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Journées équations aux dérivées partielles

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Charles L. Fefferman; Michael I. Weinstein
Waves in Honeycomb Structures
Journées équations aux dérivées partielles (2012), Exp. No. 12, 12 p., doi: 10.5802/jedp.95
Article PDF
Class. Math.: 00X99
Mots clés: Periodic structure, Dispersion relation, Dirac point, Dirac equations, Conical point, Graphene, Nonlinear Schrödinger / Gross Pitaevskii equation

Résumé - Abstract

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 $e^{-iH_Vt}\psi _0$, 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.

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