Stochastic pumping of nonlinear modulated waves

The stochastic nonlinear Schrödinger equation with time- and space-correlated forcing in the form of additive noise is used for modeling the nonlinear evolution of modulationally unstable irregular waves. The additive noise leads to the growth of the wave energy giving rise to abnormally high waves (rogue waves). In the nonlinear regime the increase of the average wave energy occurs much more slowly as compared to the linear regime, but the probability of rogue waves greatly increases during the transient stage of developing modulational instability. The stochastic noise leads to softening of the evolution of all spectral and statistical characteristics, and can significantly change the variation of the fourth statistical moment, but reduces the peak probability of extreme waves just a bit. The results are discussed in application to the wind-generated surface waves in the ocean, where the considered mechanism is responsible for stochastic fluctuations of the surface pressure.