Interactions between rarefaction waves and dispersive shock waves in the Gerdjikov--Ivanov equation with non-convex hydrodynamics

Abstract

This paper systematically investigates admissible configurations of the interaction between a rarefaction wave (RW) and a dispersive shock wave (DSW) by constructing a two-step piecewise constant initial condition in the Gerdjikov--Ivanov (GI) equation with non-convex hydrodynamics. Owing to the two-valued mapping associated with the Riemann invariants, the GI equation also supports contact DSWs (CDSWs), which are included in the interaction analysis. Using Whitham modulation theory, we elucidate the self-similar descriptions of the incoming waves, derive the boundary matching conditions, and obtain analytical expressions of the full evolution before, during, and after the interaction through generalized hodograph transformation. The results show that the interaction of a RW with either a DSW or a CDSW can exhibit two distinct scenarios: in one case, the two waves exchange spectral parameters after the interaction and fully separate, propagating independently with modified phases; in the other case, the wave structures merge upon interaction and subsequently evolve into either a soliton train or a small-amplitude harmonic wave for a classical DSW, and into an algebraic soliton train for a CDSW. The theoretical results are in good agreement with numerical simulations. This work can provide theoretical support for wave interaction phenomena in non-convex dispersive systems.