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

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 (