Nonlocal Nonlinear Schrödinger Equations as Models of Superfluidity

Condensate models for superfluid helium II with nonlocal potentials are considered. The potentials are chosen so that the models give a good fit to the Landau dispersion curve; i.e., the plot of quasi-particle energy E versus momentum p has the correct slope at the origin (giving the correct sound velocity) and the roton minimum is close to that experimentally observed. It is shown that for any such potential the condensate model has non-physical features, specifically the development of catastrophic singularities and unphysical mass concentrations. Two numerical examples are considered: the evolution of a radially symmetric mass disturbance and the flow around a solid sphere moving with constant velocity, both using the nonlocal Ginsburg–Pitaevskii model. During the evolution of the solution in time, mass concentrations develop at the origin in the radially symmetric case and along the axis of symmetry for the motion of the sphere.