Serge Aubry - Physics Department . Technion

Numerical arguments for KAM tori as limit profile
of an initially localized wavepacket
1-The participation ratio of the energy distribution does not
diverge.
2- The amplitude of the at any site does not decay to zero
though it oscillates indefinetely.
3-The largest Lyapounov exponent initially drops by 2 or 3
orders of magnitude while the wavepacket does not spread to
small amplitude. It continues to decay but slower and slower.
4- Simultaneously, the Fourier spectrum of the atomic motion
near center from broad band becomes narrow band with thin
line widths
Conclusion: The attractor of the system (limit profile)
might be a KAM tori ?
but very slow convergence (Arnol’d diffusion?)
What about the evolution for times beyond those numerically available?
Need a mathematical proof?
Serge Aubry, Aubry,
LLB, FRANCE