Title:

The geometry and algebraic structure of solitons in the chiral equation

This thesis is concerned with the behaviour of interacting solitons in the principal chiral model. It also discusses symplectic forms and Hopf algebras associated to solitons. In Chapter 1, we present some background material and definitions which will be used in the following chapters. In Chapter 2, we study the behaviour of the solitons in the principal chiral model by both pictures and calculations. Examples of soliton decay and interactions on collision are given. In Chapter 3, we give calculations to define a 2form over the phase space of the chiral solitons using the language of inverse scattering and group factorisation. We also try to write the form in terms of position and momentum. We shall conclude that the symplectic form for the sine Gordon model does not directly extend to a physically acceptable symplectic form for the chiral model. The last chapter deals with Hopf algebras and solitons. We try to construct a Hopf algebra of quantum observables for solitons, but there is a problem in the standard algebraic construction because we don't have group doublecross product. We introduce a definition for an almost group, and use this definition to define an almost Hopf algebra, which is the same as a Hopf algebra except for the rules for the unit and counit. Hopefully this might provide a framework in which the eventual quantisation of chiral solitons might fit.
