  We study the stability of Bose-Einstein condensates with superfluid currents
in a one-dimensional periodic potential. By using the Kronig-Penney model, the
condensate and Bogoliubov bands are analytically calculated and the stability
of condensates in a periodic potential is discussed. The Landau and dynamical
instabilities occur in a Kronig-Penney potential when the quasimomentum of the
condensate exceeds certain critical values as in a sinusoidal potential. It is
found that the onsets of the Landau and dynamical instabilities coincide with
the point where the perfect transmission of low energy excitations through each
potential barrier is forbidden. The Landau instability is caused by the
excitations with small $q$ and the dynamical instability is caused by the
excitations with $q=\pi/a$ at their onsets, where $q$ is the quasimomentum of
excitation and $a$ is the lattice constant. A swallow-tail energy loop appears
at the edge of the first condensate band when the mean-field energy is
sufficiently larger than the strength of the periodic potential. We find that
the upper portion of the swallow-tail is always dynamically unstable, but the
second Bogoliubov band has a phonon spectrum reflecting the positive effective
mass.
