Waves in Honeycomb Structures

We review recent work of the authors on the non-relativistic Schrödinger 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 $H_V=-Δ+V$ and (ii) the two-dimensional Dirac equations, as a large, but finite time, effective description of $e^{-iH_Vt}ψ_0$, for data $ψ_0$, 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 Schrödinger - Gross Pitaevskii equation for small amplitude initial conditions, $ψ_0$. The effective dynamics are governed by a nonlinear Dirac system.

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