CP¹ model on a sphere and on a torus
The work in this thesis is concerned with the numerical study of some stability and scattering properties of two CP¹ models in three dimensional space-time: The non-linear 0(3) model and its modified Skyrme version. Chapter 3 focuses principally on the Skyrme model on compactified plane, the topological sphere. Such model is obtained by supplementing the ordinary 0(3) lagrangian with both a Skyrme term and a potential term which, in the present work, has a rather general form. Under the numerical simulation the skyrmions behave stably and scatter either back-to-back or at 90 to the initial direction of motion, depending on the initial velocity. In the 0(3) limit the solitons are no longer stable and scatter at 90 irrespective of the speed. In the fourth chapter the 0(3) model is studied on a flat torus. Its solitons exhibit the usual instability but can be stabilised by the sole addition of a Skyrme term to the lagrangian. Scattering at right angles is observed in all cases considered, including skyrmions colliding at speeds that would bounce them back were they evolving in compactified plane. The periodic 0(3) model has no analytic solutions of degree one, so when a field configuration that resembles a single soliton is numerically evolved, it shrinks to become infinitely thin. Interestingly, such ansatz may be regarded as a soliton of unit topological charge in the context of the periodic skyrmion model. Chapter 5 closes with a summary and suggestions for future research.