Intraband discrete breathers in disordered nonlinear systems. II. Localization

Abstract

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 accurate 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, in agreement with a theorem by Albanese and Fröhlich, that the localized DBs exist provided that their frequencies belong to fat Cantor sets (i.e., with 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. Our numerical methods provide a much wider set of exact solutions which are multisite breathers and suggest conjectures extending the existing theorems. The physical implications of the existence of IDBs and possible applications for glasses and the persistent spectral hole burning are discussed.