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 Regul. Chaotic Dyn., 2019, Volume 24, Issue 1, Pages 80–89 (Mi rcd390)

Evolution of Lagrangian Manifolds and Asymptotic Solutions to the Linearized Equations of Gas Dynamics

Anna I. Alliluevaabc, Andrei I. Shafarevichcbad

a National Research Centre “Kurchatov Institute”, pl. Akademika Kurchatova 1, Moscow, 123182 Russia
b Moscow Institute of Physics and Technology, Institutskii per. 9, Dolgoprudnyi, 141700 Russia
c Institute for Problems in Mechanics, pr. Vernadskogo 101-1, Moscow, 119526 Russia
d M. V. Lomonosov Moscow State University, Leninskie Gory 1, Moscow, 119991 Russia

Abstract: We study asymptotic solution of the Cauchy problem for linearized equations of gas dynamics with rapidly oscillating initial data. We construct the formal serie, satisfying this problem. This serie is naturally divided into three parts, corresponding to the hydrodynamic mode and two acoustic modes. The summands of the serie are expressed in terms of the Maslov canonic operator on moving Lagrangian manifolds. Evolution of the manifolds is governed by the corresponding classical Hamiltonian systems.

Keywords: Lagrangian manifolds, short-wave asymptotics, equations of gas dynamics

 Funding Agency Grant Number Russian Science Foundation 16-11-10282 This work was supported by the Russian Science Foundation (grant 16-11-10282).

DOI: https://doi.org/10.1134/S1560354719010040

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MSC: 53C56, 35P20
Accepted:09.01.2019
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Citation: Anna I. Allilueva, Andrei I. Shafarevich, “Evolution of Lagrangian Manifolds and Asymptotic Solutions to the Linearized Equations of Gas Dynamics”, Regul. Chaotic Dyn., 24:1 (2019), 80–89

Citation in format AMSBIB
\Bibitem{AllSha19} \by Anna I. Allilueva, Andrei I. Shafarevich \paper Evolution of Lagrangian Manifolds and Asymptotic Solutions to the Linearized Equations of Gas Dynamics \jour Regul. Chaotic Dyn. \yr 2019 \vol 24 \issue 1 \pages 80--89 \mathnet{http://mi.mathnet.ru/rcd390} \crossref{https://doi.org/10.1134/S1560354719010040} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000457880700004} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85058161181} 

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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. Allilueva A.I., Shafarevich A.I., “Localized Asymptotic Solutions of Linearized Equations of Gas Dynamics”, Russ. J. Math. Phys., 25:4 (2018), 415–422
2. Anna I. Allilueva, Andrei I. Shafarevich, “Conic Lagrangian Varieties and Localized Asymptotic Solutions of Linearized Equations of Relativistic Gas Dynamics”, Regul. Chaotic Dyn., 24:6 (2019), 671–681
3. Allilueva A.I., Shafarevich A.I., “Short-Wave Asymptotic Solutions of a Linearized System of Relativistic Gas Dynamics”, Russ. J. Math. Phys., 26:3 (2019), 255–264