Rigorous derivation of an effective model for periodic Schrödinger equations with linear band crossing of Dirac type

In this paper we consider a family of time-dependent 1-dimensional cubic Schrödinger 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.