Linear instability and uniqueness of the peaked periodic wave in the reduced Ostrovsky equation

TL;DR

This paper proves the linear instability of peaked periodic waves in the reduced Ostrovsky equation and establishes their uniqueness among certain periodic solutions, advancing understanding of their stability properties.

Contribution

It provides the first sharp bounds on exponential growth of perturbations and proves the peaked wave's uniqueness in a specific function space.

Findings

Peaked periodic wave is linearly unstable.

Sharp bounds on exponential growth of perturbations.

Peaked wave with parabolic profile is unique in its class.

Abstract

Stability of the peaked periodic waves in the reduced Ostrovsky equation has remained an open problem for a long time. In order to solve this problem we obtain sharp bounds on the exponential growth of the $L^2$ norm of co-periodic perturbations to the peaked periodic wave, from which it follows that the peaked periodic wave is linearly unstable. We also prove that the peaked periodic wave with parabolic profile is the unique peaked wave in the space of periodic $L^2$ functions with zero mean and a single minimum per period.