  We present a series of quantum states that are characterized by dark solitons
of the nonlinear Schr\"{o}dinger equation (i.e. the Gross-Pitaevskii equation)
for the one-dimensional (1D) Bose gas interacting through the repulsive
delta-function potentials. The classical solutions satisfy the periodic
boundary conditions and we call them periodic dark solitons. Through exact
solutions we show corresponding aspects between the states and the solitons in
the weak coupling case: the quantum and classical density profiles completely
overlap with each other not only at an initial time but also at later times
over a long period of time, and they move together with the same speed in time;
the matrix element of the bosonic field operator between the quantum states has
exactly the same profiles of the square amplitude and the phase as the
classical complex scalar field of a periodic dark soliton not only at the
initial time but also at later times, and the corresponding profiles move
together for a long period of time. We suggest that the corresponding
properties hold rigorously in the weak coupling limit. Furthermore, we argue
that the lifetime of the dark soliton-like density profile in the quantum state
becomes infinitely long as the coupling constant approaches zero, by comparing
it with the quantum speed limit time. Thus, we call the quantum states quantum
dark soliton states.
