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

KAM tori in 1D random discrete nonlinear Schr\"odinger model?