Orbital stability of periodic waves in the class of reduced Ostrovsky equations

TL;DR

This paper proves the orbital stability of small-amplitude periodic waves in reduced Ostrovsky equations, which model internal waves with rotation, using conserved quantities and Lyapunov functionals, supported by numerical evidence.

Contribution

It establishes the orbital stability of periodic waves in reduced Ostrovsky equations through analytical and numerical methods, leveraging integrability and conserved quantities.

Findings

Small-amplitude waves are orbitally stable under subharmonic perturbations.

A convex Lyapunov functional is constructed and shown to be conserved.

Numerical results support convexity of the Lyapunov functional for all amplitudes.

Abstract

Periodic travelling waves are considered in the class of reduced Ostrovsky equations that describe low-frequency internal waves in the presence of rotation. The reduced Ostrovsky equations with either quadratic or cubic nonlinearities can be transformed to integrable equations of the Klein--Gordon type by means of a change of coordinates. By using the conserved momentum and energy as well as an additional conserved quantity due to integrability, we prove that small-amplitude periodic waves are orbitally stable with respect to subharmonic perturbations, with period equal to an integer multiple of the period of the wave. The proof is based on construction of a Lyapunov functional, which is convex at the periodic wave and is conserved in the time evolution. We also show numerically that convexity of the Lyapunov functional holds for periodic waves of arbitrary amplitudes.