Vortex Dynamics of the Full Time-Dependent 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 exactly 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 the renormalized energy (the free energy less the vortex self-induction energy). A rigorous study of the full time-dependent Ginzburg-Landau equations under the classical Lorentz gauge is done under the asymptotic limit κ → ∞. Under slow times the vortices remain pinned to their initial configuration. Under a fast time of order log κ the vortices move according to a steepest descent of the renormalized energy.