Transformation methods in the study of nonlinear partial differential equations
Transformation methods are perhaps the most powerful analytic tool currently available in the study of nonlinear partial differential equations. Transformations may be classified into two categories: category I includes transformations of the dependent and independent variables of a given partial differential equation and category II additionally includes transformations of the derivatives of the dependent variables. In part I of this thesis our principal attention is focused on transformations of the category I, namely point transformations. We mainly deal with groups of transformations. These groups enable us to derive similarity transformations which reduce the number of independent variables of a certain partial differential equation. Firstly, we introduce the concept of transformation groups and in the analysis which follows three methods for determining transformation groups are presented and consequently the corresponding similarity transformations are derived. We also present a direct method for determining similarity transformations. Finally, we classify all point transformations for a particular class of equations, namely the generalised Burgers equation. Bäcklund transformations belong to category II and they are investigated in part II. The first chapter is an introduction to the theory of Bäcklund transformations. Here two different classes of Bäcklund transformations are defined and appropriate example are given. These two classes are considered in the proceeding analysis, where we search for Bäcklund transformations for specific classes of partial differential equations.