Quasi 1D Bose-Einstein condensate flow past a nonlinear barrier

The problem of a quasi 1D {\it repulsive} BEC flow past through a nonlinear barrier is investigated. Two types of nonlinear barriers are considered, wide and short range ones. Steady state solutions for the BEC moving through a wide repulsive barrier and critical velocities have been found using hydrodynamical approach to the 1D Gross-Pitaevskii equation. It is shown that in contrast to the linear barrier case, for a wide {\it nonlinear} barrier an interval of velocities $0 < v < v_-$ {\it always} exists, where the flow is superfluid regardless of the barrier potential strength. For the case of the $\delta$ function-like barrier, below a critical velocity two steady solutions exist, stable and unstable one. An unstable solution is shown to decay into a gray soliton moving upstream and a stable solution. The decay is accompanied by a dispersive shock wave propagating downstream in front of the barrier.