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We propose a consideration of the properties of the two-dimensionalAblowitz-Ladik discretization of the ubiquitous nonlinear Schrodinger (NLS)model. We use singularity confinement techniques to suggest that the relevantdiscretization should not be integrable. More importantly, we identify theprototypical solitary waves of the model and examine their stability,illustrating the remarkable feature that near the continuum limit, thisdiscretization leads to the absence of collapse and complete spectral wavestability, in stark contrast to the standard discretization of the NLS. We alsobriefly touch upon the three-dimensional case and generalizations of ourconsiderations therein, and also present some more exotic solutions of themodel, such as exact line solitons and discrete vortices.

The higher-dimensional Ablowitz-Ladik model: from (non-)integrability and solitary waves to surprising collapse properties and more exotic solutions