  In this work, we highlight the correspondence between two descriptions of a
system of ultracold bosons in a one-dimensional optical lattice potential: (1)
the discrete nonlinear Schr\"{o}dinger equation, a discrete mean-field theory,
and (2) the Bose-Hubbard Hamiltonian, a discrete quantum-field theory. The
former is recovered from the latter in the limit of a product of local coherent
states. Using a truncated form of these mean-field states as initial
conditions, we build quantum analogs to the dark soliton solutions of the
discrete nonlinear Schr\"{o}dinger equation and investigate their dynamical
properties in the Bose-Hubbard Hamiltonian. We also discuss specifics of the
numerical methods employed for both our mean-field and quantum calculations,
where in the latter case we use the time-evolving block decimation algorithm
due to Vidal.
