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, λ, 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, λ), 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.