Special discontinuities depending on dispersion processes

Abstract

The simultaneous effects of dissipation and dispersion on nonlinear wave behavior in elastic media are considered when the effects are small and manifested only in narrow high gradient regions. The small-scale processes determine the structure of discontinuities (shocks) and a set of discontinuities with stationary structures. Among the discontinuities with stationary structures there are special ones on which an additional relation should be satisfied. This relation follows from the requirement for the discontinuity structure to exist. On the phase plane, the structure of such discontinuities is represented by an integral curve joining two saddles. Two models for describing dispersion processes are compared. It is shown that the set of special discontinuities can be significantly different depending on small-scale processes.