International Journal of Engineering Mathematics

Volume 2013 (2013), Article ID 746910, 5 pages

http://dx.doi.org/10.1155/2013/746910

## New Application of -Expansion Method for Generalized (2+1)-Dimensional Nonlinear Evolution Equations

Department of Mathematics, Anar Branch, Islamic Azad University, Anar, Iran

Received 20 February 2013; Accepted 27 April 2013

Academic Editor: Shouming Zhong

Copyright © 2013 Mohammad Najafi 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

We established -expansion method for (2+1)-dimensional nonlinear evolution equations. This method was used to construct travelling wave solutions of (2+1)-dimensional nonlinear evolution equations. (2+1)-Dimensional breaking soliton equation, (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff (CBS) equation, and (2+1)-dimensional Bogoyavlenskii’s Breaking soliton equation are chosen to illustrate the effectiveness of the method.

#### 1. Introduction

In this work, we will study the generalized -dimensional nonlinear evolution equations where and are parameters, for example, namely, the -dimensional Calogero-Bogoyavlenskii-Schiff (CBS) equation for which and , [1]: and the -dimensional breaking soliton equation for which and , [2]: and the -dimensional Bogoyavlenskii’s Breaking soliton equation for which and , [3]: In this paper, we solve (1) by the ()-expansion method and obtain some exact and new solutions for (2), (3), and (4).

#### 2. The ()-Expansion Method

In this section we describe the ()-expansion method for finding traveling wave solutions of nonlinear evolution equations. Suppose that a nonlinear equation, say in two independent variables , is given by where and is a polynomial of and its derivatives, in which the highest order derivatives and nonlinear terms are involved. In the following we give the main steps of the ()-expansion method.

Firstly, suppose that The traveling wave variable (6) permits reducing (5) to an ODE for

Secondly, suppose that the solution of (7) can be expressed by a polynomial in () as follows: where satisfies the second-order LODE in the form and are constants to be determined later, . The unwritten part in (8) is also a polynomial in (), the degree of which is generally equal to or less than . The positive integer can be determined by considering the homogeneous balance between the highest order derivatives and nonlinear terms appearing in (7).

Thirdly, substituting (8) into (7) and using the second-order linear ODE (9), collecting all terms with the same order of () together, the left-hand side of (7) is converted into another polynomial in (). Equating each coefficient of this polynomial to zero yields a set of algebraic equations for and .

Fourthly, assuming that the constants and can be obtained by solving the algebraic equations in *Thirdly*. Since the general solutions of the second-order LODE (9) have been well known for us, then substituting and the general solutions of (9) into (8) we have traveling wave solutions of the nonlinear evolution equation (5) (for more details see [4–7]).

#### 3. New Application of ()-Expansion Method

In this section we apply the ()-expansion method to the generalized -dimensional nonlinear evolution equations We introduce a transformation By substituting (11) into (10), we have or equivalently where prime denotes the differential with respect to . Integrating (13) with respect to and taking the integration constant as zero yields in which (14) is obtained by integrating and neglecting the constant of integration and where prime denotes the derivative with respect to the same variable and is the integration constant that can be determined later.

Suppose that the solutions of the O.D.E (14) can be expressed by a polynomial in () as follows: where are constants, satisfies the second-order LODE in the form: where and are constants.

By balancing the order of and in (11), we have , then . So we can write where , are constants to be determined later. Then it follows Substituting (18), (19), and (20) into (14) and collecting all the terms with the same power of () together, equating each coefficient to zero yields a set of simultaneous algebraic equations as follows: Solving the algebraic equations above by using the Maple, we get where ,, are arbitrary constants.

Substituting (22) into (17) and by using (11), we have where .

Substituting the general solutions of (16) into (23), we obtain the following.

*Case 1. *When
where and , , , , are arbitrary constant.

*Case 2. * When
where and , , , , are arbitrary constant.

*Case 3. *When
where and , , , , are arbitrary constant.

In particular, if we take and , then becomes and becomes When we take and , then becomes and becomes In the following Sections 4–6, we will apply solutions to -dimensional breaking soliton equation and -dimensional Calogero-Bogoyavlenskii-Schiff (CBS) equation and -dimensional Bogoyavlenskii’s Breaking soliton equation.

