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

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 10-01-0017811-01-12033-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
\Bibitem{PalSer12} \by R.~V.~Palvelev, A.~G.~Sergeev \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 \mathnet{http://mi.mathnet.ru/tm3392} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3052273} \elib{http://elibrary.ru/item.asp?id=17759407} \transl \jour Proc. Steklov Inst. Math. \yr 2012 \vol 277 \pages 191--205 \crossref{https://doi.org/10.1134/S0081543812040141} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000309232900014} \elib{http://elibrary.ru/item.asp?id=23960336} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84904044666} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
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
3. A. G. Sergeev, “On two geometric problems arising in mathematical physics”, J. Math. Sci., 223:6 (2017), 756–762
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
5. A. G. Sergeev, “Seiberg–Witten theory as a complex version of abelian Higgs model”, Sci. China-Math., 60:6, SI (2017), 1089–1100
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
7. A. G. Sergeev, “Adiabatic limit in Ginzburg–Landau and Seiberg–Witten equations”, Theoret. and Math. Phys., 203:1 (2020), 561–568
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