  A superfluid flow beyond a critical momentum in an optical lattice decays
drastically by the interplay between nonlinearity due to the interparticle
interactions in Bose-Einstein condensate and periodicity of the lattice; this
instability is called dynamical instability. The complex density modulational
profiles after the condensate becomes unstable observed experimentally is not
completely understood, while the dynamical instability has been studied
theoretically and experimentally. In this paper, we analyze the density
modulation of condensates as a precursor of the dynamical instability in the
two-dimensional Bose-Hubbard model. Our analysis has clarified the unexplored
properties of the density modulations associated with the dynamical instability
at low filling and in a wide range of interactions, while the previous works
have analyzed the density modulation on the basis of Gross-Pitaevskii equation
under the specific condition that one-dimensional optical lattice is very
shallow and the filling is very large. The numerical simulations based on the
dynamical Gutzwiller approximation elucidate that the principal mode of density
modulation highly depends on interaction strength U and the momentum
acceleration rate. We briefly discuss these features with the stability phase
diagram calculated on the basis of the Bogoliubov theory.
