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Interfacial waves with free-surface boundary conditions: An approach via a model equation. (English) Zbl 1037.76010
From the summary: The title interfacial-wave problem is reduced to a system of ordinary differential equations by using a classical perturbation method, which takes into consideration the possible resonance between short waves and “slow” solitary waves. In the past, classical Korteweg-de Vries type models have been derived but cannot deal with the resonance. All solutions of the new system of model equations, including classical as well as generalized solitary waves, are constructed. The domain of validity of the model is discussed as well. It is also shown that fronts connecting two conjugate states cannot occur for “fast” waves. For “slow” waves, fronts exist but they have ripples in their tails.

MSC:
76B55 Internal waves for incompressible inviscid fluids
76B25 Solitary waves for incompressible inviscid fluids
35Q51 Soliton equations
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[1] Sun, S.M., Existence of a generalized solitary wave solution for water with positive bond number less than 1/3, J. math. anal. appl., 156, 471-504, (1991) · Zbl 0725.76028
[2] Beale, J.T., Exact solitary water waves with capillary ripples at infinity, Comm. pure appl. math., 44, 211-257, (1991) · Zbl 0727.76019
[3] Iooss, G.; Kirchgässner, K., Water waves for small surface tension: an approach via normal form, Proc. roy. soc. Edinburgh A, 122, 267-299, (1992) · Zbl 0767.76004
[4] Sun, S.M.; Shen, M.C., Exponentially small estimate for the amplitude of capillary ripples of a generalized solitary wave, J. math. anal. appl., 172, 533-566, (1993) · Zbl 0772.76010
[5] Vanden-Broeck, J.-M., Elevation solitary waves with surface tension, Phys. fluids A, 3, 2659-2663, (1991) · Zbl 0746.76016
[6] Yang, T.S.; Akylas, T.R., Weakly nonlocal gravity – capillary solitary waves, Phys. fluids, 8, 1506-1514, (1996) · Zbl 1087.76021
[7] Benilov, E.S.; Grimshaw, R.; Kuznetsova, E.P., The generation of radiating waves in a singularly-perturbed korteweg – de Vries equation, Physica D, 69, 270-278, (1993) · Zbl 0791.35119
[8] Akylas, T.R.; Grimshaw, R.H.J., Solitary internal waves with oscillatory tails, J. fluid mech., 242, 279-298, (1992) · Zbl 0754.76014
[9] Vanden-Broeck, J.-M.; Turner, R.E.L., Long periodic internal waves, Phys. fluids A, 4, 1929-1935, (1992) · Zbl 0782.76020
[10] Peters, A.S.; Stoker, J.J., Solitary waves in liquids having non-constant density, Comm. pure appl. math., 13, 115-164, (1960) · Zbl 0090.43301
[11] Kakutani, T.; Yamasaki, N., Solitary waves on a two-layer fluid, J. phys. soc. jpn., 45, 674-679, (1978)
[12] Sun, S.M.; Shen, M.C., Exact theory of generalized solitary waves in a two-layer liquid in the absence of surface tension, J. math. anal. appl., 180, 245-274, (1993) · Zbl 0796.76019
[13] Moni, J.N.; King, A.C., Interfacial solitary waves, Q. J. mech. appl. math., 48, 21-38, (1995) · Zbl 0821.76015
[14] Michallet, H.; Dias, F., Numerical study of generalized interfacial solitary waves, Phys. fluids, 11, 1502-1511, (1999) · Zbl 1147.76461
[15] E. Părău, F. Dias, Interfacial periodic waves of permanent form with free-surface boundary conditions, J. Fluid Mech., to appear.
[16] Walker, L.R., Interfacial solitary waves in a two-fluid medium, Phys. fluids, 16, 1796-1804, (1973)
[17] Michallet, H.; Barthélemy, E., Experimental study of interfacial solitary waves, J. fluid mech., 366, 159-177, (1998) · Zbl 0967.76513
[18] Akylas, T.R.; Yang, T.-S., On short-scale oscillatory tails of long-wave disturbances, Stud. appl. math., 94, 1-20, (1995) · Zbl 0823.35155
[19] G. Iooss, M. Adelmeyer, Topics in Bifurcation Theory and Applications, World Scientific, Singapore, 1992. · Zbl 0833.34001
[20] Bona, J.L.; Sachs, R.L., The existence of internal solitary waves in a two fluid system near the KdV limit, Geophys. astrophys. fluid dyn., 48, 25-51, (1989) · Zbl 0708.76136
[21] Elphick, C.; Tirapegui, E.; Brachet, M.; Coullet, P.; Iooss, G., A simple global characterization for normal form of singular vector fields, Physica D, 29, 95-127, (1987) · Zbl 0633.58020
[22] H. Michallet, F. Dias, Nonlinear resonance between short and long waves, in: Proceedings of the Ninth International Offshore and Polar Engineering Conference, Brest, France, 1999, pp. 193-198.
[23] Lamb, K.G., Conjugate flows for a three-layer fluid, Phys. fluids, 12, 2169-2185, (2000) · Zbl 1184.76308
[24] Pennell, S., A note on exact relations for solitary waves, Phys. fluids A, 2, 281-284, (1990) · Zbl 0696.76122
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