Traveling Wave Solutions of Some Coupled Nonlinear Evolution Equations

The modified simple equation (MSE) method is executed to find the traveling wave solutions for the coupled Konno-Oono equations and the variant Boussinesq equations. The efficiency of this method for finding exact solutions and traveling wave solutions has been demonstrated. It has been shown that the proposed method is direct, effective, and can be used for many other nonlinear evolution equations (NLEEs) in mathematical physics. Moreover, this procedure reduces the large volume of calculations. 1. Introduction Nowadays NLEEs have been the subject of all-embracing studies in various branches of nonlinear sciences. A special class of analytical solutions named traveling wave solutions for NLEEs has a lot of importance, because most of the phenomena that arise in mathematical physics and engineering fields can be described by NLEEs. NLEEs are frequently used to describe many problems of protein chemistry, chemically reactive materials, in ecology most population models, in physics the heat flow and the wave propagation phenomena, quantum mechanics, fluid mechanics, plasma physics, propagation of shallow water waves, optical fibers, biology, solid state physics, chemical kinematics, geochemistry, meteorology, electricity, and so forth. Therefore, investigation, traveling wave solutions is becoming more and more attractive in nonlinear sciences day by day. However, not all equations posed of these models are solvable. As a result, many new techniques have been successfully developed by diverse groups of mathematicians and physicists, such as the modified simple equation method [1–4], the extended tanh method [5, 6], the Exp-function method [7–11], the Adomian decomposition method [12], the F-expansion method [13], the auxiliary equation method [14], the Jacobi elliptic function method [15], modified Exp-function method [16], the -expansion method [17–26], Weierstrass elliptic function method [27], the homotopy perturbation method [28–30], the homogeneous balance method [31, 32], the Hirota’s bilinear transformation method [33, 34], the tanh-function method [35, 36] and so on. The objective of this paper is to apply the MSE method to construct the exact and traveling wave solutions for nonlinear evolution equations in mathematical physics via coupled Konno-Oono equations and variant Boussinesq equations. The paper is prepared as follows. In Section 2, the MSE method is discussed. In Section 3, we apply this method to the nonlinear evolution equations pointed out above, in Section 4, physical explanations, and in Section 5 conclusions are given. 2. The MSE