Wave chaos in the non-equilibrium dynamics of the Gross-Pitaevskii equation

TL;DR

This paper demonstrates that the Gross-Pitaevskii equation, modeling Bose-Einstein condensates, exhibits chaotic wave dynamics with positive Lyapunov exponents, impacting the understanding of BEC behavior and the limits of mean-field theory.

Contribution

It reveals wave chaos in the GPE with repulsive interactions, distinguishing it from non-integrability and exploring implications for many-body physics and localization.

Findings

Positive Lyapunov exponents observed in BEC dynamics.

Wave chaos occurs in periodic, aperiodic, and disordered potentials.

Implications for the limits of GPE applicability and Anderson localization.

Abstract

The Gross-Pitaevskii equation (GPE) plays an important role in the description of Bose-Einstein condensates (BECs) at the mean-field level. The GPE belongs to the class of non-linear Schr\"odinger equations which are known to feature dynamical instability and collapse for attractive non-linear interactions. We show that the GPE with repulsive non-linear interactions typical for BECs features chaotic wave dynamics. We find positive Lyapunov exponents for BECs expanding in periodic and aperiodic smooth external potentials as well as disorder potentials. Our analysis demonstrates that wave chaos characterized by the exponential divergence of nearby initial wavefunctions is to be distinguished from the notion of non-integrability of non-linear wave equations. We discuss the implications of these observations for the limits of applicability of the GPE, the problem of Anderson localization, and the properties of the underlying many-body dynamics.