New developments of the methodology of the Modified method of simplest equation with application

TL;DR

This paper extends the modified method of simplest equation to include multiple equations, transformations, and relationships, enabling the derivation of multi-soliton solutions and solutions for nonintegrable nonlinear PDEs.

Contribution

It introduces a generalized version of the modified method of simplest equation, enhancing its capability to find exact solutions for a broader class of nonlinear PDEs.

Findings

Able to obtain multi-soliton solutions where they exist

Can find particular solutions for nonintegrable equations

Demonstrated with application examples

Abstract

We discuss an extension of the modified method of simplest equation for obtaining exact analytical solutions of nonlinear partial differential equations. The extension includes the possibility for use of: (i) more than one simplest equation; (ii) relationship that contains as particular cases the relationship used by Hirota \cite{hirota} and the relationship used in the previous version of the methodology; (iii) transformation of the solution that contains as particular case the possibility of use of the Painleve expansion; (iv) more than one balance equation. The discussed version of the methodology allows: obtaining multi-soliton solutions of nonlinear partial differential equations if such solutions do exist and obtaining particular solutions of nonintegrable nonlinear partial differential equations. Examples for the application of the methodology are discussed.