Modified method of simplest equation for obtaining exact analytical solutions of nonlinear partial differential equations: Further development of methodology with two applications

TL;DR

This paper advances a modified method of simplest equation to derive exact traveling wave solutions for nonlinear PDEs with polynomial nonlinearities, using special functions like elliptic functions, and demonstrates its application on Korteweg-de Vries equations.

Contribution

It develops a new variant of the simplest equation method employing special functions, enabling reduction of nonlinear PDEs to algebraic systems for solutions.

Findings

Derived solitary wave solutions for generalized Korteweg-de Vries equations.

Reduced complex nonlinear PDEs to algebraic systems for parameters.

Validated the methodology with higher order Korteweg-de Vries equations.

Abstract

We discuss the application of a variant of the method of simplest equation for obtaining exact traveling wave solutions of a class of nonlinear partial differential equations containing polynomial nonlinearities. As simplest equation we use differential equation for a special function that contains as particular cases trigonometric and hyperbolic functions as well as the elliptic function of Weierstrass and Jacobi. We show that for this case the studied class of nonlinear partial differential equations can be reduced to a system of two equations containing polynomials of the unknown functions. This system may be further reduced to a system of nonlinear algebraic equations for the parameters of the solved equation and parameters of the solution. Any nontrivial solution of the last system leads to a traveling wave solution of the solved nonlinear partial differential equation. The methodology is illustrated by obtaining solitary wave solutions for the generalized Korteweg-deVries equation and by obtaining solutions of the higher order Korteweg-deVries equation.