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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

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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  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    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  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    3. A. Samokhin, “Periodic boundary conditions for KdV–Burgers equation on an interval”, J. Geom. Phys., 113 (2017), 250–256  crossref  mathscinet  zmath  isi  scopus
    4. V. V. Zharinov, “Hamiltonian operators in differential algebras”, Theoret. and Math. Phys., 193:3 (2017), 1725–1736  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    5. 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
    6. A. Samokhin, “On nonlinear superposition of the KdVЦBurgers shock waves and the behavior of solitons in a layered medium”, Differ. Geom. Appl., 54:A (2017), 91–99  crossref  mathscinet  zmath  isi
    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  crossref  mathscinet  isi
    8. A. G. Kulikovskii, A. P. Chugainova, “Shock waves in anisotropic cylinders”, Proc. Steklov Inst. Math., 300 (2018), 100–113  mathnet  crossref  crossref  isi  elib
    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  mathnet  crossref  crossref  isi  elib
  • ∆урнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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