Solitary wave solutions of nonlinear partial differential equations based on the simplest equation for the function $1/\cosh^n$

TL;DR

This paper introduces a method using the simplest equation to find exact solitary wave solutions of nonlinear PDEs, including special cases like Korteweg-de Vries equations, based on the function 1/cosh^n.

Contribution

It develops a novel methodology employing the simplest equation for solving nonlinear PDEs with monomials of odd and even grades, expanding solution techniques for such equations.

Findings

Derived solitary wave solutions for specific classes of nonlinear PDEs.

Included particular solutions for Korteweg-de Vries and modified Korteweg-de Vries equations.

Demonstrated the method's applicability to equations with different monomial grades.

Abstract

The method of simplest equation is applied for obtaining exact solitary traveling-wave solutions of nonlinear partial differential equations that contain monomials of odd and even grade with respect to participating derivatives. The used simplest equation is $f_\xi^2 = n^2(f^2 -f^{(2n+2)/n})$. The developed methodology is illustrated on two examples of classes of nonlinear partial differential equations that contain: (i) only monomials of odd grade with respect to participating derivatives; (ii) only monomials of even grade with respect to participating derivatives. The obtained solitary wave solution for the case (i) contains as particular cases the solitary wave solutions of Korteweg-deVries equation and of a version of the modified Korteweg-deVries equation.