# 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

top- S. Alinhac and P. Gérard, Opérateurs pseudo-différentiels et théorème de Nash-Moser. EDP Sciences, Les Ulis, France (1991).
- B. Alvarez-Samaniego and D. Lannes, Large time existence for 3D water-waves and asymptotics. Invent. Math.171 (2008) 485–541. Zbl1131.76012
- B. Alvarez-Samaniego and D. Lannes, A Nash-Moser theorem for singular evolution equations. Application to the Serre and Green-Naghdi equations. Indiana Univ. Math. J.57 (2008) 97–131. Zbl1144.35007
- T.B. Benjamin, J.L. Bona and J.J. Mahony, Model equations for long waves in nonlinear dispersive systems. Phil. Trans. Roy. Soc. London A227 (1972) 47–78. Zbl0229.35013
- J.L. Bona, M. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I. Derivation and linear theory. J. Nonlinear Sci. 12 (2002) 283–318. Zbl1022.35044
- J.L. Bona, T. Colin and D. Lannes, Long wave approximations for water waves. Arch. Ration. Mech. Anal.178 (2005) 373–410. Zbl1108.76012
- R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons. Phys. Rev. Lett.71 (1993) 1661–1664. Zbl0972.35521
- A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Processi equations. Arch. Ration. Mech. Anal.192 (2009) 165–186. Zbl1169.76010
- W. Craig, An existence theory for water waves and the Boussinesq and the Korteweg-de Vries scaling limits. Commun. Partial Differ. Equations10 (1985) 787–1003. Zbl0577.76030
- W. Craig, T. Kappeler and W. Strauss, Gain of regularity for equations of KdV type. Ann. Institut Henri Poincaré, Anal. non linéaire9 (1992) 147–186. Zbl0764.35021
- A. Degasperis and M. Procesi, Asymptotic integrability, in Symmetry and Perturbation Theory, A. Degasperis and G. Gaeta Eds., World Scientific, Singapore (1999) 23–37. Zbl0963.35167
- M.W. Dingemans, Water waves propogation over uneven bottoms. Part 2. Advanced Series on ocean Engineering13. World Scientific, Singapore (1997).
- P.G. Drazin and R.S. Johnson, Solitons: an introduction. Cambridge University Press, Cambridge, UK (1992). Zbl0661.35001
- A.E. Green and P.M. Naghdi, A derivation of equations for wave propagation in water of variable depth. J. Fluid Mech.78 (1976) 237–246. Zbl0351.76014
- R. Grimshaw and S.R. Pudjaprasetya, Hamiltonian formulation for solitary waves propagating on a variable background. J. Engrg. Math.36 (1999) 89–98. Zbl0934.76011
- T. Iguchi, A long wave approximation for capillary-gravity waves and the Kawahara equation. Bull. Inst. Math. Acad. Sin. (N.S.)2 (2007) 179–220. Zbl1151.35084
- R.S. Johnson, A modern introduction to the mathematical theory of water waves. Cambridge University Press, Cambridge, UK (1997). Zbl0892.76001
- R.S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves. J. Fluid Mech.457 (2002) 63–82. Zbl1037.76006
- R.S. Johnson, On the development of a solitary wave moving over an uneven bottom. Proc. Cambridge Philos. Soc.73 (1973) 183–203. Zbl0255.76020
- J.J. Kirby, Nonlinear ocean surface waves. Center for Applied Coastal research, University of Delaware, USA (2004).
- D.J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves. Phil. Mag.39 (1895) 422. Zbl26.0881.02
- D. Lannes, Secular growth estimates for hyperbolic systems. J. Differ. Equ.190 (2003) 466–503. Zbl1052.35119
- D. Lannes, Well-posedness of the water waves equations. J. Amer. Math. Soc.18 (2005) 605–654. Zbl1069.35056
- D. Lannes, Sharp estimates for pseudo-differential operators with symbols of limited smoothness and commutators. J. Funct. Anal.232 (2006) 495–539. Zbl1099.35191
- D. Lannes and P. Bonneton, Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation. Phys. Fluids21 (2009). Zbl1183.76294
- J.W. Miles, On the Korteweg-de Vries equation for a gradually varying channel. J. Fluid Mech.91 (1979) 181–190. Zbl0399.76027
- V.I. Nalimov, [The Cauchy-Poisson problem]. Dinamika Splošn. Sredy Vyp. 18 Dinamika Zidkost. so Svobod. Granicami254 (1974) 104–210 (in Russian).
- D.H. Peregrine, Calculations of the development of an undular bore. J. Fluid Mech.25 (1966) 321–330.
- S.R. Pudjaprasetya and E. van Groesen, Unidirectional waves over slowly varying bottom. II. Quasi-homogeneous approximation of distorting waves. Wave Motion23 (1996) 23–38. Zbl0956.76506
- S.R. Pudjaprasetya, E. van Groesen and E. Soewono, The splitting of solitary waves running over a shallower water. Wave Motion29 (1999) 375–389. Zbl1074.76528
- G. Schneider and C. Wayne, The long-wave limit for the water wave problem I. The case of zero surface tension. Commun. Pure Appl. Math.53 (2000) 1475–1535. Zbl1034.76011
- I.A. Svendsen, A direct derivation of the KDV equation for waves on a beach, and discussion of it's implications. Prog. Rept. 39, ISVA, Tech. Univ. of Denmark (1976) 9–14.
- E. van Groesen and S.R. Pudjaprasetya, Uni-directional waves over slowly varying bottom. I. Derivation of a KdV-type of equation. Wave Motion18 (1993) 345–370. Zbl0819.76012
- S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in 2-D. Invent. Math.130 (1997) 39–72. Zbl0892.76009
- S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in 3-D. J. Amer. Math. Soc.12 (1999) 445–495. Zbl0921.76017
- S.B. Yoon and P.L.-F. Liu, A note on Hamiltonian for long water waves in varying depth. Wave Motion20 (1994) 359–370. Zbl0929.76019
- H. Yosihara, Gravity waves on the free surface of an incompressible perfect fluid of finite depth. Publ. Res. Inst. Math. Sci.18 (1982) 49–96. Zbl0493.76018

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