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Seven common errors in finding exact solutions of nonlinear differential equations © N.A. Kudryashov Communications in Nonlinear Science and Numerical Simulation (to be pubslished in 2009)

1. Introduction

During the last forty years we have been observing many publications presenting the exact solutions of nonlinear evolution equations. The emergence of these publications results from the fact that there are a lot of applications of nonlinear differential equations describing different processes in many scientific areas.

The start of this science area was given in the famous work by Zabusky and Kruskal  [1]. These authors showed that there are solitory waves with the property of the elastic particles in simple mathematical model. The result of the study of the Korteweg — de Vries equation was the discovery of the inverse scattering transform to solve the Cauchy problem for the integrable nonlinear differential equations  [2]. Now the inverse scattering transform is used to find the solution of the Cauchy problem for many nonlinear evolution equations  [345]. Later Hirota suggested the direct method  [6] now called the Hirota method, that allows us to look for the solitary wave solutions and rational solutions for the exact solvable nonlinear differential equations  [78].

As to nonintegrable nonlinear evolution equations we cannot point out the best method to look for the exact solutions of nonlinear differential equations. However we prefer to use the truncated expansion method by Weiss, Tabor and Carnevalle  [9] and the simplest equation method in finding the exact solutions. In section 1 we demonstrate that many popular methods to look for the exact solutions of nonlinear differential equations are based on the truncated expansion method. Many methods are obtained as consequence of the truncated expansion method and we are going to illustrate this fact in this paper.

Nowadays there are a lot of computer software programs like MATHEMATICA and MAPLE. Using these codes it is possible to have complicated analytical calculations to search for the different forms of the solutions for the nonlinear evolution equations and many authors use the computer codes to look for the exact solutions. However using the computer programs many investigators do not take into account some important properties of the differential equations. Therefore some authors obtain ”new” cumbersome exact solutions of the nonlinear differential equations with some errors and mistakes.

The aim of this paper is to classify and to demonstrate some common errors that we have observed studying many publications in the last years. Using some examples of the nonlinear differential equations from the recent publications we illustrate these common errors.

The outline of this paper is as follows. In section 2 we present some popular methods to search for the exact solutions of nonintegrable differential equations and we show that in essence all these approaches are equivalent. In section 3 we analyze some reductions of partial differential equations to nonlinear ordinary differential equations. We demonstrate that many reductions have the well - known general solutions and there is no need to study the solitary wave solutions for these mathematical models. In section 4 we give some examples when the authors remove the constants of integration and lose some exact solutions. In section 5 we demonstrate that many publications contain the redundant expressions for the exact solutions and these expressions can be simplified by taking arbitrary constants into account. In section 6 we discuss that some solitary wave solutions can be simplified by the authors. In section 7 we point out that we have to check some exact solutions of nonlinear differential equations because in number of cases we have ”solutions” which do not satisfy the equations studied. In section 8 we touch the solutions with redundant arbitrary constants. For many cases the redundant arbitrary constants do not lead to erroneous solutions but we believe that investigators have to take these facts into account.

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