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

While the Ablowitz-Ladik lattice is integrable, the Discrete Nonlinear Schr\"odinger equation, which is more significant for physical applications, is not. 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 small amplitude solutions of the Ablowitz-Ladik system persist in the Discrete Nonlinear Schr\"odinger one. It is shown that the closeness results are also valid in higher dimensional lattices as well as for generalized nonlinearities, the latter is exemplified by generic power-law and saturable ones. For illustration of the applicability of the approach a brief discussion of the results of numerical studies is included, showing that when 1-soliton solution of the Ablowitz-Ladik system is initiated in the Discrete Nonlinear Schr\"odinger system with cubic and saturable nonlinearity, its long-term persistence. Thereby excellent agreement of the numerical findings with the theoretical predictions are obtained.