Renormalized energy of interacting Ginzburg–Landau vortex filaments

Abstract

We consider the Ginzbug–Landau energy in a cylinder in R3, and a canonical approximation for critical points with an assembly of n 2 periodic vortex lines near the axis of the cylinder. We find a formula for the energy which, up to a large additive constant and to leading order, is the action functional of the n-body problem with a logarithmic potential in R2, the axis variable playing the role of time. A special family of rotating helicoidal critical points of the functional is found to be non-degenerate up to the invariances of the problem, and therefore persistent under small perturbations. Our analysis suggests the presence of very complex stationary configurations for vortex filaments, potentially also involving intersecting filaments.