Existence Theorems for $\frac{\pi}{n}$ Vortex Scattering

TL;DR

This paper extends the analysis of vortex-vortex scattering in the Abelian Higgs model to include $rac{ p}{n}$ scattering for all head-on collisions of $n$ vortices, establishing existence and uniqueness of solutions.

Contribution

It introduces a new framework for analyzing $rac{ p}{n}$ vortex scattering, proving the existence and uniqueness of solutions for these collision scenarios.

Findings

Proves the existence of a unique global finite-energy solution for $rac{ p}{n}$ vortex scattering.

Shows that the symmetry and local analytic solutions realize $rac{ p}{n}$ scattering.

Extends previous $90^{ ext{o}}$ scattering analysis to all head-on vortex collisions.

Abstract

The analysis of $90^{\circ}$ vortex-vortex scattering is extended to $\frac{\pi}{n}$ scattering in all head-on collisions of $n$ vortices in the Abelian Higgs model. A Cauchy problem with initial data that describe the scattering of $n$ vortices is formulated. It is shown that this Cauchy problem has a unique global finite-energy solution. The symmetry of the solution and the form of the local analytic solution then show that $\frac{\pi}{n}$ scattering is realised.