Nonlinear Schrödinger flow past an obstacle in one dimension

Abstract

The flow of a one-dimensional defocusing nonlinear Schr\odinger fluid past an obstacle is investigated. Below an obstacle-dependent critical velocity, a steady dissipationless motion is possible and the flow profile is determined analytically in several cases. At the critical velocity, the steady flow solution disappears by merging with an unstable solution in a usual saddle-node bifurcation. It is argued that this unstable solution represents the transition state for emission of gray solitons. The barrier for soliton emission is explicitly computed and vanishes at the critical velocity. Above the critical velocity, the flow becomes unsteady and its characteristics are studied numerically. It is found that gray solitons are repeatedly emitted by the obstacle and propagate downstream. Upstream propagating dispersive waves are emitted concurrently. A hydraulic approximation is used to interpret these results.