The Modified Simple Equation Method for Exact and Solitary Wave Solutions of Nonlinear Evolution Equation: The GZK-BBM Equation and Right-Handed Noncommutative Burgers Equations

The modified simple equation method is significant for finding the exact traveling wave solutions of nonlinear evolution equations (NLEEs) in mathematical physics. In this paper, we bring in the modified simple equation (MSE) method for solving NLEEs via the Generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony (GZK-BBM) equation and the right-handed noncommutative Burgers' (nc-Burgers) equations and achieve the exact solutions involving parameters. When the parameters are taken as special values, the solitary wave solutions are originated from the traveling wave solutions. It is established that the MSE method offers a further influential mathematical tool for constructing the exact solutions of NLEEs in mathematical physics. 1. Introduction The importance of nonlinear evolution equations (NLEEs) is now well established, since these equations arise in various areas of science and engineering, especially in fluid mechanics, biology, plasma physics, solid-state physics, optical fibers, biophysics and so on. As a key problem, finding their analytical solutions is of great importance and is actually executed through various efficient and powerful methods such as the Exp-function method [1–4], the tanh-function method [5, 6], the homogeneous balance method [7, 8], the -expansion method [9–16], the Hirota’s bilinear transformation method [17, 18], the Backlund transformation method [19], the inverse scattering transformation [20], the Jacobi elliptic function method [21], the modified simple equation method [22–24] and so on. The objective of this paper is to look for new study relating to the MSE method via the well-recognized GZK-BBM equation and right-handed nc-Burgers’ equation and establish the originality and effectiveness of the method. The paper is organized as follows: in Section 2, we give the description of the MSE method. In Section 3, we use this method to the nonlinear evolution equations pointed out above, and in Section 4 conclusions are given. 2. Description of the MSE Method Suppose the nonlinear evolution equation is in the following form: where ？？is a polynomial of and its partial derivatives wherein the highest order derivatives and nonlinear terms are concerned. The main steps of the MSE method [22–24] are as follows. Step 1. The traveling wave transformation permits us to reduce (1) into the following ordinary differential equation (ODE): where is a polynomial in and its total derivatives, wherein . Step 2. We suppose the solution of (3) is of the form where ？？ are arbitrary constants to be determined, such that , and is an unidentified