Solitons in Field Theory and Nonlinear Analysis
In recent decades, the study of non-linear integrable systems of partial differential equations has advanced greatly, and many interesting methods have been developed. But most of the non-linear systems of PDEs arising in modern theoretical physics are not integrable, and instead of solutions described in a closed form, general existence theorems are proved using modern tools of functional analysis. A particularly interesting feature of many non-linear systems is the existence of solitons, locally concentrated solutions of the corresponding equations. This monograph is devoted mainly to them.
Included here are many important examples of such systems – sigma models, self-dual Yang-Mills fields, generalised abelian Higgs equations, Chern-Simons equations, the Salam-Weinberg theory of electro-weak interactions and their multivortex solutions, static solutions representing dyons in various non-abelian gauge field theory models, radially symmetric solutions of a general scalar equation, strings in cosmology, vortices and anti-vortices in abelian gauge theory, and field equations arising from the classical Born-Infeld electromagnetic theory. The author mainly uses the language of theoretical physics, and these parts will be more understandable by mathematical physicists. However, many results are formulated in the language of theorems and their proofs, as is standard in the mathematical literature. Individual chapters end with notes containing open problems for research. For convenience, classical field theory is summarised in Chapter 1, and another chapter explains the classification of simple Lie algebras, presented in the language of theoretical physics. The material should be of interest to mathematicians and mathematical physicists at postgraduate level.