Periodic solutions to the self-dual Ginzburg{Landau equations

The structure of periodic solutions to the Ginzburg{Landau equations in R2 is studied in the critical case, when the equations may be reduced to the rst-order Bogomolnyi equations. We prove the existence of periodic solutions when the area of the fundamental cell is greater than 4M, M being the overall order of the vortices within the fundamental cell (the topological invariant). For smaller fundamental cell areas, it is shown that no periodic solution exists. It is then proved that as the boundaries of the fundamental cell go to innity, the periodic solutions tend to Taubes ’ arbitrary N-vortex solution.