ISRN Applied MathematicsVolume 2011 (2011), Article ID 419678, 10 pagesdoi:10.5402/2011/419678

Research Article

Exact Explicit Traveling Wave Solution for the Generalized (2+1)-Dimensional Boussinesq Equation

Libing Zeng, Keding Qin, and Shengqiang Tang

School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guangxi, Guilin 541004, China

Received 10 March 2011; Accepted 24 April 2011

Academic Editors: A.-C. Lee and M. Qatu

Copyright © 2011 Libing Zeng 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.

The sine-cosine method and the extended tanh method are used to construct exact solitary patterns solution and compactons solutions of the generalized (2+1)-dimensional Boussinesq equation. The compactons solutions and solitary patterns solutions of the generalized (2+1)-dimensional Boussinesq equation are successfully obtained. These solutions may be important and of significance for the explanation of some practical physical problems. It is shown that the sine-cosine and the extended tanh methods provide a powerful mathematical tool for solving great many nonlinear partial differential equations in mathematical physics.

1. Introduction

Studies of various physical structures of nonlinear dispersive equations had attracted much attention in connection with the important problems that arise in scientific applications. Mathematically, these physical structures have been studied by using various analytical methods, such as inverse scattering method [1], Darboux transformation method [2, 3], Hirota bilinear method [4], Lie group method [5], bifurcation method of dynamic systems [6–8], sine-cosine method [9], tanh function method [10–12], Fan-expansion method [13], and homogenous balance method [14]. Practically, there is no unified technique that can be employed to handle all types of nonlinear differential equations.

Recently, by using the sine-cosine method, Taşcan and Bekir [15] studied the following (2+1)-dimensional Boussinesq equation: More new double periodic and multiple soliton solutions are obtained for the generalized (2+1)-dimensional Boussinesq equation in [16]. Chen et al. [17] study (2+1)-dimensional Boussinesq equation by using the new generalized transformation in homogeneous balance method. Feng et al. [8] have investigated the bifurcations and global dynamic behavior of two variants of (2+1)-dimensional Boussinesq-type equations with positive and negative exponents and obtained the sufficient conditions under which solitary, kink, breaking, and periodic wave solutions appear. A mathematical method is constructed to study two variants of the two-dimensional Boussinesq water equation with positive and negative exponents in [18].

In this paper, we consider the following generalized (2+1)-dimensional Boussinesq equation: where are arbitrary constants and is nonzero integers. Specially, when system (1.2) becomes the (2+1)-dimensional Boussinesq equation.

The sine-cosine method and the tanh method will be mainly used to back up our analysis. The tanh method, developed by Malfliet and Hereman [11, 12] and Wazwaz [19], is a direct and effective algebraic method for handling many nonlinear equations. The sine-cosine method was proved to be powerful in handling nonlinear problems, with genuine nonlinear dispersion, where compactons and solitary patterns solutions are generated. The two methods will be described briefly, where details can be found in [9–12, 19] and the references therein.

Let , where is the wave speed. Then (1.2) becomes where is the derivative with respect to . Integrating (1.3) twice and neglecting constants of integration, we find

Suppose that is a continuous solution of (1.4) for and . Recall that (i) is called a solitary wave solution if and (ii) is called a kink or antikink solution if . It is well known that a limit boundary curve of a smooth family of periodic orbits gives rise to a smooth periodic traveling wave solution, that is called a compacton.

The paper is organized as follows. In Section 2, the sine-cosine method and the tanh method are briefly discussed. Section 3 represents exact analytical solutions of (1.2) by using the sine-cosine method. Section 4 represents exact analytical solutions of (1.2) by using the extended tanh method. In the last section, we conclude the paper and give some discussions.

2. Analysis of the Two Methods

The sine-cosine method and the extended tanh method have been applied for a wide variety of nonlinear problems. The main features of the two methods will be reviewed briefly.

For both methods, we first use the wave variable to carry a PDE in two independent variables into an ODE

Equation (2.2) is then integrated as long as all terms contain derivatives where integration constants are considered zeros.

2.1. The Sine-Cosine Method

The sine-cosine method admits the use of the solution in the form or in the form where , and are parameters that will be determined.

We substitute (2.3) or (2.4) into the reduced ordinary differential equation obtained above in (2.2), balance the terms of the cosine functions when (2.3) is used or balance the terms of the sine functions when (2.4) is used, and solve the resulting system of algebraic equations by using the computerized symbolic calculations to obtain all possible values of the parameters , and .

2.2. The Tanh Method and the Extended Tanh Method

The standard tanh method is introduced in [11, 12] where the tanh is used as a new variable, since all derivatives of a tanh are represented by a tanh itself. We use a new independent variable that leads to the change of derivatives:

We then apply the following finite expansion: where is a positive integer that will be determined to derive a closed form analytic solution. However, if is not an integer, then a transformation formula is usually used. Substituting (2.5) and (2.6) into the simplified ODE (2.2) results in an equation in powers of . To determine the parameter , we usually balance the linear terms of highest order in the resulting equation with the highest order nonlinear terms. With determined, we collect all coefficients of powers of in the resulting equation where these coefficients have to vanish. This will give a system of algebraic equations involving the parameters , and . Having determined these parameters, knowing that is a positive integer in most cases and using (2.7) or (2.8), we obtain an analytic solution in a closed form.

3. Using the Sine-Cosine Method

Substituting (2.3) into (1.4) yields Equation (3.1) is satisfied only if the following system of algebraic equations holds: or Solving the system (3.2) and (3.3) gives The results (3.4) and (3.5) can be easily obtained if we also use the sine method (2.4). Combining (3.4) and (3.5) with (2.3) and (2.4), the following compactons solutions are yield: However, for , we obtain the following solitary patterns solutions:

4. Using the Extended Tanh Method

In this section, we employ the extended tanh method to (1.4). Balancing with , we find

To get a closed-form analytic solution, the parameter should be an integer. A transformation formula should be used to achieve our goal. This in turn transforms (1.4) to Balancing with gives . The extended tanh method allows us to use the substitution Substite (4.4) into (4.3), collecte the coefficients of each power of , and solve the resulting system of algebraic equations with the help of Maple to find the sets of solutions:

We obtain the solitary wave solution and the solitary patterns solutions

We can obtain the following compactons solutions for (1.4):

5. Discussion

The sine-cosine method and the extended tanh method were used to investigate the generalized (2+1)-dimensional Boussinesq equation. The study revealed compactons solutions and solitary patterns solutions for some examined variants. The study emphasized the fact that the two methods are reliable in handling nonlinear problems. The obtained results clearly demonstrate the efficiency of the two methods used in this work. Moreover, the methods are capable of greatly minimizing the size of computational work. This emphasizes the fact that the two methods are applicable to a wide variety of nonlinear problems.

Acknowledgments

The authors wish to thank the anonymous reviewers for their helpful comments and suggestions. This work is supported by the National Natural Science Foundation of China (no. 11061010).

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