01605nas a2200169   4500008004100000020001400041245008000055210006900135260001500204300001400219490000800233520108300241100002401324700002101348700001901369856004701388       2022                        eng d  a1432-129700aFull description of Benjamin-Feir instability of stokes waves in deep water0 aFull description of BenjaminFeir instability of stokes waves in   c2022/11/01  a651 - 7110 v2303 a<p>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 $ 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.</p>1 aBerti, Massimiliano1 aMaspero, Alberto1 aVentura, Paolo  uhttps://doi.org/10.1007/s00222-022-01130-z