
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 TravellingWave 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 travellingwave 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 maximumamplitude limit
of a oneparameter family
of periodic smoothhump 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; OstrovskyHunter equation; Vakhnenko equation; periodic waves; solitary waves; corner waves; cuspons; loops.
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