Refraction of dispersive shock waves

El, Gennady, Khodorovskii, V.V. and Leszczyszyn, A.M. (2012) Refraction of dispersive shock waves. Physica D: Nonlinear Phenomena, 241 (18). pp. 1567-1587. ISSN 0167-2789

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Official URL: https://doi.org/10.1016/j.physd.2012.06.002

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 the defocusing nonlinear Schrödinger (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.

Item Type: Article
Uncontrolled Keywords: Dispersive shock wave, Rarefaction wave, Soliton, Nonlinear wave interactions, Whitham equations, Euler–Poisson–Darboux equation
Subjects: F100 Chemistry
G100 Mathematics
Department: Faculties > Engineering and Environment > Mathematics, Physics and Electrical Engineering
Depositing User: John Coen
Date Deposited: 14 Apr 2020 14:10
Last Modified: 14 Apr 2020 14:15
URI: http://nrl.northumbria.ac.uk/id/eprint/42757

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