  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.
