  We study here the water-waves problem for uneven bottoms in a highly
nonlinear regime where the small amplitude assumption of the Korteweg-de Vries
(KdV) equation is enforced. It is known, that for such regimes, a
generalization of the KdV equation (somehow linked to the Camassa-Holm
equation) can be derived and justified by A. Constantin, D. Lannes "The
hydrodynamical relevance of the Camassa-Holm and Degasperis-Processi equations"
when the bottom is flat. We generalize here this result with a new class of
equations taking into account variable bottom topographies. Of course, the many
variable depth KdV equations existing in the literature are recovered as
particular cases. Various regimes for the topography regimes are investigated
and we prove consistency of these models, as well as a full justification for
some of them. We also study the problem of wave breaking for our new variable
depth and highly nonlinear generalizations of the KDV equations.
