Probing quasi-integrability of the Gross-Pitaevskii equation in a harmonic-oscillator potential

TL;DR

This study investigates the quasi-integrable behavior of the one-dimensional Gross-Pitaevskii equation in a harmonic trap by analyzing the spectral properties of a truncated mode expansion, revealing conditions under which the system exhibits quasi-periodic versus ergodic dynamics.

Contribution

The paper introduces a Galerkin approximation to analyze the GPE's dynamics, demonstrating that finite-mode truncations produce quasi-discrete spectra indicative of quasi-integrability, unlike the continuous spectra in ergodic systems.

Findings

Finite-mode dynamics show quasi-discrete power spectra.

Adding random fields does not induce ergodicity.

Infinite potential box leads to continuous spectra and ergodic behavior.

Abstract

Previous simulations of the one-dimensional Gross-Pitaevskii equation (GPE) with repulsive nonlinearity and a harmonic-oscillator trapping potential hint towards the emergence of quasi-integrable dynamics -- in the sense of quasi-periodic evolution of a moving dark soliton without any signs of ergodicity -- although this model does not belong to the list of integrable equations. To investigate this problem, we replace the full GPE by a suitably truncated expansion over harmonic-oscillator eigenmodes (the Galerkin approximation), which accurately reproduces the full dynamics, and then analyze the system's dynamical spectrum. The analysis enables us to interpret the observed quasi-integrability as the fact that the finite-mode dynamics always produces a quasi-discrete power spectrum, with no visible continuous component, the presence of the latter being a necessary manifestation of ergodicity. This conclusion remains true when a strong random-field component is added to the initial conditions. On the other hand, the same analysis for the GPE in an infinitely deep potential box leads to a clearly continuous power spectrum, typical for ergodic dynamics.