Justification of the adiabatic principle for hyperbolic Ginzburg-Landau equations

Abstract

We study the adiabatic limit in hyperbolic Ginzburg-Landau equations which are the Euler-Lagrange equations for the Abelian Higgs model. By passing to the adiabatic limit in these equations, we establish a correspondence between the solutions of the Ginzburg-Landau equations and adiabatic trajectories in the moduli space of static solutions, called vortices. Manton proposed a heuristic adiabatic principle stating that every solution of the Ginzburg-Landau equations with sufficiently small kinetic energy can be obtained as a perturbation of some adiabatic trajectory. A rigorous proof of this result has been found recently by the first author.