Obstruction to ergodicity in nonlinear Schr\"{o}dinger equations with resonant potentials

TL;DR

This paper investigates how certain trapping potentials with equidistant energy spectra in nonlinear Schrödinger equations hinder ergodicity by producing narrow, evenly spaced spectral peaks, with implications for Bose-Einstein condensates.

Contribution

It identifies a class of resonant trapping potentials that prevent ergodicity in nonlinear Schrödinger equations and explains the spectral features analytically and numerically.

Findings

Resonant potentials lead to narrow, evenly spaced spectral peaks.

Spectral structures persist across weak and strong nonlinear regimes.

Relevance to Bose-Einstein condensate models like GPEs and their variants.

Abstract

We identify a class of trapping potentials in cubic nonlinear Schr\"{o}dinger equations (NLSEs) 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), which give rise to a large number of resonances enhancing the nonlinearity. In a broad range of dynamical solutions, spanning the regimes in which the nonlinearity may be either weak or strong in comparison with the linear part of the NLSE, 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 appear in 1D, 2D and 3D Gross-Pitaevskii equations (GPEs), the quintic version of these equations, and a two-component GPE system.