| Summary: | This thesis is primarily concerned with some aspects of the behaviour of interacting solitons, both classically and at the quantum level, in a large class of integrable models in two dimensions known as the affine Toda field theories, although we restrict our attention, for technical reasons, to a subclass of them based on only the simply-laced Lie algebras. The main results in the thesis are firstly in establishing the time delay formula for the classical scattering of two solitons within an algebraic construction of soliton solutions using vertex operators. This result provides partial information on the classical interaction of two solitons, it indicates that the force between two distinguishable solitons is attractive. It turns out that the time delay is proportional to the logarithm of the normal ordering coefficient <I>X<SUP>ij</SUP></I>(<I>θ</I>) of the two vertex operators associated with the two interacting solitons. This coefficient has properties reminiscent of the quantum scattering matrix for the solitons. It has poles due to the fusing of solitons in the same places as the scattering matrix, it satisfies a sort of bootstrap equation obtained by bootstrapping up these poles and it also satisfies a sort of crossing equation. It is shown that these facts are not accidental, because the full quantum scattering matrix can be extrapolated from <I>X<SUP>ij</SUP></I>(<I>θ</I>) by a sort of '<I>q</I>-deformation' process. However, in order to prove this it is necessary to review material on trigonometric quantum Yang-Baxter equations and affine quantum groups and to also review a construction of non-local charges due to Bernard and Leclair. A third result is a mathematical formulation of the classical inverse scattering method applied to sine-Gordon and some affine Toda models which can be taken to complement the algebraic construction of soliton solutions using vertex operators. As well as tidying up the method, from a mathematical and logical point of view, we see how a certain loop group factorisation can solve the system completely and define actions of operators on the classical phase space.
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