SIGMA 4 (2008), 053, 17 pages      arXiv:0806.3155      https://doi.org/10.3842/SIGMA.2008.053
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics

Periodic and Solitary Travelling-Wave Solutions of an Extended Reduced Ostrovsky Equation

E. John Parkes
Department of Mathematics, University of Strathclyde, Glasgow G1 1XH, UK

Received October 29, 2007, in final form June 16, 2008; Published online June 19, 2008

Abstract
Periodic and solitary travelling-wave solutions of an extended reduced Ostrovsky equation are investigated. Attention is restricted to solutions that, for the appropriate choice of certain constant parameters, reduce to solutions of the reduced Ostrovsky equation. It is shown how the nature of the waves may be categorized in a simple way by considering the value of a certain single combination of constant parameters. The periodic waves may be smooth humps, cuspons, loops or parabolic corner waves. The latter are shown to be the maximum-amplitude limit of a one-parameter family of periodic smooth-hump waves. The solitary waves may be a smooth hump, a cuspon, a loop or a parabolic wave with compact support. All the solutions are expressed in parametric form. Only in one circumstance can the variable parameter be eliminated to give a solution in explicit form. In this case the resulting waves are either a solitary parabolic wave with compact support or the corresponding periodic corner waves.

Key words: Ostrovsky equation; Ostrovsky-Hunter equation; Vakhnenko equation; periodic waves; solitary waves; corner waves; cuspons; loops.

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References

1. Ostrovsky L.A., Nonlinear internal waves in a rotating ocean, Oceanology 18 (1978), 119-125.
2. Stepanyants Y.A., On stationary solutions of the reduced Ostrovsky equation: Periodic waves, compactons and compound solitons, Chaos Solitons Fractals 28 (2006), 193-204.
3. Parkes E.J., Explicit solutions of the reduced Ostrovsky equation, Chaos Solitons Fractals 31 (2007), 602-610.
4. Boyd J.P., Ostrovsky and Hunter's generic wave equation for weakly dispersive waves: matched asymptotic and pseudospectral study of the paraboidal travelling waves (corner and near-corner waves), European J. Appl. Math. 16 (2005), 65-81.
5. Vakhnenko V.A., Solitons in a nonlinear model medium, J. Phys. A: Math. Gen. 25 (1992), 4181-4187.
6. Vakhnenko V.A., High-frequency soliton-like waves in a relaxing medium, J. Math. Phys. 40 (1999), 2011-2020.
7. Parkes E.J., The stability of solutions of Vakhnenko's equation, J. Phys. A: Math. Gen. 26 (1993), 6469-6475.
8. Morrison A.J., Parkes E.J., Vakhnenko V.O., The N loop soliton solution of the Vakhnenko equation, Nonlinearity 12 (1999), 1427-1437.
9. Morrison A.J., Parkes E.J., The N-soliton solution of the modified generalised Vakhnenko equation (a new nonlinear evolution equation), Chaos Solitons Fractals 16 (2003), 13-26.
10. Espinosa A., Fujioka J., Hydrodynamic foundation and Painlevé analysis of Hirota-Satsuma-type equations, J. Phys. Soc. Japan 63 (1994), 1289-1294.
11. Morrison A.J., Parkes E.J., The N-soliton solution of a generalised Vakhnenko equation, Glasg. Math. J. 43 (2001), 65-90.
12. Liu Y.-P., Li Z.-B., Wang K.-C., Symbolic computation of exact solutions for a nonlinear evolution equation, Chaos Solitons Fractals 31 (2007), 1173-1180.
13. Parkes E.J., Duffy B.R., Abbott P.C., The Jacobi elliptic-function method for finding periodic-wave solutions to nonlinear evolution equations, Phys. Lett. A 295 (2002), 280-286.
14. Vakhnenko V.O., Parkes E.J., Periodic and solitary-wave solutions of the Degasperis-Procesi equation, Chaos Solitons Fractals 20 (2004), 1059-1073.
15. Parkes E.J., Vakhnenko V.O., Explicit solutions of the Camassa-Holm equation, Chaos Solitons Fractals 26 (2005), 1309-1316.
16. Parkes E.J., Some periodic and solitary travelling-wave solutions of the short-pulse equation, Chaos Solitons Fractals 36 (2008), 154-159.
17. Zhang G.P., Qiao Z.J., Cuspons and smooth solitons of the Degasperis-Procesi equation under inhomogeneous boundary condition, Math. Phys. Anal. Geom. 10 (2007), 205-225.
18. Qiao Z.J., M-shape peakons, dehisced solitons, cuspons and new 1-peak solitons for the Degasperis-Procesi equation, Chaos Solitons Fractals 37 (2008), 501-507.
19. Qiao Z.J., Zhang G.P., On peaked and smooth solitons for the Camassa-Holm equation, Europhys. Lett. 73 (2006), 657-663.
20. Qiao Z.J., The Camassa-Holm hierarchy, N-dimensional integrable systems, and algebro-geometric solution on a symplectic submanifold, Comm. Math. Phys. 239 (2003), 309-341.
21. Byrd P.F., Friedman M.D., Handbook of elliptic integrals for engineers and scientists, Springer, Berlin, 1971.
22. Abramowitz M., Stegun I.A., Handbook of mathematical functions, Dover, New York, 1972.
23. Lenells J., Traveling wave solutions of the Camassa-Holm equation, J. Differential Equations 217 (2005), 393-430.
24. Li J.-B., Dynamical understanding of loop soliton solution for several nonlinear wave equations, Sci. China Ser. A 50 (2007), 773-785.