Symmetry methods for integrable systems

Abstract

This thesis discusses various properties of a number of differential equations which we will term "integrable". There are many definitions of this word, but we will confine ourselves to two possible characterisations — either an equation can be transformed by a suitable change of variables to a linear equation, or there exists an infinite number of conserved quantities associated with the equation that commute with each other via some Hamiltonian structure. Both of these definitions rely heavily on the concept of the symmetry of a differential equation, and so Chapters 1 and 2 introduce and explain this idea, based on a geometrical theory of p.d.e.s, and describe the interaction of such methods with variational calculus and Hamiltonian systems. Chapter 3 discusses a somewhat ad hoc method for solving evolution equations involving a series ansatz that reproduces well-known solutions. The method seems to be related to symmetry methods, although the precise connection is unclear. The rest of the thesis is dedicated to the so-called Universal Field Equations and related models. In Chapter 4 we look at the simplest two-dimensional cases, the Bateman and Born-lnfeld equations. By looking at their generalised symmetries and Hamiltonian structures, we can prove that these equations satisfy both the definitions of integrability mentioned above. Chapter Five contains the general argument which demonstrates the linearisability of the Bateman Universal equation by calculation of its generalised symmetries. These symmetries are helpful in analysing and generalising the Lagrangian structure of Universal equations. An example of a linearisable analogue of the Born-lnfeld equation is also included. The chapter concludes with some speculation on Hamiltoian properties.