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

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 upon the parameters d, the size of the superconducting slab, and kappa, the Ginzburg-Landau parameter. It was found numerically by Aftalion & Troy [1] 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)boolean OR S-3. and the bifurcation diagram is analysed in its vicinity. The triple point, corresponding to the paint at which S-1, S-2 and S-3 meet, is determined, and the bifurcation diagram and the boundaries of S-1, S-2 and S-3 are analysed in its vicinity. The results provide formal evidence for the resolution of some of the conjectures of Aftalion & Troy [1].