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Tr. Mat. Inst. Steklova, 2013, Volume 281, Pages 215–223 (Mi tm3470)  

This article is cited in 6 scientific papers (total in 6 papers)

Nonstationary solutions of a generalized Korteweg–de Vries–Burgers equation

A. P. Chugainova

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

Abstract: Nonstationary solutions of the Cauchy problem are found for a model equation that includes complicated nonlinearity, dispersion, and dissipation terms and can describe the propagation of nonlinear longitudinal waves in rods. Earlier, within this model, complex behavior of traveling waves has been revealed; it can be regarded as discontinuity structures in solutions of the same equation that ignores dissipation and dispersion. As a result, for standard self-similar problems whose solutions are constructed from a sequence of Riemann waves and shock waves with stationary structure, these solutions become multivalued. The interaction of counterpropagating (or copropagating) nonlinear waves is studied in the case when the corresponding self-similar problems on the collision of discontinuities have a nonunique solution. In addition, situations are considered when the interaction of waves for large times gives rise to asymptotics containing discontinuities with nonstationary periodic oscillating structure.

Funding Agency Grant Number
Russian Foundation for Basic Research 11-01-00034
11-01-12051-ofi-m
This work was supported by the Russian Foundation for Basic Research, project nos. 11-01-00034 and 11-01-12051-ofi-m-2011.


DOI: https://doi.org/10.1134/S0371968513020179

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English version:
Proceedings of the Steklov Institute of Mathematics, 2013, 281, 204–212

Bibliographic databases:

Document Type: Article
UDC: 519.634
Received in September 2012

Citation: A. P. Chugainova, “Nonstationary solutions of a generalized Korteweg–de Vries–Burgers equation”, Modern problems of mechanics, Collected papers. Dedicated to Academician Andrei Gennad'evich Kulikovskii on the occasion of his 80th birthday, Tr. Mat. Inst. Steklova, 281, MAIK Nauka/Interperiodica, Moscow, 2013, 215–223; Proc. Steklov Inst. Math., 281 (2013), 204–212

Citation in format AMSBIB
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\by A.~P.~Chugainova
\paper Nonstationary solutions of a~generalized Korteweg--de Vries--Burgers equation
\inbook Modern problems of mechanics
\bookinfo Collected papers. Dedicated to Academician Andrei Gennad'evich Kulikovskii on the occasion of his 80th birthday
\serial Tr. Mat. Inst. Steklova
\yr 2013
\vol 281
\pages 215--223
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
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\jour Proc. Steklov Inst. Math.
\yr 2013
\vol 281
\pages 204--212
\crossref{https://doi.org/10.1134/S0081543813040172}
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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. P. Chugainova, V. A. Shargatov, “Stability of nonstationary solutions of the generalized KdV-Burgers equation”, Comput. Math. Math. Phys., 55:2 (2015), 251–263  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    2. A. T. Il'ichev, A. P. Chugainova, V. A. Shargatov, “Spectral stability of special discontinuities”, Dokl. Math., 91:3 (2015), 347–351  mathnet  crossref  mathscinet  zmath  isi  scopus
    3. A. P. Chugainova, V. A. Shargatov, “Stability of discontinuity structures described by a generalized KdV–Burgers equation”, Comput. Math. Math. Phys., 56:2 (2016), 263–277  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    4. A. G. Kulikovskii, A. P. Chugainova, V. A. Shargatov, “Uniqueness of self-similar solutions to the Riemann problem for the Hopf equation with complex nonlinearity”, Comput. Math. Math. Phys., 56:7 (2016), 1355–1362  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    5. A. T. Il'ichev, A. P. Chugainova, “Spectral stability theory of heteroclinic solutions to the Korteweg–de Vries–Burgers equation with an arbitrary potential”, Proc. Steklov Inst. Math., 295 (2016), 148–157  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    6. A. P. Chugainova, A. T. Il'ichev, A. G. Kulikovskii, V. A. Shargatov, “Problem of arbitrary discontinuity disintegration for the generalized Hopf equation: selection conditions for a unique solution”, IMA J. Appl. Math., 82:3 (2017), 496–525  crossref  mathscinet  isi  scopus
  • Труды Математического института им. В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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