Asymptotic analysis of a secondary bifurcation of the one-dimensional Ginzburg-Landau equations of superconductivity

A. Aftalion & S. J. Chapman

The bifurcation of asymmetric superconducting solutions from the normal solution is considered for the one-dimensional Ginzburg-Landau equations by the methods of formal asymptotics. The behaviour of the bifurcating branch depends of the parameters d, the size of the superconducting slab, and kappa, the Ginzburg-Landau parameter. The secondary bifurcation in which the asymmetric solution branches reconnect with the symmetric solution branch is studied for values of (kappa,d) for which it is close to the primary bifurcation from the normal state. These values of (kappa,d) form a curve in the (kappa, d)-plane, which is determined. At one point on this curve, called the quintuple point, the primary bifurcations switch from being subcritical to supercritical, requiring a separate analysis. The results answer some of the conjectures of [Aftalion & Troy].