Discrete localized states and localization dynamics in discrete nonlinear Schrödinger equations

Dynamics of two-dimensional discrete structures is studied in the framework of the generalized two-dimensional discrete nonlinear Schrödinger equation. The nonlinear coupling in the form of the Ablowitz-Ladik non-linearity is taken into account. Stability properties of the stationary solutions are examined. The importance of the existence of stable immobile solitons in the two-dimensional dynamics of the travelling pulses is demonstrated. The process of forming narrow states from initially broad standing or moving excitations through the quasi-collapse mechanism is analyzed. The typical scenario of the two-dimensional quasi-collapse of a moving intense pulse is the formation of pinned narrow spikes.