Vortex Motion Law for the Schrödinger-Ginzburg-Landau Equations

In the Ginzburg-Landau model for superconductivity a large Ginzburg-Landau parameter κ corresponds to the formation of tight, stable vortices. These vortices are located where an applied magnetic field pierces the superconducting bulk, and each vortex induces a quantized supercurrent about the vortex. The energy of large-κ solutions blows up near each vortex which brings about difficulties in analysis. Rigorous asymptotic static theory has previously established the existence of a finite number of the vortices, and these vortices are located precisely at the critical points of a renormalized energy. We consider the motion of such vortices in a dynamic model for superconductivity that couples a U(1) gauge-invariant Schrödinger-type Ginzburg-Landau equation to a Maxwell-type equation under the limit of large Ginzburg-Landau parameter κ. It is shown that under an almost-energy-minimizing condition each vortex moves in the direction of the net supercurrent located at the vortex position, and these vortices behave like point vortices in the classical two-dimensional Euler equations.