GROUP ANALYSIS OF A GENERALIZED KDV EQUATION

 

Professor Motlatsi Molati, National University of Lesotho (Appl. Math. Inf. Sci. 8, No. 6, 2845-2848)

Abstract

In this work the Korteweg-de Vries equation which contains an arbitrary function in the nonlinear term is considered and it is referred to as a generalized KdV. This equation has applications in nonlinear solitary wave phenomena in some areas of fluid mechanics, plasma physics and quantum mechanics. The Lie group analysis approach is employed to obtain the possible forms of the arbitrary parameter.

1 Introduction

Nonlinear evolution equations are widely used as mathematical models to describe nonlinear phenomena in various fields of science and engineering. It is desirable to determine the analytical solutions to these equations in order to understand better the complexity involved. The Korteweg-de Vries (KdV) equation is one of these nonlinear evolution equations and it models the propagation of solitary waves on the shallow water surfaces. In recent studies, different types of the KdV equation have been investigated to model various situations and the analytical and numerical solution procedures have been employed to solve these types of equations.

In many real life applications the model equations contain arbitrary functions of the dependent variable or its derivatives and independent variables. In solving these equations some special forms of the model parameters (arbitrary functions) are assumed which may lead to approximation of solutions. However, the Lie group based approach known as the method of Lie group classification
is a systematic procedure that enables the specification of the possible forms of the arbitrary functions which appear in the equation of interest. Depending upon the equation being considered either the approach based upon the equivalence transformations or the direct analysis approach of the group classification method can be used. The direct analysis is more preferable when the arbitrary functions depend upon only one variable, that is, either a dependent or an independent variable.

In this paper, we consider the generalized Korteweg-de Vries (gKdV) equation
ut +F(u)ux +uxxx = 0, (1) where the first term denotes the evolution term, the second term is the nonlinear term and lastly, the third term represents dispersion. Some special cases for the function F(u) have been studied in the literature. These forms reduce Eq. (1) to the much studied KdV type equations such as the modified KdV (mKdV) and Gardner equations. The current work deals with a systematic way of specifying this arbitrary function.

The plan of this work is as follows. In Section 3 the determining equations for Lie point symmetries and the classifying relation are generated. Moreover, the direct
analysis of the classifying relation is performed in order to obtain the possible forms of the arbitrary function. In Section 3 the symmetry reduction is performed and where possible exact solutions are obtained. Section 4 presents the summary of our investigations.