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Numerical comparisons of two long-wave limit models

Published online by Cambridge University Press:  15 June 2004

Stéphane Labbé  and
Lionel Paumond
Stéphane Labbé
Affiliation:
Laboratoire de Mathématiques, Université Paris-Sud, Bâtiment 425, 91405 Orsay Cedex, France. [email protected].
Lionel Paumond
Affiliation:
Laboratoire de Mathématiques, Université Paris-Sud, Bâtiment 425, 91405 Orsay Cedex, France. [email protected].
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Abstract

The Benney-Luke equation (BL) is a model for the evolution of three-dimensional weakly nonlinear, long water waves of small amplitude. In this paper we propose a nearly conservative scheme for the numerical resolution of (BL). Moreover, it is known (Paumond, Differential Integral Equations16 (2003) 1039–1064; Pego and Quintero, Physica D132 (1999) 476–496) that (BL) is linked to the Kadomtsev-Petviashvili equation for almost one-dimensional waves propagating in one direction. We study here numerically the link between (KP) and (BL) and we point out the coupling effects emerging by considering two solitary waves propagating in two opposite directions.

Keywords

Benney-LukeKadomtsev-Petviashvilispectral methodlong wave limit.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 38 , Issue 3 , May 2004 , pp. 419 - 436
Copyright
© EDP Sciences, SMAI, 2004

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References

Ablowitz, M.J. and Segur, H., On the evolution of packets of water waves. J. Fluid Mech. 92 (1979) 691–715. CrossRef
Alexander, J.C., Pego, R.L. and Sachs, R.L., On the transverse instability of solitary waves in the Kadomtsev-Petviashvili equation. Phys. Lett. A 226 (1997) 187–192. CrossRef
Ben Youssef, W. and Colin, T., Rigorous derivation of Korteweg-de Vries-type systems from a general class of nonlinear hyperbolic systems. ESAIM: M2AN 34 (2000) 873–911. CrossRef
Ben Youssef, W. and Lannes, D., The long wave limit for a general class of 2D quasilinear hyperbolic problems. Comm. Partial Differ. Equations 27 (2002) 979–1020. CrossRef
Benney, D.J. and Luke, J.C., On the interactions of permanent waves of finite amplitude. J. Math. Phys. 43 (1964) 309–313. CrossRef
Berger, K.M. and Milewski, P.A., The generation and evolution of lump solitary waves in surface-tension-dominated flows. SIAM J. Appl. Math. 61 (2002) 731–750 (electronic). CrossRef
J.L. Bona, T. Colin and D. Lannes, Long wave approximations for water waves. Preprint Université de Bordeaux I, U-03-22 (2003).
Craig, W., An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits. Comm. Partial Differ. Equations 10 (1985) 787–1003. CrossRef
Gallay, T. and Schneider, G., KP description of unidirectional long waves. The model case. Proc. Roy. Soc. Edinburgh Sect. A 131 (2001) 885–898. CrossRef
Kano, T. and Nishida, T., A mathematical justification for Korteweg-de Vries equation and Boussinesq equation of water surface waves. Osaka J. Math. 23 (1986) 389–413.
D. Lannes, Consistency of the KP approximation. Discrete Contin. Dyn. Syst. (suppl.) (2003) 517–525. Dynam. Syst. Differ. equations (Wilmington, NC, 2002).
Milewski, P.A. and Keller, J.B., Three-dimensional water waves. Stud. Appl. Math. 97 (1996) 149–166. CrossRef
Milewski, P.A. and Tabak, E.G., A pseudospectral procedure for the solution of nonlinear wave equations with examples from free-surface flows. SIAM J. Sci. Comput. 21 (1999) 1102–1114 (electronic). CrossRef
L. Paumond, Towards a rigorous derivation of the fifth order KP equation. Submitted for publication (2002).
Paumond, L., A rigorous link between KP and a Benney-Luke equation. Differential Integral Equations 16 (2003) 1039–1064.
Pego, R.L. and Quintero, J.R., Two-dimensional solitary waves for a Benney-Luke equation. Physica D 132 (1999) 476–496. CrossRef
Schneider, G. and Wayne, C.E., The long-wave limit for the water wave problem. I. The case of zero surface tension. Comm. Pure Appl. Math. 53 (2000) 1475–1535. 3.0.CO;2-V>CrossRef