Waves in Honeycomb Structures

Fefferman, Charles L. ; Weinstein, Michael I.

Journées équations aux dérivées partielles, (2012), p. 1-12 / Harvested from Numdam

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

Détail

Publié le : 2012-01-01
DOI : https://doi.org/10.5802/jedp.95
Classification:  00X99

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