High-frequency instabilities of the Ostrovsky equation

TL;DR

This paper investigates the spectral stability of small amplitude periodic traveling waves in the Ostrovsky equation, revealing conditions under which these waves become spectrally unstable due to eigenvalue collisions.

Contribution

It provides a detailed analysis of eigenvalue collisions and Krein signatures, identifying instability mechanisms for the waves in the Ostrovsky equation.

Findings

Spectral instabilities occur from eigenvalue collisions on the imaginary axis.

All such eigenvalue collisions are classified and analyzed.

Krein signature analysis explains the instability mechanisms.

Abstract

We study spectral stability of small amplitude periodic traveling waves of the Ostrovsky equation. We prove that these waves exhibit spectral instabilities arising from a collision of pair of non-zero eigenvalues on the imaginary axis when subjected to square integrable perturbations on the whole real line. We also list all such collisions between pair of eigenvalues on the imaginary axis and do a Krein signature analysis.