Analytic model for a weakly dissipative shallow-water undular bore

El, Gennady, Grimshaw, R. H. J. and Kamchatnov, A. M. (2005) Analytic model for a weakly dissipative shallow-water undular bore. Chaos: An Interdisciplinary Journal of Nonlinear Science, 15 (3). 037102. ISSN 1054-1500

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Official URL: https://doi.org/10.1063/1.1914743

Abstract

We use the integrable Kaup–Boussinesq shallow water system, modified by a small viscous term, to model the formation of an undular bore with a steady profile. The description is made in terms of the corresponding integrable Whitham system, also appropriately modified by viscosity. This is derived in Riemann variables using a modified finite-gap integration technique for the Ablowitz–Kaup–Newell–Segur (AKNS) scheme. The Whitham system is then reduced to a simple first-order differential equation which is integrated numerically to obtain an asymptotic profile of the undular bore, with the local oscillatory structure described by the periodic solution of the unperturbed Kaup–Boussinesq system. This solution of the Whitham equations is shown to be consistent with certain jump conditions following directly from conservation laws for the original system. A comparison is made with the recently studied dissipationless case for the same system, where the undular bore is unsteady.

Item Type: Article
Subjects: F300 Physics
Department: Faculties > Engineering and Environment > Mathematics, Physics and Electrical Engineering
Depositing User: John Coen
Date Deposited: 16 Apr 2020 10:51
Last Modified: 16 Apr 2020 11:00
URI: http://nrl.northumbria.ac.uk/id/eprint/42800

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