Nonlinear Schr\"odinger Equation with Spatio-Temporal Perturbations

TL;DR

This paper studies the dynamics and stability of solitons in a perturbed nonlinear Schrödinger equation with spatio-temporal forcing, damping, and stabilization, using analytical and numerical methods to derive conditions for soliton stability and lifetime.

Contribution

It develops a collective-coordinate theory for soliton dynamics under complex perturbations and introduces a stability criterion based on the slope of a momentum-velocity curve.

Findings

Solitons perform unidirectional motion with oscillations around a mean trajectory.

Soliton stability is linked to the slope of the P(V) curve, with negative slope indicating instability.

The soliton lifetime correlates with the length of the negative slope branch.

Abstract

We investigate the dynamics of solitons of the cubic Nonlinear Schr\"odinger Equation (NLSE) with the following perturbations: non-parametric spatio-temporal driving of the form $f(x,t) = a \exp[i K(t) x]$, damping, and a linear term which serves to stabilize the driven soliton. Using the time evolution of norm, momentum and energy, or, alternatively, a Lagrangian approach, we develop a Collective-Coordinate-Theory which yields a set of ODEs for our four collective coordinates. These ODEs are solved analytically and numerically for the case of a constant, spatially periodic force $f(x)$. The soliton position exhibits oscillations around a mean trajectory with constant velocity. This means that the soliton performs, on the average, a unidirectional motion although the spatial average of the force vanishes. The amplitude of the oscillations is much smaller than the period of $f(x)$. In order to find out for which regions the above solutions are stable, we calculate the time evolution of the soliton momentum $P(t)$ and soliton velocity $V(t)$: This is a parameter representation of a curve $P(V)$ which is visited by the soliton while time evolves. Our conjecture is that the soliton becomes unstable, if this curve has a branch with negative slope. This conjecture is fully confirmed by our simulations for the perturbed NLSE. Moreover, this curve also yields a good estimate for the soliton lifetime: the soliton lives longer, the shorter the branch with negative slope is.