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Expansion shock waves in regularised shallow water theory

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journal contribution
posted on 29.04.2016 by Gennady El, M.A. Hoefer, Michael Shearer
We identify a new type of shock wave by constructing a stationary expansion shock solution of a class of regularised shallow water equations that include the Benjamin-Bona-Mahoney (BBM) and Boussinesq equations. An expansion shock exhibits divergent characteristics, thereby contravening the classical Lax entropy condition. The persistence of the expansion shock in initial value problems is analysed and justified using matched asymptotic expansions and numerical simulations. The expansion shock’s existence is traced to the presence of a non-local dispersive term in the governing equation. We establish the algebraic decay of the shock as it is gradually eroded by a simple wave on either side. More generally, we observe a robustness of the expansion shock in the presence of weak dissipation and in simulations of asymmetric initial conditions where a train of solitary waves is shed from one side of the shock.

Funding

This work was supported by the Royal Society International Exchanges Scheme IE131353 (all authors), NSF CAREER DMS-1255422 (MAH), and NSF DMS-1517291 (MS).

History

School

  • Science

Department

  • Mathematical Sciences

Published in

Proceedings of the Royal Society of London: Mathematical, Physical and Engineering Sciences

Citation

EL, G., HOEFER, M. and SHEARER, M., 2016. Expansion shock waves in regularised shallow water theory. Proceedings of the Royal Society of London. Series A, Mathematical, Physical and Engineering Sciences, 472 (2189), article 20160141.

Publisher

Royal Society (© the authors)

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

2016

Notes

This paper was accepted for publication in the journal Proceedings of the Royal Society of London. Series A, Mathematical, Physical and Engineering Sciences and the definitive published version is available at http://dx.doi.org/10.1098/rspa.2016.0141

ISSN

0962-8444

Language

en

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