinear equation and gives more solutions.

The ZK-MEW equation was first appeared in Wazwaz’s paper [13] in 2005. To my acknowledge, its many properties, such as integrability, Lax pairs, and multisoliton solutions, have not been studied. The study of these properties is a very signification work and is our task research in the future.

Acknowledgments

The author would like to thank the anonymous referees for helpful suggestions that served to improve the paper. This paper is supported by Jiaying University (2011KJZ01).

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