  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.
