International audienceWe consider a photonic crystal fiber resonator pumped by a coherent injected beam. We show that temporal cavity solitons exhibit a motion with a constant velocity. This regular drift is induced by a broken reflection symmetry mediated by a third-order dispersion. We focus the analysis on dark temporal cavity solitons. They consist of asymmetric moving dips in a uniform background of the intensity profile. The number of the moving dips and their temporal distribution are determined solely by the initial conditions. We characterize this motion by computing the velocity of the dark temporal cavity soliton. Without fourth-order dispersion, dark cavity solitons do not exist

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