On the Korteweg-de Vries approximation for uneven bottoms

Florent Chazel (1)

1 LHSV - Laboratoire d'Hydraulique Saint-Venant / Saint-Venant Laboratory for Hydraulics

In this paper we focus on the water waves problem for uneven bottoms on a two-dimensionnal domain. Starting from the symmetric Boussinesq systems derived in [Chazel, Influence of topography on long water waves, 2007], we recover the uncoupled Korteweg-de Vries (KdV) approximation justified by Schneider and Wayne for flat bottoms, and by Iguchi in the context of bottoms tending to zero at infinity at a substantial rate. The goal of this paper is to investigate the validity of this approximation for more general bathymetries. We exhibit two kinds of topography for which this approximation diverges from the Boussinesq solutions. A topographically modified KdV approximation is then proposed to deal with such bathymetries. Finally, all the models involved are numerically computed and compared.

Mots clés: Water waves; free surface flows; uneven bottoms; bottom topography; long waves; Korteweg-de Vries approximation; Boussinesq models

Domaines: Equations aux dérivées partielles [math.AP]

Florent Chazel. On the Korteweg-de Vries approximation for uneven bottoms. European Journal of Mechanics - B/Fluids, 2009, 28 (2), pp. 234-252. ⟨10.1016/j.euromechflu.2008.10.003⟩. ⟨hal-00200854⟩