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

Abstract

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 κ, the Ginzburg–Landau parameter. It was found numerically by Aftalion & Troy [1] that there are three distinct regions of the (κ, d) plane, labelled S1, S2 and S3, in which there are at most one, two and three symmetric solutions of the Ginzburg–Landau system, respectively. The curve in the (κ, d) plane across which the bifurcation switches from being subcritical to supercritical is identified, which is the boundary between S2 and S1∪S3, and the bifurcation diagram is analysed in its vicinity. The triple point, corresponding to the point at which S1, S2 and S3 meet, is determined, and the bifurcation diagram and the boundaries of S1, S2 and S3 are analysed in its vicinity. The results provide formal evidence for the resolution of some of the conjectures of Aftalion & Troy [1].