Refraction of dispersive shock waves

TL;DR

This paper analyzes how dispersive shock waves interact with rarefaction waves in nonlinear Schrödinger equations, providing asymptotic solutions for integrable and non-integrable cases and confirming results with numerical simulations.

Contribution

It offers a comprehensive asymptotic description of dispersive shock wave refraction in both integrable and non-integrable NLS equations, using novel modulation methods.

Findings

Exact solutions for integrable cubic NLS case

Main physical parameters for non-integrable saturable NLS

Numerical confirmation of theoretical predictions

Abstract

We study a dispersive counterpart of the classical gas dynamics problem of the interaction of a shock wave with a counter-propagating simple rarefaction wave often referred to as the shock wave refraction. The refraction of a one-dimensional dispersive shock wave (DSW) due to its head-on collision with the centred rarefaction wave (RW) is considered in the framework of defocusing nonlinear Schr\"odinger (NLS) equation. For the integrable cubic nonlinearity case we present a full asymptotic description of the DSW refraction by constructing appropriate exact solutions of the Whitham modulation equations in Riemann invariants. For the NLS equation with saturable nonlinearity, whose modulation system does not possess Riemann invariants, we take advantage of the recently developed method for the DSW description in non-integrable dispersive systems to obtain main physical parameters of the DSW refraction. The key features of the DSW-RW interaction predicted by our modulation theory analysis are confirmed by direct numerical solutions of the full dispersive problem.