Analytic model for a weakly dissipative shallow-water undular bore

G. A. El*, R. H.J. Grimshaw, A. M. Kamchatnov

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

27 Citations (Scopus)
21 Downloads (Pure)


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.

Original languageEnglish
Article number037102
Issue number3
Publication statusPublished - Sept 2005
Externally publishedYes


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