  We study the statics and dynamics of anisotropic, stable, bright and
dark-in-bright dipolar quasi-two-dimensional Bose-Einstein condensate (BEC)
solitons on a one-dimensional (1D) optical-lattice (OL) potential. These
solitons mobile in a plane perpendicular to the 1D OL trap can have both
repulsive and attractive contact interactions. The dark-in-bright solitons are
the excited states of the bright solitons. The solitons, when subject to a
small perturbation, exhibit sustained breathing oscillation. The dark-in-bright
solitons can be created by phase imprinting a bright soliton. At medium
velocities the collision between two solitons is found to be quasi elastic.The
results are demonstrated by a numerical simulation of the three-dimensional
mean-field Gross-Pitaevskii equation in three spatial dimensions employing
realistic interaction parameters for a dipolar $^{164}$Dy BEC.
