2026/04/30 by Ting-Yang Hsiao, Xinyang Wang
Mathematics · #math.AP #msc:76B15 #msc:35P05 #msc:35B35 #msc:35B10
fixed minor typos
arxiv created 2026/08/05 · arxiv updated 2026/08/06
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 Schrödinger 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.