Serge Aubry - Physics Department . Technion

VII Summary and concluding remarks
The possible divergency of the second moment does not implies energy spreading
as believed in early publications.
Numerical observations (over times which are not too long
for the above reasons) shows
1- There are initial conditions with practically no diffusion from the beginning
(quasiperiodic?) and others which looks (initially) chaotic.
2- the participation number does not (clearly) diverge in any case. Moreover,
it is rigorously proven it cannot diverge in DNLS models with norm conservation
and large enough amplitude wave packet.
3- The same qualitative behavior for the spreading of an initially localized
wavepacket in DNLS models with norm conservation and quartic KG models
without norm conservation
4-Conjecture: Unless the wavepacket corresponds initialy to a quasiperiodic
solution, it converges (slowly) to a limit profile which is a quasiperiodic and
spatially localized solution (KAM torus).
This profile may not have second moment or may have a second moment?
(open question)
Serge Aubry, Aubry,
LLB, FRANCE