Emergent soliton-like solutions in the parametrically driven 1-D nonlinear Schr\"odinger equation

TL;DR

This paper numerically explores long-term dynamics of breather solutions in a parametrically driven 1D nonlinear Schrödinger equation, revealing stable, soliton-like excitations with unique behaviors under dissipation.

Contribution

It demonstrates the existence of stable, soliton-like solutions in a driven NLS equation with dissipation, highlighting behaviors not seen in traditional solitons.

Findings

Robust soliton-like excitations travel with constant amplitude and velocity without dissipation.

With dissipation, solitons lose energy but gain speed, a novel behavior.

These solitons are stable against random perturbations.

Abstract

We numerically investigate the long time dynamics of spatially periodic breather solutions of the 1-D nonlinear Schr\"odinger equation under parametric forcing of the form $f(x)=f_0 \exp(iKx)$ along with dissipation. In the absence of dissipation, robust soliton-like excitations are observed that travel with constant amplitude and velocity. With dissipation, these solitons lose energy (and amplitude) yet gain speed - a characteristic not observed in an ordinary soliton. Moreover, these novel solitons are found to be stable against random perturbations.