  We present a thorough analysis of soliton solutions to the
quasi-one-dimensional nonlinear Dirac equation (NLDE) for a Bose-Einstein
condensate in a honeycomb lattice with armchair geometry. Our NLDE corresponds
to a quasi-one-dimensional reduction of the honeycomb lattice along the zigzag
direction, in direct analogy to graphene nanoribbons. Excitations in the
remaining large direction of the lattice correspond to the linear subbands in
the armchair nanoribbon spectrum. Analytical as well as numerical soliton Dirac
spinor solutions are obtained. We analyze the solution space of the
quasi-one-dimensional NLDE by finding fixed points, delineating the various
regions in solution space, and through an invariance relation which we obtain
as a first integral of the NLDE. We obtain spatially oscillating multi-soliton
solutions as well as asymptotically flat single soliton solutions using five
different methods: by direct integration; an invariance relation; parametric
transformation; a series expansion; and by numerical shooting. By tuning the
ratio of the chemical potential to the nonlinearity for a fixed value of the
energy-momentum tensor, we can obtain both bright and dark solitons over a
nonzero density background.
