Vortices in nonlocal Gross-Pitaevskii equation

We consider vortices in the nonlocal two-dimensional Gross-Pitaevskii equation with the interaction potential having the Lorentz-shaped dependence on the relative momentum. It is shown that in the Fourier series expansion with respect to the polar angle the unstable modes of the axial $n$-fold vortex have orbital numbers $l$ satisfying $0<|l|<2|n|$, similar as in the local model. Numerical simulations show that nonlocality slightly decreases the threshold rotation frequency above which the nonvortex state ceases to be the global energy minimum and decreases the frequency of the anomalous mode of the 1-vortex. In the case of higher axial vortices, nonlocality leads to the instability against splitting with creation of antivortices and gives rise to additional anomalous modes with higher orbital numbers. Despite new instability channels with creation of antivortices, for a stationary solution comprised of vortices and antivortices there always exist another vortex solution, comprised solely of vortices, with the same total vorticity but with a lower energy.