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

Abstract: We study the two-dimensional gravity-capillary water waves equations for a fluid of finite depth $\mathtt{h}>0$ under the combined effects of gravity and surface tension $\kappa \geq 0$. We analyze the linear stability and instability of small-amplitude, $2\pi$-periodic Stokes wave solutions, under the effect of longitudinal long-wave perturbations. The corresponding linearized operator has periodic coefficients and a defective zero eigenvalue of multiplicity four. Using Bloch-Floquet theory, we investigate the associated family of periodic eigenvalue problems. For all surface tension values $\kappa \geq 0$ and depths $\mathtt{h} > 0$, we establish the complete splitting of the four eigenvalues near zero when both the wave amplitude and the Floquet parameter are small. Specifically, we rigorously prove that in the regions of unstable depth and capillarity identified formally by Djordjevic-Redekopp and Ablowitz-Segur in the 1970's, the spectrum of the linearized operator near the origin depicts a "figure 8" pattern.