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We prove spatiotemporal algebraically decaying estimates for the density ofthe solutions of the linearly damped nonlinear Schr\"odinger equation withlocalized driving, when supplemented with vanishing boundary conditions. Theirderivation is made via a scheme, which incorporates suitable weighted Sobolevspaces and a time-weighted energy method. Numerical simulations examining thedynamics (in the presence of physically relevant examples of driver types anddriving amplitude/linear loss regimes), showcase that the suggested decayingrates, are proved relevant in describing the transient dynamics of thesolutions, prior their decay: they support the emergence of waveformspossessing an algebraic space-time localization (reminiscent of the Peregrinesoliton) as first events of the dynamics, but also effectively capture thespace-time asymptotics of the numerical solutions.

The linearly damped nonlinear Schr\"odinger equation with localized driving: spatiotemporal decay estimates and the emergence of extreme wave events