Equivariant stability of vortices in Manton's Chern-Simons-Schrödinger system on the hyperbolic plane

In this work we study magnetic vortices on the hyperbolic plane for a Chern-Simons-Schrödinger system introduced by Manton. The model can be thought of as the Schrödinger analogue of the Abalian-Higgs model. It consists of a system of partial differential equations, where the complex Higgs field $Φ$ evolves according to a nonlinear Schrödinger equation coupled to an electromagnetic field A. We restrict attention to the self-dual (Bogomolny) case under equivariance symmetry. For each m≥ 1 we prove the asymptotic stability of the equivariant vortex of degree m. The main novelties are unraveling the favorable structure of the equations after a nonlinear Darboux transform, and the analysis of the elliptic operator relating the original and the transformed variables.