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

TL;DR

This paper provides a complete spectral analysis of the Benjamin-Feir instability in deep water Stokes waves, confirming the eigenvalue behavior and describing the instability mechanism with a novel mathematical approach.

Contribution

It introduces a new spectral method using Kato's theory and KAM-inspired block-diagonalization to fully characterize eigenvalues related to the Benjamin-Feir instability.

Findings

Eigenvalues form a closed figure eight as Floquet exponent varies

Confirmed the conjecture about eigenvalue behavior near zero

Provided a full spectral description of the instability mechanism

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 eight, parameterized by the Floquet exponent, in full agreement with numerical simulations. Our new spectral approach to the Benjamin-Feir instability phenomenon uses Kato's theory of similarity transformation to reduce the problem to determine the eigenvalues of a $ 4 \times 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 $ Hamiltonian and reversible matrices, thus obtaining the full description of its eigenvalues.