Modulational spectrum of infinite-depth hydroelastic Stokes waves

We determine the complete local Bloch spectrum bifurcating from the origin for small-amplitude periodic hydroelastic Stokes waves in infinite depth, under the combined effects of gravity, surface tension, and elastic bending. Away from the Wilton-type resonance set, we construct a real-analytic Stokes-wave branch and analyze the four eigenvalues emerging from the defective zero eigenvalue of the linearized hydroelastic Euler system. Using analytic spectral perturbation theory and Hamiltonian-reversible reductions, we decouple them into a Benjamin--Feir pair and a long-wave pair. The long-wave pair remains purely imaginary and has the singular scale $\cO(\sqrt{|\mu|})$, whereas the Benjamin--Feir pair is governed by an explicit discriminant whose leading sign yields a sharp criterion for modulational stability and instability. We derive the exact non-resonant phase diagram in the surface-tension-bending parameter plane and identify a bounded stability island generated by elastic bending. In the unstable region, and away from a drift degeneracy, the Benjamin--Feir branches form a local figure-eight curve. In the zero-bending limit, the reduced coefficients recover the known deep-water gravity and gravity-capillary results, while the change from the finite-depth $\cO(|\mu|)$ long-wave scale to $\cO(\sqrt{|\mu|})$ shows that the infinite-depth problem is singular.