Sheets of Spectral Data of Stokes Waves in Weakly Nonlinear Models

We study the spectral stability of small-amplitude Stokes waves in a family of weakly nonlinear, unidirectional models of the form uₜ + L u + (u²)ₓ = 0 u t + L u + ( u 2 ) x = 0 . We introduce a perturbation method to expand the spectral data in wave amplitude near flat-state eigenvalue collisions, with the ratio of the colliding modes as a free parameter. This yields sheets of spectral data whose slices at fixed amplitude give isolas of instability. The same perturbation framework treats both high-frequency and Benjamin–Feir instabilities, extends to discontinuous dispersion relations (including the Akers–Milewski equation), and, for the first time, provides an analytic approximation of the Benjamin–Feir spectrum for this model and a direct comparison of high-frequency and Benjamin–Feir growth rates across the full family of models. Asymptotic predictions are validated against numerical spectra computed by Floquet–Fourier–Hill and quasi-Newton methods.