Instability of Ginzburg—Landau vortices on manifolds

Abstract

We investigate two settings of Ginzburg-Landau posed on a manifold where vortices are unstable. The first is an instability result for critical points with vortices of the Ginzburg-Landau energy posed on a simply connected, compact, closed 2-manifold. The second is a vortex annihilation result for the Ginzburg-Landau heat flow posed on certain surfaces of revolution with boundary.

Stable solutions to the abelian Yang--Mills--Higgs equations on $S^2$ and $T^2$

“…For the Ginzburg-Landau equation ε 2 ∆u = (1 − |u| 2 )u on complex-valued functions, a similar result was proved by Jimbo-Morita [JM94], assuming the homogeneous Neumann condition. Other related results on the classification of stable solutions to equations similar to (1.1) can be found, for instance, in [CH78,Mat79,Ser05,Che13].…”

The generalized point-vortex problem and rotating solutions to the Gross–Pitaevskii equation on surfaces of revolution

“…In Section 2, we introduce a conformal map mapping from R 2 {∞} to M as in [3] and identify the explicit formula for f in (1.5). Then we find rotating periodic solutions to (1.1) having two rings, C ± , of n equally spaced vortices with degrees ±1 such that the total degree is zero.…”

Dynamics Of Ginzburg-Landau And Gross-Pitaevskii Vortices On Manifolds

“…Vortex annihilation results in the plane for (1.2) were first established in [2] for any finite ε and later the previously mentioned investigations [4], [5], [6] carried this out on bounded planar domains in the regime ε ≪ 1 under very mild assumptions on the initial data. In [8], the first author addresses the question of whether there can exist stable vortex configurations in the sense of second variation for the energy E ε on a closed manifold without boundary when ε is small and she also presents an annihilation result valid for any ε for the flow (1.2) augmented with a Dirichlet condition on a manifold with boundary.…”