q-Breathers in Discrete Nonlinear Schroedinger arrays with weak disorder

TL;DR

This paper extends the concept of q-breathers to weakly disordered nonlinear lattices, demonstrating their localized nature, analyzing stability thresholds, and proposing methods to control energy flow in such systems.

Contribution

It generalizes q-breathers to disordered systems, analyzes their stability, and introduces a way to manipulate energy transfer through inhomogeneities.

Findings

q-breathers remain exponentially localized with weak disorder

stability thresholds depend sensitively on disorder realization

energy flow can be controlled by arranging inhomogeneities

Abstract

Nonlinearity and disorder are key players in vibrational lattice dynamics, responsible for localization and delocalization phenomena. $q$-Breathers -- periodic orbits in nonlinear lattices, exponentially localized in the reciprocal linear mode space -- is a fundamental class of nonlinear oscillatory modes, currently found in disorder-free systems. In this paper we generalize the concept of $q$-breathers to the case of weak disorder, taking the Discrete Nonlinear Schr\"{o}dinger chain as an example. We show that $q$-breathers retain exponential localization near the central mode, provided that disorder is sufficiently small. We analyze statistical properties of the instability threshold and uncover its sensitive dependence on a particular realization. Remarkably, the threshold can be intentionally increased or decreased by specifically arranged inhomogeneities. This effect allows us to formulate an approach to controlling the energy flow between the modes. The relevance to other model arrays and experiments with miniature mechanical lattices, light and matter waves propagation in optical potentials is discussed.