Waves in Honeycomb Structures

We review recent work of the authors on the non-relativistic Schr"odinger equation with a honeycomb lattice potential, $V$. In particular, we summarize results on (i) the existence of Dirac points, conical singularities in dispersion surfaces of $H_V=-\mathrm{\Delta }+V$ and (ii) the two-dimensional Dirac equations, as a large, but finite time, effective description of $e^{-i{H}_{V}t}{\psi }_0$, for data ${\psi }_0$, which is spectrally localized at a Dirac point. We conclude with a formal derivation and discussion of the effective large time evolution for the nonlinear Schr"odinger - Gross Pitaevskii equation for small amplitude initial conditions, ${\psi }_0$. The effective dynamics are governed by a nonlinear Dirac system.