KAM tori in 1D random discrete nonlinear Schrödinger model?

Details KAM tori in 1D random discrete nonlinear Schrödinger model? KAM tori in 1D random discrete nonlinear Schrödinger model?

2010-07-12

arxiv.org/abs/1007.1912v1

EPL 91 (2010) 50001

Nonlinear Sciences

Pattern Formation and Solitons

7 pages, 3 figures, submitted to EPL

Scientific paper

10.1209/0295-5075/91/50001

We suggest that KAM theory could be extended for certain infinite-dimensional systems with purely discrete linear spectrum. We provide empirical arguments for the existence of square summable infinite-dimensional invariant tori in the random discrete nonlinear Schr\"odinger equation, appearing with a finite probability for a given initial condition with sufficiently small norm. Numerical support for the existence of a fat Cantor set of initial conditions generating almost-periodic oscillations is obtained by analyzing (i) sets of recurrent trajectories over successively larger time scales, and (ii) finite-time Lyapunov exponents. The norm region where such KAM-like tori may exist shrinks to zero when the disorder strength goes to zero and the localization length diverges.

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