Stability of Abrikosov lattices under gauge-periodic perturbations

We consider Abrikosov-type vortex lattice solutions of the Ginzburg-Landau equations of superconductivity, consisting of single vortices, for magnetic fields close to the second critical magnetic field H_c2 = κ² and for superconductors filling the entire {R}^2 . Here κ is the Ginzburg-Landau parameter. The lattice shape, parametrized by τ, is allowed to be arbitrary (not just triangular or rectangular). Within the context of the time-dependent Ginzburg-Landau equations, called the Gorkov-Eliashberg-Schmid equations, we prove that such lattices are asymptotically stable under gauge-periodic perturbations for \kappa^2 > \frac{1}{2}(1 - \frac{1}{\beta(\tau)}) and unstable for \kappa^2 < \frac{1}{2}(1 - \frac{1}{\beta(\tau)}) , where β(τ) is the Abrikosov constant depending on the lattice shape τ. This result goes against the common belief among physicists and mathematicians that Abrikosov-type vortex lattice solutions are stable only for triangular lattices and \kappa^2 > \frac{1}{2} . (There is no real contradiction though as we consider very special perturbations.)