  Dark soliton solutions in the one-dimensional classical nonlinear
Schr\"odinger equation has been considered to be related to the yrast states
corresponding to the type-II excitations in the Lieb-Liniger model. However,
the relation is nontrivial and remains unclear because a dark soliton localized
in space breaks the translation symmetry, while yrast states are
translationally invariant. In this work, we construct a symmetry-broken quantum
soliton state and investigate the relation to the yrast states. By interpreting
a quantum dark soliton as a Bose-Einstein condensation to the wave function of
a classical dark soliton, we find that the quantum soliton state has a large
weight only on the yrast states, which is analytically proved in the free-boson
limit and numerically verified in the weak-coupling regime. By extending these
results, we derive a parameter-free expression of a quantum soliton state that
is written as a superposition of yrast states with Gaussian weights. The
density profile of this quantum soliton state excellently agrees to that of the
classical dark soliton. The dynamics of a quantum dark soliton is also studied,
and it turns out that the density profile of a dark soliton decays, but the
decay time increases as the inverse of the coupling constant in the
weak-coupling limit.
