Nonstationary solutions of a generalized Korteweg–de Vries–Burgers equation

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.