Nonlinear dynamics and chaos in multidimensional disordered Hamiltonian systems

TL;DR

This study investigates how nonlinearity affects chaos and localization in multidimensional disordered Hamiltonian systems, revealing persistent chaos and fluctuations that influence thermalization.

Contribution

It provides detailed analysis of chaos evolution in 1D and 2D disordered nonlinear Schrödinger systems using Lyapunov exponents and deviation vectors, highlighting persistent chaos without crossover to regular motion.

Findings

Lyapunov exponent decreases as a power law over time.

No crossover to regular motion behavior observed.

Chaotic hotspots fluctuate and aid in system thermalization.

Abstract

We study the chaotic behavior of multidimensional Hamiltonian systems in the presence of nonlinearity and disorder. It is known that any localized initial excitation in a large enough linear disordered system spreads for a finite amount of time and then halts forever. This phenomenon is called Anderson localization (AL). What happens to AL when nonlinearity is introduced is an interesting question which has been considered in several studies over the past decades. However, the characteristics and the asymptotic fate of such evolutions still remain an issue of intense debate due to their computational difficulty, especially in systems of more than one spatial dimension. As the spreading of initially localized wave packets is a non-equilibrium thermalization process related to the ergodic and chaotic properties of the system, in our work we investigate the properties of chaos studying the behavior of observables related to the system's tangent dynamics. In particular, we consider the disordered discrete nonlinear Schr\"odinger (DDNLS) equation of one (1D) and two (2D) spatial dimensions. We present detailed computations of the time evolution of the system's maximum Lyapunov exponent (MLE--$\Lambda$), and the related deviation vector distribution (DVD). We find that although the systems' MLE decreases in time following a power law $t^{\alpha_\Lambda}$ with $\alpha_\Lambda <0$ for both the weak and strong chaos regimes, no crossover to the behavior $\Lambda \propto t^{-1}$ (which is indicative of regular motion) is observed. In addition, the analysis of the DVDs reveals the existence of random fluctuations of chaotic hotspots with increasing amplitudes inside the excited part of the wave packet, which assist in homogenizing chaos and contribute to the thermalization of more lattice sites.