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 ${\mathtt{h}}_{WB}\approx 1.363$. In this paper we completely describe, for any 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 ${\mathtt{h}}_{WB}$, which coincides with the one predicted by Whitham and Benjamin, such that, for any $0<\mathtt{h}<{\mathtt{h}}_{WB}$, the eigenvalues close to zero remain purely imaginary and, for any $\mathtt{h}>{\mathtt{h}}_{WB}$, a pair of non-purely imaginary eigenvalues depicts a closed figure "8", parameterized by the Floquet exponent. As $\mathtt{h}\to {\mathtt{h}}_{WB}^{+}$ this figure "8" collapses to the origin of the complex plane. The proof combines a symplectic version of Kato's perturbative theory to compute the eigenvalues of a $4\times 4$ Hamiltonian and reversible matrix, and KAM inspired transformations to block-diagonalize it. The four eigenvalues have all the same size $O(\mu )$ - unlike the infinitely deep water case in [6]- and the correct Benjamin-Feir phenomenon appears only after one non-perturbative block-diagonalization step. In addition one has to track, along the whole proof, the explicit dependence of the entries of the $4\times 4$ reduced matrix with respect to the depth $\mathtt{h}$.

47 pages, 5 figures, revised introduction. arXiv admin note: substantial text overlap with arXiv:2109.11852