Bolley C., Helffer B.
Change of stability for symmetric bifurcating solutions in the Ginzburg-Landau
equations.
(75K, LATEX 2e)
ABSTRACT. We consider the bifurcating solutions
for the Ginzburg-Landau equations when the superconductor is a film of
thickness $d$ submitted to an external magnetic field.
We refine some results obtained in our article \cite{BoHe1997}
on the stability of bifurcating solutions starting from normal
solutions.\\
We prove, in particular, the existence of curves $d\ar \kappa_0(d)$,
defined for large $d$ and tending to $2^{-1/2}$ when $d\ar +\infty$ and
$\kappa \ar d_1(\kappa)$, defined for small $\kappa$
and tending to $\sqrt{5}$ when $\kappa \ar 0$, which
separate the sets of pairs $(\kappa,d)$ corresponding to different
behaviors of the symmetric bifurcating solutions. By this way, we give
in particular a complete answer to the question of stability of symmetric
bifurcating solutions in the
asymptotics $\kappa$ fixed-$d$ large or $d$ fixed-$\kappa$ small.