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 functi ons of the parameters (kappa, d), where d is the thickness of the film and kappa is the Ginzburg-Landau parameter characterizing the materia l. Although kappa and d play independent roles in the determination of these properties, we will exhibit the dominant role taken up by the p roduct nd in the existence and uniqueness of bifurcating solutions as much as in their stability. Using the semi-classical analysis develope d in our previous papers for getting the existence of asymmetric solut ions and asymptotics for the supercooling field, we prove in particula r that the symmetric bifurcating solutions are stable for (kappa, d) s uch that kappa d is small and d less than or equal to root 5 - eta (fo r 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) suc h that, for kappa less than or equal to Sigma(0) - eta and kappa d lar ge, the asymmetric solutions are unstable, while, for kappa greater th an or equal to Sigma(0) + eta and kappa d large, the asymmetric soluti ons are stable. Finally, we also analyze the symmetric problem which l eads to other stability results.