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Variable depth KdV equations and generalizationsto more nonlinear regimes

Published online by Cambridge University Press:  27 January 2010

Samer Israwi
Samer Israwi*
Affiliation:
Laboratoire de Mathématiques Appliquées de Bordeaux, Université Bordeaux 1, 351 Cours de la Libération, 33405 Talence Cedex, France. [email protected]
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Abstract

We study here the water waves problem for uneven bottoms in a highly nonlinear regime wherethe small amplitude assumption of the Korteweg-de Vries (KdV) equation is enforced. It is knownthat, 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 isflat. We generalize here this resultwith 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 justificationfor some of them. We also study the problem of wave breaking for our newvariable depth and highly nonlinear generalizations of the KdV equations.

Keywords

Water wavesKdV equationstopographic effects
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 44 , Issue 2 , March 2010 , pp. 347 - 370
Copyright
© EDP Sciences, SMAI, 2010

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