  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
Schr\"odinger 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 und M. Spitz, Local wellposedness of quasilinear Maxwell equations
with conservative interface conditions, \textit{Commun. Math. Sci.}, accepted,
2022] and extends this to asymptotically large time intervals for small data
using an involved bootstrapping argument.
