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Zh. Vychisl. Mat. Mat. Fiz., 2015, Volume 55, Number 2, Pages 253–266 (Mi zvmmf10155)  

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

Stability of nonstationary solutions of the 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 nonstationary solutions to the Cauchy problem for a model equation with a complex nonlinearity, dispersion, and dissipation is analyzed. The equation describes the propagation of nonlinear longitudinal waves in rods. Previously, complex behavior of traveling waves was found, which can be treated as discontinuity structures in solutions of the same equation without dissipation and dispersion. As a result, the solutions of standard self-similar problems constructed as a sequence of Riemann waves and shocks with a stationary structure become multivalued. The multivaluedness of the solutions is attributed to special discontinuities caused by the large effect of dispersion in conjunction with viscosity. The stability of special discontinuities in the case of varying dispersion and dissipation parameters is analyzed numerically. The computations performed concern the stability analysis of a special discontinuity propagating through a layer with varying dispersion and dissipation parameters.

Key words: generalized KdV-Burgers equation, stability of nonstationary solutions, difference solution method.

Funding Agency Grant Number
Russian Foundation for Basic Research 13-01-12047


DOI: https://doi.org/10.7868/S0044466915020076

Full text: PDF file (357 kB)
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English version:
Computational Mathematics and Mathematical Physics, 2015, 55:2, 251–263

Bibliographic databases:

Document Type: Article
UDC: 519.634
Received: 16.07.2014

Citation: A. P. Chugainova, V. A. Shargatov, “Stability of nonstationary solutions of the generalized KdV-Burgers equation”, Zh. Vychisl. Mat. Mat. Fiz., 55:2 (2015), 253–266; Comput. Math. Math. Phys., 55:2 (2015), 251–263

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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 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
    2. A. K. Volkov, N. A. Kudryashov, “Nonlinear waves described by a fifth-order equation derived from the Fermi–Pasta–Ulam system”, Comput. Math. Math. Phys., 56:4 (2016), 680–687  mathnet  crossref  crossref  mathscinet  isi  elib
    3. 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
    4. 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
    5. 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
    6. A. Samokhin, “Nonlinear waves in layered media: solutions of the kdv-Burgers equation”, J. Geom. Phys., 130 (2018), 33–39  crossref  mathscinet  zmath  isi
    7. 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
    8. A. V. Samokhin, “Reflection and refraction of solitons by the $KdV$–Burgers equation in nonhomogeneous dissipative media”, Theoret. and Math. Phys., 197:1 (2018), 1527–1533  mathnet  crossref  crossref  adsnasa  isi  elib
    9. You Sh., Guo B., “A Non-Homogeneous Boundary-Value Problem For the Kdv-Burgers Equation Posed on a Finite Domain”, Appl. Math. Lett., 94 (2019), 155–159  crossref  mathscinet  zmath  isi  scopus
  • Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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