Affine Toda solitons and fusing rules

Abstract

This thesis is concerned with various soliton solutions to some of the affine Toda field theories. These are field theories in 1+1 dimensions that possess a rich underlying Lie algebraic structure and they are known to be integrable. The soliton solutions occur as a result of the multi-vacua that appear in the field theory when the coupling constant is taken to be purely imaginary. In chapter one a review of the affine Toda field theories is undertaken. This is meant to be a relatively complete and exhaustive survey of the literature that has appeared on the subject in recent years. A brief introduction to the theory of solitons and the methods of obtaining such solutions in field theory is given in chapter two, resulting in the construction of the relevant machinery for the Toda theories. In chapter three, Hi rota's method is used to construct single and double soliton solutions to these theories. As a consequence of these explicit formulae the fusing structure of the solitons may be investigated and shown to be equivalent to that found in the classical particle regime, supplemented by further 'annihilations' of 'soliton-antisoliton'. The calculations of the double soliton solutions are claimed to be original in this context. The fusing has also been examined by Olive, Turok and Underwood(^16) through an abstract group-theoretical approach to the affine Toda field theories, however very few explicit formulae are given by them, and hence all the solutions given here are important in their own right. An algebra-independent analysis of such phenomena is undertaken in chapter four where a vertex operator construction is given for the relevant interaction functions. Some properties of these functions are noted; (some of these facts correspond with those in [16] concerning the fusing structure of the solitons).