Diffractive wave transmission in dispersive media

The aim of this paper is to study the reflection–transmission of diffractive geometrical optic rays described by semi-linear symmetric hyperbolic systems such as the Maxwell–Lorentz equations with the anharmonic model of polarization.The framework is that of P. Donnat's thesis [P. Donnat, Quelques contributions mathématiques en optique non linéaire, chapters 1 and 2, thèse, 1996] and V. Lescarret [V. Lescarret, Wave transmission in dispersive media, M3AS 17 (4) (2007) 485–535]: we consider an infinite WKB expansion of the wave over long times/distances O(1/ε) and because of the boundary, we decompose each profile into a hyperbolic (purely oscillating) part and elliptic (evanescent) part as in M. William [M. William, Boundary layers and glancing blow-up in nonlinear geometric optics, Ann. Sci. École Norm. Sup. 33 (2000) 132–209].Then to get the usual sublinear growth on the hyperbolic part of the profiles, for every corrector, we consider E, the space of bounded functions decomposing into a sum of pure transports and a “quasi compactly” supported part. We make a detailed analysis on the nonlinear interactions on E which leads us to make a restriction on the set of resonant phases.We finally give a convergence result which justifies the use of “quasi compactly” supported profiles.