Bifurcation of vortex and boundary-vortex solutions in a Ginzburg-Landau model

We consider a simplified Ginzburg-Landau model of superconductivity in a two-dimensional infinite strip domain under the assumption of the periodicity in the infinite direction. This model equation has two physical parameters,., h, coming from the Ginzburg-Landau parameter and the strength of an applied magnetic field, respectively. We study the bifurcation of non-trivial solutions in the parameter space (h, lambda), in particular through a bifurcation of the existence of a vortex solution, that is, a solution with isolated zeros. We first observe that in the parameter space there is a smooth (bifurcation) curve on which a solution with k-mode in the periodic direction takes place. This bifurcating solution, however, is vortexless. Then analysing the local bifurcation structure around the critical point at which two bifurcation curves for k and m(> k) intersect, we prove the existence of vortex solutions under a generic condition. Moreover, we show that the solutions have vortices lying on a boundary if the parameters belong to a certain curve emanating from the critical point. The stability of such solutions is also discussed.