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

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.

40 pages, 6 figures. arXiv admin note: substantial text overlap with arXiv:1904.03481, arXiv:1906.08053

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Cited by in corpus (1)

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