Exact Solutions of the BBM and MBBM Equations by the Generalized (G'/G )-expansion Method Equations

In this article, we establish exact solutions for the BBM and the MBBM equations by using a generalized (G'/G )-expansion method. The generalized (G'/G )-expansion method was used to construct solitary wave and periodic wave solutions of nonlinear evolution equations. This method is developed for searching exact travelling wave solutions of nonlinear partial differential equations. It is shown that the (G'/G )-expansion method is straightforward and powerful mathematical tool for solving nonlinear problems.

(1.1) is well known in physical applications. This equation models long waves in a nonlinear dispersive system and as a result, t he BBM equation incorporates dispersive effects. The solutions of the BBM equation exhibit definite soliton-like behavior that is not explainable by any known theory [21]. Both KdV and BBM equations cover cases of the following type [22]: surface waves of long wavelength in liquids, acoustic-gravity waves in compressible fluids, hydromagnetics waves in cold plasm, acoustic waves in anharmonic crystals. Motivated by the rich treasure of the Benjamin-Bona-Mahony equation in science, we will study nonlinear dispersive the modified Benjamin-Bona-Mahony equation (MBBM) [24] (1.2) which was first derived to describe an approximation for surface long waves in nonlinear dispersive media [25]. The equation can also characterize the hydromagnetic waves in cold plasma, acoustic waves in anharmonic crystals and acoustic-gravity waves in compressible fluids [26,27]. The article is organized as follows: In Section 2, first we briefly give the steps of the m ethod and apply it to solve nonlinear partial differential equations. In Sections 3 and 4, we employ this technique to the BBM and the MBBM equations. Also a conclusion is given in Section 5.

BASIC IDEA OF (G'/G)-EXPANSION METHOD
We give the detailed description of method which first presented by Wang [20].
Step 1: For a given nonlinear partial differential equations (NLPDEs) with independent variables X = (x,y,z,t) and dependent variable u can be converted to an ODE (2.2) which transformation ξ = x+y-ct is wave variable. Also, c is constant to be determined later.
Step 2: We seek its solutions in the more general polynomial form as follows (2.3) where G(ξ) satisfies the second order linear ordinary differential equation (LODE) in the form where, a 0 , a k (k = 1,2,…,m), λ and µ are constants to be determined later a m = 0 but the degree of which is generally equal to or less than m-1, the positive integer m can be determined by considering the homogeneous balance between the highest order derivatives and nonlinear terms appearing in Eq. (2.2).
Step 3: Substituting (2.3) and Eq. (2.4) into Eq. (2.2) with the value of m obtained in Step 1. Collectingthe coefficients of then setting each coefficient to zero, we can get a set of over-determined partial differential equations for a 0 , a i (k = 1,2,…,m), λ, c and µ with the aid of symbolic computation Maple 12.

CONCLUSION
In this article, we investigated the BBM and the modified BBM equations using the generalized (G'/G)-expansion method. This method is applied for finding travelling wave solutions of nonlinear evolution equations. This method has been successfully applied to obtain some new generalized solitonary solutions to the BBM and the MBBM equations. These exact solutions include three types hyperbolic function solution, trigonometric function solution and rational solution. The generalized (G'/G)-expansion method is more powerful in searching for exact solutions of NLPDEs. Can be seen that the results are the same, with comparing results [24]. Also, new results are formally developed in this article. It can be concluded that the this method is a very powerful and efficient technique in finding exact solutions for wide classes of problems.that the this method is a very powerful and efficient technique in finding exact solutions for wide classes of problems.