  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.
