##
**Nonstationary solutions of a generalized Korteweg-de Vries-Burgers equation.**
*(English.
Russian original)*
Zbl 1292.35263

Proc. Steklov Inst. Math. 281, 204-212 (2013); translation from Tr. Mat. Inst. Steklova 281, 215-223 (2013).

This paper concerns nonstationary solutions of a generalized Korteweg-de Vries-Burgers equation. Numerical simulations of the effect of dispersion terms on the formation of asymptotics for the model equation were carried out. The author presents in Sections 1 and 2 the results of previous research that are necessary to understand the new results described in Sections 3–6. Nonstationary solutions of the Cauchy problem are found for a model equation that includes complicated nonlinearity, dispersion, and dissipation terms and can describe the propagation of nonlinear longitudinal waves in rods. Earlier, within this model, complex behavior of traveling waves has been revealed; it can be regarded as discontinuity structures in solutions of the same equation that ignores dissipation and dispersion. As a result, for standard self-similar problems whose solutions are constructed from a sequence of Riemann waves and shock waves with stationary structure, these solutions become multivalued. The interaction of counterpropagating (or copropagating) nonlinear waves is studied in the case when the corresponding self-similar problems on the collision of discontinuities have a nonunique solution. In addition, situations are considered when the interaction of waves for large times gives rise to asymptotics containing discontinuities with nonstationary periodic oscillating structure.

Reviewer: Ahmed Lesfari (El Jadida)

### MSC:

35Q53 | KdV equations (Korteweg-de Vries equations) |

37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |

PDF
BibTeX
XML
Cite

\textit{A. P. Chugainova}, Proc. Steklov Inst. Math. 281, 204--212 (2013; Zbl 1292.35263); translation from Tr. Mat. Inst. Steklova 281, 215--223 (2013)

Full Text:
DOI

### References:

[1] | I. M. Gel’fand, ”Some problems in the theory of quasilinear equations,” Usp. Mat. Nauk 14(2), 87–158 (1959) [Am. Math. Soc. Transl., Ser. 2, 29, 295–381 (1963)]. · Zbl 0096.06602 |

[2] | S. K. Godunov, ”On nonunique ’blurring’ of discontinuities in solutions of quasilinear systems,” Dokl. Akad. Nauk SSSR 136(2), 272–273 (1961) [Sov. Math., Dokl. 2, 43–44 (1961)]. · Zbl 0117.06401 |

[3] | S. K. Godunov and E. I. Romenskii, Elements of Continuum Mechanics and Conservation Laws (Nauchn. Kniga, Novosibirsk, 1998; Kluwer, New York, 2003). · Zbl 1031.74004 |

[4] | A. G. Kulikovskii, N. V. Pogorelov, and A. Yu. Semenov, Mathematical Aspects of Numerical Solution of Hyperbolic Systems (Fizmatlit, Moscow, 2001; Chapman & Hall/CRC, Boca Raton, FL, 2001). · Zbl 0965.35001 |

[5] | A. G. Kulikovskii, ”A possible effect of oscillations in the structure of a discontinuity on the set of admissible discontinuities,” Dokl. Akad. Nauk SSSR 275(6), 1349–1352 (1984) [Sov. Phys., Dokl. 29, 283–285 (1984)]. |

[6] | A. G. Kulikovskii, ”Surfaces of discontinuity separating two perfect media of different properties: Recombination waves in magnetohydrodynamics,” Prikl. Mat. Mekh. 32(6), 1125–1131 (1968) [J. Appl. Math. Mech. 32, 1145–1152 (1968)]. · Zbl 0187.25206 |

[7] | A. G. Kulikovskii and E. I. Sveshnikova, Nonlinear Waves in Elastic Media (Mosk. Litsei, Moscow, 1998; CRC Press, Boca Raton, FL, 1995). · Zbl 0865.73004 |

[8] | A. G. Kulikovskii and A. P. Chugainova, ”Classical and non-classical discontinuities in solutions of equations of non-linear elasticity theory,” Usp. Mat. Nauk 63(2), 85–152 (2008) [Russ. Math. Surv. 63, 283–350 (2008)]. |

[9] | A. G. Kulikovskii and A. P. Chugainova, ”On the steady-state structure of shock waves in elastic media and dielectrics,” Zh. Eksp. Teor. Fiz. 137(5), 973–985 (2010) [J. Exp. Theor. Phys. 110, 851–862 (2010)]. |

[10] | A. G. Kulikovskii and N. I. Gvozdovskaya, ”The effect of dispersion on the set of admissible discontinuities in continuum mechanics,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 223, 63–73 (1998) [Proc. Steklov Inst. Math. 223, 55–65 (1998)]. · Zbl 1122.74411 |

[11] | A. G. Kulikovskii and A. P. Chugainova, ”Simulation of the influence of small-scale dispersion processes in a continuum on the formation of large-scale phenomena,” Zh. Vychisl. Mat. Mat. Fiz. 44(6), 1119–1126 (2004) [Comput. Math. Math. Phys. 44, 1062–1068 (2004)]. |

[12] | O. A. Oleinik, ”Uniqueness and stability of the generalized solution of the Cauchy problem for a quasi-linear equation,” Usp. Mat. Nauk 14(2), 165–170 (1959) [Am. Math. Soc. Transl., Ser. 2, 33, 285–290 (1963)]. |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.