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Analytic model for a frictional shallow-water undular bore

G. A. El·R. H. J. Grimshaw·A. M. Kamchatnov
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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 friction. This is derived in Riemann variables using a modified finite-gap integration technique for the 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.

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G. A. El, R. H. J. Grimshaw, A. M. Kamchatnov. 2004-12-22. Analytic model for a frictional shallow-water undular bore. https://doi.org/10.1063/1.1914743

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