On the stability of the Ginzburg-Landau vortex

TL;DR

This paper develops a new functional framework to analyze the stability of the Ginzburg-Landau vortex, proving its minimality and orbital stability within the context of the Gross-Pitaevskii equation.

Contribution

It introduces a tailored functional framework and establishes the vortex as a unique global minimizer and orbitally stable solution.

Findings

Renormalized Ginzburg-Landau energy is well-defined in the new framework.

The vortex is the unique global minimizer up to invariances.

Orbital stability of the vortex under Gross-Pitaevskii evolution.

Abstract

We introduce a functional framework taylored to investigate the minimality and stability properties of the Ginzburg-Landau vortex of degree one on the whole plane. We prove that a renormalized Ginzburg-Landau energy is well-defined in that framework and that the vortex is its unique global minimizer up to the invariances by translation and phase shift. Our main result is a nonlinear coercivity estimate for the renormalized energy around the vortex, from which we can deduce its orbital stability as a solution to the Gross-Pitaevskii equation, the natural Hamiltonian evolution equation associated to the Ginzburg-Landau energy.