  We present relativistic linear stability equations (RLSE) for
quasi-relativistic cold atoms in a honeycomb optical lattice. These equations
are derived from first principles and provide a method for computing
stabilities of arbitrary localized solutions of the nonlinear Dirac equation
(NLDE), a relativistic generalization of the nonlinear Schr\"odinger equation.
We present a variety of such localized solutions: skyrmions, solitons,
vortices, and half-quantum vortices, and study their stabilities via the RLSE.
When applied to a uniform background, our calculations reveal an experimentally
observable effect in the form of Cherenkov radiation. Remarkably, the Berry
phase from the bipartite structure of the honeycomb lattice induces a
boson-fermion transmutation in the quasi-particle operator statistics.
