  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.
