Excitation of Peregrine-type waveforms from vanishing initial conditions in the presence of periodic forcing

Abstract

We show by direct numerical simulations that spatiotemporally localized wave forms, strongly reminiscent of the Peregrine rogue wave, can be excited by vanishing initial conditions for the periodically driven nonlinear Schr\"odinger equation. The emergence of the Peregrine-type waveforms can be potentially justified, in terms of the existence and modulational instability of spatially homogeneous solutions of the model, and the continuous dependence of the localized initial data for small time intervals. We also comment on the persistence of the above dynamics, under the presence of small damping effects, and justify, that this behavior should be considered as far from approximations of the corresponding integrable limit.