The twistor description of integrable systems

Abstract

The theory of twistors and the theory of integrable models have, for many years, developed independently of each other. However, in recent years it has been shown that there is considerable overlap between these two apparently disparate areas of mathematical physics. The aim of this thesis is twofold; firstly to show how many known integrable models may be given a natural geometrical/twistorial interpretation, and secondly to show how this leads to new integrable models, and in particular new higher dimensional models. After reviewing those elements of twistor theory that are needed in the thesis, a generalisation of the Yang-Mills self-duality equations is constructed. This is the framework into which many known examples of integrable models may be naturally fitted, and it also provides a simple way to construct higher dimensional generalisations of such models. Having constructed new examples of (2 + l)-dimensional integrable models, one of these is studied in more detail. Embedded within this system are the sine-Gordon and Non-Linear Schrodinger equations. Some solutions of this (2 + l)-dimensional integrable model are found using the 'Riemann Problem with Zeros' method, and these include the sohton solutions of the SG and NLS equations. The relation between this approach and one based the Atiyah-Ward ansatze is dicussed briefly. Scattering of localised structures in integrable models is very different from scattering in non-integrable models, and to illustrate this the scattering of vortices in a modified Abelian-Higgs model is considered. The scattering is studied, for small speeds, using the 'slow motion approximation' which involves the calculation of a moduli space metric. This metric is found for a general TV-lump vortex configuration. Various examples of scattering processes are discussed, and compared with scattering in an integrable model. Finally this geometrical approach is compared with other approaches to the study of integrable systems, such as the Hirota method. The thesis closes with some suggestions for how the KP equation may be fitted into this geometrical/twistorial scheme.