Adiabatic limit in Ginzburg–Landau equations

Hyperbolic Ginzburg–Landau equations are the Euler–Lagrange equations for the (2+1)-dimensional Abelian Higgs model, arising in gauge field theory. Static solutions of these equations are called vortices and their moduli space is described by Taubes. The structure of the moduli space of dynamic solutions is far from being understood but there is an heuristic method, due to Manton, allowing to construct solutions of Ginzburg–Landau equations with small kinetic energy. The idea is that in the adiabatic limit dynamic solutions should converge to geodesics on the moduli space of vortices in the metric, generated by kinetic energy functional. According to Manton's adiabatic principle, any solution of dynamic equations with a sufficiently small kinetic energy can be obtained as a perturbation of some geodesic of this type. Our talk is devoted to the mathematical justification of this principle.