Extended nonlinear waves in multidimensional dynamical lattices

We explore spatially extended dynamical states in the discrete nonlinear Schrödinger lattice in two- and three-dimensions, starting from the anti-continuum limit. We first consider the "core" of the relevant states (either a two-dimensional "tile" or a three-dimensional "stone"), and examine its stability analytically. The predictions are corroborated by numerical results. When the core is stable, we propose a method allowing the extension of the structure to as many sites as may be desired. In this way, various patterns of excited sites can be formed. The stability of the full extended nonlinear structures is studied numerically, which yields instability thresholds for such structures, which are attained with the increase of the lattice coupling constant. Finally, in cases of instability, direct numerical simulations are used to elucidate the evolution of the pattern; it is found that, typically, the unstable extended nonlinear pattern breaks up in an oscillatory way, leading to "lattice turbulence".