Soliton Generation and Multiple Phases in Dispersive Shock and Rarefaction Wave Interaction

TL;DR

This paper investigates the complex interactions between dispersive shock waves and rarefaction waves in the Korteweg-de Vries equation, revealing multiphase dynamics and soliton formation in six canonical interaction scenarios.

Contribution

It provides a detailed analysis of six canonical DSW-RW interaction cases, identifying the resulting multiphase solutions and soliton structures, with some outcomes determined analytically.

Findings

Two-phase solutions transiently appear in DSW-DSW interactions.

Certain interactions evolve into a single-phase DSW or a pure RW.

Some cases result in a finite number of solitons or small wave trains.

Abstract

Interactions of dispersive shock (DSWs) and rarefaction waves (RWs) associated with the Korteweg-de Vries equation are shown to exhibit multiphase dynamics and isolated solitons. There are six canonical cases: one is the interaction of two DSWs which exhibit a transient two-phase solution, but evolve to a single phase DSW for large time; two tend to a DSW with either a small amplitude wave train or a finite number of solitons, which can be determined analytically; two tend to a RW with either a small wave train or a finite number of solitons; finally, one tends to a pure RW.