Document Type

Article

Rights

This item is available under a Creative Commons License for non-commercial use only

Disciplines

Applied mathematics, Fluids and plasma physics

Publication Details

Wave Motion (2009), vol.46 (2009), 389-396

Abstract

In this contribution we describe the role of several two-component integrable systems in the classical problem of shallow water waves. The starting point in our derivation is the Euler equation for an incompressible fluid, the equation of mass conservation, the simplest bottom and surface conditions and the constant vorticity condition. The approximate model equations are generated by introduction of suitable scalings and by truncating asymptotic expansions of the quantities to appropriate order. The so obtained equations can be related to three different integrable systems: a two component generalization of the Camassa-Holm equation, the Zakharov-Ito system and the Kaup-Boussinesq system. The significance of the results is the inclusion of vorticity, an important feature of water waves that has been given increasing attention during the last decade. The presented investigation shows how – up to a certain order – the model equations relate to the shear flow upon which the wave resides. In particular, it shows exactly how the constant vorticity affects the equations.