Unstable Stokes waves in constant vorticity flows of infinite depth

Abstract

We study the spectral stability of small-amplitude Stokes waves with constant vorticity in infinite depth. We describe the spectrum near the origin and near the first nonzero double eigenvalue. For nonzero vorticity $ω<2\sqrt{g/3}$, where $g$ is the gravitational constant, the eigenvalues near the origin form a figure eight, whereas for $ω>2\sqrt{g/3}$ the local spectrum remains purely imaginary. Away from the origin, the interaction of two Fourier modes whose indices differ by two produces an unstable isola when $ω>-\sqrt{g/6}$, $ω\ne0$, and the resonant wave numbers are noninteger. This curve is a real analytic oval with an explicit leading ellipse and maximal growth of quadratic order in the amplitude. We also prove instability at the integer resonances needed to cover the remaining vorticities. Together with the classical irrotational result, these two mechanisms show that sufficiently small nonzero Stokes waves are spectrally unstable for every fixed constant vorticity. The proof uses finite-dimensional spectral reductions of the full Euler linearization. Near the origin, the finite-dimensional linearized operator has a one-sided limit as the Floquet exponent vanishes, which differs from the periodic linearization. We use the Kato transformation to construct a three-dimensional reduction with biorthogonal bases and use all three eigenvectors of the reference operator to calculate the Benjamin--Feir coefficient. At the nonzero collision, a real two-dimensional reduction and an explicit second-order calculation determine the unstable curve.