Beyond the tanh method-looking for explicit travelling wave solutions to partial differential equations

Abstract

In this work, we focus on a general procedure for finding exact travelling wave solutions for evolution equations with polynomial nonlinearites. Mathematically, looking for travelling wave solutions is asking the question whether a given PDE has solutions invariant under a Galilean transformation; in such a case, it can be reduced to an ODE. We discuss the existence of travelling wave solutions by using phase plane analysis. We show that popular methods such as the tanh-method, G0/G-method and many more are special cases of the presented approach. Analytical solutions to several examples of nonlinear equations are illustrated. In the application, we use the Maple program to compute solutions to nonlinear systems of equations.