Justification of the adiabatic principle for hyperbolic Ginzburg-Landau equations
type: Item
instance of: scholarly article
title: Justification of the adiabatic principle for hyperbolic Ginzburg-Landau equations
zbMATH Open document ID: 1311.35304
DOI: 10.1134/S0081543812040141
author: R. V. Palvelev
author: Armen Sergeev
published in: Proceedings of the Steklov Institute of Mathematics
publication date: 2014-09-18T00:00:00Z
review text: The paper is devoted to the mathematical justification of the heuristic adiabatic principle proposed by Manton for hyperbolic analogues of the parabolic Ginzburg-Landau (GL) equations, arising in superconductivity theory. These hyperbolic GL equations are the Euler-Lagrange (EL) equations for the action functional given by the GL Lagrangian. Before Manton formulated the adiabatic principle stating that any dynamic solution of the GL equations, which are close to static ones, can be obtained as a perturbation of some adiabatic trajectory (geodesic of a Riemannian metric on the space of vortices -- static solutions of the GL equations). First, in the paper, the hyperbolic GL equations are introduced as the EL equations for the \((2+1)\)-dimensional Abelian Higgs model arising in gauge field theory. With this aim, the GL action functional is defined through kinetic energy and potential energy terms with variables of \(\mathrm{U}(1)\)-connection and Higgs field. The GL equations are the EL equations for the GL action being invariant under the gauge transformation. The solutions of the EL equations are dynamic solutions. Static solutions of the GL equations realize local minima of the potential energy and they have an integer-valued topological invariant, the vortex number \(N\). By considering dynamic solutions of the GL equations, the authors introduce an adiabatic limit with limiting adiabatic trajectory in the configuration space. Every point of the adiabatic trajectory is a static solution, at the same time the whole trajectory cannot be a dynamic solution. The approximate description of `slow' dynamic solutions in a term of moduli space of static solutions was proposed at a heuristic level by Manton. The authors give a rigorous formulation of the adiabatic principle by considering the tangent structure of the moduli space of \(N\)-vortices. Then a linearized vortex operator and infinitesimal gauge transformations are introduced. The main theorem of the paper is dev…
Mathematics Subject Classification ID: 35Q56
Mathematics Subject Classification ID: 82D55
Mathematics Subject Classification ID: 35L30
zbMATH DE Number: 6345586
zbMATH Keywords: Ginzburg-Landau equations
zbMATH Keywords: Euler-Lagrange equations
zbMATH Keywords: Abelian Higgs model
zbMATH Keywords: adiabatic trajectory
zbMATH Keywords: moduli space
zbMATH Keywords: vortices
zbMATH Keywords: perturbation
reviewed by: Ivan A. Parinov
MaRDI profile type: Publication
cites work: Q3906754
cites work: A remark on the scattering of BPS monopoles.
cites work: Dynamics of Abelian Higgs vortices in the near Bogomolny regime
description: scientific article; zbMATH DE number 6345586