  We investigate the band structure of a Bose-Einstein condensate in a
one-dimensional periodic potential by calculating stationary solutions of the
Gross-Pitaevskii equation which have the form of Bloch waves. We demonstrate
that loops ("swallow tails") in the band structure occur both at the Brillouin
zone boundary and at the center of the zone, and they are therefore a generic
feature. A physical interpretation of the swallow tails in terms of periodic
solitons is given. The linear stability of the solutions is investigated as a
function of the strength of the mean-field interaction, the magnitude of the
periodic potential, and the wave vector of the condensate. The regions of
energetic and dynamical stability are identified by considering the behavior of
the Gross-Pitaevskii energy functional for small deviations of the condensate
wave function from a stationary state. It is also shown how for long-wavelength
disturbances the stability criteria may be obtained within a hydrodynamic
approach.
