Be careful with the Exp-function method

TL;DR

This paper critically examines the Exp-function method for solving nonlinear differential equations, highlighting common mistakes, emphasizing the need for solution simplification, and discussing alternative approaches like the singular manifold and simple equation methods.

Contribution

It identifies typical errors in applying the Exp-function method and introduces the modified simple equation method for finding exact solutions.

Findings

Common mistakes in Exp-function method application are demonstrated.

Simplification of obtained solutions is often necessary.

Alternative methods like the singular manifold and simple equation methods are discussed.

Abstract

An application of the Exp-function method to search for exact solutions of nonlinear differential equations is analyzed. Typical mistakes of application of the Exp-function method are demonstrated. We show it is often required to simplify the exact solutions obtained. Possibilities of the Exp-function method and other approaches in mathematical physics are discussed. The application of the singular manifold method for finding exact solutions of the Fitzhugh - Nagumo equation is illustrated. The modified simple equation method is introduced. This approach is used to look for exact solutions of the generalized Korteweg - de Vries equation.