Methods of symmetry reduction and their application

In this thesis methods of symmetry reduction are applied to several physically relevant partial differential equations.

The first chapter serves to acquaint the reader with the symmetry methods used in this thesis. In particular the classical method of Lie, an extension of it by Bluman and Cole [1969], known as the nonclassical method, and the direct method of Clarkson and Kruskal [1989] are described. Other known extensions of these methods are outlined, including potential symmetries, introduced by Bluman, Kumei and Reid [1988]. Also described are the tools used in practice to perform the calculations. The remainder of the thesis is split into two parts.

In Part One the classical and nonclassical methods are applied to three classes of scalar equation: a generalised Boussinesq equation, a class of third order equations and a class of fourth order equations. Many symmetry reductions and exact solutions are found.

In Part Two each of the classical, nonclassical and direct methods are applied to various systems of partial differential equations. These include shallow water wave systems, six representations of the Boussinesq equation and a reaction-diffusion equation written as a system. In Chapters Five and Six both the actual application of these methods and their results is compared and contrasted. In such applications, remarkable phenomena can occur, in both the nonclassical and direct methods. In particular it is shown that the application of the direct method to systems of equations is not as conceptually straightforward as previously thought, and a way of completing the calculations of the nonclassical method via hodograph transformations is introduced. In Chapter Seven it is shown how more symmetry reductions may be found via nonclassical potential symmetries, which are a new extension on the idea of potential symmetries.

In the final chapter the relationship between the nonclassical and direct methods is investigated in the light of the previous chapters. The thesis is concluded with some general remarks on its findings and on possible future work.