On the One-Dimensional Ginzburg-Landau BVPs

We study the one-dimensional system of Ginzburg-Landau equations that models a thin film of superconductor subjected to a tangential magnetic field. We prove that the bifurcation curve for the symmetric problem is the graph of a continuous function of the supremum of the order parameter. We also prove the existence of a critical magnetic field. In general, there is more than one positive solution to the symmetric boundary value problem. Our numerical experiments have shown cases with three solutions. It is still an open question whether only one of these corresponds to the physical solution that minimizes the Gibbs free energy. We establish uniqueness for a related boundary value problem. AMS(MOS) Subject Classification. Primary 34B15. Secondary 82D55. Key Words and Phrases. Ginzburg-Landau systems, superconductivity, boundary value problem, uniqueness. This work was supported by the Office of Scientific Computing, U.S. Department of Energy, under Contract W-31-109-Eng-38. 1 Intro..