Abstract
The method of multiple scales is used to derive the fourth-order nonlinear Schrödinger equation (NSEIV) that describes the amplitude modulations of the fundamental harmonic of Stokes waves on the surface of a medium-and large-depth (compared to the wavelength) fluid layer. The new terms of this equation describe the third-order linear dispersion effect and the nonlinearity dispersion effects. As the nonlinearity and the dispersion decrease, the equation uniformly transforms into the nonlinear Schrödinger equation for Stokes waves on the surface of a finite-depth fluid that was first derived by Hasimoto and Ono. The coefficients of the derived equation are given in an explicit form as functions of kh (h is the fluid depth, and k is the wave number). As kh tends to infinity, these coefficients transform into the coefficients of the NSEIV that was first derived by Dysthe for an infinite depth.
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References
V. E. Zakharov and E. A. Kuznetsov, Zh. Éksp. Teor. Fiz. 113, 1892 (1998) [JETP 86, 1035 (1998)].
A. Hasegawa and Y. Kodama, Solitons in Optical Communication (Oxford Univ. Press, London, 1995); Y. Kodama and A. Hasegawa, IEEE J. Quantum Electron. 23, 510 (1987).
E. M. Gromov and V. I. Talanov, Zh. Éksp. Teor. Fiz. 110, 137 (1996) [JETP 83, 73 (1996)].
K. B. Dysthe, Proc. R. Soc. London, Ser. A 369, 105 (1979); K. Trulsen and K. B. Dysthe, Wave Motion 24, 281 (1996); K. Trulsen and K. B. Dysthe, J. Fluid Mech. 352, 359_(1997); K. Trulsen, I. Kliakhandler, K. B. Dysthe, and M. G. Velarde, Phys. Fluids 12, 2432 (2000).
V. P. Lukomskii, Zh. Éksp. Teor. Fiz. 108, 567 (1995) [JETP 81, 306 (1995)].
F. Dias and K. Kharif, Annu. Rev. Fluid Mech. 31, 301 (1999).
J. R. Stocker and D. H. Peregrine, J. Fluid Mech. 399, 335 (1999).
U. Brinch-Nielsen and I. G. Jonsson, Wave Motion 8, 455 (1986).
D. Clamond and J. P. Germain, Eur. J. Mech. B (Fluids) 18, 67 (1999).
Yu. G. Rapoport, C. E. Zaspel, J. H. Mantha, and V. V. Grimalsky, Phys. Rev. B 65, 024423-1 (2002).
V. P. Lukoms’kii and Yu. V. Sedlets’kii, Zh. Fiz. Dosl. 5, 107 (2001).
A. G. Litvak and V. I. Talanov, Izv. Vyssh. Uchebn. Zaved., Radiofiz. 10, 539 (1967).
V. I. Karpman, J. J. Rasmussen, and A. G. Shagalov, Phys. Rev. E 64, 026614-1 (2001).
A. Mahalingam and K. Porsezian, Phys. Rev. E 64, 046608-1 (2001).
V. E. Zakharov, Prikl. Mekh. Tekh. Fiz., No. 2, 86 (1968).
H. C. Yuen and B. M. Lake, Phys. Fluids 18, 956 (1975); H. Yuen and B. M. Lake, Nonlinear Dynamics of Deep-Water Gravity Waves (Academic, New York, 1982; Mir, Moscow, 1987).
D. J. Benney and G. J. Roskes, Stud. Appl. Math. 48, 377 (1969).
V. E. Zakharov and V. G. Kharitonov, Prikl. Mekh. Tekh. Fiz., No. 5, 45 (1970).
V. H. Chu and C. C. Mei, J. Fluid. Mech. 41, 873 (1970).
H. Hasimoto and H. Ono, J. Phys. Soc. Jpn. 33, 805 (1972).
M. Stiasnie and L. Shemer, J. Fluid Mech. 143, 47 (1984).
A. Devey and K. Stewartson, Proc. R. Soc. London, Ser. A 338, 101 (1974).
T. Kawahara, J. Phys. Soc. Jpn. 38, 265 (1975).
P. Liu and M. Dingemans, Wave Motion 11, 41 (1989).
P. Christodoulides and F. Dias, Phys. Fluids 7, 3013 (1995).
M. Stiassnie, Wave Motion 6, 431 (1984).
P. A. E. M. Janssen, J. Fluid Mech. 126, 1 (1983).
S. J. Hogan, Proc. R. Soc. London, Ser. A 402, 359 (1985).
E. Lo and C. C. Mei, J. Fluid Mech. 150, 395 (1985).
M. P. Tulin and T. Waseda, J. Fluid Mech. 378, 197 (1999).
L. Shemer, Haiyuing Jiao, E. Kit, and Y. Agnon, J. Fluid Mech. 427, 107 (2001).
M. J. Ablowitz, J. Hammack, D. Henderson, and C. M. Schober, Phys. Rev. Lett. 84, 887 (2000); Physica D (Amsterdam) 152–153, 416 (2001).
Bhattacharyya Sudebi and K. P. Das, J. Aust. Math. Soc. B, Appl. Math. 39, 214 (1997).
E. Kit and L. Shemer, J. Fluid Mech. 450, 201 (2002).
G. B. Whitham, J. Fluid Mech. 27, 399 (1967).
T. B. Benjamin, Proc. R. Soc. London, Ser. A 299, 59 (1967).
M. Tajiri and Y. Watanabe, Phys. Rev. E 57, 3510 (1998); M. Tajiri and T. Arai, Phys. Rev. E 60, 2297 (1999); M. Tajiri, T. Arai, and K. Takenahi, Phys. Rev. E 64, 056622-1 (2001).
R. Grimshaw, D. Pelinovsky, E. Pelinovsky, and T. Talipova, Physica D (Amsterdam) 159, 35 (2001).
R. Kh. Zeitunyan, Usp. Fiz. Nauk 165, 1403 (1995) [Phys. Usp. 38, 1333 (1995)].
R. S. Johnson, Proc. R. Soc. London, Ser. A 357, 131 (1977).
T. Kakutani and K. Michihiro, J. Phys. Soc. Jpn. 52, 4129 (1983).
L. Shemer, E. Kit, Haiyuing Jiao, and O. Eitan, J. Waterway, Port, Coastal, Ocean Eng. 124, 320 (1998).
V. E. Zakharov and E. A. Kuznetsov, Usp. Fiz. Nauk 167, 1137 (1997) [Phys. Usp. 40, 1087 (1997)].
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Translated from Zhurnal Éksperimental’no\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{l}\) i Teoretichesko\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{l}\) Fiziki, Vol. 124, No. 1, 2003, pp. 200–213.
Original Russian Text Copyright © 2003 by Sedletsky.
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Sedletsky, Y.V. The fourth-order nonlinear Schrödinger equation for the envelope of Stokes waves on the surface of a finite-depth fluid. J. Exp. Theor. Phys. 97, 180–193 (2003). https://doi.org/10.1134/1.1600810
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DOI: https://doi.org/10.1134/1.1600810