Numerical comparisons of two long-wave limit models
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 38 (2004) no. 3, p. 419-436

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 Equations 16 (2003) 1039-1064; Pego and Quintero, Physica D 132 (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.

DOI : https://doi.org/10.1051/m2an:2004020
Classification:  35L05,  35Q51,  35Q53,  65J15,  65M70
Keywords: Benney-Luke, Kadomtsev-Petviashvili, spectral method, long wave limit
@article{M2AN_2004__38_3_419_0,
     author = {Labb\'e, St\'ephane and Paumond, Lionel},
     title = {Numerical comparisons of two long-wave limit models},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {38},
     number = {3},
     year = {2004},
     pages = {419-436},
     doi = {10.1051/m2an:2004020},
     zbl = {1130.76324},
     mrnumber = {2075753},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2004__38_3_419_0}
}
Labbé, Stéphane; Paumond, Lionel. Numerical comparisons of two long-wave limit models. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 38 (2004) no. 3, pp. 419-436. doi : 10.1051/m2an:2004020. http://www.numdam.org/item/M2AN_2004__38_3_419_0/

[1] M.J. Ablowitz and H. Segur, On the evolution of packets of water waves. J. Fluid Mech. 92 (1979) 691-715. | Zbl 0413.76009

[2] J.C. Alexander, R.L. Pego and R.L. Sachs, On the transverse instability of solitary waves in the Kadomtsev-Petviashvili equation. Phys. Lett. A 226 (1997) 187-192. | Zbl 0962.35505

[3] W. Ben Youssef and T. Colin, Rigorous derivation of Korteweg-de Vries-type systems from a general class of nonlinear hyperbolic systems. ESAIM: M2AN 34 (2000) 873-911. | Numdam | Zbl 0962.35152

[4] W. Ben Youssef and D. Lannes, The long wave limit for a general class of 2D quasilinear hyperbolic problems. Comm. Partial Differ. Equations 27 (2002) 979-1020. | Zbl 1072.35572

[5] D.J. Benney and J.C. Luke, On the interactions of permanent waves of finite amplitude. J. Math. Phys. 43 (1964) 309-313. | Zbl 0128.44601

[6] K.M. Berger and P.A. Milewski, The generation and evolution of lump solitary waves in surface-tension-dominated flows. SIAM J. Appl. Math. 61 (2002) 731-750 (electronic). | Zbl 1136.76325

[7] J.L. Bona, T. Colin and D. Lannes, Long wave approximations for water waves. Preprint Université de Bordeaux I, U-03-22 (2003). | MR 2196497 | Zbl 1050.76006

[8] W. Craig, An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits. Comm. Partial Differ. Equations 10 (1985) 787-1003. | Zbl 0577.76030

[9] T. Gallay and G. Schneider, KP description of unidirectional long waves. The model case. Proc. Roy. Soc. Edinburgh Sect. A 131 (2001) 885-898. | Zbl 1015.76015

[10] T. Kano and T. Nishida, A mathematical justification for Korteweg-de Vries equation and Boussinesq equation of water surface waves. Osaka J. Math. 23 (1986) 389-413. | Zbl 0622.76021

[11] D. Lannes, Consistency of the KP approximation. Discrete Contin. Dyn. Syst. (suppl.) (2003) 517-525. Dynam. Syst. Differ. equations (Wilmington, NC, 2002). | Zbl 1066.35017

[12] P.A. Milewski and J.B. Keller, Three-dimensional water waves. Stud. Appl. Math. 97 (1996) 149-166. | Zbl 0860.35115

[13] P.A. Milewski and E.G. Tabak, 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). | Zbl 0953.65073

[14] L. Paumond, Towards a rigorous derivation of the fifth order KP equation. Submitted for publication (2002). | Zbl 1137.35424

[15] L. Paumond, A rigorous link between KP and a Benney-Luke equation. Differential Integral Equations 16 (2003) 1039-1064. | Zbl 1056.76014

[16] R.L. Pego and J.R. Quintero, Two-dimensional solitary waves for a Benney-Luke equation. Physica D 132 (1999) 476-496. | Zbl 0935.35139

[17] G. Schneider and C.E. Wayne, The long-wave limit for the water wave problem. I. The case of zero surface tension. Comm. Pure Appl. Math. 53 (2000) 1475-1535. | Zbl 1034.76011