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

We study the propagation of narrow solitons through various profiles of dispersive shock waves (DSW) for the generalized Korteweg-de Vries equation. We consider situations in which the soliton passes through the DSW region quickly enough and does not get trapped in it. The idea is to consider the motion of such solitons through DSW as motion along some smooth effective profile. In the case of KdV and modified KdV, based on the law of conservation of momentum and the equation of motion, this idea is proven rigorously; for other cases of generalized KdV, we take this as a natural generalization. In specific cases of self-similar decays for KdV and modified KdV, a method for selecting an effective field is demonstrated. For the case of generalized KdV, a hypothesis is proposed for selecting an effective field for any wave pulse that is not very large compared to the soliton. All proposed suggestions are numerically tested and demonstrate a high accuracy of reliability.