Full Benjamin-Feir instability of capillary-gravity Stokes waves in finite depth

We consider the 2D capillary-gravity water wave equations for a fluid layer of finite depth h>0, subject to the combined influences of gravity and surface tension \kappa ≥ 0. Our focus is on the linear stability and instability of small-amplitude, 2\pi-periodic Stokes-wave solutions with respect to longitudinal, long-wave perturbations. The associated linearized operator possesses periodic coefficients and a defective zero eigenvalue of algebraic multiplicity four. Employing Bloch-Floquet theory, we analyze the resulting family of periodic eigenvalue problems. For every \kappa ≥ 0 and h>0, we demonstrate the complete splitting of the four eigenvalues near zero when both the wave amplitude and the Floquet parameter are small. In particular, we rigorously confirm that, within the instability regimes of depth and surface tension first identified formally by Djordjevic-Redekopp and Ablowitz-Segur in the 1970s, the spectrum of the linearized operator near the origin exhibits a characteristic ``figure-8'' configuration.