Bifurcation of vortex solutions to a Ginzburg-Landau equation in an annulus

We consider a simplified Ginzburg-Landau model of superconductivity in an annulus domain. This model equation has two physical parameters lambda and h which are related to the Ginzburg-Landau parameter and strength of an applied magnetic field respectively. Then a solution with k-mode in the polar angle bifurcates from the trivial solution at appropriate parameter values lambda and h. This solution is vortexless, that is, it has no zeros. We study the bifurcation near the critical point on which bifurcation curves corresponding to two different modes intersect in the parameter space (h, lambda). We then investigate the local bifurcation structure around the critical point and prove the existence of a vortex solution under a generic condition. In particular we show that the solution has zeros on a boundary if the parameters is on some curve emanating from the critical point. The stability of the vortex solution is also discussed by applying the center manifold theorem.