  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.
