  We study quantum dynamics of a dark soliton in a one-dimensional Bose gas in
an optical lattice within the truncated Wigner approximation. A previous work
has revealed that in the absence of quantum fluctuations, dynamical stability
of the dark soliton significantly depends on whether its phase kink is located
at a lattice site or a link of two neighboring sites. It has also shown that
the dark soliton is unstable in a regime of strong quantum fluctuations
regardless of the phase-kink position. To bridge the gap between the classical
and strongly quantum regimes, we investigate the dynamical stability of the
dark soliton in a regime of weak quantum fluctuations. We find that the
position dependence of the dynamical stability gradually diminishes and
eventually vanishes as the strength of quantum fluctuations increases. This
classical-to-quantum crossover of the soliton stability remains even in the
presence of a parabolic trapping potential. We suggest that the crossover
behavior can be used for experimentally diagnosing whether the instability of a
dark soliton is due to quantum fluctuations or classical dynamical instability.
