Dispersive shock waves in systems with nonlocal dispersion of Benjamin–Ono type

El, Gennady, Nguyen, Lu Trong Nguyen and Smyth, Noel (2018) Dispersive shock waves in systems with nonlocal dispersion of Benjamin–Ono type. Nonlinearity, 31 (4). pp. 1392-1416. ISSN 0951-7715

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Official URL: https://doi.org/10.1088/1361-6544/aaa10a

Abstract

We develop a general approach to the description of dispersive shock waves (DSWs) for a class of nonlinear wave equations with a nonlocal Benjamin–Ono type dispersion term involving the Hilbert transform. Integrability of the governing equation is not a pre-requisite for the application of this method which represents a modification of the DSW fitting method previously developed for dispersive-hydrodynamic systems of Korteweg-de Vries (KdV) type (i.e. reducible to the KdV equation in the weakly nonlinear, long wave, unidirectional approximation). The developed method is applied to the Calogero–Sutherland dispersive hydrodynamics for which the classification of all solution types arising from the Riemann step problem is constructed and the key physical parameters (DSW edge speeds, lead soliton amplitude, intermediate shelf level) of all but one solution type are obtained in terms of the initial step data. The analytical results are shown to be in excellent agreement with results of direct numerical simulations.

Item Type: Article
Uncontrolled Keywords: dispersive shock wave, undular bore, modulation theory, Benjamin–Ono, soliton
Subjects: G100 Mathematics
Department: Faculties > Engineering and Environment > Mathematics, Physics and Electrical Engineering
Depositing User: Paul Burns
Date Deposited: 12 Nov 2018 17:40
Last Modified: 11 Oct 2019 10:11
URI: http://nrl.northumbria.ac.uk/id/eprint/36638

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