## 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].