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

2019/08/02 by Vitanov, Nikolay K., Dimitrova, Zlatinka I.
#Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences

paper · doi:10.48550/arxiv.1908.01075

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 \citehirota 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_ξ=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.

Related

• New developments of the methodology of the Modified method of simplest equation with application
• The Simple Equations Method (SEsM) and the use of exponential functions for obtaining simple and multisoliton solutions of some nonlinear partial differential equations
• Multisoliton solutions via a separation of variables
• Modified method of simplest equation for obtaining exact solutions of nonlinear partial differential equations: past and present
• Simple equations method (SEsM) and some of its numerous particular cases