Refraction of dispersive shock waves

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.Comment: 45 pages, 23 figures, minor revisio

Transformation of a shoaling undular bore

“…We note that a similar phase shift arises in the interactions of dispersive shock waves with rarefaction waves (Ablowitz et al (2009);El et al (2011)), which is to some degree analogous to the present problem of the transformation of the undular over sloping bottom (see the end of this Section). Importantly, the mentioned phase shifts do not affect relationships (3.5) between the modulation parameters in the periodic solution, which implies that amplitude of the leading solitary wave of the transformed bore would then be 2U 0 , that is, unchanged from the value before the variable depth region is encountered.…”

Incoherent Dispersive Shocks in the Spectral Evolution of Random Waves

“…2. In this case, features analogous to those widely studied for coherent DSWs [30,31] are exhibited by the incoherent wave in the spectral domain. The evolving spectrum gives rise indeed to an expansion (rarefaction) wave on the leading edge and a gradient catastrophe on the trailing edge, which is resolved by an expanding dispersive wave train, both features being fully captured by our theory based on SIDKEs.…”

Dispersive shock waves and modulation theory

“…where µ(m) = E(m)/K(m). It is not difficult to show using the representation (2.125) that the NLS-Whitham system (2.124), (2.126) is genuinely nonlinear and strictly hyperbolic (see [102]). The general proof of these properties for the multiphase case can be found in [103] (see also [104]) and is analogous to the original proof in [90] for the KdV-Whitham system.…”

“…The head-on interaction of a NLS DSW with a centered rarefaction wave has been considered in [102] as a dispersive counterpart of the classical gas dynamics problem of shock wave refraction (see, e.g., [141,58]). When a one-dimensional viscous shock wave undergoes a headon collision with a rarefaction wave, the parameters of the two waves alter so that the long-time output of such an interaction consists of a new pair of shock and rarefaction waves propagating in opposite directions.…”

“…The refraction shift d of the DSW soliton edge quantifies the intensity of the interaction. Reprinted from[102] Copyright (2012), with permission from Elsevier.…”