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

TL;DR

This paper establishes algebraic decay estimates for solutions of a damped nonlinear Schrödinger equation with localized driving, revealing transient waveforms similar to Peregrine solitons and analyzing their long-term behavior through numerical simulations.

Contribution

It introduces a novel analytical framework for decay estimates in damped NLS equations with localized driving, connecting theoretical results with numerical observations.

Findings

Decay estimates accurately describe transient dynamics

Emergence of Peregrine-like waveforms as initial events

Numerical results confirm theoretical decay rates and wave localization

Abstract

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