On the uniqueness of solutions of the Ginzburg-Landau system for thin films

In this paper we investigate the uniqueness of solutions of the Ginzburg-La ndau system for superconductivity in the regime where the thickness 2a of t he film is small. We analyse the equations of first variation with respect to two shooting parameters and obtain estimates on all relevant quantities at the end of the film where the boundary conditions are prescribed. Using these estimates, we prove that the bifurcation curve of symmetric solutions is given by a decreasing function of the order parameter beta when the fil m is thin enough. In addition, we prove that there is no curve of asymmetri c solutions branching from the symmetric curve.