  The properties of the superfluid phase of ultra cold bosonic atoms loaded in
a circular array are investigated in the framework of the Bose-Hubbard model
and the Bogoliubov theory. We derive and solve the Gross-Pitaevskii equation of
the model to find that the atoms condense in states of well-defined
quasimomentum. A detailed analysis of the coupling structure in the effective
quadratic grand-canonical Hamiltonian shows that only pairs of distinct and
identical quasimomenta are coupled. Solving the corresponding Bogoliubov-de
Gennes equations we see that each pair of distinct quasimomenta gives raise to
doublets in the excitation energy spectrum and that the quasimomenta of the
zero-energy mode and of the occupied state in the condensates are identical.
The dynamical and energetic stabilities of the condensates are determined by
studying the behavior of the elementary excitations in the control parameters
space. Our investigation establishes that superflow condensates exists only in
the central region of the first Brillouin zone whereas there is none in the
last quarters since they are energetically unstable, independently of the
control parameters.
