98-622 Bolley C., Helffer B.

Change of stability for symmetric bifurcating solutions in the Ginzburg-Landau equations. (75K, LATEX 2e) Sep 30, 98

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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.

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