Evolution and scattering of excited topological defects: Interaction between internal modes

TL;DR

This thesis analyzes how topological solitons in (1+1) and (2+1) dimensions emit radiation and interact when internal modes are excited, revealing the dynamics of energy exchange and decay mechanisms through analytical and numerical methods.

Contribution

It provides a detailed study of internal mode excitation effects on topological solitons, including radiation emission and scattering, extending analytical techniques to higher dimensions.

Findings

Internal modes influence radiation emission in kinks and vortices.

Resonant energy exchange occurs during kink/antikink collisions.

Internal mode decay in vortices is characterized by radiation loss.

Abstract

This thesis presents an extensive analysis of the behavior of topological solitons when one or more of their internal modes are activated. The first part of this manuscript is devoted to the study of the simplest topological solitons in (1+1) dimensions: kinks. Specifically, we investigate how these solutions emit radiation when one of their internal modes is initially excited, within the framework of the double $\phi^4$ model. The simplest kink solution in this theory exhibits a complex internal mode structure that depends on a coupling constant appearing in the potential governing the dynamics. We will show how the amplitude and frequency of the emitted radiation are affected by changes in this coupling constant. We also examine the dynamics of wobbling kink/antikink scattering when the kinks possess more than one internal mode. To this end, we study kink/antikink collisions in the context of the simplest kink solution arising in the MSTB model. This analysis sheds light on the resonant energy exchange mechanism, allowing energy transfer between internal modes and the translational mode. The second part of this thesis focuses on excited vortex solutions in (2+1) dimensions. We begin with a detailed study of the internal mode structure associated with vortex solutions in the Abelian-Higgs model. We demonstrate how the problem can be significantly simplified by choosing an appropriate angular dependence for the eigenfunctions. Furthermore, we investigate the radiation emitted by a vortex when its internal mode is initially activated. To achieve this, we extend the analytical techniques used in (1+1) dimensions to field theories defined in two spatial dimensions. This enables us to compute the radiation amplitude, its frequency, and the decay of the internal mode amplitude due to energy loss via radiation. All analytical results are contrasted with data from numerical simulations.