Symmetry analysis and exact solutions of some Ostrovsky equations

  • M. L. GandariasEmail author
  • M. S. Bruzón


We apply the classical Lie method and the nonclassical method to a generalized Ostrovsky equation (GOE) and to the integrable Vakhnenko equation (VE), which Vakhnenko and Parkes proved to be equivalent to the reduced Ostrovsky equation. Using a simple nonlinear ordinary differential equation, we find that for some polynomials of velocity, the GOE has abundant exact solutions expressible in terms of Jacobi elliptic functions and consequently has many solutions in the form of periodic waves, solitary waves, compactons, etc. The nonclassical method applied to the associated potential system for the VE yields solutions that arise from neither nonclassical symmetries of the VE nor potential symmetries. Some of these equations have interesting behavior such as “nonlinear superposition.”


classical symmetry exact solution partial differential equation 


  1. 1.
    L. A. Ostrovsky, Oceanology, 18, 119–125 (1978).Google Scholar
  2. 2.
    L. A. Ostrovsky and Yu. A. Stepanyants, “Nonlinear waves in a rotating fluid [in Russian],” in: Nonlinear Waves: Physics and Astrophysics (A. V. Gaponov-Grekhov and M. I. Rabinovich, eds.), Nauka, Moscow (1993), pp. 132–153.Google Scholar
  3. 3.
    R. H. J. Grimshaw, L. A. Ostrovsky, V. I. Shrira, and Yu. A. Stepanyants, Surv. Geophys., 19, 289–338 (1998).ADSCrossRefGoogle Scholar
  4. 4.
    O. A. Gilman, R. Grimshaw, and Yu. A. Stepanyants, Stud. Appl. Math., 95, 115–126 (1995).MathSciNetzbMATHGoogle Scholar
  5. 5.
    O. A. Gilman, R. Grimshaw, and Yu. A. Stepanyants, Dynam. Atmos. Oceans, 23, 403–411 (1996).ADSCrossRefGoogle Scholar
  6. 6.
    A. I. Leonov, “The effect of the earth’s rotation on the propagation of weak nonlinear surface and internal long oceanic waves,” in: Fourth Intl. Conf. on Collective Phenomena (J. L. Lebowitz, eds.), Acad. Sci., New York (1981), pp. 150–159.Google Scholar
  7. 7.
    V. M. Galkin and Yu. A. Astepanyants, J. Appl. Math. Mech., 55, 939–943 (1991).MathSciNetCrossRefGoogle Scholar
  8. 8.
    Yu. A. Stepanyants, Chaos Solitons Fractals, 28, 193–204 (2006).MathSciNetADSzbMATHCrossRefGoogle Scholar
  9. 9.
    V. O. Vakhnenko and E. J. Parkes, Nonlinearity, 11, 1457–1464 (1998).MathSciNetADSzbMATHCrossRefGoogle Scholar
  10. 10.
    E. Yusufoğlu and A. Bekir, Chaos Solitons Fractals, 38, 1126–1133 (2008).MathSciNetADSzbMATHCrossRefGoogle Scholar
  11. 11.
    L. Tian and J. Yin, Chaos Solitons Fractals, 35, 991–995 (2008).MathSciNetADSzbMATHCrossRefGoogle Scholar
  12. 12.
    G. W. Bluman and J. D. Cole, J. Math. Mech., 18, 1025–1042 (1968/69).MathSciNetGoogle Scholar
  13. 13.
    P. A. Clarkson and E. L. Mansfield, SIAM J. Appl. Math., 54, 1693–1719 (1994); arXiv:solv-int/9401002v2 (1994).MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    N. A. Kudryashov, Chaos Solitons Fractals, 24, 1217–1231 (2005); arXiv:nlin/0406007v1 (2004).MathSciNetADSzbMATHCrossRefGoogle Scholar
  15. 15.
    N. A. Kudryashov and N. B. Loguinova, Appl. Math. Comput., 205, 396–402 (2008).MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    I. S. Krasil’shchik and A. M. Vinogradov, Acta Appl. Math., 2, 79–96 (1984).MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    I. S. Krasil’shchik and A. M. Vinogradov, Acta Appl. Math., 15, 161–209 (1989).MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    A. M. Vinogradov, ed., Symmetries of Partial Differential Equations: Conservation Laws, Applications, Algorithms, Kluwer, Dordrecht (1989).Google Scholar
  19. 19.
    I. Sh. Akhatov, R. K. Gazizov, and N. Kh. Ibragimov, J. Sov. Math., 55, 1401–1450 (1991).CrossRefGoogle Scholar
  20. 20.
    G. W. Bluman, G. J. Reid, and S. Kumei, J. Math. Phys., 29, 806–811 (1988).MathSciNetADSzbMATHCrossRefGoogle Scholar
  21. 21.
    G. W. Bluman and Z. Yan, European J. Appl. Math., 16, 239–261 (2005).MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    M. L. Gandarias and M. S. Bruzon, Nonlinear Anal., 71, e1826–e1834 (2009).MathSciNetCrossRefGoogle Scholar
  23. 23.
    M. L. Gandarias, “New potential symmetries,” in: SIDE III: Symmetries and Integrability of Difference Equations (CRM Proc. Lect. Notes, Vol. 25, D. Levi and O. Ragnisco, eds.), Amer. Math. Soc., Providence, R. I. (2000), pp. 161–165.Google Scholar
  24. 24.
    M. L. Gandarias, J. Phys. A, 29, 607–633 (1996).MathSciNetADSzbMATHCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2011

Authors and Affiliations

  1. 1.Departamento de MatematicasUniversidad de CadizCadizSpain

Personalised recommendations