Simple Equations methodology (SEsM) for searching of multisolitons and other exact solutions of nonlinear partial differential equations

TL;DR

The paper introduces an enhanced methodology for finding exact solutions, including multisolitons, of nonlinear PDEs by utilizing multiple simplest equations, transformations, and relationships, enabling solutions for both integrable and nonintegrable cases.

Contribution

It presents a novel version of the Simple Equations Methodology that incorporates multiple simplest equations and transformations, expanding the scope of solvable nonlinear PDEs.

Findings

Able to obtain multisoliton solutions for certain nonlinear PDEs

Can find particular solutions for nonintegrable equations

Introduces a special simplest equation for fractional power solutions

Abstract

We discuss a version the methodology for obtaining exact solutions of nonlinear partial differential equations based on 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: (i) obtaining multi-soliton solutions of nonlinear partial differential equations if such solutions do exist; (ii) obtaining particular solutions of nonintegrable nonlinear partial differential equations. Several examples for the application of the methodology are discussed. Special attention is devoted to the use of the simplest equation $f_\xi =n[f^{(n-1)/n} - f^{(n+1)/n}]$ where $n$ is a positive real number. This simplest equation allows us to obtain exact solutions of nonlinear partial differential equations containing fractional powers.