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Refraction of dispersive shock waves

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journal contribution
posted on 2015-03-10, 16:33 authored by Gennady El, V.V. Khodorovskii, Antin M. Leszczyszyn
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.

History

School

  • Science

Department

  • Mathematical Sciences

Published in

Physica D: Nonlinear Phenomena

Volume

241

Issue

18

Pages

1567 - 1587

Citation

EL, G.A., KHODOROVSKII, V.V. and LESZCZYSZYN, A.M., 2012. Refraction of dispersive shock waves. Physica D: Nonlinear Phenomena, 241 (18), pp.1567-1587.

Publisher

© Elsevier

Version

  • AM (Accepted Manuscript)

Publisher statement

This work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/

Publication date

2012

Notes

This is the author’s version of a work that was accepted for publication in Physica D: Nonlinear Phenomena. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published at: http://dx.doi.org/10.1016/j.physd.2012.06.002

ISSN

0167-2789

Other identifier

S0167278912001601

Language

  • en