  We address the stability of superfluid currents in a system of interacting
lattice bosons. We consider various Gutzwiller trial states for the quantum
phase model which provides a good approximation for the Bose-Hubbard model in
the limit of large interactions and boson populations. We thoroughly analyze
the current-carrying stationary states of the dynamics ensuing from a Gaussian
ansatz, and derive analytical results for the critical lines signaling their
modulational and energetic instability, as well as the maximum of the carried
current. We show that these analytical results are in good qualitative
agreement with those obtained numerically in previous works on the Bose-Hubbard
model, and in the present work for the quantum phase model.
