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