×

zbMATH — the first resource for mathematics

On stationary solutions of the reduced Ostrovsky equation: periodic waves, compactons and compound solitons. (English) Zbl 1088.35531
Summary: Detailed analysis of solitary and periodic stationary solutions of the reduced Ostrovsky equation is presented. The full Ostrovsky equation \[ (u_{t} + c_{0}u_{x} + \alpha uu_{x} + \beta u_{xxx})_{x} = \gamma u \] was derived in 1978 for oceanic waves affected by Earth rotation, and its reduced form without small-scale dispersion, \[ (u_{t} + c_{0}u_{x} + \alpha uu_{x})_{x} = \gamma u, \] was pointed out in his original paper and then, derived later by many authors in different physical contexts. However, some aspects of stationary solutions to the last equation remained unclear so far. An attempt to introduce clarity to this problem is undertaken in this paper.

MSC:
35Q35 PDEs in connection with fluid mechanics
86A05 Hydrology, hydrography, oceanography
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Ablowitz, M.J.; Segur, H., Solitons and the inverse scattering transform, (1981), SIAM Philadelphia · Zbl 0299.35076
[2] Benilov, E.S.; Pelinovsky, E.N., On the theory of wave propagation in nonlinear fluctuating media without dispersion, Zhetf, 94, 175-185, (1988), (in Russian) (Engl transl: Sov Phys JETP 1988;67:98-103)
[3] Boyd, J.P.; Chen, G.Y., Five regimes of the quasi-cnoidal, steadily translating waves of the rotation-modified Korteweg-de Vries (Ostrovsky) equation, Wave motion, 35, 141-155, (2002) · Zbl 1163.74322
[4] Fraunie, P.; Stepanyants, Y., Decay of cylindrical and spherical solitons in rotating media, Phys lett A, 293, 166-172, (2002) · Zbl 1002.74050
[5] Galkin, V.M.; Stepanyants, Yu.A., On the existence of stationary solitary waves in a rotating fluid, Prikl matamat i mekhanika, 55, 1051-1055, (1991), (in Russian) (Engl transl: J Appl Maths Mechs 1991;55:939-43) · Zbl 0786.76016
[6] Gilman, O.A.; Grimshaw, R.; Stepanyants, Yu.A., Approximate analytical and numerical solutions of the stationary Ostrovsky equation, Stud appl math, 95, 115-126, (1995) · Zbl 0843.76008
[7] Gilman, O.A.; Grimshaw, R.; Stepanyants, Yu.A., Dynamics of internal solitary waves in a rotating fluid, Dyn atm oceans, 23, 403-411, (1996), [Special issue. Stratified flows, pt. A]
[8] Grimshaw, R.H.J., Adjustment processes and radiating solitary waves in a regularised Ostrovsky equation, Eur J mech B/fluids, 18, 535-543, (1999) · Zbl 0929.76025
[9] Grimshaw, R.H.J.; He, J.-M.; Ostrovsky, L.A., Terminal damping of a solitary wave due to radiation in rotational systems, Stud appl math, 101, 197-210, (1998) · Zbl 1136.76334
[10] Grimshaw, R.; Ostrovsky, L.A.; Shrira, V.I.; Stepanyants, Yu.A., Long nonlinear surface and internal gravity waves in a rotating Ocean, Surv geophys, 19, 289-338, (1998)
[11] Konno, K.; Jeffrey, A., Some remarkable properties of two loop soliton solutions, J phys soc jpn, 52, 1-3, (1983)
[12] Leonov, A.I., The effect of Earth rotation on the propagation of weak nonlinear surface and internal long oceanic waves, Ann New York acad sci, 373, 150-159, (1981) · Zbl 0575.76033
