Abstract
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.”
Similar content being viewed by others
References
L. A. Ostrovsky, Oceanology, 18, 119–125 (1978).
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.
R. H. J. Grimshaw, L. A. Ostrovsky, V. I. Shrira, and Yu. A. Stepanyants, Surv. Geophys., 19, 289–338 (1998).
O. A. Gilman, R. Grimshaw, and Yu. A. Stepanyants, Stud. Appl. Math., 95, 115–126 (1995).
O. A. Gilman, R. Grimshaw, and Yu. A. Stepanyants, Dynam. Atmos. Oceans, 23, 403–411 (1996).
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.
V. M. Galkin and Yu. A. Astepanyants, J. Appl. Math. Mech., 55, 939–943 (1991).
Yu. A. Stepanyants, Chaos Solitons Fractals, 28, 193–204 (2006).
V. O. Vakhnenko and E. J. Parkes, Nonlinearity, 11, 1457–1464 (1998).
E. Yusufoğlu and A. Bekir, Chaos Solitons Fractals, 38, 1126–1133 (2008).
L. Tian and J. Yin, Chaos Solitons Fractals, 35, 991–995 (2008).
G. W. Bluman and J. D. Cole, J. Math. Mech., 18, 1025–1042 (1968/69).
P. A. Clarkson and E. L. Mansfield, SIAM J. Appl. Math., 54, 1693–1719 (1994); arXiv:solv-int/9401002v2 (1994).
N. A. Kudryashov, Chaos Solitons Fractals, 24, 1217–1231 (2005); arXiv:nlin/0406007v1 (2004).
N. A. Kudryashov and N. B. Loguinova, Appl. Math. Comput., 205, 396–402 (2008).
I. S. Krasil’shchik and A. M. Vinogradov, Acta Appl. Math., 2, 79–96 (1984).
I. S. Krasil’shchik and A. M. Vinogradov, Acta Appl. Math., 15, 161–209 (1989).
A. M. Vinogradov, ed., Symmetries of Partial Differential Equations: Conservation Laws, Applications, Algorithms, Kluwer, Dordrecht (1989).
I. Sh. Akhatov, R. K. Gazizov, and N. Kh. Ibragimov, J. Sov. Math., 55, 1401–1450 (1991).
G. W. Bluman, G. J. Reid, and S. Kumei, J. Math. Phys., 29, 806–811 (1988).
G. W. Bluman and Z. Yan, European J. Appl. Math., 16, 239–261 (2005).
M. L. Gandarias and M. S. Bruzon, Nonlinear Anal., 71, e1826–e1834 (2009).
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.
M. L. Gandarias, J. Phys. A, 29, 607–633 (1996).
Author information
Authors and Affiliations
Corresponding author
Additional information
Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 168, No. 1, pp. 49–64, July, 2011.
Rights and permissions
About this article
Cite this article
Gandarias, M.L., Bruzón, M.S. Symmetry analysis and exact solutions of some Ostrovsky equations. Theor Math Phys 168, 898–911 (2011). https://doi.org/10.1007/s11232-011-0073-3
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11232-011-0073-3