Low-dimensional q-Tori in FPU Lattices: Dynamics and Localization Properties

TL;DR

This paper investigates low-dimensional q-tori in FPU lattices, employing Poincare-Lindstedt series to analyze energy localization and propagation patterns in both alpha and beta models, revealing exponential localization profiles and extensive properties.

Contribution

It introduces explicit quasi-periodic representations of q-tori in the alpha FPU model and analyzes localization patterns for various initial excitations, extending previous beta model studies.

Findings

Exponential energy localization profiles for low-frequency mode packets.

Propagation of non-zero terms explains localization patterns.

Some q-tori solutions exhibit properties independent of system size.

Abstract

This is a continuation of our study concerning q-tori, i.e. tori of low dimensionality in the phase space of nonlinear lattice models like the Fermi-Pasta-Ulam (FPU) model. In our previous work we focused on the beta FPU system, and we showed that the dynamical features of the q-tori serve as an interpretational tool to understand phenomena of energy localization in the FPU space of linear normal modes. In the present paper i) we employ the method of Poincare - Lindstedt series, for a fixed set of frequencies, in order to compute an explicit quasi-periodic representation of the trajectories lying on q-tori in the alpha model, and ii) we consider more general types of initial excitations in both the alpha and beta models. Furthermore we turn into questions of physical interest related to the dynamical features of the q-tori. We focus on particular q-tori solutions describing low-frequency `packets' of modes, and excitations of a small set of modes with an arbitrary distribution in q-space. In the former case, we find formulae yielding an exponential profile of energy localization, following an analysis of the size of the leading order terms in the Poincare - Lindstedt series. In the latter case, we explain the observed localization patterns on the basis of a rigorous result concerning the propagation of non-zero terms in the Poincare - Lindstedt series from zeroth to subsequent orders. Finally, we discuss the extensive (i.e. independent of the number of degrees of freedom) properties of some q-tori solutions.