#### 4. Exact Solution of (2)

In this section, we investigate explicit formula of solutions of the following -dimensional Calogero-Bogoyavlenskii-Schiff (CBS) equation given in [1] By using Section 3, we have following exact solution.

*Case 1. *When
where and , , , , are arbitrary constant.

*Case 2. *When
where and , , , , are arbitrary constant.

*Case 3. *When
where and , , , , are arbitrary constant.

In particular, if we take and , then becomes and becomes when we take and , then becomes and becomes

#### 5. Exact Solution of (3)

In this section, we investigate explicit formula of solutions of the following -dimensional Breaking soliton equation given in [2] By using Section 3, we have following exact solution:

*Case 1. *When
where and , , , , are arbitrary constant.

*Case 2. *When
where and , , , , are arbitrary constant.

*Case 3. * When
where and , , , , are arbitrary constant.

In particular, if we take and , then becomes and becomes When we take and , then becomes and becomes

#### 6. Exact Solution of (4)

In this section, we investigate explicit formula of solutions of the following -dimensional Bogoyavlenskii’s Breaking soliton equation given in [3]: By using Section 3, we have following exact solutions.

*Case 1. *When
where and , , , , are arbitrary constant.

*Case 2. *When
where and , , , , are arbitrary constant.

*Case 3. *When
where and , , , , are arbitrary constant.

In particular, if we take and , then becomes and becomes When we take and , then becomes and becomes

#### 7. Conclusions

In this paper, by using the ()-expansion method, we obtained some explicit formulas of solutions for the generalized -dimensional nonlinear evolution equations. We chose -dimensional breaking soliton equation and -dimensional Calogero-Bogoyavlenskii-Schiff (CBS) equation and -dimensional Bogoyavlenskii’s Breaking soliton equation to illustrate the effectiveness of the method. Those solutions were similar to the solutions obtained in other paper. The study reveals the power of the method.

#### References

- A. M. Wazwaz, “The (2+1)-dimensional CBS equations: multiple soliton solutions and multiple singular soliton solutions,”
*Zeitschrift fur Naturforschung A*, vol. 65, no. 3, pp. 173–181, 2010. View at Google Scholar · View at Scopus - A. M. Wazwaz, “Integrable (2+1)-dimensional breaking soliton equations,”
*Physica Scripta*, vol. 81, pp. 1–5, 2010. View at Google Scholar - M. T. Darvishi and M. Najafi, “Some exact solutions of the (2+1)-dimensional breaking soliton equation using the three-wave method,”
*International Journal of Computational and Mathematical Sciences*, vol. 6, no. 1, pp. 13–16, 2012. View at Google Scholar · View at Scopus - M. T. Darvishi, M. Najafi, and M. Najafi, “Traveling wave solutions for the (3+1)-dimensional breaking soliton equation by $\left({G}^{\prime}/G\right)$-expansion method and modified F-expansion method,”
*International Journal of Computational and Mathematical Sciences*, vol. 6, no. 2, pp. 64–69, 2012. View at Google Scholar - E. M. E. Zayed and S. Al-Joudi, “Applications of an Improved $\left({G}^{\prime}/G\right)$-expansion method to nonlinear PDEs in mathematical physics,”
*AIP Conference Proceedings*, vol. 1168, no. 1, pp. 371–376, 2009. View at Google Scholar - Z. Y. Bin and L. Chao, “Application of modified $\left({G}^{\prime}/G\right)$-expansion method to traveling wave solutions for Whitham-Broer-Kaup-like equations,”
*Communications in Theoretical Physics*, vol. 51, no. 4, pp. 664–670, 2009. View at Publisher · View at Google Scholar · View at Scopus - I. Aslan and T. Özis, “On the validity and reliability of the $\left({G}^{\prime}/G\right)$-expansion method by using higher-order nonlinear equations,”
*Applied Mathematics and Computation*, vol. 211, no. 2, pp. 531–536, 2009. View at Publisher · View at Google Scholar · View at Scopus