High-Frequency Instabilities of a Boussinesq-Whitham System: A Perturbative Approach

TL;DR

This paper investigates high-frequency instabilities in small-amplitude periodic solutions of a Boussinesq-Whitham system, combining perturbation analysis with numerical validation to understand their spectral stability.

Contribution

It introduces a perturbative method to estimate the asymptotic behavior of high-frequency instabilities in the system, complementing numerical results.

Findings

Small-amplitude solutions exhibit high-frequency instabilities.

Perturbation estimates align with numerical computations.

This work advances understanding of spectral stability in wave models.

Abstract

We analyze the spectral stability of small-amplitude, periodic, traveling-wave solutions of a Boussinesq-Whitham system. These solutions are shown numerically to exhibit high-frequency instabilities when subject to bounded perturbations on the real line. We use a formal perturbation method to estimate the asymptotic behavior of these instabilities in the small-amplitude regime. We compare these asymptotic results with direct numerical computations. This is the second paper in a series of three that investigates high-frequency instabilities of Stokes waves.