Waves in Honeycomb Structures

Fefferman, Charles L.; Weinstein, Michael I.

Journées équations aux dérivées partielles (2012), article no. 12, 12 p.

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 $H_V=-\Delta +V$ and (ii) the two-dimensional Dirac equations, as the large (but finite) time effective system of equations governing the evolution $e^{-i{H}_{V}t}{\psi }_0$, for data ${\psi }_0$, which is spectrally localized near 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, ${\psi }_0$. The effective dynamics are governed by a nonlinear Dirac system.

Classification : 00X99
Mots clés : Periodic structure, Dispersion relation, Dirac point, Dirac equations, Conical point, Graphene, Nonlinear Schrödinger / Gross Pitaevskii equation

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