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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 first-order Bogomolnyi equations. We prove the existence of periodic solutions when the area of the fundamental cell is greater than 4πM, 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 infinity, the periodic solutions tend to Taubes' arbitrary N-vortex solution.