Abstract
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.
Highlights
Integrability, relaxation, and thermalization of many-body systems are intricately-linked key topics of the modern theory of non-equilibrium dynamical systems
Previous numerical simulations based on the 1D Gross-Pitaevskii equation (GPE) have revealed shuttle oscillations of dark solitons in the harmonic-oscillator potential
The comparison of results produced by the Galerkin approximation to those of full GPE simulations shows that the Galerkin approximation for the model with the harmonic-oscillator potential, with M = 16 modes, reproduces the full solutions virtually exactly for indefinitely long evolution times
Summary
Integrability, relaxation, and thermalization of many-body systems are intricately-linked key topics of the modern theory of non-equilibrium dynamical systems. Such an equation is known to be integrable in the 1D free space (including the case of periodic boundary conditions) [12, 13, 14, 15], but not in the presence of the harmonic-oscillator confining potential, which is relevant for modeling actual experiments Even in this case, long-time simulations of the 1D GPE have revealed no conclusive evidence of chaotization [16, 17], which is believed to originate in the experiment from the coupling to transverse degrees of freedom, beyond the limits of the 1D approximation [6]. And perhaps somewhat unexpectedly, we find in this case that the power spectrum of generic trajectories is continuous, in direct contrast to the quasi-discrete spectrum found in the harmonic-oscillator potential, which clearly suggests chaotization (ergodicity) of the dynamics in the box, rather than evolution guided by invariant tori Such behaviour is wholly captured by our finite-mode expansion (without the need for using the known exact box eigenstates of the nonlinear equation [51]).
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