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

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 11-01-0003411-01-12051-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

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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• http://mi.mathnet.ru/eng/tm3470
• https://doi.org/10.1134/S0371968513020179
• http://mi.mathnet.ru/eng/tm/v281/p215

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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
2. A. T. Il'ichev, A. P. Chugainova, V. A. Shargatov, “Spectral stability of special discontinuities”, Dokl. Math., 91:3 (2015), 347–351
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
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
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
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
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