On the Zeros of Solutions to Ginzburg–Landau Type Systems

Abstract

The authors consider minimizers of a nonlinear functional whose Euler–Lagrange equation includes the Ginzburg–Landau system. For a certain class of Dirichlet data, it is proved that a minimizes has exactly one zero which necessarily has winding number $ \pm 1$. Moreover, the same result holds for solutions of the corresponding parabolic system at all sufficiently large, fixed values of time, under certain conditions on the initial and boundary values. Their result on minimizers supports several theories from physics (concerning interacting bosons, for example). These theories predict that stable solutions with isolated zeros (called vortices) exist, and each zero of a stable solution has winding number $ \pm 1$.