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 h is larger than a critical threshold hWB≈ 1.363 . In this paper, we completely describe, for any finite value of h> 0 , the four eigenvalues close to zero of the linearized equations at the Stokes wave, as the Floquet exponent μ is turned on. We prove, in particular, the existence of a unique depth hWB , which coincides with the one predicted by Whitham and Benjamin, such that, for any 0 < h< hWB , the eigenvalues close to zero are purely imaginary and, for any h> hWB , a pair of non-purely imaginary eigenvalues depicts a closed figure “8”, parameterized by the Floquet exponent. As h→hWB+ 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 h→ + ∞ (deep water) and μ→ 0 (long waves) do not commute. In finite depth, the four eigenvalues have all the same size O(μ) , 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 h is carefully tracked.