Skip Navigation Links
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


On Stationary Solutions of the Reduced Gardner–Ostrovsky Equation

Discontinuity, Nonlinearity, and Complexity 3(4) (2014) 445--456 | DOI:10.5890/DNC.2014.12.007

Maria Obregon$^{1}$; Yury Stepanyants$^{2}$,$^{3}$

$^{1}$ E.T.S. Ingeniería Industrial, University of Malaga, Dr Ortiz Ramos s/n, 29071, Malaga, Spain

$^{2}$ Nizhny Novgorod State Technical University n.a. R.E. Alexeev, 24 Minin St., Nizhny Novgorod, 603950, Russia

$^{3}$ University of Southern Queensland, Faculty of Health, Engineering and Sciences, West St., Toowoomba, QLD,4350, Australia

Download Full Text PDF



The detailed analysis of stationary solutions of the reduced Gardner– Ostrovsky (GO) equation is presented. The GO equation (ut + c0ux + αuux + α1u2ux + βuxxx )x = γu is the popular model for the description of large-amplitude internal oceanic waves affected by Earth’s rotation. Its reduced version in which the small-scale dispersion is neglected ( β = 0 ) is used when very long internal waves are considered. The equation is also applicable to other types of nonlinear waves in various media (plasma, optical media, relaxing media, etc.) when the large-scale dispersion ∼ γ plays a dominant role in comparison with the small-scale dispersion ∼ β. Balancing the nonlinear effect such dispersion gives rise to existence of stationary waves, both periodic and non-periodic. It is shown that only smooth periodic waves make physical sense. Systematic analysis of stationary solutions to the GO equation and their categorisation is presented.


Research of Maria Obregon was supported by the Ministerio de Ciencia e Innovaci´on of Spain, Grant No ENE2010-16851, and research of Yury Stepanyants was supported by the State Project of Russian Federation in the field of scientific activity (Task5.30.2014/K). The authors are grateful to Prof. R. Fernandez-Feria for his critical remarks and comments, as well as to two anonymous Referees for their constructive criticism.


  1. [1]  Ostrovsky, L.A. (1978), Nonlinear internal waves in a rotating ocean, Okeanologiya 18, 181-191 (in Russian). (Engl. Transl. Oceanology, 1978, 18, 119-125.)
  2. [2]  Stepanyants, Y.A. (2006), On stationary solutions of the reduced Ostrovsky equation: Periodic waves, compactons and compound solitons, Chaos, Solitons and Fractals, 28, 193-204.
  3. [3]  Ostrovsky, L.A. and Stepanyants Yu.A. (1990), Nonlinear surface and internal waves in rotating fluids, in Nonlinear Waves 3. Proc. 1989 Gorky School on Nonlinear Waves, eds. A.V. Gaponov-Grekhov, M.I. Rabinovich and J. Engelbrecht. Springer-Verlag, Berlin-Heidelberg, 106-128.
  4. [4]  Grimshaw, R., Ostrovsky, L.A., Shrira, V.I., and Stepanyants, Yu.A. (1998), Long nonlinear surface and internal gravity waves in a rotating ocean, Surveys in Geophys, 19, 289-338.
  5. [5]  Ostrovsky, L.A. and Stepanyants, Yu.A. (1989), Do internal solitons exist in the ocean? Reviews of Geophysics 27(3), 293-310.
  6. [6]  Apel, J., Ostrovsky, L.A., Stepanyants, Y.A., and Lynch, J.F. (2007), Internal solitons in the ocean and their effect on underwater sound, The Journal of the Acoustical Society of America, 121(2), 695-722.
  7. [7]  Holloway, P., Pelinovsky. E., and Talipova, T. (1999), A generalised Korteweg-de Vries model of internal tide transformation in the coastal zone, Journal of Geophysical Research, 104, 18,333-18,350.
  8. [8]  Ablowitz, M.J. and Segur, H. (1981), Solitons and the Inverse Scattering Transform, SIAM, Philadelphia.
  9. [9]  Muzylev, S.V. (1982), Nonlinear equatorial waves in the ocean, in Digest of Reports, 2-nd All-Union Congress of Oceanographers, Sebastopol, USSR 2, 26-27 (in Russian).
  10. [10]  Nikitenkova, S.P., Stepanyants, Yu.A., and Chikhladze, L.M. (2000), Solutions of the modified Ostrovskii equation with cubic non-linearity, Prikl. Matamat. i Mekhanika 64, 276-284 (in Russian). (Engl. Transl. Journal of Applied Mathematics and Mechanics, 64, 267-274.)
  11. [11]  Schäfer, T. and Wayne, C.E. (2004), Propagation of ultra-short optical pulses in cubic nonlinear media, Physica D, 196, 90-105.
  12. [12]  Chung, Y., Jones, C.K.R.T., Schäfer, T., and Wayne, C.E. (2005), Ultra-short pulses in linear and nonlinear media, Nonlinearity, 18, 1351-1374.
  13. [13]  Parkes, E.J. (2007), Explicit solutions of the reduced Ostrovsky equation, Chaos, Solitons and Fractals, 31, 602-610.
  14. [14]  Benilov, E.S. and Pelinovsky, E.N. (1988), On the theory of wave propagation in nonlinear fluctuating media without dispersion, ZhETF 94, 175-185 (in Russian). (Engl. Transl. Soviet Physics - JETP , 67, 98-103.)
  15. [15]  Grimshaw, R.H.J, Helfrich, K., and Johnson, E.R. (2012), The reduced Ostrovsky equation: Integrability and breaking, Studies in Applied Mathematics , 129(4), 414-436.
  16. [16]  Johnson, E.R. and Grimshaw, R.H.J. (2013), Modified reduced Ostrovsky equation: Integrability and breaking, Physical Revive E, 88(2), 1539-3755.
  17. [17]  Sakovich, A.and Sakovich, S. (2005), The short pulse equation is integrable, Journal of the Physical Society of Japan, 74, 239-241.
  18. [18]  Sakovich, A.and Sakovich, S. (2006), Solitary wave solution of the short pulse equation, Journal of Physics A: Mathematical and General, 39, L361-L367.
  19. [19]  Talipova, T.G., Pelinovsky, E.N., Lamb, K., Grimshaw, R., and Holloway, P. (1999), Cubic nonlinearity effects in the propagation of intense internal waves, Doklady Akademii Nauk 365(6), 824-827 (in Russian; Engl. Transl., Doklady Earth Sciences, 365(2), 241-244).
  20. [20]  Grimshaw, R.H.J. (1999), Adjustment processes and radiating solitary waves in a regularised Ostrovsky equation, European Journal of Mechanics - B/Fluids, 18, 535-543.