Study on the Behavior of Weakly Nonlinear Water Waves in the Presence of Random Wind Forcing

Specific solutions of the nonlinear Schr"odinger equation, such as the Peregrine breather, are considered to be prototypes of extreme or freak waves in the oceans. An important question is, whether these solutions also exist in the presence of gusty wind. Using the method of multiple scales, a nonlinear Schr"odinger equation is obtained for the case of wind forced weakly nonlinear deep water waves. Thereby, the wind forcing is modeled as a stochastic process. This leads to a stochastic nonlinear Schr"odinger equation, which is calculated for different wind regimes. For the case of wind forcing which is either random in time or random in space, it is shown that breather type solutions such as the Peregrine breather occur even in strong gusty wind conditions.