Variable depth KdV equations and generalizations to more nonlinear regimes

Samer Israwi

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 44, Issue: 2, page 347-370
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Israwi, Samer. "Variable depth KdV equations and generalizations to more nonlinear regimes." ESAIM: Mathematical Modelling and Numerical Analysis 44.2 (2010): 347-370. <http://eudml.org/doc/250833>.

@article{Israwi2010,
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. },
author = {Israwi, Samer},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Water waves; KdV equations; topographic effects; water waves},
language = {eng},
month = {3},
number = {2},
pages = {347-370},
publisher = {EDP Sciences},
title = {Variable depth KdV equations and generalizations to more nonlinear regimes},
url = {http://eudml.org/doc/250833},
volume = {44},
year = {2010},
}

TY - JOUR
AU - Israwi, Samer
TI - Variable depth KdV equations and generalizations to more nonlinear regimes
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 44
IS - 2
SP - 347
EP - 370
AB - 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.
LA - eng
KW - Water waves; KdV equations; topographic effects; water waves
UR - http://eudml.org/doc/250833
ER -

References

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