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Mathematical Problems in Engineering
Volume 2013 (2013), Article ID 923408, 7 pages
http://dx.doi.org/10.1155/2013/923408
Research Article

Variable-Coefficient Exact Solutions for Nonlinear Differential Equations by a New Bernoulli Equation-Based Subequation Method

School of Business, Shandong University of Technology, Zibo, Shandong 255049, China

Received 20 January 2013; Revised 10 March 2013; Accepted 11 March 2013

Academic Editor: Ebrahim Momoniat

Copyright © 2013 Chunxia Qi and Shunliang Huang. 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 new Bernoulli equation-based subequation method is proposed to establish variable-coefficient exact solutions for nonlinear differential equations. For illustrating the validity of this method, we apply it to the asymmetric (2 + 1)-dimensional NNV system and the Kaup-Kupershmidt equation. As a result, some new exact solutions with variable functions coefficients for them are successfully obtained.

1. Introduction

Nonlinear differential equations (NLDEs) can be used to describe many nonlinear phenomena such as fluid mechanics, plasma physics, optical fibers, biology, solid state physics, chemical kinematics, and chemical physics. Recently, research for seeking exact analytical solutions of NLDEs has been a hot topic, and many powerful and efficient methods to find exact solutions have been presented so far. For example, these methods include the known homogeneous balance method [1, 2], the tanh method [35], the inverse scattering transform [6], the Backlund transform [7, 8], the Hirotas bilinear method [9, 10], the generalized Riccati subequation method [11, 12], the Jacobi elliptic function expansion [13, 14], the -expansion method [15], the exp-function expansion method [16, 17], and the -expansion method [18, 19]. However, we notice that most of the existing methods are accompanied with constant coefficients, while very few methods are concerned with variable coefficients.

In this paper, by introducing a new ansatz, we develop a new Bernoulli equation-based sub equation method for obtaining variable-coefficient exact solutions for NLDEs. First we give the description of the Bernoulli equation-based subequation method. Then we apply the method to solve the asymmetric (2+1)-dimensional NNV system and the Kaup-Kupershmidt equation. Some conclusions are presented at the end of the paper.

2. Description of the Bernoulli Equation-Based Subequation Method

We consider the following Bernoulli equation: where is a complex number and . The solutions of (1) are denoted by where is an arbitrary constant. In particular, when is a real number and , we obtain

When , , where is a real number and is the unit of imaginary number, we obtain

Suppose that a nonlinear equation, say in two or three independent variables , , , is given by where is an unknown function and is a polynomial in and its various partial derivatives, in which the highest order derivatives and nonlinear terms are involved.

Step 1. Suppose that and then (5) can be turned into the following form:

Step 2. Suppose that the solution of (7) can be expressed by a polynomial in as follows: where satisfies (1) and are all unknown functions to be determined later with . The positive integer can be determined by considering the homogeneous balance between the highest order derivatives and nonlinear terms appearing in (7).

Step 3. Substituting (8) into (7) and using (1), collecting all terms with the same order of together, the left-hand side of (7) is converted to another polynomial in . Equating each coefficient of this polynomial to zero yields a set of partial differential equations for .

Step 4. Solving the equations system in Step 3, and using the solutions of (1), we can construct exact coefficient function solutions of (7).

Remark 1. As the partial differential equations in Step 3 are usually overdetermined, we may choose some special forms of as done in the following.

3. Application of the Bernoulli Equation-Based Subequation Method to Some NLDEs

3.1. Asymmetric (2+1)-Dimensional NNV System

We consider the (2+1)-dimensional asymmetric NNV system [2022]: where and are arbitrary nonzero constants.

Assume that , where , and then (9) can be turned into Suppose that the solutions of (10)-(11) can be expressed by a polynomial in as follows: where , are underdetermined functions and satisfies (1). Balancing the order of and in (10) and the order of and in (11), we can obtain , , . So we have Substituting (13) into (10)-(11) and collecting all the terms with the same power of together, equating each coefficient to zero yields a set of underdetermined partial differential equations for , , , and . Solving these equations yields the following.

Case 1. Consider where is an arbitrary constant and and are two arbitrary functions with respect to the variables and , respectively.

Case 2. Consider where , , and are arbitrary constants and and are two arbitrary functions with respect to the variables and , respectively.

Case 3. Consider where , , and are arbitrary constants and is an arbitrary function.

Case 4. Consider where is an arbitrary constant with and is an arbitrary function.

Substituting the results in the four cases mentioned previously into (13), and combining with the solutions of (1) as denoted in (2), we can obtain a rich variety of exact solutions to the asymmetric (2+1)-dimensional NNV system as follows.

Family 1. One has

Family 2. One has where

Family 3. One has where .

Family 4. Consider where .

Particularly, by a combination between Cases 14 and (3) we can obtain some special hyperbolic function solutions as follows: where .

Consider where

Consider where

Consider where .

By a combination with (4) we can obtain some special trigonometric function solutions as follows: where , is a real number and .

One has where .  

One has where where .

Remark 2. In [2022], some exact solutions for the asymmetric (2+1)-dimensional NNV system are established using different methods. We note that the established solutions mentioned previously are different from them essentially as they are new exact solutions with variable functions coefficients and have been reported by other authors in the literature.

3.2. Kaup-Kupershmidt Equation

We consider the following Kaup-Kupershmidt equation [2325]: Suppose that , , and, furthermore, by balancing the order in (31), we can suppose that where satisfies (1). Similar to the process mentioned before, we obtain Then we can obtain a kind of exact solutions of (31) which is unrelated to time variable as follows: In particular, by the combination with (3)-(4) we can obtain hyperbolic function and trigonometric function solutions as follows: where , is a real number and .

Remark 3. The previous established solutions for the Kaup-Kupershmidt equation cannot be obtained by the methods in [2325] and are new exact solutions to our best knowledge.

4. Conclusions

We have proposed a Bernoulli equation-based subequation method for solving nonlinear differential equations and applied it to find exact solutions with variable functions coefficients of the asymmetric (2+1)-dimensional asymmetric NNV system and the Kaup-Kupershmidt equation. As a result, some new exact solutions for them have been successfully found. Finally, we note that the proposed method can be applied to solve other nonlinear evolution equations.

Acknowledgments

This work is partially supported by Humanity and Social Science Youth Foundation of Ministry of Education of China (11YJCZH070). The authors would like to thank the reviewers very much for their valuable suggestions on this paper.

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