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 TMF, 2011, Volume 168, Number 1, Pages 49–64 (Mi tmf6663)

Symmetry analysis and exact solutions of some Ostrovsky equations

M. L. Gandarias, M. S. Bruzón

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”.

Keywords: classical symmetry, exact solution, partial differential equation

DOI: https://doi.org/10.4213/tmf6663

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English version:
Theoretical and Mathematical Physics, 2011, 168:1, 898–911

Bibliographic databases:

Citation: M. L. Gandarias, M. S. Bruzón, “Symmetry analysis and exact solutions of some Ostrovsky equations”, TMF, 168:1 (2011), 49–64; Theoret. and Math. Phys., 168:1 (2011), 898–911

Citation in format AMSBIB
\Bibitem{GanBru11} \by M.~L.~Gandarias, M.~S.~Bruz\'on \paper Symmetry analysis and exact solutions of some Ostrovsky equations \jour TMF \yr 2011 \vol 168 \issue 1 \pages 49--64 \mathnet{http://mi.mathnet.ru/tmf6663} \crossref{https://doi.org/10.4213/tmf6663} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2021731} \transl \jour Theoret. and Math. Phys. \yr 2011 \vol 168 \issue 1 \pages 898--911 \crossref{https://doi.org/10.1007/s11232-011-0073-3} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-79961163722} 

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• https://doi.org/10.4213/tmf6663
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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. Hashemi M.S., Nucci M.C., Abbasbandy S., “Group Analysis of the Modified Generalized Vakhnenko Equation”, Commun. Nonlinear Sci. Numer. Simul., 18:4 (2013), 867–877
2. Kaur L., Gupta R.K., “Some Invariant Solutions of Field Equations With Axial Symmetry For Empty Space Containing An Electrostatic Field”, Appl. Math. Comput., 231 (2014), 560–565
3. Najafi R. Bahrami F. Hashemi M.S., “Classical and nonclassical Lie symmetry analysis to a class of nonlinear time-fractional differential equations”, Nonlinear Dyn., 87:3 (2017), 1785–1796
4. A. V. Bochkarev, A. I. Zemlyanukhin, “The geometric series method for constructing exact solutions to nonlinear evolution equations”, Comput. Math. Math. Phys., 57:7 (2017), 1111–1123
5. Bahrami F. Najafi R. Hashemi M.S., “On the Invariant Solutions of Space/Time-Fractional Diffusion Equations”, Indian J. Phys., 91:12 (2017), 1571–1579
6. Bruzon M.S., Recio E., de la Rosa R., Gandarias M.L., “Local Conservation Laws, Symmetries, and Exact Solutions For a Kudryashov-Sinelshchikov Equation”, Math. Meth. Appl. Sci., 41:4 (2018), 1631–1641
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