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

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

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English version:
Computational Mathematics and Mathematical Physics, 2015, 55:2, 251–263

Bibliographic databases:

Document Type: Article
UDC: 519.634

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

Citation in format AMSBIB
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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
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
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
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
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
6. A. Samokhin, “Nonlinear waves in layered media: solutions of the kdv-Burgers equation”, J. Geom. Phys., 130 (2018), 33–39
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
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
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
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