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.
Mots clés : water waves, KdV equations, topographic effects
@article{M2AN_2010__44_2_347_0, author = {Israwi, Samer}, title = {Variable depth {KdV} equations and generalizations to more nonlinear regimes}, journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique}, pages = {347--370}, publisher = {EDP-Sciences}, volume = {44}, number = {2}, year = {2010}, doi = {10.1051/m2an/2010005}, mrnumber = {2655953}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2010005/} }
TY - JOUR AU - Israwi, Samer TI - Variable depth KdV equations and generalizations to more nonlinear regimes JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2010 SP - 347 EP - 370 VL - 44 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2010005/ DO - 10.1051/m2an/2010005 LA - en ID - M2AN_2010__44_2_347_0 ER -
%0 Journal Article %A Israwi, Samer %T Variable depth KdV equations and generalizations to more nonlinear regimes %J ESAIM: Modélisation mathématique et analyse numérique %D 2010 %P 347-370 %V 44 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2010005/ %R 10.1051/m2an/2010005 %G en %F M2AN_2010__44_2_347_0
Israwi, Samer. Variable depth KdV equations and generalizations to more nonlinear regimes. ESAIM: Modélisation mathématique et analyse numérique, Tome 44 (2010) no. 2, pp. 347-370. doi : 10.1051/m2an/2010005. http://www.numdam.org/articles/10.1051/m2an/2010005/
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