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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.

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