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

by T Bland, N G Parker, N P Proukakis, B A Malomed
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.
Reference:
Probing quasi-integrability of the GrosstextendashPitaevskii equation in a harmonic-oscillator potential,
T Bland, N G Parker, N P Proukakis, B A Malomed,
Journal of Physics B: Atomic, Molecular and Optical Physics, 51, 205303, 2018.
Bibtex Entry:
@article{Bland_2018,
	doi = {10.1088/1361-6455/aae0ba},
	url = {https://doi.org/10.1088/1361-6455/aae0ba},
	year = 2018,
	month = {sep},
	publisher = {{IOP} Publishing},
	volume = {51},
	number = {20},
	pages = {205303},
	author = {T Bland and N G Parker and N P Proukakis and B A Malomed},
	title = {Probing quasi-integrability of the Gross{textendash}Pitaevskii equation in a harmonic-oscillator potential},
	journal = {Journal of Physics B: Atomic, Molecular and Optical Physics},
	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.}
}