Remarks on dispersion-improved shallow water equations with uneven bottom

TL;DR

This paper investigates the limitations of dispersion-improved shallow water equations on uneven bottoms, revealing potential issues with ill-posedness and proposing well-posed modifications for constant slopes.

Contribution

It identifies failures of existing dispersion modifications on uneven bottoms and introduces new well-posed equations with improved dispersive properties for specific bottom slopes.

Findings

Dispersion improvements can fail on uneven bottoms.

Modifications may cause ill-posed equations.

New well-posed equations are derived for constant slopes.

Abstract

It is shown that asymptotically consistent modifications of (Boussinesq-like) shallow water approximations, in order to improve their dispersive properties, can fail for uneven bottoms (i.e., the dispersion is actually not improved). It is also shown that these modifications can lead to ill-posed equations when the water depth is not constant. These drawbacks are illustrated with the (fully nonlinear, weakly dispersive) Serre equations. We also derive asymptotically consistent, well-posed, modified Serre equations with improved dispersive properties for constant slopes of the bottom.