Methods for integrable systems

This work concerns the study of certain methods for investigating integrable systems, and the application of these methods to specific problems and examples. After introducing the notion of integrability in chapters 1 and 2, we go on, in chapter 3, to develop a novel type of discrete integrable equation by considering ways of enforcing Leibniz's rule for finite difference operators. We look at several approaches to the problem, derive some solutions and study several examples. Chapter 4 describes a numerical implementation of a method for solving initial value problems for an integrable equation in 2+1 dimensions, exploiting the integrability of the equation. The introduction of twisters enables a powerful scheme to be developed. In chapter 5 Darboux transformations derived from the factorisation of a scattering problem are examined, and a general operator form considered. The topic of chapter 6 is the relationship between the Darboux transform for the sine-Gordon and related equations and certain ansatze established by twistor methods. Finally in chapter 7 a geometric setting for partial differential equations is introduced and used to investigate the structure of Bäcklund transformations and generalised symmetries.