Stability of bifurcating solutions for the Ginzburg-Landau equations

This paper is concerned with superconducting solutions of the Ginzburg-Landau equations for a film. We study the structure and the stability of the bifurcating solutions starting from normal solutions as functions of the parameters (kappa, d), where d is the thickness of the film and kappa is the Ginzburg-Landau parameter characterizing the material. Although kappa and d play independent roles in the determination of these properties, we will exhibit the dominant role taken up by the product nd in the existence and uniqueness of bifurcating solutions as much as in their stability. Using the semi-classical analysis developed in our previous papers for getting the existence of asymmetric solutions and asymptotics for the supercooling field, we prove in particular that the symmetric bifurcating solutions are stable for (kappa, d) such that kappa d is small and d less than or equal to root 5 - eta (for any eta > 0) and unstable for (kappa, d) such that kappa d is large. We also show the existence of an explicit critical value Sigma(0) such that, for kappa less than or equal to Sigma(0) - eta and kappa d large, the asymmetric solutions are unstable, while, for kappa greater than or equal to Sigma(0) + eta and kappa d large, the asymmetric solutions are stable. Finally, we also analyze the symmetric problem which leads to other stability results.