First isola of modulational instability of Stokes waves in deep water

We prove high-frequency modulational instability of small-amplitude Stokes waves in deep water under longitudinal perturbations, providing the first isola of unstable eigenvalues branching off from $\mathtt{i}\frac{3}{4}$. Unlike the finite depth case this is a degenerate problem and the real part of the unstable eigenvalues has a much smaller size than in finite depth. By a symplectic version of Kato theory we reduce to search the eigenvalues of a $2\times 2$ Hamiltonian and reversible matrix which has eigenvalues with non-zero real part if and only if a certain analytic function is not identically zero. In deep water we prove that the Taylor coefficients up to order three of this function vanish, but not the fourth-order one.