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

Accurate one-dimensional effective description of realistic matter-wave gap solitons