Waves in Honeycomb Structures

Fefferman, Charles L.¹  ; Weinstein, Michael I.² 

¹ Department of Mathematics Princeton University Princeton, NJ 08540 USA
² Department of Applied Physics and Applied Mathematics Columbia University New York, NY 10027 USA

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

Résumé

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.

Détail

DOI : 10.5802/jedp.95

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

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