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Traveling waves and undercompressive shocks in solutions of the generalized Korteweg–de Vries–Burgers equation with a time-dependent dissipation coefficient distribution

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Abstract

Solutions of the generalized KdV–Burgers equation are analyzed in the case when the dissipation coefficient depends on the spatial coordinate and time. Solutions in the form of traveling waves describing the structure of discontinuities, including special discontinuities, are studied. In the frame of this problem, nonstationary solutions of the generalized KdV–Burgers equation including the special discontinuity are numerically found.

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References

  1. A.G. Kulikovskii, Soviet. Phys. Dokl. 29, 283 (1984)

  2. A.G. Kulikovskii, N.V. Pogorelov, AYu. Semenov, Mathematical Aspects of Numerical Solution of Hyperbolic Systems, Monographs and Surveys in Pure and Applied Mathematics (Chapman & Hall/CRC, Boca Raton, FL, 2001)

    Google Scholar 

  3. P.G. LeFloch, Hyperbolic Systems of Conservation Laws: The Theory of Classical and Nonclassical Shock Waves Lectures in Mathematics (ETH Zurich, Birkhauser, 2002)

  4. A.G. Kulikovskii, A.P. Chugainova, Russ. Math. Surv. 63, 2 (2008)

    Article  Google Scholar 

  5. A.P. Chugainova, Proc. Steklov Inst. Math. 281, 204–212 (2013)

  6. A.L. Bertozzi, A. Munch, M. Shearer, Phys. D 134, 431–464 (1999)

  7. I.V. Lomonosov, N.A. Tahir, Appl. Phys. Lett. 92, 101905 (2008)

    Article  ADS  Google Scholar 

  8. I.V. Lomonosov, Laser Particle Beams 25 (2007)

  9. I.B. Bakholdin, Fluid Dyn. 34, 4 (1990)

    MathSciNet  Google Scholar 

  10. I.B. Bakholdin, J. Appl. Math. Mech. 75, 2 (2011)

    Article  MathSciNet  Google Scholar 

  11. I.B. Bakholdin, J. Appl. Math. Mech. 78, 6 (2014)

    Article  MathSciNet  Google Scholar 

  12. G.A. El, M.A. Hoefer, M. Shearer, SIAM Rev. 59, 1 (2016)

    Google Scholar 

  13. B. Hayes, M. Shearer, Proc. Roy. Soc. Edinb. A 129 (1999)

  14. D. Jacobs, B. McKinney, M. Shearer, Journ. Diff. Equations 116, (1995)

  15. P.G. LeFloch, M. Shearer, Proc. R. Soc. Edinb. A 134 (2004)

  16. B. Hayes, M. Shearer, Q. Appl. Math. 59 (2001)

  17. A.G. Kulikovskii, A.P. Chugainova, Comput. Math. Math. Phys. 44, 6 (2004)

    MathSciNet  Google Scholar 

  18. A.P. Chugainova, V.A. Shargatov, Comput. Math. Math. Phys. 56, 2 (2016)

    Article  Google Scholar 

  19. A.T. Il’ichev, A.P. Chugainova, V.A. Shargatov, Dokl. Math. 91 (2015)

  20. A.G. Kulikovskii, A.P. Chugainova, V.A. Shargatov, Comput. Math. Math. Phys. 56, 7 (2016)

    Google Scholar 

  21. A.P. Chugainova, V.A. Shargatov, Commun. Nonlinear Sci. Numer. Simul. 66 (2019)

  22. L.D. Landau, E.M. Lifshits, Course of Theoretical Physics, Vol. 6: Fluid Mechanics. Pergamon, Oxford (1987)

  23. P.D. Lax, Commun. Pure Appl. Math. 10, 537 (1957)

  24. A.G. Kulikovskii, J. Appl. Math. Mech. 32 (1968)

  25. A.P. Chugainova, A.T. Il’ichev, A.G. Kulikovskii, V.A. Shargatov, J. Appl. Math. 82, 3 (2017)

    Google Scholar 

  26. A.V. Samokhin, J. Geom. Phys. 113 (2017)

  27. A.V. Samokhin, Theor. Math. Phys. 197, 1 (2018)

    Article  Google Scholar 

  28. A.V. Samokhin, J. Geom. Phys. 130 (2018)

  29. A.P. Chugainova, V.A. Shargatov, Comput. Math. Math. Phys. 55, 2 (2015)

    Article  Google Scholar 

  30. A.P. Chugainova, V.A. Shargatov, AIP Conf. Proc. 2164, 050002 (2019)

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported by the RFBR Grant 20-01-00071.

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Authors and Affiliations

  1. Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina St., Moscow, 119991, Russia

    A. P. Chugainova

  2. National Research Nuclear University “MePHI”, Kashirskoye shosse 31, Moscow, 115409, Russia

    V. A. Shargatov

  3. N. N. Semenov Institute of Chemical Physics (ICP), Russian Academy of Sciences, Moscow, Russia

    V. A. Shargatov

Authors
  1. A. P. Chugainova
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  2. V. A. Shargatov
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Correspondence to A. P. Chugainova.

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Chugainova, A.P., Shargatov, V.A. Traveling waves and undercompressive shocks in solutions of the generalized Korteweg–de Vries–Burgers equation with a time-dependent dissipation coefficient distribution. Eur. Phys. J. Plus 135, 635 (2020). https://doi.org/10.1140/epjp/s13360-020-00659-3

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