High-Order Nonlinear Schr¨odinger Equation for the Envelope of Slowly Modulated Gravity Waves on the Surface of Finite-Depth Fluid and its Quasi-Soliton Solutions

  • I. S. Gandzha Institute of Physics, Nat. Acad. of Sci. of Ukraine
  • Yu. V. Sedletsky Institute of Physics, Nat. Acad. of Sci. of Ukraine
  • D. S. Dutykh Universit´e de Savoie Mont Blanc
Keywords: nonlinear Schr¨odinger equation, gravity waves, finite depth, slow modulations, wave envelope, wave envelopemultiple-scale expansions

Abstract

We consider the high-order nonlinear Schr¨odinger equation derived earlier by Sedletsky [Ukr. J. Phys. 48(1), 82 (2003)] for the first-harmonic envelope of slowly modulated gravity waves on the surface of finite-depth irrotational, inviscid, and incompressible fluid with flat bottom. This equation takes into account the third-order dispersion and cubic nonlinear dispersive terms. We rewrite this equation in dimensionless form featuring only one dimensionless parameter kℎ, where k is the carrier wavenumber and ℎ is the undisturbed fluid depth. We show that one-soliton solutions of the classical nonlinear Schr¨odinger equation are transformed into quasi-soliton solutions with slowly varying amplitude when the high-order terms are taken into consideration. These quasi-soliton solutions represent the secondary modulations of gravity waves.

References

M.J. Ablowitz, J. Hammack, D. Henderson, and C.M. Schober, PRL 84(5), 887 (2000).

https://doi.org/10.1103/PhysRevLett.84.887

M.J. Ablowitz, J. Hammack, D. Henderson, and C.M. Schober, Physica D 152–153, 416 (2001).

https://doi.org/10.1016/S0167-2789(01)00183-X

T.R. Akylas, J. Fluid Mech. 198, 387 (1989).

https://doi.org/10.1017/S0022112089000182

U. Bandelow and N. Akhmediev, PRE 86, 026606 (2012).

https://doi.org/10.1103/PhysRevE.86.026606

D.J. Benney and A.C. Newell, J. Math. Phys. 46, 133 (1967).

https://doi.org/10.1002/sapm1967461133

D.J. Benney and G.J. Roskes, Stud. Appl. Math. 48(4), 377 (1969).

https://doi.org/10.1002/sapm1969484377

V. Bespalov and V. Talanov, JETP Lett. 3, 307 (1966).

U. Brinch-Nielsen and I.G. Jonsson, Wave Motion 8, 455 (1986).

https://doi.org/10.1016/0165-2125(86)90030-2

A. Chabchoub, N.P. Hoffmann, and N. Akhmediev, PRL 106, 204502 (2011).

https://doi.org/10.1103/PhysRevLett.106.204502

M. Chen, J.M. Nash, and C.E. Patton, J. Appl. Phys. 73, 3906 (1993).

https://doi.org/10.1063/1.352878

D. Clamond, M. Francius, J. Grue, and C. Kharif, Eur. J. Mech. B/Fluids 25, 536 (2006).

https://doi.org/10.1016/j.euromechflu.2006.02.007

S.H. Crutcher, A. Osei, and A. Biswas, Optics & Laser Techn. 44, 1156 (2012).

https://doi.org/10.1016/j.optlastec.2011.09.027

A. Davey and K. Stewartson, Proc. R. Soc. Lond. A 338, 101 (1974).

https://doi.org/10.1098/rspa.1974.0076

S. Debsarma and K.P. Das, Phys. Fluids 17, 104101 (2005).

https://doi.org/10.1063/1.2046714

F. Dias and C. Kharif, Annu. Rev. Fluid Mech. 31, 301 (1999).

https://doi.org/10.1146/annurev.fluid.31.1.301

R.K. Dodd, J.C. Eilbeck, J.D. Gibbon, and H.C. Morris, Solitons and Nonlinear Wave Equations (Academic Press, London, 1984).

