The Solitonic Waltz: Abelian Higgs Vortex Dynamics

This thesis investigates soliton dynamics in non-integrable field theories, with a primary focus on the Abelian Higgs model for vortices.

In the Baby Skyrme model, we introduce a novel family of solutions exhibiting nested ring structures with dihedral symmetry. In addition, we examine periodic solutions in cylindrical domains, enhancing the understanding of soliton solutions in two dimensions.

For the Abelian Higgs model, we develop robust numerical methods to study vortex dynamics. Our results reveal rich dynamical phenomena, such as quasi-bound states in vortex scattering and the emergence of spectral walls; a non-linear effect arising when internal modes transition to the continuous spectrum, altering vortex trajectories. Beyond critical coupling, we explore vortex interactions in Type I and Type II superconductivity, identifying attractive and repulsive regimes, uncovering non-trivial quasi-stationary states influenced by excited normal modes. Furthermore, we investigate orbiting vortex solutions, including vortex-antivortex pairs and 2-vortex systems, demonstrating sustained rotational motion induced by linear perturbations.

A significant finding is the observation of spectral walls not only in critically coupled vortices but also away from critical coupling, suggesting their broader relevance across topological field theories. These results deepen the understanding of soliton dynamics, bridging one-dimensional kinks and vortex interactions in gauge theories. The thesis concludes by proposing future research directions, including multi-vortex scattering, the role of impurities, and extensions to cosmic strings and Chern-Simons systems, laying a foundation for further exploration of soliton phenomena in theoretical and experimental contexts.