Stability limits for modes held in alternating trapping-expulsive potentials

TL;DR

This paper investigates the stability limits of two-dimensional states in systems with periodically switching trapping and expulsive potentials, using numerical simulations and variational approximation, to understand collapse thresholds and parametric resonance effects.

Contribution

It introduces a trapping-expulsion management scheme and predicts stability boundaries and collapse thresholds in nonlinear systems with periodic potential modulation.

Findings

Stability limited by parametric resonance below collapse threshold g<5.85

VA accurately predicts stability boundaries including for g<0

Collapse threshold slightly increased by ~1.5% above known limit

Abstract

We elaborate a scheme of trapping-expulsion management (TEM), in the form of the quadratic potential periodically switching between confinement and expulsion, as a means of stabilization of two-dimensional dynamical states against the backdrop of the critical collapse driven by the cubic self-attraction with strength g. The TEM scheme may be implemented, as spatially or temporally periodic modulations, in optics or BEC, respectively. The consideration is carried out by dint of numerical simulations and variational approximation (VA). In terms of the VA, the dynamics amounts to a nonlinear Ermakov equation, which, in turn, is tantamount to a linear Mathieu equation. Stability boundaries are found as functions of g and parameters of the periodic modulation of the trapping potential. Below the usual collapse threshold, which is known, in the numerical form, as g < 5.85 (in the standard notation), the stability is limited by the onset of the parametric resonance. This stability limit, including the setup with the self-repulsive sign of the cubic term (g < 0), is accurately predicted by the VA. At g > 5.85, the collapse threshold is found with the help of full numerical simulations. The relative increase of the critical value of g above 5.85 is ~ 1.5%, which is a meaningful result, even if its size is small, because the collapse threshold is a universal constant, which is difficult to change.