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

TL;DR

This paper investigates periodic and solitary travelling-wave solutions of an extended reduced Ostrovsky equation, classifying wave types based on parameter combinations and providing parametric solutions, including explicit forms for special cases.

Contribution

It introduces a systematic classification of wave solutions for the extended reduced Ostrovsky equation and derives explicit parametric and some explicit solutions.

Findings

Wave solutions include smooth humps, cuspons, loops, and parabolic waves.

Maximum amplitude waves are parabolic corner waves.

Explicit solutions are obtained in special parameter cases.

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.