Analytic model for a weakly dissipative shallow-water undular bore

• Published in: Chaos (Woodbury, N.Y.) Vol. 15; no. 3; p. 37102
• Main Authors: El, G A, Grimshaw, R H J, Kamchatnov, A M
• Format: Journal Article
• Language: English
• Published: United States 01-09-2005
• Subjects: Algorithms; Computer Simulation; Energy Transfer; Models, Biological; Models, Statistical; Motion; Nonlinear Dynamics; Rheology - methods; Water
• ISSN: 1054-1500
• DOI: 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 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.