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We consider the Josephson vortex motion in a long one--dimensional Josephson junction in a thin film. We show that this Josephson vortex is similar to a mesoscopic capacitor. We demonstrate that a single Cooper pair tunneling results in nonlinear Bloch--type oscillations of a Josephson vortex in a current-biased Josephson junction. We find the frequency and the amplitude of this motion.doi:10.1209/0295-5075/28/5/011 fatcat:igytqm3qhvghlcj73onx7zvpaa