Extreme wave events at the initial stage of the modulational instability in a forced and damped wave medium

The modulational instability of waves in a medium under the action of an external monochromatic force and dissipation is considered. The model which describes the nonlinear stage of the modulation instability was constructed with using methods of so-called $S$-theory. Following this theory, we take into account only interactions between modes with slowly varying relative phases. Within framework of this model, it is shown that the waves satisfying the criterion of abnormality appears as a rule at the initial stage of the instability. With further development of the process, the abnormal waves can be also observed but their amplitudes decrease gradually with decreasing of the average wave height.