Small equivariant perturbations of the degree-one vortex decay with radiation rate, while the internal mode damps like epsilon squared over one plus Gamma epsilon squared t, proving asymptotic stability.
Equivariant stability of vortices in Manton's Chern-Simons-Schr\"odinger system on the hyperbolic plane
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abstract
In this work we study magnetic vortices on the hyperbolic plane for a Chern-Simons-Schr\"odinger system introduced by Manton. The model can be thought of as the Schr\"odinger analogue of the Abalian-Higgs model. It consists of a system of partial differential equations, where the complex Higgs field $\Phi$ evolves according to a nonlinear Schr\"odinger equation coupled to an electromagnetic field $A$. We restrict attention to the self-dual (Bogomolny) case under equivariance symmetry. For each $m\geq 1$ 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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Asymptotic stability of the degree-one vortex in the abelian Yang-Mills-Higgs model
Small equivariant perturbations of the degree-one vortex decay with radiation rate, while the internal mode damps like epsilon squared over one plus Gamma epsilon squared t, proving asymptotic stability.