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

The bifurcation of asymmetric superconducting solutions from the normal sol ution is considered for the one-dimensional Ginzburg Landau equations by th e methods of formal asymptotics. The behavior of the bifurcating branch dep ends on the parameters d, the size of the superconducting slab, and k, the Ginzburg Landau parameter. The secondary bifurcation in which the asymmetri c solution branches reconnect with the symmetric solution branch is studied for values of (k,d) for which it is close to the primary bifurcation from the normal state. These values of (k,d) form a curve in the kd-plane, which is determined. At one point on this curve, called the quintuple point, the primary bifurcations switch from being subcritical to supercritical, requi ring a separate analysis. The results answer some of the conjectures of [ A . Aftalion and W. C. Troy, Phys. D, 132 (1999), pp. 214-232].