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

Lookup NU author(s): Dr Tom Bland, Professor Nick ParkerORCiD, Professor Nikolaos ProukakisORCiD



This work is licensed under a Creative Commons Attribution 4.0 International License (CC BY 4.0).


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.

Publication metadata

Author(s): Bland T, Parker NG, Proukakis NP, Malomed BA

Publication type: Article

Publication status: Published

Journal: Journal of Physics B

Year: 2018

Volume: 51

Pages: 205303

Print publication date: 28/09/2018

Online publication date: 28/09/2018

Acceptance date: 04/09/2018

Date deposited: 28/09/2018

ISSN (print): 0953-4075

ISSN (electronic): 1361-6455

Publisher: Institute of Physics


DOI: 10.1088/1361-6455/aae0ba


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Funder referenceFunder name
EP/M005127/1Commonwealth Scholarship Commission in the United Kingdom