Rigorous derivation of an effective model for periodic Schr\"odinger equations with linear band crossing of Dirac type

TL;DR

This paper derives an effective nonlinear Dirac equation from a periodic Schrödinger equation with a potential that exhibits linear band crossing, providing a rigorous link between the two models in a semiclassical regime.

Contribution

The paper provides a rigorous derivation of an effective nonlinear Dirac equation for 1D cubic Schrödinger equations with periodic potentials near Dirac points, using semiclassical and multiscale analysis.

Findings

Effective nonlinear Dirac equation accurately describes dynamics near Dirac points.

Spectral localization around Dirac points is crucial for the derivation.

The approach bridges Schrödinger and Dirac models in a periodic setting.

Abstract

In this paper we consider a family of time-dependent 1-dimensional cubic Schr\"odinger equation (NLS) with periodic potential. Exploiting semiclassical scaling and multiscale analysis, we derive an effective nonlinear Dirac equation, which describes the dynamics of solutions to NLS spectrally localized around Dirac points.