On the stability of the Ginzburg-Landau vortex

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.

A uniqueness result for travelling waves in the Gross-Pitaevskii equation

“…For n = 1, the function ρ 1 is strictly increasing and concave. Also, it has been recently shown in [26] that the vortices of degree ±1 are orbitally stable, and it is conjectured that higher degree vortices should be unstable.…”