Asymptotic analysis of the bifurcation diagram for symmetric one-dimensional solutions of the Ginzburg-Landau equations

A. Aftalion & S. J. Chapman

The bifurcation of symmetric 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. It was found numerically in [Aftalion & Troy] that there are three distinct regions of the (kappa, d) plane, labelled S_1, S_2 and S_3, in which there are at most one, two and three symmetric solutions of the Ginzburg-Landau system respectively.

The curve in the (kappa, d) plane across which the bifurcation switches from being subcritical to supercritical is identified, which is the boundary between S_2 and S_1 union S_3, and the bifurcation diagram is analysed in its vicinity.

The results provide formal evidence for the resolution of some of the conjecture s of [Aftalion & Troy].