Full description of Benjamin-Feir instability of stokes waves in deep water

Abstract

Small-amplitude, traveling, space periodic solutions –called Stokes waves– of the 2 dimensional gravity water waves equations in deep water are linearly unstable with respect to long-wave perturbations, as predicted by Benjamin and Feir in 1967. We completely describe the behavior of the four eigenvalues close to zero of the linearized equations at the Stokes wave, as the Floquet exponent is turned on. We prove in particular the conjecture that a pair of non-purely imaginary eigenvalues depicts a closed figure “8”, parameterized by the Floquet exponent, in full agreement with numerical simulations. Our new spectral approach to the Benjamin-Feir instability phenomenon uses a symplectic version of Kato’s theory of similarity transformation to reduce the problem to determine the eigenvalues of a $$ 4 \times 4 $$ 4 × 4 complex Hamiltonian and reversible matrix. Applying a procedure inspired by KAM theory, we block-diagonalize such matrix into a pair of $$2 \times 2 $$ 2 × 2 Hamiltonian and reversible matrices, thus obtaining the full description of its eigenvalues.

A high-order asymptotic analysis of the Benjamin–Feir instability spectrum in arbitrary depth

“…Some of the lower-order details of our method are also presented by Akers (2015) for the Benjamin-Feir instability in infinite depth, although this work uses different conventions for the water wave equations and underlying Stokes waves. In contrast, our expressions are in one-to-one correspondence with those reported by Berti et al (2021Berti et al ( , 2022, giving confidence in the rigorous results as well as in our asymptotic calculations.…”

“…Our method to obtain these high-order asymptotic approximations is a modification of that developed for high-frequency instabilities in Creedon, Deconinck & Trichtchenko (2021a,b) and Creedon et al (2022). Although the method is formal, it offers a more direct approach to the Benjamin-Feir figure-eight curve and produces results consistent with numerical computations (for sufficiently small ε) as well as with rigorous results reported by Berti et al (2021Berti et al ( , 2022. The method loses validity for sufficiently large ε, when the Benjamin-Feir instability spectrum separates from the origin and changes its topology, see figure 3.…”

“…The pairs (0, −1), (0, 0) and (0, 1), however, correspond to the eigenvalue at the origin. This eigenvalue generates the Benjamin-Feir instability in sufficiently deep water for 0 < ε 1 (Bridges & Mielke 1995;Hur & Yang 2022;Nguyen & Strauss 2020;Berti et al 2021Berti et al , 2022. REMARK 2.6.…”

“…The eigenvalue λ 0 is non-simple with algebraic multiplicity 4 and geometric multiplicity 3 (Bridges & Mielke 1995;Nguyen & Strauss 2020;Berti et al 2021Berti et al , 2022Hur & Yang 2022). The corresponding eigenspace is spanned by…”

“…In recent months, seminal work by Berti, Maspero & Ventura (2021 has confirmed the existence of the Benjamin-Feir figure-eight in finite and infinite depth, provided κh > α BW and ε is sufficiently small. The proof of both cases relies on the Hamiltonian and reversibility properties of the water wave equations together with Kato's In this work, we obtain high-order asymptotic expansions of the Benjamin-Feir figure-eight curve in finite and infinite depth.…”