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The Drastic Role of Beyond Nearest-Neighbor Interactions on Two-Dimensional Dynamical Lattices: A Case Example

In the present work, we highlight the significant effect that the simplest beyond nearest neighbor interactions can have on two-dimensional dynamical lattices. To do so, we select as our case example the closest further neighbor, namely the diagonal one, and a prototypical nonlinear lattice, the discrete nonlinear Schrodinger equation. Varying solely the strength of this extra neighbor interaction, we see examples of (a) destabilization of states that start out as stable in the nearest neighbor limit; (b) stabilization of states that start out as unstable in that limit; (c) bifurcation of novel states that do not exist in the nearest neighbor case. These dramatic changes are first theoretically highlighted through an analysis of a reduction of the problem to a few excited sites and the associated set of conditions that govern their existence and their dynamical stability. Then, they are corroborated numerically through fixed point computations, spectral analysis and nonlinear dynamical evolution simulations.