Remarks on the Simple Equations Method (SEsM) for Obtaining Exact Solutions of Nonlinear Differential Equations: Selected Simple Equations

We present a short review of the methodology and applications of the Simple Equations Method (SEsM) for obtaining exact solutions to nonlinear differential equations. The applications part of the review is focused on the simple equations used, with examples of the use of the differential equations for exponential functions, for the function 1p+exp(qξ)r, for the function 1/coshn, and for the function tanhn. We list several propositions and theorems that are part of the SEsM methodology. We show how SEsM can lead to multisoliton solutions of integrable equations. Furthermore, we note that each exact solution to a nonlinear differential equation can, in principle, be obtained by the methodology of SEsM. The methodology of SEsM can be based on different simple equations. Numerous methods exist for obtaining exact solutions to nonlinear differential equations, which are based on the construction of a solution using certain known functions. Many of these methods are specific cases of SEsM, where the simple differential equation used in SEsM is the equation whose solution is the corresponding function used in these methodologies. We note that the exact solutions obtained by SEsM can be used as a basis for further research on exact solutions to corresponding differential equations by the application of methods that use the symmetries of the solved equation.