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We point out a class of trapping potentials in nonlinear Schrödinger equations that make them non-integrable, but prevent the emergence of power spectra associated with ergodicity. The potentials are characterized by equidistant energy spectra (e.g., the harmonic-oscillator trap), and therefore by a large number of resonances enhancing the nonlinearity. In a broad range of dynamical solutions, spanning the regimes of both weak and strong nonlinearity, the power spectra are shaped as narrow (quasi-discrete) evenly spaced spikes, unlike generic truly continuous (ergodic) spectra. We develop an analytical explanation for the emergence of these spectral features in the case of weak nonlinearity. In the strongly nonlinear regime, the presence of such structures is tracked numerically by performing simulations with random initial conditions. Some potentials that prevent ergodicity in this manner are of direct relevance to Bose-Einstein condensates: they naturally enter 1D, 2D and 3D Gross-Pitaevskii equations (GPEs), the quintic version of these equations, and a two-component GPE system.

We numerically investigate spherically symmetric collapses in the Gross-Pitaevskii equation with attractive nonlinearity in a harmonic potential. Even below threshold for direct collapse, the wave function bounces off from the origin and may eventually become singular after a number of oscilla- tions in the trapping potential. This is reminiscent of the evolution of Einstein gravity sourced by a scalar field in Anti-de Sitter space where collapse corresponds to black hole formation. We carefully examine the long time evolution of the wave function for continuous families of initial states in order to sharpen out this qualitative coincidence which may bring new insights in both directions. On one hand, we comment on possible implications for the so-called Bosenova collapses in cold atom Bose-Einstein condensates. On the other hand, Gross-Pitaevskii provides a toy model to study the relevance of either the resonance conditions or the nonlinearity for the problem of Anti-de Sitter instability., Comment: 9 pages, 7 figures, published version, extended discussions