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We study theoretically the formation and dynamics of phase-locked temporal localized states (TLSs) and frequency combs that can be generated from a high finesse Fabry-Perot microcavity containing a Kerr medium that is coupled to a external cavity in the presence of optical injection. These TLSs possess a strongly asymmetrical oscillating tail which results from third order dispersion induced by the quasi-resonant cavity. Using a first principle model based upon delay algebraic equations and applying a combination of direct numerical simulations and path continuation methods, we disclose sets of multistable dark and bright TLSs coexisting on their respective bistable homogeneous backgrounds. In particular, we show that the detuning of the injection with respect to the microcavity resonance controls the amount of normal dispersion as well as the region of existence of TLSs. We identify a scenario of period-doubling route to chaos. Further we discuss a transformation of the original delay model to a normal form given by a partial differential equation using a rigorous multiple time scale analysis and a functional mapping approach. The analysis is performed in two experimentally sound situations. The first one is adapted to cavities with large losses and we consider the onset of bistability for normal dispersion. There, one obtain a real order parameter equation akin to the Swift-Hohenberg model that is modified by the presence of third order dispersion and self-steepening. The second case consists in the weakly nonlinear regime obtained in the good cavity limit for abnormal dispersion. This situation leads to the generalized Lugiato-Lefever model with third order dispersion and self-steepening. We finally discuss the lossless case and the more general question of how to link the conservative, time reversible, nonlinear Schrodinger equation with the dynamics of nonlinear time delayed systems.
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