Classical sigma models in 2+1 dimensions

Abstract

The work in this thesis is concerned with the study of dynamics, scattering and stability of solitons in planar models, i.e. where spacetime is (2+l)-dimensionaI. We consider both integrable models, where exact solutions can be written in closed form, and nonintegrable models where approximations and numerical methods must be employed. For theories that possess a topological lower bound on the energy, there is a useful approximation in which the kinetic energy is assumed to remain small. All these approaches are used at various stages of the thesis. Chapters 1 and 2 review the planar models which are the subjects of this thesis. Chapters 3 and 4 are concerned with integrable chiral equations. First we exhibit an infinite sequence of well-defined conserved quantities and then we construct exact soliton and soliton-antisoliton solutions using analytical methods. We find that there exist solitons that scatter in a different way to those previously found in integrable models. Furthermore, this soliton scattering resembles very closely that found in nonintegrable models, thereby providing a link between the two classes. Chapter 5 develops a numerical simulation based on topological arguments, which is used in a study of soliton stability in the (unmodified) 0(3) model. This confirms that the sohtons are unstable, in the sense that their size is subject to large changes. The same results are obtained by using the slow-motion approximation.