Dynamics of vortices in Ginzburg-Landau theories with applications to superconductivity

We study the dynamics of vortices in time-dependent Ginzburg-Landau theories in the asymptotic limit when the vortex core size is much smaller than the inter-vortex distance. We derive reduced systems of ODEs governing the evolution of these vortices. We then extend these to study the dynamics of vortices in extremely type-II superconductors. Dynamics of vortex lines is also considered. For the simple Ginzburg-Landau equation without the magnetic field, we find that the vortices are stationary in the usual diffusive scaling, and obey remarkably simple dynamic laws when time is speeded up by a logarithmic factor. For columnar vortices in superconductors, we find a similar dynamic law with a potential which is screened by the current. For curved vortex lines in curerconductors, we find that the vortex lines move in the direction of the normal with a speed proportional to the curvature. Comparisons are made with the previous results of John Neu.