Waves in Honeycomb Structures

Charles L. Fefferman[1]; Michael I. Weinstein[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)

  • page 1-12
  • ISSN: 0752-0360

Abstract

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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 - i H 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.

How to cite

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Fefferman, Charles L., and Weinstein, Michael I.. "Waves in Honeycomb Structures." Journées Équations aux dérivées partielles (2012): 1-12. <http://eudml.org/doc/275480>.

@article{Fefferman2012,
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.},
affiliation = {Department of Mathematics Princeton University Princeton, NJ 08540 USA; Department of Applied Physics and Applied Mathematics Columbia University New York, NY 10027 USA},
author = {Fefferman, Charles L., Weinstein, Michael I.},
journal = {Journées Équations aux dérivées partielles},
keywords = {Periodic structure; Dispersion relation; Dirac point; Dirac equations; Conical point; Graphene; Nonlinear Schrödinger / Gross Pitaevskii equation},
language = {eng},
pages = {1-12},
publisher = {Groupement de recherche 2434 du CNRS},
title = {Waves in Honeycomb Structures},
url = {http://eudml.org/doc/275480},
year = {2012},
}

TY - JOUR
AU - Fefferman, Charles L.
AU - Weinstein, Michael I.
TI - Waves in Honeycomb Structures
JO - Journées Équations aux dérivées partielles
PY - 2012
PB - Groupement de recherche 2434 du CNRS
SP - 1
EP - 12
AB - 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.
LA - eng
KW - Periodic structure; Dispersion relation; Dirac point; Dirac equations; Conical point; Graphene; Nonlinear Schrödinger / Gross Pitaevskii equation
UR - http://eudml.org/doc/275480
ER -

References

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