  We investigate the flow of a one-dimensional nonlinear Schrodinger model with
periodic boundary conditions past an obstacle, motivated by recent experiments
with Bose--Einstein condensates in ring traps. Above certain rotation
velocities, localized solutions with a nontrivial phase profile appear. In
striking difference from the infinite domain, in this case there are many
critical velocities. At each critical velocity, the steady flow solutions
disappear in a saddle-center bifurcation. These interconnected branches of the
bifurcation diagram lead to additions of circulation quanta to the phase of the
associated solution. This, in turn, relates to the manifestation of persistent
current in numerous recent experimental and theoretical works, the connections
to which we touch upon. The complex dynamics of the identified waveforms and
the instability of unstable solution branches are demonstrated.
