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

TL;DR

This paper investigates vortex dynamics in Ginzburg-Landau and Gross-Pitaevskii models on Riemannian surfaces, showing their evolution follows gradient and Hamiltonian flows, with specific results on spheres including vortex annihilation and energy identities.

Contribution

It establishes the limiting vortex dynamics as gradient and Hamiltonian flows on manifolds, and provides new results for vortex behavior on spheres, including annihilation and energy identities.

Findings

Vortices evolve according to gradient and Hamiltonian flows.

On spheres, vortex annihilation occurs under heat flow.

Weighted energy identities are derived for Ginzburg-Landau heat flow.

Abstract

We consider the dissipative heat flow and conservative Gross-Pitaevskii dynamics associated with the Ginzburg-Landau energy posed on a Riemannian 2-manifold M. We show the limiting vortices of the solutions to these two problems evolve according to the gradient flow and Hamiltonian point-vortex flow respectively, associated with the renormalized energy on M. For the heat flow, we then specialize to the case where M is a sphere and study the limiting system of ODE's and establish an annihilation result. Finally, for the Ginzburg-Landau heat flow on a sphere, we derive some weighted energy identities.