Topological solitons and their dynamics

Topological solitons are particle-like solutions of nonlinear field equations with important applications in physics. This thesis presents four research projects concerning topological solitons and their dynamics. We investigate solitons in (1+1)- and (2+1)-dimensions, and develop numerical methods to obtain static solutions and simulate soliton scattering.

We first study kink collisions in a model with two scalar fields in the presence of false vacua. We find a variety of scattering outcomes depending on the initial velocity and vacuum structure. Kinks can either repel, form a true or false domain wall, annihilate, or collide and escape to infinity. These behaviours occur in alternating windows of initial velocity. When the kinks escape to infinity, there are a number of oscillations or ``bounces" before the kinks escape, and this bounce number is conserved in each of the windows.

In the second project we design new baby Skyrme models that do not require a potential term to allow topological soliton solutions. We raise the Skyrme and sigma terms to fractional powers, which enables us to evade Derrick's theorem. We calculate topological energy bounds for our models and numerically obtain minimal energy solutions for solitons of charge 1, 2, and 3. For each charge, the minimal energy solution is a ring.

The last two projects concern vortices in the Ginzburg-Landau model. In the first of these, we numerically investigate the scattering of multi-vortex rings. When two 2-vortex rings collide, there are two distinct scattering outcomes. In both cases, one pair of vortices will scatter at right angles and escape along the $y$-axis. The remaining two vortices will either form a bound state or escape along an axis after colliding a number of times.

Finally, we study vortices scattering with magnetic impurities of the form $\sigma(r)=ce^{-dr^2}$. An impurity will attract or repel a vortex depending on the coupling constant $\lambda$ and the parameters $c$ and $d$. We scatter critically coupled vortices with two different impurities and explore the relationship between the scattering angle and impact parameter. We also find that a 2-vortex ring will break up in a head-on collision with an impurity.