Generation of vortices in the Ginzburg–Landau heat flow

Abstract

We consider the Ginzburg–Landau heat flow on the flat two-dimensional torus, starting from initial data with a finite number of nondegenerate zeros – but very high initial energy. We show that the initial zeros are conserved, while away from these zeros the modulus quickly grows close to 1, and the flow rapidly enters a logarithmic energy regime, from which the evolution of vortices can be described by the works of Bethuel, Orlandi and Smets.