2D solutions of the hyperbolic discrete nonlinear Schr\"odinger equation

TL;DR

This paper derives and analyzes various stationary solutions of the two-dimensional hyperbolic discrete nonlinear Schrödinger equation, exploring their bifurcations, stability, and dynamical behavior.

Contribution

It introduces a systematic method to find and classify multiple solution types and their bifurcations in the 2D hyperbolic DNLS equation, including stability analysis.

Findings

Identification of nine primary solution types.

Discovery of four main bifurcation events.

Analysis of stability and dynamical evolution of solutions.

Abstract

We derive stationary solutions to the two-dimensional hyperbolic discrete nonlinear Schr\"odinger (HDNLS) equation by starting from the anti-continuum limit and extending solutions to include nearest-neighbor interactions in the coupling parameter. We use pseudo-arclength continuation to capture the relevant branches of solutions and explore their corresponding stability and dynamical properties (i.e., their fate when unstable). We focus 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 nonzero sites. The chosen configurations are found to merge into four distinct bifurcation events. We unveil the nature of the bifurcation phenomena and identify the critical points associated with these states and also explore the consequences of the termination of the branches on the dynamical phenomenology of the model.