Obstruction to ergodicity in nonlinear Schrödinger equations with resonant potentials

Area of Science:

• Physics
• Nonlinear Dynamics
• Quantum Mechanics

Background:

• Nonlinear Schrödinger equations (NLSEs) describe various physical phenomena, including wave propagation and Bose-Einstein condensates.
• Ergodicity in dynamical systems is often associated with continuous power spectra, indicating chaotic behavior.
• Trapping potentials can significantly alter the dynamics of NLSEs.

Purpose of the Study:

• To identify specific trapping potentials in NLSEs that lead to nonintegrability without inducing ergodicity.
• To understand the spectral properties of systems with these potentials.
• To explore the relevance of these findings to Bose-Einstein condensates.

Main Methods:

• Analysis of cubic nonlinear Schrödinger equations (NLSEs) with specific trapping potentials.
• Investigation of systems with equidistant energy spectra, such as harmonic-oscillator traps.
• Analytical explanation for spectral features in the weak nonlinearity regime.
• Numerical simulations with random initial conditions for the strongly nonlinear regime.

Main Results:

• A class of trapping potentials was identified that results in nonintegrable NLSEs but prevents ergodic power spectra.
• These potentials exhibit equidistant energy spectra, leading to numerous resonances that enhance nonlinearity.
• Dynamical solutions show narrow, evenly spaced spikes in power spectra, deviating from continuous ergodic spectra.
• An analytical theory explains these spectral features for weak nonlinearity, and numerical simulations confirm them for strong nonlinearity.

Conclusions:

• Equidistant energy spectra in trapping potentials create unique spectral signatures in NLSEs, hindering ergodicity.
• These findings have direct implications for understanding Bose-Einstein condensates (BECs) described by Gross-Pitaevskii equations (GPEs).
• The identified potentials are relevant to 1D, 2D, and 3D GPEs, including quintic and two-component systems.