Benjamin–Feir Instability of Stokes Waves in Finite Depth

Abstract

Whitham and Benjamin predicted in 1967 that small-amplitude periodic traveling Stokes waves of the 2d-gravity water waves equations are linearly unstable with respect to long-wave perturbations, if the depth {mathtt h} is larger than a critical threshold texttt{h}_{scriptscriptstyle {textsc {WB}}}approx 1.363 . In this paper, we completely describe, for any finite value of mathtt h >0 , the four eigenvalues close to zero of the linearized equations at the Stokes wave, as the Floquet exponent mu is turned on. We prove, in particular, the existence of a unique depth texttt{h}_{scriptscriptstyle {textsc {WB}}}, which coincides with the one predicted by Whitham and Benjamin, such that, for any 0< mathtt h < texttt{h}_{scriptscriptstyle {textsc {WB}}}, the eigenvalues close to zero are purely imaginary and, for any mathtt h > texttt{h}_{scriptscriptstyle {textsc {WB}}}, a pair of non-purely imaginary eigenvalues depicts a closed figure “8”, parameterized by the Floquet exponent. As {mathtt h} rightarrow texttt{h}_{scriptscriptstyle {textsc {WB}}}^{, +} the “8” collapses to the origin of the complex plane. The complete bifurcation diagram of the spectrum is not deduced as in deep water, since the limits texttt{h}rightarrow +infty (deep water) and mu rightarrow 0 (long waves) do not commute. In finite depth, the four eigenvalues have all the same size mathcal {O}(mu ), unlike in deep water, and the analysis of their splitting is much more delicate, requiring, as a new ingredient, a non-perturbative step of block-diagonalization. Along the whole proof, the explicit dependence of the matrix entries with respect to the depth texttt{h} is carefully tracked.