Waves in Honeycomb Structures
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 =-Δ+V and (ii) the two-dimensional Dirac equations, as the large (but finite) time effective system of equations governing the evolution e -iH V t ψ 0 , for data ψ 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, ψ 0 . The effective dynamics are governed by a nonlinear Dirac system.

DOI : 10.5802/jedp.95
Classification : 00X99
Mots clés : Periodic structure, Dispersion relation, Dirac point, Dirac equations, Conical point, Graphene, Nonlinear Schrödinger / Gross Pitaevskii equation
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
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Fefferman, Charles L.; Weinstein, Michael I. Waves in Honeycomb Structures. Journées équations aux dérivées partielles (2012), article  no. 12, 12 p. doi : 10.5802/jedp.95. http://www.numdam.org/articles/10.5802/jedp.95/

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