[13] Morrison, A.J.; Parkes, E.J.; Vakhnenko, V.O., The N loop soliton solution of the Vakhnenko equation, Nonlinearity, 12, 1427-1437, (1999) · Zbl 0935.35129
[14] Muzylev SV. Nonlinear equatorial waves in the ocean. Digest of Reports, 2nd All-Union Congress of oceanographers, Sebastopol, USSR, vol. 2, 1982. p. 26-7 [in Russian].
[15] New AL, Esteban M. A new Korteweg-de Vries-type theory for internal solitary waves in a rotating continuously-stratified ocean. In: Pelinovsky EN, Talanov VI, editors. Near-surface ocean layer, vol. 1. Physical processes and remote sensing. Collection of scientific papers, Nizhny Novgorod, IAP RAS, 1999. p. 173-203.
[16] Nikitenkova, S.P.; Stepanyants, Yu.A.; Chikhladze, L.M., Solutions of the modified ostrovskii equation with cubic non-linearity, Prikl matamat i mekhanika, 64, 276-284, (2000), (in Russian) (Engl transl: J Appl Maths Mechs 2000;64:267-74) · Zbl 0982.35086
[17] Obregon, M.A.; Stepanyants, Yu.A., Oblique magneto-acoustic solitons in a rotating plasma, Phys lett A, 249, 315-323, (1998)
[18] Ostrovsky, L.A., Nonlinear internal waves in a rotating Ocean, Okeanologiya, 18, 181-191, (1978), (in Russian) (Engl transl: Oceanology 1978;18:119-25)
[19] Ostrovsky LA. Nonlinear internal waves in the ocean. In: Nonlinear waves. Proceedings of IV Gorky school on nonlinear waves. 1979. p. 292-329 [in Russian].
[20] Ostrovsky, L.A.; Stepanyants, Yu.A., Nonlinear surface and internal waves in rotating fluids, (), 106-128 · Zbl 0719.76021
[21] Parkes, E.J., The stability of solutions of vakhnenko’s equation, J phys A: math gen, 26, 6469-6475, (1993) · Zbl 0809.35086
[22] Ramirez, C.; Renouard, D.; Stepanyants, Yu.A., Propagation of cylindrical waves in a rotating fluid, Fluid dyn res, 30, 169-196, (2002)
[23] Rosenau, P.; Hyman, N.M., Compactons: solitons with finite wavelength, Phys rev lett, 70, 564-567, (1993) · Zbl 0952.35502
[24] Rybak, S.A.; Skrynnikov, Yu.I., A solitary wave in a uniformly curved thin rod, Akust zhurn, 36, 730-732, (1990), (in Russian) (Engl transl: A single wave in a thin rod of constant curvature. Sov Phys Acoustics 1990;36:410-1) · Zbl 0825.76729
[25] Vakhnenko VA. Periodic short-wave perturbations in a relaxing medium. Inst Geophys Ukrainian Acad Sci, Kiev, 1991. Preprint.
[26] Vakhnenko, V.A., Solitons in a nonlinear model medium, J phys A: math gen, 25, 4181-4187, (1992) · Zbl 0754.35132
[27] Vakhnenko, V.O., High-frequency soliton-like waves in a relaxing medium, J math phys, 40, 2011-2020, (1999) · Zbl 0946.35094
[28] Vakhnenko, V.O.; Parkes, E.J., The two loop soliton solution of the Vakhnenko equation, Nonlinearity, 11, 1457-1464, (1998) · Zbl 0914.35115
[29] Vakhnenko, V.O.; Parkes, E.J., The calculation of multi-soliton solutions of the Vakhnenko equation by the inverse scattering method, Chaos, solitons & fractals, 13, 1819-1826, (2002) · Zbl 1067.37106
[30] Vakhnenko, V.O.; Parkes, E.J.; Michtchenko, A.V., The Vakhnenko equation from the viewpoint of the inverse scattering method for the KdV equation, Int J diff eqns appl, 1, 429-449, (2000)
[31] Whitham, G.B., Linear and nonlinear waves, (1974), Wiley-Interscience Publication New York · Zbl 0373.76001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.