Rigorous derivation of nonlinear Dirac equations for wave propagation in honeycomb structures

TL;DR

This paper rigorously derives a nonlinear Dirac equation as an effective model for wave propagation in honeycomb lattice structures, capturing the behavior of weakly nonlinear waves near Dirac points.

Contribution

It introduces a mathematically rigorous derivation of a nonlinear Dirac equation from a nonlinear Schrödinger equation in honeycomb potentials, advancing theoretical understanding.

Findings

Effective nonlinear Dirac model derived with error estimates

Model describes wave behavior near Dirac points in honeycomb structures

Provides a foundation for analyzing nonlinear wave dynamics in such media

Abstract

We consider a nonlinear Schroedinger equation in two spatial dimensions subject to a periodic honeycomb lattice potential. Using a multi-scale expansion together with rigorous error estimates, we derive an effective model of nonlinear Dirac type. The latter describes the propagation of slowly modulated, weakly nonlinear waves spectrally localized near a Dirac point.