On the Korteweg–de Vries approximation for uneven bottoms

In this paper we focus on the water waves problem for uneven bottoms on a two-dimensional domain. Starting from the symmetric Boussinesq systems derived in [F. Chazel, Influence of bottom topography on long water waves, M2AN 41 (4) (2007) 771–799], we recover the uncoupled Korteweg–de Vries (KdV) approximation justified in [G. Schneider, C.E. Wayne, The long-wave limit for the water-wave problem. I. The case of zero surface tension, Comm. Pure Appl. Math. 162 (3) (2002) 247–285] for flat bottoms, and in [T. Iguchi, A long wave approximation for capillary-gravity waves and an effect of the bottom, Comm. Partial Differential Equations 32 (2007) 37–85] 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, where topography-dependent terms are added to the solutions of the KdV equations. Finally, all the models involved are numerically computed and compared.