Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

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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