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Tr. Mat. Inst. Steklova, 2012, Volume 277, Pages 199–214 (Mi tm3392)  

This article is cited in 7 scientific papers (total in 7 papers)

Justification of the adiabatic principle for hyperbolic Ginzburg–Landau equations

R. V. Palvelev, A. G. Sergeev

Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia

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.

Funding Agency Grant Number
Russian Foundation for Basic Research 10-01-00178
11-01-12033-ofi-m
Ministry of Education and Science of the Russian Federation NSh-7675.2010.1
Russian Academy of Sciences - Federal Agency for Scientific Organizations
The work was supported in part by the Russian Foundation for Basic Research (project nos. 10-01-00178 and 11-01-12033-ofi-m-2011), by a grant of the President of the Russian Federation (project no. NSh-7675.2010.1), and by the scientific program "Nonlinear Dynamics" of the Presidium of the Russian Academy of Sciences.


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English version:
Proceedings of the Steklov Institute of Mathematics, 2012, 277, 191–205

Bibliographic databases:

UDC: 514.763.43+514.83
Received in February 2012

Citation: R. V. Palvelev, A. G. Sergeev, “Justification of the adiabatic principle for hyperbolic Ginzburg–Landau equations”, Mathematical control theory and differential equations, Collected papers. In commemoration of the 90th anniversary of Academician Evgenii Frolovich Mishchenko, Tr. Mat. Inst. Steklova, 277, MAIK Nauka/Interperiodica, Moscow, 2012, 199–214; Proc. Steklov Inst. Math., 277 (2012), 191–205

Citation in format AMSBIB
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\paper Justification of the adiabatic principle for hyperbolic Ginzburg--Landau equations
\inbook Mathematical control theory and differential equations
\bookinfo Collected papers. In commemoration of the 90th anniversary of Academician Evgenii Frolovich Mishchenko
\serial Tr. Mat. Inst. Steklova
\yr 2012
\vol 277
\pages 199--214
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. G. Sergeev, “Adiabatic limit in the Ginzburg–Landau and Seiberg–Witten equations”, Proc. Steklov Inst. Math., 289 (2015), 227–285  mathnet  crossref  crossref  isi  elib
    2. R. V. Palvelev, “Rasseyanie vikhrei v abelevykh modelyakh Khiggsa na kompaktnykh rimanovykh poverkhnostyakh”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 19:2 (2015), 293–310  mathnet  crossref  zmath  elib
    3. A. G. Sergeev, “On two geometric problems arising in mathematical physics”, J. Math. Sci., 223:6 (2017), 756–762  mathnet  crossref  mathscinet  elib
    4. A. G. Sergeev, “Adiabatic limit in Ginzburg–Landau and Seiberg–Witten equations”, Geometric Methods in Physics, Trends in Mathematics, eds. P. Kielanowski, S. Ali, P. Bieliavsky, A. Odzijewicz, M. Schlichenmaier, T. Voronov, Springer Int Publishing Ag, 2016, 321–330  crossref  mathscinet  zmath  isi
    5. A. G. Sergeev, “Seiberg–Witten theory as a complex version of abelian Higgs model”, Sci. China-Math., 60:6, SI (2017), 1089–1100  crossref  mathscinet  zmath  isi
    6. A. G. Sergeev, “Adiabatic limit in abelian Higgs model with application to Seiberg–Witten equations”, Phys. Part. Nuclei Lett., 14:2 (2017), 341–346  crossref  isi
    7. A. G. Sergeev, “Adiabatic limit in Ginzburg–Landau and Seiberg–Witten equations”, Theoret. and Math. Phys., 203:1 (2020), 561–568  mathnet  crossref  crossref  isi  elib
  • Труды Математического института им. В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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