  In the present work, we consider the self-focusing discrete nonlinear
Schrodinger equation on hexagonal and honeycomb lattice geometries. Our
emphasis is on the study of the effects of anisotropy, motivated by the
tunability afforded in recent optical and atomic physics experiments. We find
that important classes of solutions, such as the so-called discrete vortices,
undergo destabilizing bifurcations as the relevant anisotropy control parameter
is varied. We quantify these bifurcations by means of explicit analytical
calculations of the solutions, as well as of their spectral linearization
eigenvalues. Finally, we corroborate the relevant stability picture through
direct numerical computations. In the latter, we observe the prototypical
manifestation of these instabilities to be the spontaneous rearrangement of the
solution, for larger values of the coupling, into localized waveforms typically
centered over fewer sites than the original unstable structure. For weak
coupling, the instability appears to result in a robust breathing of the
relevant waveforms.
