  We present the theoretical and mathematical foundations of stability analysis
for a Bose-Einstein condensate (BEC) at Dirac points of a honeycomb optical
lattice. The combination of s-wave scattering for bosons and lattice
interaction places constraints on the mean-field description, and hence on
vortex configurations in the Bloch-envelope function near the Dirac point. A
full derivation of the relativistic linear stability equations (RLSE) is
presented by two independent methods to ensure veracity of our results.
Solutions of the RLSE are used to compute fluctuations and lifetimes of vortex
solutions of the nonlinear Dirac equation, which include Mermin-Ho and
Anderson-Toulouse skyrmions, with lifetime $\approx 4$ seconds. Beyond vortex
stabilities the RLSE provide insight into the character of collective
superfluid excitations, which we find to encode several established theories of
physics. In particular, the RLSE reduce to the Andreev equations, in the
nonrelativistic and semiclassical limits, the Majorana equation, inside vortex
cores, and the Dirac-Bogoliubov-de Gennes equations, when nearest-neighbor
interactions are included. Furthermore, by tuning a mass gap, relative
strengths of various spinor couplings, for the small and large quasiparticle
momentum regimes, we obtain weak-strong Bardeen-Cooper-Schrieffer
superconductivity, as well as fundamental wave equations such as Schr\"odinger,
Dirac, Klein-Gordon, and Bogoliubov-de Gennes equations. Our results apply
equally to a strongly spin-orbit coupled BEC in which the Laplacian
contribution can be neglected.
