Spectral stability of periodic waves in the generalized reduced Ostrovsky equation

TL;DR

This paper proves spectral stability of all smooth periodic traveling waves in the generalized reduced Ostrovsky equation using an energy convexity argument, without relying on coordinate transformations, thus simplifying the stability analysis.

Contribution

It introduces a straightforward energy convexity method to establish spectral stability of periodic waves, independent of the nonlinearity power, avoiding complex coordinate transformations.

Findings

All smooth periodic traveling waves are spectrally stable.

The stability proof is simplified and does not depend on the nonlinearity power.

The method avoids transformations to Klein-Gordon type equations.

Abstract

We consider stability of periodic travelling waves in the generalized reduced Ostrovsky equation with respect to co-periodic perturbations. Compared to the recent literature, we give a simple argument that proves spectral stability of all smooth periodic travelling waves independently on the nonlinearity power. The argument is based on the energy convexity and does not use coordinate transformations of the reduced Ostrovsky equations to the semi-linear equations of the Klein--Gordon type.