Rigorous envelope approximation for interface wave packets in maxwell's equations with two dimensional localization

We study transverse magnetic (vector valued) wave packets in the time dependent Kerr nonlinear Maxwell's equations at the interface of two inhomogeneous dielectrics with an instantaneous material response. The resulting model is quasilinear. The problem is solved on each side of the interface and the fields are coupled via natural interface conditions. The wave packet is localized at the interface and propagates in the tangential direction. For a slowly modulated envelope approximation the nonlinear Schro"\dinger equation is formally derived as an amplitude equation for the envelope. We rigorously justify the approximation in a Sobolev space norm on the corresponding asymptotically large time intervals. The well-posedness result for the quasilinear Maxwell problem builds on the local theory of [R. Schnaubelt and M. Spitz, Commun. Math. Sci., 20 (2022), pp. 2265--2313] and extends this to asymptotically large time intervals for small data using an involved bootstrapping argument.