The Simple Equations Method (SEsM) and the use of exponential functions for obtaining simple and multisoliton solutions of some nonlinear partial differential equations

TL;DR

The paper introduces the Simple Equations Method (SEsM), a versatile approach for deriving exact solutions, including multisoliton solutions, of various nonlinear partial differential equations, both integrable and nonintegrable.

Contribution

It presents SEsM as a generalization of existing methods, capable of producing exact solutions for a wide class of nonlinear PDEs, including multisoliton solutions.

Findings

SEsM can generate multisoliton solutions for integrable equations.

SEsM applies to nonintegrable equations, providing exact solutions.

The method encompasses many existing solution techniques.

Abstract

We discuss the last version as well as applications of a method for obtaining exact solutions of nonlinear partial differential equations. As this version is based on more than one simple equation we call it Simple Equations Method (SEsM). SEsM contains as particular case the Modified Method of Simplest Equation (MMSE) for the case when we use one simple equation and the solution is searched as power series of the solution of the simple equation. SEsM contains as particular cases many other methodologies for obtaining exact solutions of non-linear partial differential equations. We demonstrate that SEsM can lead to multisoliton solutions of integrable nonlinear partial differential equations and in addition we demonstrate that SEsM keeps the property of the Modified Method of Simplest Equation to lead to exact solutions of nonitegrable nonlinear partial differential equations.