Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-24T11:04:34.605Z Has data issue: false hasContentIssue false
  • English
  • Français

Rigorous derivation of Korteweg-de Vries-type systems from a general class of nonlinear hyperbolic systems

Published online by Cambridge University Press:  15 April 2002

Walid Ben Youssef  and
Thierry Colin
Walid Ben Youssef
Affiliation:
Mathématiques Appliquées de Bordeaux, Université Bordeaux 1 et CNRS UMR 5466, 351 Cours de la Libération, 33405 Talence Cedex, France. (benyou@math.u-bordeaux.fr)
Thierry Colin
Affiliation:
Mathématiques Appliquées de Bordeaux, Université Bordeaux 1 et CNRS UMR 5466, 351 Cours de la Libération, 33405 Talence Cedex, France. (colin@math.u-bordeaux.fr)
Get access

Abstract

In this paper, we study the long wave approximation for quasilinear symmetric hyperbolic systems. Using the technics developed by Joly-Métivier-Rauch for nonlinear geometrical optics, we prove that under suitable assumptions the long wave limit is described by KdV-type systems. The error estimate if the system is coupled appears to be better. We apply formally our technics to Euler equations with free surface and Euler-Poisson systems. This leads to new systems of KdV-type.

Keywords

Hyperbolic systemssystems of KdV-typeEuler-Poissonwater-wavesasymptotic expansionlong-wave approximation.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 34 , Issue 4 , July 2000 , pp. 873 - 911
Copyright
© EDP Sciences, SMAI, 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

S. Alinhac and P. Gerard, Opérateurs pseudo-différentiels et thérorème de Nash-Moser. Interéditions/Éditions du CNRS (1991).
Benjamin, T.B., Bona, J.L. and Mahony, J.J., Model equations for long waves in nonlinear, dispersive systems. Philos. Trans. Roy. Soc. Lond. A 272 (1972) 47-78. CrossRef
W. Ben Youssef, The global Cauchy problem for Korteweg-de Vries type systems describing counter-propagating waves. MAB, Université Bordeaux I, preprint (1999).
W. Ben Youssef, Conservative, high order schemes and numerical study of a coupled system of Korteweg-de Vries type. Université de Bordeaux I, preprint (1999).
J.L. Bona and H. Chen, Lecture notes in Austin. Texas Institute for Computational and Applied Mathematics (1997).
Bona, J.L. and Chen, H., Boussinesq, A system for two-way propagation of nonlinear dispersive waves. Physica D 116 (1998) 191-224. CrossRef
J.L. Bona, H. Chen and J.C. Saut, personal communications.
Bona, J.L. and Smith, R., The initial-value problem for the Korteweg-de Vries equation. Philos. Trans. Roy. Soc. Lond. A 278 (1975) 555-604. CrossRef
Bourgain, J., Fourier transform restriction phenomena for certain lattice substes and applications to nonlinear evolution equations. II. The Korteweg-de Vries equation. Geom. Funct. Anal. 3 (1993) 209-262. CrossRef
S. Cordier and E. Grenier, Quasineutral limit of Euler-Poisson system arising from plasma physics. Université de Paris VI, preprint (1997).
Craig, W., An existence theory for water waves and the Boussinesq and the Korteweg-de Vries scaling limits. Comm. Partial Differential Equations 10 (1985) 787-1003. CrossRef
T. Colin, Rigorous derivation of the nonlinear Schrodinger equation and Davey-Stewartson systems from quadratic hyperbolic systems. Université de Bordeaux I, preprint No. 99001 (1999).
R.K. Dodd, J.C. Eilbeck, J.D. Gibbon and H.C. Morris, Solitons and nonlinear wave equations. Academic Press (1982).
P. Donnat, J.L. Joly, G. Metivier and J. Rauch, Diffractive nonlinear geometric optics. I. Séminaire équations aux dérivées partielles. École Polytechnique, Palaiseau, exposé No. XVII-XVIII (1995-1996).
Joly, J.L., Metivier, G. and Rauch, J., Diffractive nonlinear geometric optics with rectification. Indiana Univ. Math. J. 47 (1998) 1167-1241. CrossRef
Joly, J.L., Metivier, G. and Rauch, J., Generic rigorous asymptotic expansions for weakly nonlinear multidimensional oscillatory waves. Duke Math. J. 70 (1993) 373-404. CrossRef
T. Kato, Perturbation theory for linear operators. Grundlehren Math. Wiss. 132 (1966).
C.E. Kenig, G. Ponce and L. Vega, Well posedness and scaterring results for the generalized Korteweg-de Vries equation via the contraction principle. Comm. Pure Appl. Math. XLVI (1993) 527-620.
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 wave. Phil. Mag. 39 (1895) 422-443. CrossRef
Lannes, D., Dispersive effects for nonlinear geometrical optics with rectification. Asymptot. Anal. 18 (1998) 111-146.
G. Schneider and C.E. Wayne, The long wave limit for the water wave problem. I. The case of zero surface tension. University of Bayreuth, preprint (1999).
G.B. Whitham, Linear and nonlinear waves. J. Wiley, New York (1974).
Zabusky, N.J. and Kruskal, M.D., Interaction of solitons in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15 (1965) 240. CrossRef