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Zh. Vychisl. Mat. Mat. Fiz., 2017, Volume 57, Number 6, Pages 1023–1032 (Mi zvmmf10542)  

This article is cited in 5 scientific papers (total in 5 papers)

Special discontinuities in nonlinearly elastic media

A. P. Chugainova

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: Solutions of a nonlinear hyperbolic system of equations describing weakly nonlinear quasitransverse waves in a weakly anisotropic elastic medium are studied. The influence of small-scale processes of dissipation and dispersion is investigated. The small-scale processes determine the structure of discontinuities (shocks) and a set of discontinuities with a stationary structure. Among the discontinuities with a stationary structure, there are special ones that, in addition to relations following from conservation laws, satisfy additional relations required for the existence of their structure. In the phase plane, the structure of such discontinuities is represented by an integral curve joining two saddles. Special discontinuities lead to nonunique self-similar solutions of the Riemann problem. Asymptotics of non-self-similar problems for equations with dissipation and dispersion are found numerically. These asymptotics correspond to self-similar solutions of the problems.

Key words: special discontinuities, generalized KdV-Burgers equation, self-similar Riemann problem, nonuniqueness of solutions.

Funding Agency Grant Number
Russian Science Foundation 14-50-00005
This work was supported by the Russian Science Foundation under grant 14-50-00005.


DOI: https://doi.org/10.7868/S0044466917060072

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English version:
Computational Mathematics and Mathematical Physics, 2017, 57:6, 1013–1021

Bibliographic databases:

UDC: 519.634
Received: 06.09.2016

Citation: A. P. Chugainova, “Special discontinuities in nonlinearly elastic media”, Zh. Vychisl. Mat. Mat. Fiz., 57:6 (2017), 1023–1032; Comput. Math. Math. Phys., 57:6 (2017), 1013–1021

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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. Kulikovskii, A. P. Chugainova, “Long nonlinear waves in anisotropic cylinders”, Comput. Math. Math. Phys., 57:7 (2017), 1194–1200  mathnet  crossref  crossref  mathscinet  isi  elib
    2. A. G. Kulikovskii, A. P. Chugainova, “Shock waves in anisotropic cylinders”, Proc. Steklov Inst. Math., 300 (2018), 100–113  mathnet  crossref  crossref  isi  elib
    3. A. G. Kulikovskii, E. I. Sveshnikova, “Problem of the motion of an elastic medium formed at the solidification front”, Proc. Steklov Inst. Math., 300 (2018), 86–99  mathnet  crossref  crossref  isi  elib
    4. V. A. Shargatov, A. P. Chugainova, S. V. Gorkunov, S. I. Sumskoi, “Flow structure behind a shock wave in a channel with periodically arranged obstacles”, Proc. Steklov Inst. Math., 300 (2018), 206–218  mathnet  crossref  crossref  isi  elib
    5. Chugaynova A., 14Th International Conference on Vibration Engineering and Technology of Machinery (Vetomac Xiv), Matec Web of Conferences, 211, eds. Maia N., Dimitrovova Z., E D P Sciences, 2018  crossref  isi
  • Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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