  Through an exact method we numerically solve the time evolution of the
density profile for an initially localized state in the one-dimensional bosons
with repulsive short-range interactions. We show that a localized state with a
density notch is constructed by superposing one-hole excitations. The initial
density profile overlaps the plot of the squared amplitude of a dark soliton in
the weak coupling regime. We observe the localized state collapsing into a flat
profile in equilibrium for a large number of particles such as N=1000. The
relaxation time increases as the coupling constant decreases, which suggests
the existence of off-diagonal long-range order. We show a recurrence phenomenon
for a small number of particles such as N=20.
