# Variable depth KdV equations and generalizations to more nonlinear regimes

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

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

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topIsrawi, 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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