Propagation of narrow and fast solitons through dispersive shock waves in hydrodynamics of simple waves

Abstract

We study the propagation of narrow and fast solitons through various profiles of dispersive shock waves (DSWs) in the framework of the generalized Korteweg-de Vries (gKdV) equation. The idea of considering such a motion as a propagation along a smooth effective field is proposed. In the case of KdV and modified KdV this idea is proven rigorously; for other cases, we take this as a natural hypothesis. For cases of self-similar breaking for KdV and mKdV, a specific method for selecting the effective field is proposed, demonstrating high agreement with the numerical solution. For the breaking of a smooth pulse into the resting medium in gKdV case, we propose using the pulse’s maximum value as an approximation of the effective field. In the considered special cases, this proposal demonstrates good agreement with the numerical solution only for fast solitons.