Kolmogorov turbulence, Anderson localization and KAM integrability

TL;DR

This paper investigates the emergence of Kolmogorov turbulence in finite systems, proposing an analogy with Anderson localization in disordered systems, and analyzes how nonlinear interactions influence the transition between integrability and chaos.

Contribution

It introduces an analytical and numerical framework linking turbulence, Anderson localization, and KAM theory, highlighting the role of nonlinear interactions in energy flow and chaos onset.

Findings

Existence of a chaos border separating KAM integrability and turbulence regimes.

Nonlinear wave interactions can suppress or facilitate energy transfer across scales.

Finite size effects influence the transition between localized and turbulent states.

Abstract

The conditions for emergence of Kolmogorov turbulence, and related weak wave turbulence, in finite size systems are analyzed by analytical methods and numerical simulations of simple models. The analogy between Kolmogorov energy flow from large to small spacial scales and conductivity in disordered solid state systems is proposed. It is argued that the Anderson localization can stop such an energy flow. The effects of nonlinear wave interactions on such a localization are analyzed. The results obtained for finite size system models show the existence of an effective chaos border between the Kolmogorov-Arnold-Moser (KAM) integrability at weak nonlinearity, when energy does not flow to small scales, and developed chaos regime emerging above this border with the Kolmogorov turbulent energy flow from large to small scales.