Dresden 2009 – wissenschaftliches Programm

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SYAL: Symposium Anderson Localization in Nonlinear and Many-Body Systems

SYAL 1: Anderson Localization in Nonlinear and Many-Body Systems

SYAL 1.2: Hauptvortrag

Montag, 23. März 2009, 14:30–15:00, BAR SCHÖ

Absence of Diffusion in a Fröhlich-Spencer-Wayne model for nonlinear random systems — •Serge Aubry — Laboratoire Leon Brillouin, CEA Saclay, 91191 Gif-sur-Yvette, France

In linear random models with Anderson localization, there is absence of diffusion for any initially localized wavepacket. When nonlinearity is present, the localized Anderson eigenmodes interact by higher order nonlinear terms and then, it is often believed, on the basis of certain numerical observations, that wavepackets exhibit a (slowly) diffuse behavior (subdiffusion). However, there is a great debate to determine whether this behavior is only transient or lasts forever. We shall describe new numerical results on a variation of the standard nonlinear Anderson model, called Fröhlich-Spencer-Wayne (FSW) model, which consists of a random DNLS model with purely nonlinear nearest neighbor interactions (and norm conservation). For that kind of models, a theorem predicts the existence of many non-diffusive quasiperiodic (KAM) solutions at small enough nonlinearity. In agreement with this prediction, we numerically observe that at small nonlinearity, an initial single-site wavepacket does evolve quasiperiodically over very long time and does not spread at all. At larger nonlinearity, the single site initial wavepacket begins to spread chaotically, possibly over a large number of sites, but later the spreading always slows down and seems to stop. Moreover, in that model, for even larger nonlinearity, norm conservation forbids the complete spreading of any wavepacket. Our results support (at least in that family of models) the absence of diffusion for any initially localized wavepacket.