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We study transverse magnetic (vector valued) wave-packets in the timedependent Kerr nonlinear Maxwell's equations at the interface of twoinhomogeneous dielectrics with an instantaneous material response. Theresulting model is quasilinear. The problem is solved on each side of theinterface and the fields are coupled via natural interface conditions. Thewave-packet is localized at the interface and propagates in the tangentialdirection. For a slowly modulated envelope approximation the nonlinearSchr\"odinger equation is formally derived as an amplitude equation for theenvelope. We rigorously justify the approximation in a Sobolev space norm onthe corresponding asymptotically large time intervals. The well-posednessresult for the quasilinear Maxwell problem builds on the local theory of [R.Schnaubelt und M. Spitz, Local wellposedness of quasilinear Maxwell equationswith conservative interface conditions, \textit{Commun. Math. Sci.}, accepted,2022] and extends this to asymptotically large time intervals for small datausing an involved bootstrapping argument.

Rigorous Envelope Approximation for Interface Wave-Packets in Maxwell's Equations with 2D Localization