A refined empirical stability criterion for nonlinear Schroedinger solitons under spatiotemporal forcing

TL;DR

This paper refines the stability criterion for nonlinear Schrödinger solitons under spatiotemporal forcing, extending its applicability and confirming the results through direct simulations and phase portrait analysis.

Contribution

A new, generally applicable stability criterion for NLS solitons under complex forcing, validated by simulations and phase space analysis.

Findings

The refined criterion applies to harmonic and biharmonic forcing.

Soliton instability correlates with negative slope in the stability curve p(v).

Negative rotation in phase portraits indicates soliton instability.

Abstract

We investigate the dynamics of travelling oscillating solitons of the cubic NLS equation under an external spatiotemporal forcing of the form $f(x,t) = a \exp[iK(t)x]$. For the case of time-independent forcing a stability criterion for these solitons, which is based on a collective coordinate theory, was recently conjectured. We show that the proposed criterion has a limited applicability and present a refined criterion which is generally applicable, as confirmed by direct simulations. This includes more general situations where $K(t)$ is harmonic or biharmonic, with or without a damping term in the NLS equation. The refined criterion states that the soliton will be unstable if the "stability curve" $p(\v)$, where $p(t)$ and $\v(t)$ are the normalized momentum and the velocity of the soliton, has a section with a negative slope. Moreover, for the case of constant $K$ and zero damping we use the collective coordinate solutions to compute a "phase portrait" of the soliton where its dynamics is represented by two-dimensional projections of its trajectories in the four-dimensional space of collective coordinates. We conjecture, and confirm by simulations, that the soliton is unstable if a section of the resulting closed curve on the portrait has a negative sense of rotation.