Variable depth KdV equations and generalizations to more nonlinear regimes

Laboratoire de Mathématiques Appliquées de Bordeaux, Université Bordeaux 1, 351 Cours de la Libération, 33405 Talence Cedex, France. Samer.Israwi@math.u-bordeaux1.fr

Received: 14 January 2009

Abstract

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 [Constantin and Lannes, Arch. Ration. Mech. Anal. 192 (2009) 165–186] when the bottom is flat. We generalize here this result with a new class of equations taking into account variable bottom topographies. Of course, 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.