High-Order Nonlinear Schrödinger Equation for the Envelope of Slowly Modulated Gravity Waves on the Surface of Finite-Depth Fluid and its Quasi-Soliton Solutions
DOI:
https://doi.org/10.15407/ujpe59.12.1201Keywords:
gravity waves, finite depth, slow modulations, wave envelope, wave envelopemultiple-scale expansions, nonlinear Schrödinger equationAbstract
We consider the high-order nonlinear Schrödinger 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 Schrödinger 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).
Downloads
Published
How to Cite
Issue
Section
License
Copyright Agreement
License to Publish the Paper
Kyiv, Ukraine
The corresponding author and the co-authors (hereon referred to as the Author(s)) of the paper being submitted to the Ukrainian Journal of Physics (hereon referred to as the Paper) from one side and the Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, represented by its Director (hereon referred to as the Publisher) from the other side have come to the following Agreement:
1. Subject of the Agreement.
The Author(s) grant(s) the Publisher the free non-exclusive right to use the Paper (of scientific, technical, or any other content) according to the terms and conditions defined by this Agreement.
2. The ways of using the Paper.
2.1. The Author(s) grant(s) the Publisher the right to use the Paper as follows.
2.1.1. To publish the Paper in the Ukrainian Journal of Physics (hereon referred to as the Journal) in original language and translated into English (the copy of the Paper approved by the Author(s) and the Publisher and accepted for publication is a constitutive part of this License Agreement).
2.1.2. To edit, adapt, and correct the Paper by approval of the Author(s).
2.1.3. To translate the Paper in the case when the Paper is written in a language different from that adopted in the Journal.
2.2. If the Author(s) has(ve) an intent to use the Paper in any other way, e.g., to publish the translated version of the Paper (except for the case defined by Section 2.1.3 of this Agreement), to post the full Paper or any its part on the web, to publish the Paper in any other editions, to include the Paper or any its part in other collections, anthologies, encyclopaedias, etc., the Author(s) should get a written permission from the Publisher.
3. License territory.
The Author(s) grant(s) the Publisher the right to use the Paper as regulated by sections 2.1.1–2.1.3 of this Agreement on the territory of Ukraine and to distribute the Paper as indispensable part of the Journal on the territory of Ukraine and other countries by means of subscription, sales, and free transfer to a third party.
4. Duration.
4.1. This Agreement is valid starting from the date of signature and acts for the entire period of the existence of the Journal.
5. Loyalty.
5.1. The Author(s) warrant(s) the Publisher that:
– he/she is the true author (co-author) of the Paper;
– copyright on the Paper was not transferred to any other party;
– the Paper has never been published before and will not be published in any other media before it is published by the Publisher (see also section 2.2);
– the Author(s) do(es) not violate any intellectual property right of other parties. If the Paper includes some materials of other parties, except for citations whose length is regulated by the scientific, informational, or critical character of the Paper, the use of such materials is in compliance with the regulations of the international law and the law of Ukraine.
6. Requisites and signatures of the Parties.
Publisher: Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine.
Address: Ukraine, Kyiv, Metrolohichna Str. 14-b.
Author: Electronic signature on behalf and with endorsement of all co-authors.
.png){kind=small}
