# Temporal dissipative solitons in time-delay feedback systems

## TL;DR

This paper develops a theoretical framework for understanding temporal dissipative solitons in systems with time-delayed feedback, including their classification, stability, and examples in laser and neural models.

## Contribution

It introduces a novel theory for temporal dissipative solitons with an advanced argument and classifies their spectral properties, expanding understanding beyond spatial systems.

## Key findings

- Derived a system with an advanced argument for TDS profile determination.
- Classified TDS spectrum into interface and pseudo-continuous spectrum.
- Identified destabilization mechanisms leading to delocalization and modulational instability.

## Abstract

Localized states are a universal phenomenon observed in spatially distributed dissipative nonlinear systems. Known as dissipative solitons, auto-solitons, spot or pulse solutions, these states play an important role in data transmission using optical pulses, neural signal propagation, and other processes. While this phenomenon was thoroughly studied in spatially extended systems, temporally localized states are gaining attention only recently, driven primarily by applications from fiber or semiconductor lasers. Here we present a theory for temporal dissipative solitons (TDS) in systems with time-delayed feedback. In particular, we derive a system with an advanced argument, which determines the profile of the TDS. We also provide a complete classification of the spectrum of TDS into interface and pseudo-continuous spectrum. We illustrate our theory with two examples: a generic delayed phase oscillator, which is a reduced model for an injected laser with feedback, and the FitzHugh-Nagumo neuron with delayed feedback. Finally, we discuss possible destabilization mechanisms of TDS and show an example where the TDS delocalizes and its pseudo-continuous spectrum develops a modulational instability.