# 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

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topFefferman, 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

top- M. Ablowitz, C. Curtis, and Y. Zhu. On tight-binding approximations in optical lattices. Stud. Appl. Math, 129(4):362—388, 2012. Zbl1297.35212MR2993124
- M.J. Ablowitz and Y. Zhu. Nonlinear waves in shallow honeycomb lattices. SIAM J. Appl. Math., 72(240–260), 2012. Zbl1258.41012MR2888342
- O. Bahat-Treidel, O. Peleg, and M. Segev. Symmetry breaking in honeycomb photonic lattices. Optics Letters, 33(2251–2253), 2008.
- M.V. Berry and M.R. Jeffrey. Conical Diffraction: Hamilton’s diabolical point at the heart of crystal optics. Progress in Optics. 2007.
- L. Erdös, B. Schlein, and H.T. Yau. Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems. Invent. Math., 167:515–614, 2007. Zbl1123.35066MR2276262
- C.L. Fefferman and M.I. Weinstein. Dynamics of wave packets in honeycomb structures and two-dimensional Dirac equations. submitted, http://arxiv.org/abs/1212.6072. Zbl1292.35195
- C.L. Fefferman and M.I. Weinstein. Honeycomb lattice potentials and Dirac points. J. Amer. Math. Soc., 25:1169–1220, 2012. Zbl1316.35214MR2947949
- V. V. Grushin. Multiparameter perturbation theory of Fredholm operators applied to Bloch functions. Mathematical Notes, 86(6):767–774, 2009. Zbl1197.47025MR2643450
- F.D.M. Haldane and S. Raghu. Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry. Phys. Rev. Lett., 100:013904, 2008.
- R. A. Indik and A. C. Newell. Conical refraction and nonlinearity. Optics Express, 14(22):10614–10620, 2006.
- P. Kuchment and O. Post. On the spectra of carbon nano-structures. Comm. Math. Phys., 275:805–826, 2007. Zbl1145.81032MR2336365
- J.V. Maloney and A.C. Newell. Nonlinear Optics. Westview Press, 2003. Zbl1054.78001
- A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, and A.K. Geim. The electronic properties of graphene. Reviews of Modern Physics, 81:109–162, 2009.
- K. S. Novoselov. Nobel lecture: Graphene: Materials in the flatland. Reviews of Modern Physics, 837–849, 2011.
- O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D.N. Christodoulides. Conical diffraction and gap solitons in honeycomb photonic lattices. Phys. Rev. Lett., 98:103901, 2007.
- L. P. Pitaevskii and S. Stringari. Bose Einstein Condensation. Oxford University Press, 2003. Zbl1110.82002MR2012737
- M.C. Rechtsman, J.M. Zeuner, Y. Plotnik, Y. Lumer, S. Nolte, M. Segev, and A. Szameit. Photonic Floquet topological insulators. http://arxiv.org/abs/1212.3146.
- P.R. Wallace. The band theory of graphite. Phys. Rev., 71:622, 1947. Zbl0033.14304

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