A.I. Dyachenko and V.E. Zakharov, JETP Lett. 93(12), 701 (2011).

https://doi.org/10.1134/S0021364011120058

A.I. Dyachenko and V.E. Zakharov, Eur. J. Mech. B/Fluids 32, 17 (2012).

https://doi.org/10.1016/j.euromechflu.2011.08.001

K.B. Dysthe, Proc. R. Soc. Lond. A 369, 105 (1979).

https://doi.org/10.1098/rspa.1979.0154

F. Fedele and D. Dutykh, JETP Lett. 94(12), 840 (2011).

https://doi.org/10.1134/S0021364011240039

F. Fedele and D. Dutykh, JETP Lett. 95(12), 622 (2012).

https://doi.org/10.1134/S0021364012120041

F. Fedele and D. Dutykh, J. Fluid Mech. 712, 646 (2012).

https://doi.org/10.1017/jfm.2012.447

F. Fedele and D. Dutykh, ArXiv:1110.3605 (2012).

I.S. Gandzha, Ukr. J. Phys. Rev. 8(1), 3 (2013) [in Ukrainian].

O. Gramstad and K. Trulsen, Phys. Fluids 23, 062102 (2011).

https://doi.org/10.1063/1.3598316

O. Gramstad and K. Trulsen, J. Fluid Mech. 670, 404 (2011).

https://doi.org/10.1017/S0022112010005355

O. Gramstad, J. Fluid Mech. 740, 254 (2014).

https://doi.org/10.1017/jfm.2013.649

C. Gilson, J. Hietarinta, J. Nimmo, and Y. Ohta, PRE 68, 016614 (2003).

https://doi.org/10.1103/PhysRevE.68.016614

R.H.J. Grimshaw and S.Y. Annenkov, Stud. Appl. Math. 126, 409 (2011).

https://doi.org/10.1111/j.1467-9590.2010.00508.x

H. Hasimoto and H. Ono, J. Phys. Soc. Jpn. 33, 805 (1972).

https://doi.org/10.1143/JPSJ.33.805

S.J. Hogan, Proc. R. Soc. Lond. A 402, 359 (1985).

https://doi.org/10.1098/rspa.1985.0122

S.J. Hogan, Phys. Fluids 29, 3479 (1986).

https://doi.org/10.1063/1.865816

E. Infeld and G. Rowlands, Nonlinear Waves, Solitons and Chaos (Cambridge Univ. Press, Cambridge, 2000).

https://doi.org/10.1017/CBO9781139171281

P.A.E.M. Janssen, J. Fluid Mech. 126, 1 (1983).

https://doi.org/10.1017/S0022112083000014

R.S. Johnson, Proc. R. Soc. Lond. A 357, 131 (1977).

https://doi.org/10.1098/rspa.1977.0159

R.S. Johnson, J. Fluid Mech. 455, 63 (2002).

https://doi.org/10.1017/S0022112001007224

T. Kakutani and K. Michihiro, J. Phys. Soc. Jpn. 52, 4129 (1983).

https://doi.org/10.1143/JPSJ.52.4129

V.I. Karpman, J.J. Rasmussen, and A.G. Shagalov, PRE 64, 026614 (2001).

https://doi.org/10.1103/PhysRevE.64.026614

C. Kharif and E. Pelinovsky, Eur. J. Mech. B/Fluids 22, 603 (2003).

https://doi.org/10.1016/j.euromechflu.2003.09.002

E. Lo and C.C. Mei, J. Fluid Mech. 150, 395 (1985).

https://doi.org/10.1017/S0022112085000180

V.P. Lukomski˘ı, JETP 81(2), 306 (1995).

V.P. Lukomsky and I.S. Gandzha, Ukr. J. Phys. 54(1-2), 207 (2009).

G.M. Muslu and H.A. Erbay, Math. Comput. Simul. 67, 581 (2005).

https://doi.org/10.1016/j.matcom.2004.08.002

M. Onorato, S. Residori, U. Bortolozzo, A. Montina, and F.T. Arecchi, Phys. Reports 528(2), 47 (2013).

https://doi.org/10.1016/j.physrep.2013.03.001

E.J. Parkes, J. Phys. A: Math. Gen. 20, 2025 (1987).

https://doi.org/10.1088/0305-4470/20/8/021

G.J. Roskes, Phys. Fluids 20, 1576 (1977).

https://doi.org/10.1063/1.862025

Yu.V. Sedletsky, Ukr. J. Phys. 48(1), 82 (2003) [in Ukrainian].

