  We study multiple-period Bloch states of a Bose-Einstein condensate with
spatially periodic interactomic interaction. Solving the Gross-Pitaevskii
equation for the continuum model, and also using a simplified discrete version
of it, we investigate the energy-band structures and the corresponding
stability properties. We observe a new "attraction-induced dynamical stability"
mechanism caused by the localization of the density distribution in the
attractive domains of the system and the isolation of these higher-density
regions. This makes the superfluid stable near the zone boundary, and also
enhances the stability of higher-periodic states if the nonlinear interaction
strength is sufficiently high.
