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 Zh. Vychisl. Mat. Mat. Fiz., 2016, Volume 56, Number 2, Pages 259–274 (Mi zvmmf10343)

This article is cited in 9 scientific papers (total in 9 papers)

Stability of discontinuity structures described by a generalized KdV–Burgers equation

A. P. Chugainovaa, V. A. Shargatovb

a Steklov Mathematical Institute, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia
b National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Kashirskoe sh. 31, Moscow, 115409, Russia

Abstract: The stability of discontinuities representing solutions of a model generalized KdV–Burgers equation with a nonmonotone potential of the form $\varphi(u)=u^4-u^2$ is analyzed. Among these solutions, there are ones corresponding to special discontinuities. A discontinuity is called special if its structure represents a heteroclinic phase curve joining two saddle-type special points (of which one is the state ahead of the discontinuity and the other is the state behind the discontinuity).The spectral (linear) stability of the structure of special discontinuities was previously studied. It was shown that only a special discontinuity with a monotone structure is stable, whereas special discontinuities with a nonmonotone structure are unstable. In this paper, the spectral stability of nonspecial discontinuities is investigated. The structure of a nonspecial discontinuity represents a phase curve joining two special points: a saddle (the state ahead of the discontinuity) and a focus or node (the state behind the discontinuity). The set of nonspecial discontinuities is examined depending on the dispersion and dissipation parameters. A set of stable nonspecial discontinuities is found.

Key words: generalized KdV–Burgers equation, spectral (linear) stability of stationary solutions, special discontinuities.

 Funding Agency Grant Number Russian Science Foundation 14-50-00005 Chugainova acknowledges the support of the Russian Science Foundation, project no. 14-50-00005.

DOI: https://doi.org/10.7868/S0044466916020058

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English version:
Computational Mathematics and Mathematical Physics, 2016, 56:2, 263–277

Bibliographic databases:

Document Type: Article
UDC: 519.634
Received: 18.05.2015

Citation: A. P. Chugainova, V. A. Shargatov, “Stability of discontinuity structures described by a generalized KdV–Burgers equation”, Zh. Vychisl. Mat. Mat. Fiz., 56:2 (2016), 259–274; Comput. Math. Math. Phys., 56:2 (2016), 263–277

Citation in format AMSBIB
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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, 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
2. 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
3. A. Samokhin, “Periodic boundary conditions for KdV–Burgers equation on an interval”, J. Geom. Phys., 113 (2017), 250–256
4. V. V. Zharinov, “Hamiltonian operators in differential algebras”, Theoret. and Math. Phys., 193:3 (2017), 1725–1736
5. A. G. Kulikovskii, A. P. Chugainova, “Long nonlinear waves in anisotropic cylinders”, Comput. Math. Math. Phys., 57:7 (2017), 1194–1200
6. A. Samokhin, “On nonlinear superposition of the KdV–Burgers shock waves and the behavior of solitons in a layered medium”, Differ. Geom. Appl., 54:A (2017), 91–99
7. 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
8. A. G. Kulikovskii, A. P. Chugainova, “Shock waves in anisotropic cylinders”, Proc. Steklov Inst. Math., 300 (2018), 100–113
9. 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
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