  We consider stationary matter-wave gap solitons realized in Bose--Einstein
condensates loaded in one-dimensional (1D) optical lattices and investigate
whether the effective 1D equation proposed in [Phys. Rev. A \textbf{77}, 013617
(2008)] can be a reliable alternative to the three-dimensional treatment of
this kind of system in terms of the Gross--Pitaevskii equation (GPE). Our
results demonstrate that, unlike the standard 1D GPE (which is not applicable
in most realistic situations), the above effective model is able to correctly
predict the distinctive trajectories characterizing the different gap soliton
families as well as the corresponding axial wavefunctions along the entire band
gaps. It can also predict the stability properties of the different gap soliton
families as follows from both a linear stability analysis and a representative
set of numerical computations. In particular, by numerically solving the
corresponding Bogoliubov--de Gennes equations we show that the effective 1D
model gives the correct spectrum of complex eigenfrequencies responsible for
the dynamical stability of the system, thus providing us with a useful tool for
the physical description of stationary matter-wave gap solitons in 1D optical
lattices.
