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

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.