Geodesic
Symmetry analysis and exact solutions of some Ostrovsky equations
Teoretičeskaâ i matematičeskaâ fizika, Tome 168 (2011) no. 1, pp. 49-64

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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, partial differential equation.
Mots-clés : exact solution 
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     author = {M. L. Gandarias and M. S. Bruz\'on},
     title = {Symmetry analysis and exact solutions of some {Ostrovsky} equations},
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M. L. Gandarias; M. S. Bruzón. Symmetry analysis and exact solutions of some Ostrovsky equations. Teoretičeskaâ i matematičeskaâ fizika, Tome 168 (2011) no. 1, pp. 49-64. http://geodesic.mathdoc.fr/item/TMF_2011_168_1_a4/