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 of permanent form and constant velocity, in water of finite and infinite depth. We develop a perturbation method to describe to high-order accuracy the unstable spectral elements associated with this instability, obtained by linearizing Euler's equations about the small-amplitude Stokes waves. These unstable elements form a figure-eight curve centred at the origin of the complex spectral plane, which is parametrized by a Floquet exponent. Our asymptotic expansions of this figure-eight are in excellent agreement with numerical computations as well as recent rigorous results by Berti et al. (Full description of Benjamin-Feir instability of Stokes waves in deep water, 2021, arXiv:2109.11852) and Berti et al. (Benjamin-Feir instability of Stokes waves in finite depth, 2022, arXiv:2204.00809). From our expansions, we derive high-order estimates for the growth rates of the Benjamin-Feir instability and for the parametrization of the Benjamin-Feir figure-eight curve with respect to the Floquet exponent. We are also able to compare the Benjamin-Feir and high-frequency instability spectra analytically for the first time, revealing three different regimes of the Stokes waves, depending on the predominant instability.