Spectral structure of the Benjamin-Feir instability in deep-water gravity-capillary Stokes waves

TL;DR

This paper provides a rigorous spectral analysis of the Benjamin-Feir instability in deep-water gravity-capillary Stokes waves, confirming classical predictions and identifying stability regions based on surface tension.

Contribution

It offers the first complete spectral description at the Euler equation level, validating asymptotic predictions through rigorous analysis.

Findings

Identified eigenvalue pairs with non-zero real parts forming a figure-eight pattern.

Mapped sharp stability and instability regions depending on surface tension.

Provided a rigorous justification of classical modulational instability predictions.

Abstract

We investigate the Benjamin-Feir instability of small-amplitude gravity-capillary Stokes waves in deep water for the full water wave equations. While modulational instability has been classically predicted by formal asymptotic approaches, such as nonlinear Schr\"odinger approximations, a complete spectral description at the level of the Euler equations has remained open. We perform a rigorous Bloch-Floquet spectral analysis of the linearized operator and describe the splitting of the multiple eigenvalues at the origin. In the unstable regime, we identify a pair of eigenvalues with non-zero real part forming the characteristic ``figure-eight'' pattern in the complex plane. As a consequence, we recover sharp instability and stability regions in terms of the surface tension parameter, thereby providing a fully rigorous justification of the classical predictions in the gravity-capillary setting.