Intraband discrete breathers in disordered nonlinear systems. II: Localization
type: Item
instance of: scholarly article
title: Intraband discrete breathers in disordered nonlinear systems. II: Localization
zbMATH Open document ID: 0974.70018
DOI: 10.1016/S0167-2789(99)00220-1
author: Georgios Kopidakis
author: S. Aubry
published in: Physica D
publication date: 2001-12-12T00:00:00Z
review text: [For part I see the authors, ibid. 130, No.~3-4, 155-186 (1999; Zbl 0943.70016).] From the summary: We find spatially localized, time-periodic solutions (discrete breathers or DBs) in disordered nonlinear systems with frequency inside the linear phonon spectrum under conditions that strictly prohibit their existence in their periodic counterparts. For that purpose, we develop a new in situ method for the calculation of these solutions which does not make use of any continuation from an anticontinuous limit. Using this method, we demonstrate that intraband localized modes (intraband discrete breathers or IDBs) at a given site with frequencies inside the discrete linear spectrum do exist, provided these frequencies do not belong to forbidden resonance gaps. Since there is a dense set of resonant frequencies, we illustrate numerically that the localized DBs exist provided that their frequencies belong to fat Cantor sets (i.e., with a finite measure). Such a set contains as accumulation points the linear frequency of the normal mode at the occupied site. We check that many of these solutions are linearly stable, and conjecture that their frequency belongs to another smaller fat Cantor set.
Mathematics Subject Classification ID: 70K99
zbMATH DE Number: 1515736
zbMATH Keywords: finite measure
zbMATH Keywords: intraband discrete breathers
zbMATH Keywords: disordered nonlinear systems
zbMATH Keywords: linear phonon spectrum
zbMATH Keywords: resonant frequencies
zbMATH Keywords: fat Cantor sets
zbMATH Keywords: accumulation points
MaRDI profile type: Publication
cites work: Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators
cites work: Intraband discrete breathers in disordered nonlinear systems. I: Delocalization
cites work: Perturbation theory for periodic orbits in a class of infinite dimensional Hamiltonian systems
cites work: Breathers in nonlinear lattices: numerical calculation from the anticontinuous limit
cites work: Breathers in nonlinear lattices: existence, linear stability and quantization
cites work: Q4400439
cites work: Localization in disordered, nonlinear dynamical systems
description: scientific article; zbMATH DE number 1515736