2D solutions of the hyperbolic discrete nonlinear Schrödinger equation

Abstract

Stationary solutions to the two-dimensional hyperbolic discrete nonlinear Schrödinger equation are derived by starting from the anti-continuum limit and extending solutions to include nearest-neighbor interactions in the coupling parameter. Pseudo-arclength continuation is used to capture the relevant most fundamental branches of localized solutions, and their corresponding stability and dynamical properties (i.e. their fate when unstable) are explored. The focus is on nine primary types of solutions: single site, double site in- and out-of-phase, squares with four sites in-phase and out-of phase in each of the vertical and horizontal directions, four sites out-of-phase arranged in a line horizontally, and two additional solutions having respectively six and eight excited sites. The chosen configurations are found to merge into four distinct bifurcation events. The nature of the bifurcation phenomena is unveiled, typically involving saddle-center collisions and pairwise disappearances of the branches. Finally, the consequences of the termination of the branches on the dynamical phenomenology of the model are explored. When the branches are unstable for small coupling values, they may often dynamically lead into a single site branch. For larger coupling values where no stable branches exist, the solutions are typically found to lead to dispersion involving one or more ‘masses’ dispersing the energy around a central core.