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We perform a numerical study of the initial-boundary value problem, withvanishing boundary conditions, of a driven nonlinear Schr\"odinger equation(NLS) with linear damping and a Gaussian driver. We identify Peregrine-likerogue waveforms, excited by two different types of vanishing initial datadecaying at an algebraic or exponential rate. The observed extreme eventsemerge on top of a decaying support. Depending on the spatial/temporal scalesof the driver, the transient dynamics -- prior to the eventual decay of thesolutions -- may resemble the one in the semiclassical limit of the integrableNLS, or may, e.g., lead to large-amplitude breather-like patterns. The effectsof the damping strength and driving amplitude, in suppressing or enhancingrespectively the relevant features, as well as of the phase of the driver inthe construction of a diverse array of spatiotemporal patterns, are numericallyanalyzed.

Extreme wave events for a nonlinear Schr\"odinger equation with linear damping and Gaussian driving