Vortex motion laws for dynamic Ginzburg-Landau models in two dimensions

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 the renormalized energy (the free energy less the vortex self-induction energy). We consider three different types of dynamic Ginzburg-Landau model equations invariant under the same U(1) gauge and study the large κ limit.; First, a rigorous study of the full time-dependent Ginzburg-Landau equations is undertaken. Under a slow time-scale of order less than log κ, we prove that the vortices are pinned to their initial configuration. Under a fast time of order log κ it is shown that the vortices move according to a steepest descent of the renormalized energy.; Second, a Schrödinger-type Ginzburg-Landau equation coupled to a Maxwell-type equation is considered under the limit of large Ginzburg-Landau parameter κ. These equations can be transformed into a system of conservation laws, similar to the Madelung transformation found in the study of the nonlinear Schrödinger equation. Under the large κ limit the equations converge to a set of forced, incompressible Euler equations, and under unit time-scale the vortices move in the direction of the net supercurrent at the vortex location, which is a symplectic action on the gradient of the renormalized energy.; Finally, a nonlinear gauge-invariant wave operator coupled to Maxwell's equations is considered under the limit of large Ginzburg-Landau parameter κ. Under a fast time of order $logk$ it is shown that the vortices accelerate in the direction of the gradient of the renormalized energy.