Periodic solutions to the self-dual Ginzburg–Landau equations

Abstract

The structure of periodic solutions to the Ginzburg-Landau equations in R 2 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.

Existence and stability of periodic solution to the 3D Ginzburg-Landau equation in weighted Sobolev spaces

“…Equation (1.2) in 3D plus a damping term u and an external force f but without derivative in the nonlinear terms was investigated by Li-Guo [17] and periodic solution in Sobolev spaces was obtained with the help of Faedo-Schauder fix point theorem and the standard compactness arguments. For more results concerning the periodic solution to the Ginzburg-Landau equation, one can refer to [1,4,20,21,26,27]. For results with respect to other aspects to the Ginzburg-Landau equation, one can refer to [2,3,6,13,14,18,25].…”