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We show analytically and numerically that time delayed nonlocal responseinduces traveling localized states in bistable systems. These states resultfrom fronts interaction. We illustrate this mechanism in a generic bistablemodel with a nonlocal delayed response. Analytical expression of the width andthe speed of traveling localized states are derived. Without time delayednonlocal response traveling localized states are excluded. Finally, we consideran experimentally relevant system, the fiber cavity with the non-instantaneousRaman response, and show evidence of traveling localized state. In addition, wepropose realistic parameters and perform numerical simulations of the governingmodel equation.

Time-delayed nonlocal response induces traveling localized structures