Landau damping and coherent structures in narrow-banded 1+1 deep water gravity waves

We study the modulational instability in surface gravity waves with random phase spectra. Starting from the nonlinear Schrödinger equation and using the Wigner-Moyal transform, we study the stability of the narrow-banded approximation of a typical wind-wave spectrum, i.e., the JONSWAP spectrum. By performing numerical simulations of the nonlinear Schrödinger equation we show that in the unstable regime, the nonlinear stage of the modulational instability is responsible for the formation of coherent structures. Furthermore, a Landau-type damping, due to the incoherence of the waves, whose role is to provide a stabilizing effect against the modulational instability, is both analytically and numerically discussed.