A high-order asymptotic analysis of the Benjamin–Feir instability spectrum in arbitrary depth

We investigate the Benjamin-Feir (or modulational) instability of Stokes waves, i.e., small-amplitude, one-dimensional periodic gravity waves permanent form and constant velocity, in water finite infinite depth. develop a perturbation method to describe high-order accuracy unstable spectral elements associated with this instability, obtained by linearizing Euler's equations about small-amplitude waves. These figure-eight curve centered at origin complex plane, which is parameterized Floquet exponent. Our asymptotic expansions are excellent agreement numerical computations as well recent rigorous results Berti, Maspero, Ventura. From our expansions, we derive estimates for growth rates parameterization respect also able compare high-frequency spectra analytically first time, revealing three different regimes depending on predominant instability.