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We consider in this work the different possible stationary flows of a onedimensional quantum fluid in the mean-field regime. We focus on the supersonicregime where a transition from a time dependent flow to a stationarydiffractive flow occurs at a given critical velocity. We give nonperturbativeresults for this critical velocity in the presence of a localised obstacle ofarbitrary size and strength. In addition, we discuss the existence ofsuperfluid-like solution in the supersonic regime due to resonant transport andprovide a complete map of the different regimes of stationary transport of aquantum fluid.

Stationary transport above the critical velocity in a one-dimensional superflow past an obstacle