Quasi-integrable models in (2+1) dimensions

Abstract

Recently σ-models have received a lot of attention for many reasons. One interesting aspect of the CP(^n) sigma models is the fact they are the simplest Lorentz invariant models which possess topologically stable (minimum of the action) solutions in (2+0) dimensions. Unfortunately, it appears that Lorentz covariance and integrability are incompatable in (2+1) dimensions. In the literature a few integrable models were constructed in (2+1) dimensions at the expense of Lorentz invariance (e.g. modified chiral model,...). An alternative way to proceed is to retain Lorentz invariance and relax the property of integrability by replacing it with a new property of quasi-integrability. Zakrzewski and others have constructed an example of such quasi-integrable models. Their example is based on the CP(^1) model modified by the addition of two stabilising terms (the first called the "Skyrme-like" term and the second the "potential-like" term) to the basic Lagrangian. In this thesis we have addressed the following relevant questions: How unique is this model? What are the properties of its static structures (skyrmions)? Is it possible to generalise this model? Is quasi-integrabilty, as a property, shared by all CP(^2) models, or it is only restricted to the CP(^1) model? It turns out that the first stabilising term [i.e the Skyrme-like term) is only unique for CP(^1) model and this uniqueness does not survive the generalisations to larger coset spaces, say, CP(^n). The second stabilising term is not unique. By taking advantage of this observation, i.e arbitrariness of the potential term, a generalisation of Zakrzewski's model has become possible. Most important of all is the fact that all the CP" models are quasi-integrable provided one incurs the size instabilities of their soliton solutions.