# Structured light entities, chaos and nonlocal maps

## TL;DR

This paper introduces nonlocal nonlinear maps derived from Green functions that emulate complex spatiotemporal phenomena like solitons and vortex lattices in optical resonators, linking iterative maps to PDEs of Ginzburg-Landau type.

## Contribution

It generalizes classical maps to include convolution integrals, connecting iterative maps with PDEs, and demonstrates their application in modeling optical phenomena.

## Key findings

- Nonlocal maps are equivalent to Ginzburg-Landau PDEs.
- Maps can simulate spatial solitons and vortex lattices.
- Noise helps stabilize entities and reduce artifacts.

## Abstract

Spatial chaos as a phenomenon of ultimate complexity requires the efficient numerical algorithms. For this purpose iterative low-dimensional maps have demonstrated high efficiency. Natural generalization of Feigenbaum and Ikeda maps may include convolution integrals with kernel in a form of Green function of a relevant linear physical system. It is shown that such iterative $nonlocal$ $nonlinear$ $maps$ are equivalent to ubiquitous class of nonlinear partial differential equations of Ginzburg-Landau type. With a Green functions relevant to generic optical resonators these $nonlocal$ $maps$ emulate the basic spatiotemporal phenomena as spatial solitons, vortex eigenmodes breathing via relaxation oscillations mediated by noise, vortex-vortex and vortex-antivortex lattices with periodic location of vortex cores. The smooth multimode noise addition facilitates the selection of stable entities and elimination of numerical artifacts.