Yu.V. Sedletsky, JETP 97(1), 180 (2003).

https://doi.org/10.1134/1.1600810

I. Selezov, O. Avramenko, C. Kharif, and K. Trulsen, C. R. Mecanique 331, 197 (2003).

https://doi.org/10.1016/S1631-0721(03)00043-3

L. Shemer, A. Sergeeva, and A. Slunyaev, Phys. Fluids 22, 016601 (2010).

https://doi.org/10.1063/1.3290240

L. Shemer and L. Alperovich, Phys. Fluids 25, 051701 (2013).

https://doi.org/10.1063/1.4807055

A.V. Slunyaev, JETP 101(5), 926 (2005).

https://doi.org/10.1134/1.2149072

A.V. Slunyaev, JETP 109(4), 676 (2009).

https://doi.org/10.1134/S1063776109100148

A. Slunyaev, E. Pelinovsky, A. Sergeeva, A. Chabchoub, N. Hoffmann, M. Onorato, and N. Akhmediev, PRE 88, 012909 (2013).

https://doi.org/10.1103/PhysRevE.88.012909

A. Slunyaev, G.F. Clauss, M. Klein, and M. Onorato, Phys. Fluids 25, 067105 (2013).

https://doi.org/10.1063/1.4811493

M. Stiasnie, Wave Motion 6, 431 (1984).

https://doi.org/10.1016/0165-2125(84)90043-X

M. Stiasnie and L. Shemer, J. Fluid Mech. 143, 47 (1984).

https://doi.org/10.1017/S0022112084001257

J.J. Stoker, Water Waves: The Mathematical Theory with Applications (Wiley, New York, 1992).

https://doi.org/10.1002/9781118033159

G. Strang, SIAM J. Numer. Anal. 5(3), 506 (1968).

https://doi.org/10.1137/0705041

M.-Y. Su, Phys. Fluids 25(12), 2167 (1982).

https://doi.org/10.1063/1.863708

R. Thomas, C. Kharif, and M. Manna, Phys. Fluids 24, 127102 (2012).

https://doi.org/10.1063/1.4768530

K. Trulsen and K.B. Dysthe, Wave Motion 24, 281 (1996).

https://doi.org/10.1016/S0165-2125(96)00020-0

K. Trulsen, I. Kliakhandler, K.B. Dysthe, and M.G. Velarde, Phys. Fluids 12(10), 2432 (2000).

https://doi.org/10.1063/1.1287856

S.K. Turitsyn, B.G. Bale, and M.P. Fedoruk, Phys. Reports 521, 135 (2012).

https://doi.org/10.1016/j.physrep.2012.09.004

H. Yoshida, Phys. Lett. A 150, 262 (1990).

https://doi.org/10.1016/0375-9601(90)90092-3

H.C. Yuen and B.M. Lake, Phys. Fluids 18(8), 956 (1975).

https://doi.org/10.1063/1.861268

H. Yuen and B. Lake, Adv. Appl. Mech. 22, 229 (1982).

https://doi.org/10.1016/S0065-2156(08)70066-8

V.E. Zakharov, J. Appl. Mech. and Tech. Phys., 9(2), 190 (1968).

https://doi.org/10.1007/BF00913182

V.E. Zakharov and A.B. Shabat, JETP 34(1), 62 (1972).

V.E. Zakharov and E.A. Kuznetsov, JETP 86(5), 1035 (1998).

https://doi.org/10.1134/1.558551

V.E. Zakharov and L.A. Ostrovsky, Physica D 238, 540 (2009).

https://doi.org/10.1016/j.physd.2008.12.002

V.E. Zakharov and A.I. Dyachenko, Eur. J. Mech. B/Fluids 29, 127 (2010).

https://doi.org/10.1016/j.euromechflu.2009.10.003

V.E. Zakharov and E.A. Kuznetsov, Physics-Uspekhi 55(6), 535 (2012).

https://doi.org/10.3367/UFNe.0182.201206a.0569

Published
2018-10-28
How to Cite
Gandzha, I., Sedletsky, Y., & Dutykh, D. (2018). High-Order Nonlinear Schr¨odinger Equation for the Envelope of Slowly Modulated Gravity Waves on the Surface of Finite-Depth Fluid and its Quasi-Soliton Solutions. Ukrainian Journal of Physics, 59(12), 1201. https://doi.org/10.15407/ujpe59.12.1201
Section
Nonlinear processes