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

## TL;DR

This paper numerically investigates a driven nonlinear Schrödinger equation with damping, revealing rogue waveforms and diverse spatiotemporal patterns influenced by damping, driving, and initial data.

## Contribution

It introduces a detailed numerical analysis of rogue wave formation and pattern diversity in a damped, driven NLS with various initial conditions and parameters.

## Key findings

- Identification of Peregrine-like rogue waves.
- Dependence of dynamics on driver scales and initial data.
- Effects of damping and driving amplitude on pattern formation.

## Abstract

We perform a numerical study of the initial-boundary value problem, with vanishing boundary conditions, of a driven nonlinear Schr\"odinger equation (NLS) with linear damping and a Gaussian driver. We identify Peregrine-like rogue waveforms, excited by two different types of vanishing initial data decaying at an algebraic or exponential rate. The observed extreme events emerge on top of a decaying support. Depending on the spatial/temporal scales of the driver, the transient dynamics -- prior to the eventual decay of the solutions -- may resemble the one in the semiclassical limit of the integrable NLS, or may, e.g., lead to large-amplitude breather-like patterns. The effects of the damping strength and driving amplitude, in suppressing or enhancing respectively the relevant features, as well as of the phase of the driver in the construction of a diverse array of spatiotemporal patterns, are numerically analyzed.