Non-Nearest-Neighbor Interactions in Nonlinear Dynamical Lattices

TL;DR

This paper analyzes how non-nearest-neighbor interactions influence the existence, phase structure, and stability of solutions in nonlinear lattice models, providing analytical insights supported by numerical validation.

Contribution

It introduces a systematic analytical framework for studying non-nearest-neighbor effects in nonlinear lattices, revealing new solution structures and stability properties.

Findings

Next-nearest-neighbor interactions enable nontrivial phase solutions.

Non-nearest-neighbor interactions can destabilize certain vortex solutions.

Analytical predictions are confirmed by numerical bifurcation analysis.

Abstract

We revisit the theme of non-nearest-neighbor interactions in nonlinear dynamical lattices, in the prototypical setting of the discrete nonlinear Schrodinger equation. Our approach offers a systematic way of analyzing the existence and stability of solutions of the system near the so-called anti-continuum limit of zero coupling. This affords us a number of analytical insights such as the fact that, for instance, next-nearest-neighbor interactions allow for solutions with nontrivial phase structure in infinite one-dimensional lattices; in the case of purely nearest-neighbor interactions, such phase structure is disallowed. On the other hand, such non-nearest-neighbor interactions can critically affect the stability of unstable structures, such as topological charge S=2 discrete vortices. These analytical predictions are corroborated by numerical bifurcation and stability computations.