Justification of the adiabatic principle for hyperbolic Ginzburg--Landau equations

R. V. Palvelev ; A. G. Sergeev

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Mathematical control theory and differential equations, Tome 277 (2012), pp. 199-214

Résumé

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.