Analytic model for a frictional shallow-water undular bore

Details Analytic model for a frictional shallow-water undular bore Analytic model for a frictional shallow-water undular bore

2004-12-22

arxiv.org/abs/nlin/0412061v1

Chaos, 15, 037102 (2005)

Nonlinear Sciences

Pattern Formation and Solitons

24 pages

Scientific paper

10.1063/1.1914743

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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