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

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.