Exciting extreme events in the damped and AC-driven NLS equation through plane-wave initial conditions

We investigate, by direct numerical simulations, the dynamics of damped and forced nonlinear Schr\"odinger (NLS) equation in presence a time periodic forcing for certain parametric regimes. It is thus revealed, that wave-number plane-wave initial condition dictates number emerged Peregrine type rogue waves at early stages modulation instability. The formation these events gives rise to same transient "triangular" spatio-temporal patterns, each which reminiscent one emerging integrable NLS its semiclassical limit, when supplemented with vanishing conditions. find $L^2$-norm spatial derivative $L^4$-norm detect appearance as local extrema their evolution. impact various parameters noisy perturbations affecting above behavior also discussed. long behaviour, regimes where extreme wave are observable, explained terms global attractor possessed system asymptotic orbital stability spatially uniform continuous solutions.