Nonlinear dynamics of modulated waves on graphene like quantum graphs

Abstract

We consider cubic Klein-Gordon equations on infinite two‐dimensional periodic metric graphs having for instance the form of graphene. At non‐Dirac points of the spectrum, with a multiple scaling expansion Nonlinear Schrödinger (NLS) equations can be derived in order to describe slow modulations in time and space of traveling wave packets. Here we justify this reduction by proving error estimates between solutions of the cubic Klein–Gordon equations and the associated NLS approximations. Moreover, we discuss the validity of the modulation equations appearing by the same procedure at the Dirac points of the spectrum.