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Justification of the adiabatic principle for hyperbolic Ginzburg-Landau equations

  • R. V. Palvelev
  • A. G. Sergeev
Article

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.

Keywords

Modulus Space Gauge Transformation STEKLOV Institute Landau Equation Auxiliary System 
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References

  1. 1.
    A. Jaffe and C. Taubes, Vortices and Monopoles: Structure of Static Gauge Theories (Birkhäuser, Boston, 1980).MATHGoogle Scholar
  2. 2.
    N. S. Manton, “A Remark on the Scattering of BPS Monopoles,” Phys. Lett. B 110, 54–56 (1982).MathSciNetMATHCrossRefGoogle Scholar
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    R. V. Pal’velev, “Justification of the Adiabatic Principle in the Abelian Higgs Model,” Tr. Mosk. Mat. Obshch. 72(2), 281–314 (2011) [Trans. Moscow Math. Soc. 2011, 219–244 (2011)].Google Scholar
  4. 4.
    D. Stuart, “Dynamics of Abelian Higgs Vortices in the Near Bogomolny Regime,” Commun. Math. Phys. 159, 51–91 (1994).MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2012

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

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