High-Frequency Instabilities of the Kawahara Equation: A Perturbative Approach

TL;DR

This paper investigates high-frequency instabilities in small-amplitude periodic solutions of the Kawahara equation, introducing a perturbative method to analyze their spectral stability and verify results numerically.

Contribution

It presents a novel perturbation approach to determine asymptotic growth rates of instabilities in the Kawahara equation's solutions.

Findings

High-frequency instabilities are characterized for small-amplitude solutions.

A formal perturbation method effectively predicts instability growth rates.

Numerical computations confirm the asymptotic analysis.

Abstract

We analyze the spectral stability of small-amplitude, periodic, traveling-wave solutions of the Kawahara equation. These solutions exhibit high-frequency instabilities when subject to bounded perturbations on the whole real line. We introduce a formal perturbation method to determine the asymptotic growth rates of these instabilities, among other properties. Explicit numerical computations are used to verify our asymptotic results.