On Benjamin-Feir instability and evolution of a nonlinear wave with finite-amplitude sidebands

Abstract. In the past decade it became customary to relate the probability of appearance of extremely steep (the so-called freak, or rogue waves) to the value of the Benjamin-Feir Index (BFI) that represents the ratio of wave nonlinearity to the spectral width. This ratio appears naturally in the cubic Schrödinger equation that describes evolution of unidirectional narrow-banded wave field. The notion of this index stems from the Benjamin-Feir linear stability analysis of Stokes wave. The application of BFI to evaluate the evolution of wave fields, with non-vanishing amplitudes of sideband disturbances, is investigated using the Zakharov equation as the theoretical model. The present analysis considers a 3-wave system for which the exact analytical solution of the model equations is available.