# Spatiotemporal algebraically localized waveforms for a nonlinear Schr\"odinger model with gain and loss

## TL;DR

This paper analyzes the behavior of solutions to a nonlinear Schrödinger model with gain and loss, revealing the formation of algebraically decaying waveforms similar to rogue waves across various regimes.

## Contribution

It provides an analytical description of dynamical regimes and demonstrates the emergence of spatiotemporal algebraically decaying waveforms in a nonintegrable NLS model with gain and loss.

## Key findings

- Identification of regimes for finite-time collapse, decay, and global existence.
- Discovery of algebraically decaying waveforms resembling rogue waves.
- Differences in dynamics between vanishing and non-vanishing initial conditions.

## Abstract

We consider the asymptotic behavior of the solutions of a nonlinear Schr\"odinger (NLS) model incorporating linear and nonlinear gain/loss. First, we describe analytically the dynamical regimes (depending on the gain/loss strengths), for finite-time collapse, decay, and global existence of solutions in the dynamics. Then, for all the above parametric regimes, we use direct numerical simulations to study the dynamics corresponding to algebraically decaying initial data. We identify crucial differences between the dynamics of vanishing initial conditions, and those converging to a finite constant background: in the former (latter) case we find strong (weak) collapse or decay, when the gain/loss parameters are selected from the relevant regimes. One of our main results, is that in all the above regimes, non-vanishing initial data transition through spatiotemporal, algebraically decaying waveforms. While the system is nonintegrable, the evolution of these waveforms is reminiscent to the evolution of the Peregrine rogue wave of the integrable NLS limit. The parametric range of gain and loss for which this phenomenology persists is also touched upon.