On the modified method of simplest equation and the nonlinear Schr\"odinger equation

TL;DR

This paper extends the modified method of simplest equation to use two equations simultaneously, enabling the derivation of exact solutions for nonlinear Schrödinger equations related to deep water waves.

Contribution

It introduces an extended methodology that applies two simplest equations, broadening the solution techniques for nonlinear Schrödinger equations.

Findings

Successfully obtained exact solutions for nonlinear Schrödinger equations

Extended methodology applicable to other nonlinear Schrödinger-type equations

Demonstrated effectiveness for deep water wave models

Abstract

We consider an extension of the methodology of the modified method of simplest equation to the case of use of two simplest equations. The extended methodology is applied for obtaining exact solutions of model nonlinear partial differential equations for deep water waves: the nonlinear Schr\"odinger equation. It is shown that the methodology works also for other equations of the nonlinear Schr\"odinger kind.