Infinitely many isolas of modulational instability for Stokes waves

We prove the long-standing conjecture regarding the existence of infinitely many high-frequency modulational instability ``isolas" for a Stokes wave in arbitrary depth h > 0, subject to longitudinal perturbations. We completely describe the spectral bands with non-zero real part away from the origin of the L²(R)-spectrum of the water waves system linearized at a Stokes waves of small amplitude ε > 0. The unstable spectrum is the union of isolas of elliptical shape, parameterized by integers p≥ 2, with semiaxis of size |β₁⁽ᵖ⁾ (h)| εᵖ+ O(εᵖ⁺¹ ) where β₁( p) (h) is a nonzero analytic function of the depth h that depends on the Taylor coefficients of the Stokes waves up to order p.