KAM tori in 1D random discrete nonlinear Schr\"odinger model?

TL;DR

This paper explores the potential extension of KAM theory to infinite-dimensional systems, providing empirical evidence for the existence of invariant tori in a random discrete nonlinear Schrödinger model, supported by numerical analysis.

Contribution

It offers the first empirical evidence suggesting KAM tori can exist in infinite-dimensional, disordered quantum systems, extending classical theory.

Findings

Existence of invariant tori with finite probability in the model.

Presence of a fat Cantor set of initial conditions leading to almost-periodic oscillations.

The region where KAM-like tori exist shrinks as disorder decreases.

Abstract

We suggest that KAM theory could be extended for certain infinite-dimensional systems with purely discrete linear spectrum. We provide empirical arguments for the existence of square summable infinite-dimensional invariant tori in the random discrete nonlinear Schr\"odinger equation, appearing with a finite probability for a given initial condition with sufficiently small norm. Numerical support for the existence of a fat Cantor set of initial conditions generating almost-periodic oscillations is obtained by analyzing (i) sets of recurrent trajectories over successively larger time scales, and (ii) finite-time Lyapunov exponents. The norm region where such KAM-like tori may exist shrinks to zero when the disorder strength goes to zero and the localization length diverges.