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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. Samer.Israwi@math.u-bordeaux1.fr
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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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References

S. Alinhac and P. Gérard, Opérateurs pseudo-différentiels et théorème de Nash-Moser. EDP Sciences, Les Ulis, France (1991).
Alvarez-Samaniego, B. and Lannes, D., Large time existence for 3D water-waves and asymptotics. Invent. Math. 171 (2008) 485–541. CrossRef
Alvarez-Samaniego, B. and Lannes, D., Nash-Moser, A theorem for singular evolution equations. Application to the Serre and Green-Naghdi equations. Indiana Univ. Math. J. 57 (2008) 97–131.
Benjamin, T.B., Bona, J.L. and Mahony, J.J., Model equations for long waves in nonlinear dispersive systems. Phil. Trans. Roy. Soc. London A 227 (1972) 47–78. CrossRef
Bona, J.L., Chen, M. and Saut, J.-C., 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. CrossRef
Bona, J.L., Colin, T. and Lannes, D., Long wave approximations for water waves. Arch. Ration. Mech. Anal. 178 (2005) 373–410. CrossRef
Camassa, R. and Holm, D., An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71 (1993) 1661–1664. CrossRef
Constantin, A. and Lannes, D., The hydrodynamical relevance of the Camassa-Holm and Degasperis-Processi equations. Arch. Ration. Mech. Anal. 192 (2009) 165–186. CrossRef
Craig, W., An existence theory for water waves and the Boussinesq and the Korteweg-de Vries scaling limits. Commun. Partial Differ. Equations 10 (1985) 787–1003. CrossRef
W. Craig, T. Kappeler and W. Strauss, Gain of regularity for equations of KdV type. Ann. Institut Henri Poincaré, Anal. non linéaire 9 (1992) 147–186.
A. Degasperis and M. Procesi, Asymptotic integrability, in Symmetry and Perturbation Theory, A. Degasperis and G. Gaeta Eds., World Scientific, Singapore (1999) 23–37.
M.W. Dingemans, Water waves propogation over uneven bottoms. Part 2. Advanced Series on ocean Engineering 13. World Scientific, Singapore (1997).
P.G. Drazin and R.S. Johnson, Solitons: an introduction. Cambridge University Press, Cambridge, UK (1992).
Green, A.E. and Naghdi, P.M., A derivation of equations for wave propagation in water of variable depth. J. Fluid Mech. 78 (1976) 237–246. CrossRef
Grimshaw, R. and Pudjaprasetya, S.R., Hamiltonian formulation for solitary waves propagating on a variable background. J. Engrg. Math. 36 (1999) 89–98.
Iguchi, T., A long wave approximation for capillary-gravity waves and the Kawahara equation. Bull. Inst. Math. Acad. Sin. (N.S.) 2 (2007) 179–220.
R.S. Johnson, A modern introduction to the mathematical theory of water waves. Cambridge University Press, Cambridge, UK (1997).
Johnson, R.S., Camassa-Holm, Korteweg-de Vries and related models for water waves. J. Fluid Mech. 457 (2002) 63–82.
Johnson, R.S., On the development of a solitary wave moving over an uneven bottom. Proc. Cambridge Philos. Soc. 73 (1973) 183–203. CrossRef
J.J. Kirby, Nonlinear ocean surface waves. Center for Applied Coastal research, University of Delaware, USA (2004).
Korteweg, D.J. and de Vries, G., 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. CrossRef
Lannes, D., Secular growth estimates for hyperbolic systems. J. Differ. Equ. 190 (2003) 466–503. CrossRef
Lannes, D., Well-posedness of the water waves equations. J. Amer. Math. Soc. 18 (2005) 605–654. CrossRef
Lannes, D., Sharp estimates for pseudo-differential operators with symbols of limited smoothness and commutators. J. Funct. Anal. 232 (2006) 495–539. CrossRef
D. Lannes and P. Bonneton, Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation. Phys. Fluids 21 (2009).
Miles, J.W., On the Korteweg-de Vries equation for a gradually varying channel. J. Fluid Mech. 91 (1979) 181–190. CrossRef
Nalimov, V.I., [The Cauchy-Poisson problem]. Dinamika Splošn. Sredy Vyp. 18 Dinamika Zidkost. so Svobod. Granicami 254 (1974) 104–210 (in Russian).
Peregrine, D.H., Calculations of the development of an undular bore. J. Fluid Mech. 25 (1966) 321–330. CrossRef
Pudjaprasetya, S.R. and van Groesen, E., Unidirectional waves over slowly varying bottom. II. Quasi-homogeneous approximation of distorting waves. Wave Motion 23 (1996) 23–38. CrossRef
Pudjaprasetya, S.R., van Groesen, E. and Soewono, E., The splitting of solitary waves running over a shallower water. Wave Motion 29 (1999) 375–389. CrossRef
Schneider, G. and Wayne, C., The long-wave limit for the water wave problem I. The case of zero surface tension. Commun. Pure Appl. Math. 53 (2000) 1475–1535. 3.0.CO;2-V>CrossRef
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.
van Groesen, E. and Pudjaprasetya, S.R., Uni-directional waves over slowly varying bottom. I. Derivation of a KdV-type of equation. Wave Motion 18 (1993) 345–370. CrossRef
Well-posedness, S. Wu in Sobolev spaces of the full water wave problem in 2-D. Invent. Math. 130 (1997) 39–72.
Well-posedness, S. Wu in Sobolev spaces of the full water wave problem in 3-D. J. Amer. Math. Soc. 12 (1999) 445–495.
Yoon, S.B. and Liu, P.L.-F., A note on Hamiltonian for long water waves in varying depth. Wave Motion 20 (1994) 359–370. CrossRef
Yosihara, H., Gravity waves on the free surface of an incompressible perfect fluid of finite depth. Publ. Res. Inst. Math. Sci. 18 (1982) 49–96. CrossRef