  In this paper, we study the nonequilibrium dynamics of the Bose-Hubbard model
with the nearest-neighbor repulsion by using time-dependent Gutzwiller (GW)
methods. In particular, we vary the hopping parameters in the Hamiltonian as a
function of time, and investigate the dynamics of the system from the density
wave (DW) to the superfluid (SF) crossing a first-order phase transition and
vice-versa. From the DW to SF, we find scaling laws for the correlation length
and vortex density with respect to the quench time. This is a reminiscence of
the Kibble-Zurek scaling for continuous phase transitions and contradicts the
common expectation. We give a possible explanation for this observation. On the
other hand from the SF to DW, the system evolution depends on the initial SF
state. When the initial state is the ground-state obtained by the static GW
methods, a coexisting state of the SF and DW domains forms after passing
through the critical point. Coherence of the SF order parameter is lost as the
system evolves. This is a phenomenon similar to the glass transition in
classical systems. When the state starts from the SF with small local phase
fluctuations, the system obtains a large-size DW-domain structure with thin
domain walls.
