Seven common errors in finding exact solutions of nonlinear differential equations

TL;DR

This paper identifies and classifies seven common errors in the literature on finding exact solutions to nonlinear differential equations, highlighting issues like redundant solutions and misuse of arbitrary constants.

Contribution

It systematically analyzes and illustrates common mistakes in recent publications, clarifying the equivalence of solution methods and emphasizing correct solution practices.

Findings

Many solution methods are equivalent.

Authors often neglect general solutions.

Incorrect solutions are frequently published.

Abstract

We analyze the common errors of the recent papers in which the solitary wave solutions of nonlinear differential equations are presented. Seven common errors are formulated and classified. These errors are illustrated by using multiple examples of the common errors from the recent publications. We show that many popular methods in finding of the exact solutions are equivalent each other. We demonstrate that some authors look for the solitary wave solutions of nonlinear ordinary differential equations and do not take into account the well - known general solutions of these equations. We illustrate several cases when authors present some functions for describing solutions but do not use arbitrary constants. As this fact takes place the redundant solutions of differential equations are found. A few examples of incorrect solutions by some authors are presented. Several other errors in finding the exact solutions of nonlinear differential equations are also discussed.