Analytic model for a weakly dissipative shallow-water undular bore.
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.
El
G A
GA
Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3T, United Kingdom. g.el@lboro.ac.uk
Grimshaw
R H J
RH
Kamchatnov
A M
AM
D000465:Algorithms; D003198:Computer Simulation; D004735:Energy Transfer; D008954:Models, Biological; D015233:Models, Statistical; D009038:Motion; D017711:Nonlinear Dynamics; D012212:Rheology / Q000379:methods*; D014867:Water
D014867:Water