Travelling Wave Solutions of a Generalized Variable-Coefficient Gardner Equation

Abstract

In this paper, a simple way to construct exact solutions by using equivalence transformations is shown. We consider a generalized variable-coefficient Gardner equation from the point of view of Lie symmetries in partial differential equations. We obtain the continuous equivalence transformations of the equation in order to reduce the number of arbitrary functions and give a clearer formulation of the results. Furthermore, we calculate Lie symmetries of the reduced equation. Then, we determine the similarity variables and the similarity solutions which allow us to reduce our equation into an ordinary differential equation. Finally, we obtain some exact travelling wave solutions of the equation by using the simplest equation method.