# Time-delayed nonlocal response induces traveling localized structures

## TL;DR

This paper demonstrates that time-delayed nonlocal responses can induce traveling localized states in bistable systems, supported by analytical, numerical, and experimental evidence, highlighting a new mechanism for localized structure dynamics.

## Contribution

The study introduces a novel mechanism where time-delayed nonlocal responses generate traveling localized states in bistable systems, supported by analytical derivations and experimental validation.

## Key findings

- Traveling localized states are induced by time-delayed nonlocal responses.
- Analytical expressions for width and speed of these states are derived.
- Experimental evidence is provided in fiber cavity systems with Raman response.

## Abstract

We show analytically and numerically that time delayed nonlocal response induces traveling localized states in bistable systems. These states result from fronts interaction. We illustrate this mechanism in a generic bistable model with a nonlocal delayed response. Analytical expression of the width and the speed of traveling localized states are derived. Without time delayed nonlocal response traveling localized states are excluded. Finally, we consider an experimentally relevant system, the fiber cavity with the non-instantaneous Raman response, and show evidence of traveling localized state. In addition, we propose realistic parameters and perform numerical simulations of the governing model equation.