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While the Ablowitz-Ladik lattice is integrable, the Discrete NonlinearSchr\"odinger equation, which is more significant for physical applications, isnot. We prove closeness of the solutions of both systems in the sense of a"continuous dependence" on their initial data in the $l^2$ and $l^{\infty}$metrics. The most striking relevance of the analytical results is that smallamplitude solutions of the Ablowitz-Ladik system persist in the DiscreteNonlinear Schr\"odinger one. It is shown that the closeness results are alsovalid in higher dimensional lattices as well as for generalised nonlinearities.For illustration of the applicability of the approach, a brief numerical studyis included, showing that when the 1-soliton solution of the Ablowitz-Ladiksystem is initiated in the Discrete Nonlinear Schr\"odinger system with cubicand saturable nonlinearity, it persists for long-times. Thereby excellentagreement of the numerical findings with the theoretical predicti ions isobtained.

The closeness of the Ablowitz-Ladik lattice to the Discrete Nonlinear Schr\"odinger equation