Uniqueness of topological solutions for a class of self-dual vortex theories

We study the solutions of topological type for a class of self-dual vortex theories in two dimensions. We consider the regime corresponding to the limit of small vortex core size with respect to the separation distance between vortices, namely as the scaling parameter $\delta>0$ tends to zero. Using a gluing technique for the corresponding nonlinear elliptic equation on the plane, with any number (finite or countable) of prescribed singular sources, we prove the existence of multi-vortex solutions which behave as a single vortex solution near each vortex point, up to an error exponentially small, as $\delta\to 0$. Moreover, in the physically relevant cases, namely when the vortex points are either finite or periodically arranged in the plane, we prove that the multi-vortex solution satisfying a ‘topological condition' is unique, for $\delta>0$ sufficiently small.