Analytic model for a frictional 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 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. Undular bores are nonlinear wave-like structures, which are generated in the breaking profiles of large-scale nonlinear waves propagating in dispersve media. A general theory, based on the Whitham modulation equations, has been previously developed for dissipationless, unsteady, undular bores on the basis of completely integrable models such as the Korteweg-de Vries equation, nonlinear Schrödinger equation etc. The introduction of physically important small dissipation in the system dramatically changes its properties, allowing in some cases for the presence of steady solutions. The most explored model