KAM tori and absence of diffusion of a wave packet in the 1D random DNLS model
LLB, CEA Saclay and MPIPKS, Dresden
Mon, Jun. 14th 2010, 14:00
Salle Claude Itzykson, Bât. 774, Orme des Merisiers
When nonlinearity is added to an infinite system with purely discrete linear spectrum, the Anderson modes become coupled one with each other by terms of higher order than linear allowing energy exchange between them. It is generally believed, on the basis of numerical simulations in such systems, that any initial wave packet with finite energy spreads down chaotically to zero amplitude with second moment diverging as a power law of time, slower than standard diffusion (subdiffusion). We present results contrary to this general belief which suggest that despite the fact that there may exist initially some substantial spreading, there is no (sub)diffusion, at least in many cases. \par
In the random Discrete Nonlinear Schrödinger (DNLS) model often considered as a prototype for studying this problem, we prove that if the wave packet has a large enough initial amplitude, a non-vanishing part of the initial energy must remain localized forever (self-trapping). Otherwise, we conjecture on the base of both empirical and numerical investigations that KAM theory may still hold in infinite systems under two conditions which are (1) the linearized spectrum is purely discrete and (2) the considered solutions are square summable not too large in amplitude. There are initial conditions which can be found with finite probability which generate (nonspreading) infinite dimension tori (almost periodic solutions) forming a fat Cantor set in (projected) phase space. \par
There are however initial wave packets chosen outside the fat Cantor set of KAM tori which look initially chaotic and start to spread. We present arguments suggesting that self-organisation may occur after a sufficiently long time, so that spreading stops. The limit state could be a strange attractor with long spatial tail, marginal chaos and singular continuous spectrum.