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arXiv · 1007.1912

KAM tori in 1D random discrete nonlinear Schrödinger model?

Magnus Johansson·Georgios Kopidakis·Serge Aubry
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Abstract

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 Schrödinger 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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Magnus Johansson, Georgios Kopidakis, Serge Aubry. 2010-07-12. KAM tori in 1D random discrete nonlinear Schrödinger model?. https://doi.org/10.1209/0295-5075%2F91%2F50001

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