Chaos and localization in the Discrete Nonlinear Schr\"odinger Equation

TL;DR

This paper investigates the chaotic behavior and localization phenomena in the one-dimensional discrete nonlinear Schrödinger equation, revealing how localized states influence chaos and long-term dynamics across different energy densities.

Contribution

It provides a detailed analysis of Lyapunov spectra and covariant vectors, highlighting the role of discrete breathers in chaos and relaxation processes in the model.

Findings

Maximal Kolmogorov-Sinai entropy occurs at infinite temperature.

Localized states induce very small Lyapunov exponents, indicating long time-scales.

Discrete breathers affect the interaction with system degrees of freedom.

Abstract

We analyze the chaotic dynamics of a one-dimensional discrete nonlinear Schr\"odinger equation. This nonintegrable model, ubiquitous in several fields of physics, describes the behavior of an array of coupled complex oscillators with a local nonlinear potential. We explore the Lyapunov spectrum for different values of the energy density, finding that the maximal value of the Kolmogorov-Sinai entropy is attained at infinite temperatures. Moreover, we revisit the dynamical freezing of relaxation to equilibrium, occurring when large localized states (discrete breathers) are superposed to a generic finite-temperature background. We show that the localized excitations induce a number of very small, yet not vanishing, Lyapunov exponents, which signal the presence of extremely long characteristic time-scales. We widen our analysis by computing the related Lyapunov covariant vectors, to investigate the interaction of a single breather with the various degrees of freedom.