Simple equations method (SEsM): Review and new results

Abstract

We discuss the Simple Equations Method (SEsM) for obtaining exact solutions of nonlinear partial differential equations. We show that specific case of SEsM can be used in order to reproduce the methodology of the Inverse Scattering Transform Method for the case of the Burgers equation and Korteweg - de Vries equation. This specific case is connected to use of a specific case of Step. 2 of SEsM: representation of the solution of the solved nonlinear partial differential equation as expansion as power series containing powers of a ”small” parameter ε, solving the differential equations occurring from this representation by means of Fourier series and transition from the obtained solution for small values of ε to solution for arbitrary finite values of ε. Next, we discuss the application of composite functions in SEsM. We proof two propositions connected to obtaining solutions of nonlinear differential equations with polynomial nonlinearities by means of use of composite functions. We present several examples of applications of this methodology and obtain exact solutions of the generalized Korteweg - deVries equation, Olver equation, and several other equations. Next we discuss the most simple version of SEsM: the Modified Method of Simplest Equation (MMSE). We start with the role of the simplest equation and discuss the several cases of simplest equations such as nonlinear ordinary differential equations called Riccati equation and Bernoulli equation. The theory is illustrated by obtaining exact solution of various nonlinear partial differential equations such as Newel-Whitehead equation, FitzHugh-Nagumo equation, etc. MMSE is further illustrated by obtaining exact solutions of many equations such as Swift-Hohenberg equation, Rayleigh equation, Huxley equation. Special attention is given to the process of obtaining of balance equations in the MMSE. This process is illustrated by obtaining balance equations for several model nonlinear differential equations from the area of ecology and population dynamics. Among the discussed examples are the reaction-diffusion equation with density-dependent diffusion as well as the reaction-telegraph equation. Finally we obtain exact solution of two nonlinear model differential equations connected to the water wave propagation. These are the extended Korteweg-de Vries equation and the generalized Camassa-Holm equation. We close the discussion by several remark on the methodology and about the future plans connected to our research in this area.