  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.
