Several threads of the last 25 years' developments in nonlinear wave theory that stem from the classical Korteweg–de Vries (KdV) equation are surveyed. The focus is on various generalizations of the KdV equation which include higher-order nonlinearity, large-scale dispersion, and a non-local integral dispersion. We also discuss how relatively simple models can capture strongly nonlinear dynamics and how various modifications of the KdV equation lead to qualitatively new, non-trivial solutions and regimes of evolution observable in the laboratory and in nature. As the main physical example, we choose internal gravity waves in the ocean for which all these models are applicable and have genuine importance. We also briefly outline the authors' view of the future development of the chosen lines of nonlinear wave theory.
References
1.
2.
M. J.
Ablowitz
and H.
Segur
, Solitons and the Inverse Scattering Transform
(SIAM
, Philadelphia
, 1981
).3.
R. K.
Dodd
, J. C.
Eilbeck
, J. D.
Gibbon
, and H. C.
Morris
, Solitons and Nonlinear Wave Equations
(Academic Press
, London
, 1982
);A. C.
Newell
, Solitons in Mathematics and Physics
(SIAM, University of Arizona
, 1985
).4.
A. R.
Osborne
, Nonlinear Ocean Waves and the Inverse Scattering Transform
(Academic Press
, Boston
, 2010
).5.
R.
Grimshaw
, E.
Pelinovsky
, and O.
Poloukhina
, Nonlinear Processes Geophys.
9
, 221
(2002
);A.
Karczewska
, P.
Rozmej
, and E.
Infeld
, Phys. Rev. E
90
, 012907
(2014
).6.
A. S.
Fokas
and Q. M.
Liu
, Phys. Rev. Lett.
77
, 2347
(1996
).7.
T.
Kawahara
, J. Phys. Soc. Jpn.
33
, 260
(1972
).8.
K. A.
Gorshkov
, L. A.
Ostrovsky
, V. V.
Papko
, and A. S.
Pikovsky
, Phys. Lett. A
74
, 177
(1979
);K. A.
Gorshkov
, L. A.
Ostrovsky
, and Yu. A.
Stepanyants
, “Dynamics of soliton chains: From simple to complex and chaotic motions
,” in Long-Range Interactions, Stochasticity and Fractional Dynamics
, edited by A. C. J.
Luo
and V.
Afraimovich
(Springer
, 2010
), pp. 177
–218
.9.
J. P.
Boyd
, Physica D
48
, 129
(1991
).10.
V.
Voronovich
, I.
Sazonov
, and V.
Shrira
, J. Fluid Mech.
568
, 273
(2006
).11.
S.
Kichenassamy
and P. J.
Olver
, SIAM J. Math. Anal.
23
, 1141
(1992
);T. R.
Akylas
and R. H. J.
Grimshaw
, J. Fluid Mech.
242
, 279
(1992
);V. I.
Karpman
, Phys. Rev. E
47
, 2073
(1993
);E. S.
Benilov
, R.
Grimshaw
, and E. P.
Kuznetsova
, Physica D
69
, 270
(1993
);D. J.
Kaup
and B. A.
Malomed
, Physica D
184
, 153
(2003
).12.
R.
Grimshaw
, E.
Pelinovsky
, and T.
Talipova
, Physica D
132
, 40
(1999
).13.
T. L.
Perelman
, A. Kh.
Fridman
, and M. M.
El'yashevich
, Sov. Phys. JETP
39
, 643
(1974
).14.
A.
Slyunyaev
and E.
Pelinovskii
, JETP
89
, 173
(1999
);A. V.
Slyunyaev
, JETP
92
, 529
(2001
).15.
L. A.
Ostrovsky
and K. A.
Gorshkov
, “Perturbation theories for nonlinear waves
,” in Nonlinear Science at the Dawn of the XXI Century
, edited by P.
Christiansen
and M.
Soerensen
(Elsevier
, Amsterdam
, 2000
), pp. 47
–66
.16.
D. E.
Pelinovsky
and R. H. J.
Grimshaw
, Phys. Lett. A
229
, 165
(1997
).17.
K. A.
Gorshkov
, L. A.
Ostrovsky
, I. A.
Soustova
, and V. G.
Irisov
, Phys. Rev. E
69
, 016614
(2004
).18.
19.
V. E.
Zakharov
, Stud. Appl. Math.
122
, 219
(2009
).20.
E. N.
Pelinovsky
, E. G.
Shurgalina
, A. V.
Sergeeva
, T. G.
Talipova
, G. A.
El
, and R. H. J.
Grimshaw
, Phys. Lett. A
377
, 272
(2013
);E. N.
Pelinovsky
and E. G.
Shurgalina
, Radiophys. Quantum Electron.
57
, 737
(2015
).21.
E.
Pelinovsky
and A.
Sergeeva
, Eur. J. Mech. - B/Fluids
25
, 425
(2006
).22.
