Mathematics

Small-amplitude, traveling, space periodic solutions –called Stokes waves– of the gravitywater waves equations with infinite depth are linearly unstable with respect to long-waveperturbations, as predicted by Benjamin and Feir in 1967. We completely describe thebehavior 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 ofnon-purely imaginary eigenvalues depicts a closed figure eight, parameterized by the Floquetexponent, in full agreement with numerical simulations. Our new spectral approach to theBenjamin-Feir instability phenomenon uses Kato’s theory of similarity transformation toreduce the problem to determine the eigenvalues of a 4×4 complex Hamiltonian and reversible matrix. Applying a procedure inspired by KAM theory we block-diagonalize such matrix into a pair of 2 × 2 Hamiltonian and reversible matrices, thus obtaining the full description of its eigenvalues.