Three-Dimensional Nonlinear Lattices: From Oblique Vortices and Octupoles to Discrete Diamonds and Vortex Cubes

TL;DR

This paper introduces new 3D localized topological states in the discrete nonlinear Schrödinger equation, including vortices, dipoles, and skyrmions, with analysis of their stability and evolution.

Contribution

It presents novel 3D localized states with topological structures, including vortex cubes and diamonds, and analyzes their stability in the discrete nonlinear Schrödinger framework.

Findings

Identification of stable and unstable regions for 3D solutions.

Construction of vortex cubes and diamonds with topological features.

Comparison of 3D solutions with 2D counterparts.

Abstract

We construct a variety of novel localized states with distinct topological structures in the 3D discrete nonlinear Schr{\"{o}}dinger equation. The states can be created in Bose-Einstein condensates trapped in strong optical lattices, and crystals built of microresonators. These new structures, most of which have no counterparts in lower dimensions, range from purely real patterns of dipole, quadrupole and octupole types to vortex solutions, such as "diagonal" and "oblique" vortices, with axes oriented along the respective directions $(1,1,1)$ and $(1,1,0)$. Vortex "cubes" (stacks of two quasi-planar vortices with like or opposite polarities) and "diamonds" (discrete skyrmions formed by two vortices with orthogonal axes) are constructed too. We identify stability regions of these 3D solutions and compare them with their 2D counterparts, if any. An explanation for the stability/instability of most solutions is proposed. The evolution of unstable states is studied as well.