Ginzburg-Landau vortex dynamics driven by an applied boundary current

TL;DR

This paper investigates the dynamics of vortices in the time-dependent Ginzburg-Landau equations with boundary currents, deriving energy bounds and a novel vortex motion law including a Lorentz force effect.

Contribution

It introduces a new analysis of vortex dynamics under boundary currents, including a derived law of vortex motion with a Lorentz force term.

Findings

Energy growth bounds for solutions with boundary currents

Vortex motion law includes a novel Lorentz force term

Analysis of vortex behavior on different time scales

Abstract

In this paper we study the time-dependent Ginzburg-Landau equations on a smooth, bounded domain $\Omega \subset \Rn{2}$, subject to an electrical current applied on the boundary. The dynamics with an applied current are non-dissipative, but via the identification of a special structure in an interaction energy, we are able to derive a precise upper bound for the energy growth. We then turn to the study of the dynamics of the vortices of the solutions in the limit $\ep \to 0$. We first consider the original time scale, in which the vortices do not move and the solutions undergo a "phase relaxation." Then we study an accelerated time scale in which the vortices move according to a derived dynamical law. In the dynamical law, we identify a novel Lorentz force term induced by the applied boundary current.