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October 12, 2025Open Access

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

Key Points

The study demonstrates the asymptotic stability of equivariant vortices of degree m on the hyperbolic plane.
Each vortex exhibits stability often seen in models with nonlinear interactions, particularly between the Higgs field and an electromagnetic field.
A nonlinear Darboux transform reveals a favorable structure in the equations, enhancing understanding of the system's dynamics.
Analysis focuses on the elliptic operator, linking original and transformed variables for deeper insights on stability.

Abstract

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.

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