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We are interested in the spatiotemporal dynamics of cavity solitons in a broad area vertical-cavity surface-emitting laser with saturable absorption subjected to time-delayed optical feedback. We adopt the mean-field approach assuming a single longitudinal mode operation. Using a combination of analytical, numerical, and path continuation methods, we identify the possible bifurcations and map them in a plane of feedback parameters. We show that for both the homogeneous (CW) and localized stationary lasing solutions in one spatial dimension, the time-delayed feedback induces complex spatiotemporal dynamics, in particular a period doubling route to chaos, quasiperiodic oscillations, and multistability of the stationary solutions. In addition, we analyze the bifurcation structure of stationary and moving cavity solitons and identify two different types of traveling localized solutions, corresponding to slow and fast motion. We show that the delay impacts both stationary and moving solutions either causing drifting and wiggling dynamics of initially stationary cavity solitons or leading to stabilization of intrinsically moving structures.
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