Sheets of Spectral Data of Stokes Waves in Weakly Nonlinear Models

Abstract

We study the spectral stability of small-amplitude Stokes waves in a family of weakly nonlinear, unidirectional models of the form $u_t + L u + (u^2)_x = 0$. 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.