Uniqueness of radial Abelian Higgs vortices away from the BPS point

Establish uniqueness of radial solutions to the singular second-order Abelian Higgs vortex boundary-value problem with winding number n, coupling parameter β≠1, regular origin data [?] and boundary conditions φ(∞)=1 and a(∞)=0; equivalently, prove that the associated two-parameter shooting map has at most one admissible zero.

Background

The paper studies radial minimizers of the quartic Abelian Higgs energy for large winding number. At the BPS value β=1, a Bogomolny reduction and Taubes's uniqueness theorem are available. Away from β=1, the radial equations remain a singular second-order boundary-value problem with two local parameters at the origin and two conditions at infinity.

The authors do not require uniqueness for their asymptotic results, because their arguments apply to every radial minimizer. Nevertheless, they explicitly identify uniqueness away from the BPS point as an unresolved shooting problem.