  We consider Abrikosov-type vortex lattice solutions of the Ginzburg-Landau
equations of superconductivity, consisting of single vortices, for magnetic
fields below but close to the second critical magnetic field H_{c2} = kappa^2
and for superconductors filling the entire R^2. Here kappa is the
Ginzburg-Landau parameter. The lattice shape, parameterized by tau, is allowed
to be arbitrary (not just triangular or rectangular). Within the context of the
time-dependent Ginzburg-Landau equations, called the Gorkov-Eliashberg-Schmidt
equations, we prove that such lattices are asymptotically stable under gauge
periodic perturbations for kappa^2 > (1/2)(1 - (1/beta(tau)) and unstable for
kappa^2 < (1/2)(1 - (1/beta(tau)), where beta(tau) is the Abrikosov constant
depending on the lattice shape tau. 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 > 1/2. (There is
no real contradiction though as we consider very special perturbations.)
