  We discuss the long time behaviour of a finite energy wave packet in
nonlinear Hamiltonians on infinite lattices at arbitrary dimension, exhibiting
linear Anderson localization. Strong arguments both mathematical and numerical,
suggest for infinite models that small amplitude wave packets may generate
stationary quasiperiodic solutions (KAM tori) almost undistinguishable from
linear wave packets. The probability of this event is non vanishing at small
enough amplitude and goes to unity at amplitude zero. Most other wave packets
(non KAM tori) are chaotic. We discuss the Arnold diffusion conjecture
(recently partially proven) and propose a modified Boltzmann statistics for
wave packets valid in generic models. The consequence is that the probability
that a chaotic wave packet spreads to zero amplitude is zero. It must always
remain focused around one or few chaotic spots which moves randomly over the
whole system and generates subdiffusion.We study a class of Ding Dong models
also generating subdiffusion where the nonlinearities are replaced by hard core
potentials. Then we prove rigorously that spreading is impossible for any
initial wave packet.
