Waves in Honeycomb Structures

Charles L. Fefferman; Michael I. Weinstein

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arXiv · 1212.6684

Waves in Honeycomb Structures

Charles L. Fefferman·Michael I. Weinstein

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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 $H_V=-Δ+V$ and (ii) the two-dimensional Dirac equations, as a large, but finite time, effective description of $e^{-iH_Vt}ψ_0$, for data $ψ_0$, which is spectrally localized at 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.

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Charles L. Fefferman, Michael I. Weinstein. 2012-12-30. Waves in Honeycomb Structures. https://arxiv.org/abs/1212.6684

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