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 Tr. Mat. Inst. Steklova, 2012, Volume 277, Pages 7–21 (Mi tm3390)

Asymptotic expansion of solutions in a rolling problem

I. Ya. Aref'eva, I. V. Volovich

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

Abstract: Asymptotic methods in the theory of differential equations and in nonlinear mechanics are commonly used to improve perturbation theory in the small oscillation regime. However, in some problems of nonlinear dynamics, in particular for the Higgs equation in field theory, it is important to consider not only small oscillations but also the rolling regime. In this article we consider the Higgs equation and develop a hyperbolic analogue of the averaging method. We represent the solution in terms of elliptic functions and, using an expansion in hyperbolic functions, construct an approximate solution in the rolling regime. An estimate of accuracy of the asymptotic expansion in an arbitrary order is presented.

 Funding Agency Grant Number Russian Foundation for Basic Research 11-01-00894-a11-01-00828-a11-01-12114-ofi-m Ministry of Education and Science of the Russian Federation NSh-4612.2012.1NSh-2928.2012.1 The work was partially supported by the Russian Foundation for Basic Research (project nos. 11-01-00894-a (I.A.) and 11-01-00828-a, 11-01-12114-ofi-m-2011 (I.V.)) and by grants of the President of the Russian Federation (project nos. NSh-4612.2012.1 (I.A.) and NSh-2928.2012.1 (I.V.)).

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English version:
Proceedings of the Steklov Institute of Mathematics, 2012, 277, 1–15

Bibliographic databases:

UDC: 517.925

Citation: I. Ya. Aref'eva, I. V. Volovich, “Asymptotic expansion of solutions in a rolling problem”, 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, 7–21; Proc. Steklov Inst. Math., 277 (2012), 1–15

Citation in format AMSBIB
\Bibitem{AreVol12} \by I.~Ya.~Aref'eva, I.~V.~Volovich \paper Asymptotic expansion of solutions in a~rolling problem \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 7--21 \publ MAIK Nauka/Interperiodica \publaddr Moscow \mathnet{http://mi.mathnet.ru/tm3390} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3052260} \elib{https://elibrary.ru/item.asp?id=17759394} \transl \jour Proc. Steklov Inst. Math. \yr 2012 \vol 277 \pages 1--15 \crossref{https://doi.org/10.1134/S0081543812040013} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000309232900001} \elib{https://elibrary.ru/item.asp?id=23960333} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84904034920} 

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This publication is cited in the following articles:
1. E. V. Piskovskii, “Rezhim skatyvaniya v modeli Khiggsa s treniem”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 2(31) (2013), 127–130
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