Nonlocal nonlinear Schrodinger equations are considered as models of liquid
helium II. The models contain a nonlocal interaction potential that leads
to a phonon-roton-like dispersion relation. Also, a higher-order term in th
e local density approximation for the correlation energy is introduced into
the model to overcome nonphysical mass concentrations. These equations are
solved for straight-line vortices. It is demonstrated that the parameters
of the equation can be chosen to bring into agreement the vortex core param
eter and the healing length. The structure of vortex rings of large radius
is studied. The family of the vortex rings of different radii propagating w
ith different velocities is found numerically. As the velocity of the vorte
x ring reaches the Landau critical velocity the sequence of rings terminate
s.