23.
A. M.
Kamchatnov
, Y.-H.
Kuo
, T.-C.
Lin
, T.-L.
Horng
, S.-C.
Gou
, R.
Clift
, G. A.
El
, and R. H. J.
Grimshaw
, Phys. Rev. E
86
, 036605
(2012
).24.
F. S.
Henyey
and A.
Hoering
, J. Geophys. Res. C
102
, 3323
(1997
).25.
J. H.
Lee
, I.
Lozovatsky
, S.-T.
Jang
, Ch. J.
Jang
, Ch. S.
Hong
, and H. J. S.
Fernando
, Geophys. Res. Lett.
33
, L18601
, doi: (2006
).26.
K. G.
Lamb
, O.
Polukhina
, T.
Talipova
, E.
Pelinovsky
, W.
Xiao
, and A.
Kurkin
, Phys. Rev. E
75
, 046306
(2007
).27.
R.
Grimshaw
, E.
Pelinovsky
, and T.
Talipova
, Surv. Geophys.
28
, 273
(2007
).28.
29.
R. H. J.
Grimshaw
, Eur. J. Mech. - B/Fluids
18
, 535
(1999
).30.
J. P.
Boyd
and G. Y.
Chen
, Wave Motion
35
, 141
(2002
).31.
R.
Grimshaw
and K.
Helfrich
, Stud. Appl. Math.
121
, 71
(2008
);R.
Grimshaw
and K.
Helfrich
, IMA J. Appl. Math.
77
, 326
(2012
).32.
33.
M. A.
Obregon
and Yu. A.
Stepanyants
, Phys. Lett. A
249
, 315
(1998
).34.
A. I.
Leonov
, Ann. N.Y. Acad. Sci.
373
, 150
(1981
);V. M.
Galkin
and Yu. A.
Stepanyants
, J. Appl. Math. Mech.
55
, 939
(1991
).35.
L. A.
Ostrovsky
and Yu. A.
Stepanyants
, “Nonlinear surface and internal waves in rotating fluids
,” in Nonlinear Waves 3, Proceedings of 1989 Gorky School on Nonlinear Waves
, edited by A. V.
Gaponov-Grekhov
, M. I.
Rabinovich
, and J.
Engelbrecht
(Springer-Verlag
, Berlin–Heidelberg
, 1990
), pp. 106
–128
.36.
Y. A.
Stepanyants
, Chaos, Solitons Fractals
28
, 193
(2006
);E. J.
Parkes
, Chaos, Solitons Fractals
31
, 602
(2007
).37.
V. O.
Vakhnenko
and E. J.
Parkes
, Chaos, Solitons Fractals
13
, 1819
(2002
).38.
R. H. J.
Grimshaw
, K. R.
Helfrich
, and E. R.
Johnson
, Stud. Appl. Math.
129
, 414
(2012
).39.
D.
Farmer
, Q.
Li
, and J.-H.
Park
, Atmos.-Ocean
47
, 267
(2009
);Q.
Li
and D. M.
Farmer
, J. Phys. Oceanogr.
41
, 1345
(2011
).40.
O. A.
Gilman
, R.
Grimshaw
, and Yu. A.
Stepanyants
, Stud. Appl. Math.
95
, 115
(1995
).41.
G.-Y.
Chen
and J. P.
Boyd
, Physica D
155
, 201
(2001
).42.
R. H. J.
Grimshaw
, J.-M.
He
, and L. A.
Ostrovsky
, Stud. Appl. Math.
101
, 197
(1998
).43.
O. A.
Gilman
, R.
Grimshaw
, and Yu. A.
Stepanyants
, Dyn. Atmos. Oceans
23
, 403
(1996
).44.
D.
Yagi
and T.
Kawahara
, Wave Motion
34
, 97
(2001
).45.
R. H. J.
Grimshaw
, K. R.
Helfrich
, and E. R.
Johnson
, Phys. Fluids
25
, 056602
(2013
).46.
M.
Obregon
, N.
Raj
, and Y.
Stepanyants
, “Numerical study of nonlinear wave processes by means of discrete chain models
,” in Proceedings of 4th International Conference on Computational Methods (ICCM2012)
, 25–27 November 2012
, Gold Coast, Australia
, see www.ICCM-2012.org.47.
S. A.
Kozlov
and C. V.
Sazonov
, JETP
84
, 221
(1997
);S. P.
Nikitenkova
, Yu. A.
Stepanyants
, and L. M.
Chikhladze
, J. Appl. Math. Mech.
64
, 267
(2000
);A. A.
Balakin
, A. G.
Litvak
, V. A.
Mironov
, and S. A.
Skobelev
, JETP
104
, 363
(2007
).48.
E. R.
Johnson
and R. H. J.
Grimshaw
, Phys. Rev. E
88
, 021201
(2013
).49.
T.
Gerkema
, J. Mar. Res.
54
, 421
(1996
);R.
