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.


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.)
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