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

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.

Keywords

• Boussinesq–Whitham systems
• traveling waves
• spectral stability
• high-frequency instabilities
• perturbation methods