Grimshaw
, E.
Pelinovsky
, Y.
Stepanyants
, and T.
Talipova
, Mar. Freshwater Res.
57
, 265
(2006
).50.
P.
Holloway
, E.
Pelinovsky
, and T.
Talipova
, J. Geophys. Res. C
104
, 18,333
(1999
).51.
R. H. J.
Grimshaw
, C.
Guo
, K. R.
Helfrich
, and V.
Vlasenko
, J. Phys. Oceanogr.
44
, 1116
(2014
).52.
R. R.
Long
, Tellus
5
, 42
(1953
).53.
C. J.
Amick
and R. E. L.
Turner
, Trans. Am. Math. Soc.
298
, 431
(1986
);R. E. L.
Turner
and J.-M.
Vanden-Broeck
, Phys. Fluids
31
, 2486
(1988
).54.
W. A. B.
Evans
and M. J.
Ford
, Phys. Fluids
8
, 2032
(1996
).55.
H.
Sandström
and C.
Quon
, Fluid Dyn. Res.
11
, 119
(1993
);N.
Zahibo
, A.
Slunyaev
, T.
Talipova
, E.
Pelinovsky
, A.
Kurkin
, and O.
Polukhina
, Nonlinear Processes Geophys.
14
, 247
(2007
).56.
L. A.
Ostrovsky
and J.
Grue
, Phys. Fluids
15
, 2934
(2003
).57.
L. A.
Ostrovsky
and K. R.
Helfrich
, Nonlinear Processes Geophys.
18
, 91
(2011
).58.
G. B.
Whitham
, Proc. R. Soc. London, Ser. A
299
, 6
(1967
).59.
A. E.
Green
and P. M.
Naghdi
, J. Fluid Mech.
78
, 237
(1976
).60.
M.
Miyata
, “Long internal waves of large amplitude
,” in Nonlinear Water Waves
, edited by K.
Horikawa
and H.
Maruo
(Springer-Verlag
, 1988
), pp. 399
–406
.61.
W.
Choi
and R.
Camassa
, J. Fluid Mech.
396
, 1
(1999
).62.
W.
Choi
and R.
Camassa
, Phys. Rev. Lett.
77
, 1759
(1996
).63.
W.
Choi
, Phys. Fluids
18
, 036601
(2006
).64.
A. G.
Voronovich
, J. Fluid Mech.
474
, 85
(2003
).65.
L. A.
Ostrovsky
and Yu. A.
Stepanyants
, Rev. Geophys.
27
, 293
, doi: (1989
);J.
Apel
, L. A.
Ostrovsky
, Y. A.
Stepanyants
, and J. F.
Lynch
, J. Acoust. Soc. Am.
121
, 695
(2007
).
[PubMed]
66.
T. P.
Stanton
and L. A.
Ostrovsky
, Geophys. Res. Lett.
25
, 2695
, doi: (1998
).67.
V.
Vlasenko
, N.
Stashchuk
, and K.
Hutter
, Baroclinic Tides: Theoretical Modeling and Observational Evidence
(Cambridge University Press
, Cambridge
, 2005
).68.
R. H. J.
Grimshaw
, in Without Bonds: A Scientific Canvas of Nonlinearity and Complex Dynamics (Springer
, 2013), pp. 317–334;A.
Alias
, R. H. J.
Grimshaw
, and K. R.
Khusnutdinova
, Phys. Fluids
26
, 126603
(2014
).69.
M.
Onorato
, D.
Ambrosi
, A.
Osborne
, and M.
Serio
, Phys. Fluids
15
, 3871
(2003
);A. J.
Whitefield
and E. R.
Johnson
, Chaos
25
, 023109
(2014
).70.
C. F. F.
Karney
, A.
Sen
, and F. Y. F.
Chu
, Phys Fluids
22
, 940
(1979
);O. B.
Gorbacheva
and L. A.
Ostrovsky
, Physica D
8
, 223
(1983
);S. P.
Nikitenkova
, N.
Raj
, and Y. A.
Stepanyants
, Commun. Nonlinear Sci. Numer. Simul.
20
, 731
(2015
).71.
S.
Badulin
, V.
Shrira
, C.
Kharif
, and M.
Ioualalen
, J. Fluid Mech.
303
, 297
(1995
).72.
B. S.
Kerner
and V. V.
Osipov
, Autosolitons: A New Approach to Problems of Self-Organization and Turbulence
(Springer
, 1994
).73.
S.
Novikov
, S. V.
Manakov
, L. P.
Pitaevskii
, and V. E.
Zakharov
, Theory of Solitons: The Inverse Scattering Method
(Springer
, 1984
).74.
L. A.
Ostrovsky
, Asymptotic Perturbation Theory of Waves
(Imperial College Press
, 2015
).© 2015 AIP Publishing LLC.
2015
AIP Publishing LLC
You do not currently have access to this content.
