Energy localization on q-tori, long term stability and the interpretation of FPU recurrences

TL;DR

This paper investigates the FPU paradox by introducing the concept of q-tori, which are exponentially localized solutions, to unify existing explanations involving natural packets and q-breathers, enhancing understanding of energy localization and stability.

Contribution

It introduces q-tori as a new class of solutions that unify previous approaches, offering a comprehensive framework for understanding energy localization in FPU lattices.

Findings

Q-tori are exponentially localized solutions on low-dimensional tori.

Q-tori stability properties reconcile natural packets and q-breathers.

The approach provides a more complete explanation of the FPU paradox.

Abstract

We focus on two approaches that have been proposed in recent years for the explanation of the so-called FPU paradox, i.e. the persistence of energy localization in the `low-q' Fourier modes of Fermi-Pasta-Ulam nonlinear lattices, preventing equipartition among all modes at low energies. In the first approach, a low-frequency fraction of the spectrum is initially excited leading to the formation of `natural packets' exhibiting exponential stability, while in the second, emphasis is placed on the existence of `q-breathers', i.e periodic continuations of the linear modes of the lattice, which are exponentially localized in Fourier space. Following ideas of the latter, we introduce in this paper the concept of `q-tori' representing exponentially localized solutions on low-dimensional tori and use their stability properties to reconcile these two approaches and provide a more complete explanation of the FPU paradox.