# Excitable wave patterns in temporal systems with two long delays

## TL;DR

This paper demonstrates that excitable wave patterns similar to those in spatial systems can emerge in time-delay systems, specifically in a semiconductor laser with two feedback loops, revealing new causality-related features.

## Contribution

It introduces the concept that complex excitable wave phenomena can be encoded in the dynamics of systems with two long delays, bridging spatial and temporal pattern formation.

## Key findings

- Experimental observation of wave pattern transitions in a laser system
- Numerical simulations confirming the observed phenomena
- Identification of causality-related features unique to delayed systems

## Abstract

Excitable waves arise in many spatially-extended systems of either biological, chemical, or physical nature due to the interplay between local reaction and diffusion processes. Here we demonstrate that similar phenomena are encoded in the time-dynamics of an excitable system with two, hierarchically long delays. The transition from 1D localized structures to curved wave-segments is experimentally observed in an excitable semiconductor laser with two feedback loops and reproduced by numerical simulations of a prototypical model. While closely related to those found in 2D excitable media, wave patterns in delayed systems exhibit unobserved features originating from causality-related constraints. An appropriate dynamical representation of the data uncovers these phenomena and permits to interpret them as the result of an effective 2D advection-reaction-diffusion process.