  In a one-dimensional shallow optical lattice, in the presence of both cubic
and quintic nonlinearity, a superfluid density wave is identified in a
Bose-Einstein condensate. Interestingly, it ceases to exist when only one of
these interactions is operative. We predict the loss of superfluidity through a
classical dynamical phase transition, where modulational instability leads to
the loss of phase coherence. In a certain parameter domain, the competition
between lattice potential and the interactions is shown to give rise to a
stripe phase, where atoms are confined in finite domains. In a pure two-body
case, apart from the known superfluid and insulating phases, a density wave
insulating phase is found to exist, possessing two frequency modulations
commensurate with the lattice potential.
