$q$-Breathers in finite lattices: nonlinearity and weak disorder

TL;DR

This paper extends the concept of $q$-breathers to weakly disordered nonlinear lattices, demonstrating their stability and localization properties, and proposes methods to control energy flow through impurity design.

Contribution

It generalizes $q$-breathers to disordered lattices, analyzing their stability and localization, and introduces a control approach for energy transfer via impurities.

Findings

$q$-breathers remain exponentially localized in disordered lattices.

Stability of $q$-breathers depends on disorder realization and can be controlled.

Impurities can modify the instability threshold and energy flow.

Abstract

Nonlinearity and disorder are the recognized ingredients of the lattice vibrational dynamics, the factors that could be diminished, but never excluded. We generalize the concept of $q$-breathers -- periodic orbits in nonlinear lattices, exponentially localized in the reciprocal linear mode space -- to the case of weak disorder, taking the Fermi-Pasta-Ulan chain as an example. We show, that these nonlinear vibrational modes remain exponentially localized near the central mode and stable, provided the disorder is sufficiently small. The instability threshold depends sensitively on a particular realization of disorder and can be modified by specifically designed impurities. Basing on it, an approach to controlling the energy flow between the modes is proposed. The relevance to other model lattices and experimental miniature arrays is discussed.