  The nonlinear Dirac equation for Bose-Einstein condensates in honeycomb
optical lattices gives rise to relativistic multi-component bright and dark
soliton solutions. Using the relativistic linear stability equations, the
relativistic generalization of the Boguliubov-de Gennes equations, we compute
soliton lifetimes against quantum fluctuations and classify the different
excitation types. For a Bose-Einstein condensate of $^{87}\mathrm{Rb}$ atoms,
we find that our soliton solutions are stable on time scales relevant to
experiments. Excitations in the bulk region far from the core of a soliton and
bound states in the core are classified as either spin waves or as a
Nambu-Goldstone mode. Thus, solitons are topologically distinct
pseudospin-$1/2$ domain walls between polarized regions of $S_z = \pm 1/2$.
Numerical analysis in the presence of a harmonic trap potential reveals a
discrete spectrum reflecting the number of bright soliton peaks or dark soliton
notches in the condensate background. For each quantized mode the chemical
potential versus nonlinearity exhibits two distinct power law regimes
corresponding to the free-particle (weakly nonlinear) and soliton (strongly
nonlinear) limits.
