  A model of the Bose-Einstein condensate (BEC) of dipolar atoms, confined in a
combination of a cigar-shaped trap and optical lattice acting in the axial
direction, is studied in the framework of the one-dimensional (1D)
nonpolynomial Schr\"{o}dinger equation (NPSE) with additional terms describing
long-range dipole-dipole (DD) interactions. The NPSE makes it possible to
describe the collapse of localized modes, which was experimentally observed in
the self-attractive BEC confined in tight traps, in the framework of the 1D
description. We study the influence of the DD interactions on the dynamics of
bright solitons, especially as concerns their collapse-induced instability.
Both attractive and repulsive contact and DD interactions are considered. The
results are summarized in the form of stability/collapse diagrams in a
respective parametric space. In particular, it is shown that the attractive DD
interactions may prevent the collapse instability in the condensate with
attractive contact interactions.
