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The aim of this paper is to study the reflection-transmission of geometrical optic rays described by semi-linear symmetric hyperbolic systems such as the Maxwell–Lorentz equations with the anharmonic model of polarisation. The framework is both that of Donnat and Williams since we consider dispersive media and profiles with hyperbolic (imaginary) phases and elliptic phases (complex with non-null real part). We first give hypothesis close to the Maxwell equation. Then we introduce a decomposition for both profile into boundary (tangential) and normal part and we solve the so-called "microscopic" equation of the small scales for each boundary frequency. Then we show that the non-linearities generate harmonics which interact at the boundary and generate new resonant profiles with harmonic tangential frequency. Lastly we make a WKB expansion at any order and give a precise description of the correctors.

We develop finite difference numerical schemes for a model arising in multi-body structures, previously analyzed by H. Koch and E. Zuazua [13], constituted by two n-dimensional wave equations coupled with a (n − 1)-dimensional one along a flexible interface.That model, under suitable assumptions on the speed of propagation in each media, is well-posed in asymmetric spaces in which the regularity of solutions differs by one derivative from one medium to the other.Here we consider a flat interface and analyze this property at a discrete level, for finite difference and mixed finite element methods on regular meshes parallel to the interface. We prove that those methods are well-posed in such asymmetric spaces uniformly with respect to the mesh-size parameters and we prove the convergence of the numerical solutions towards the continuous ones in these spaces.In other words, these numerical methods that are well-behaved in standard energy spaces, preserve the convergence properties in these asymmetric spaces too.These results are illustrated by several numerical experiments.

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.

Abstract. The propagation of modulated light in a 2d nonlinear photonic waveguide is investigated in the framework of diffractive optics. It is shown that the dynamics obeys a nonlinear Schrödinger equation at leading order. We compute the first and second corrector and show that the latter may describe some dispersive radiation through the structure. We prove the validity of the approximation in the interval of existence of the leading term. IntroductionPeriodic media such as metallic crystals and semi-conductors have drawn scientists' attention because of their band-gap spectrum which accounts for their strong dispersive properties. In the range of optical wavelengths we find the Photonic Crystals (PhC) whose popularity is partly due to the small sizes and the possibility to slow down the light. We refer the reader to the quite exhaustive report [6] on PhC. Because the spectrum gives a good hint on the way waves propagate, the research on these materials mostly focuses on this aspect (see [13,14,15,17,18]). However, as pointed out in [31] the spectrum alone does not explain the wave resonance and wave radiation in a medium. Such effects are not harmonic in time and are indeed observed in nonlinear media when one lets a wave evolve from an initial state close but not equal to an eigenvalue. With this idea in mind we address the issue of computing and analyzing the propagation of an electromagnetic pulse in 2d, straight, nonlinear PhC waveguides (see Figure 1.1). The guides are made of a PhC from which a finite number of rows is removed. The spatial extent of the pulse envelope may be of the order of a few cells of periodicity of the PhC or much larger along the direction of the guide. We denote by η the periodicity of the cell divided by the spatial length of the pulse envelope. The nonlinear response of the PhC makes possible the existence of solitons whose shape and energy are preserved and is responsible for harmonics (third harmonics in centro-symmetric media) which may radiate part of the energy. We consider this issue in the framework of diffractive optics (see [11]), that 2 Vincent Lescarret and Guido Schneider is, for times of order 1/η 2 for which one can observe the soliton propagating and the third harmonic radiating. The main goal of this study is to mathematically describe this harmonic radiation and the way it disturbs the soliton.This situation is quite academic but brings new results in (mathematical) nonlinear optics for PhC. The case of linear optics in homogeneous PhC was addressed in [9]. Then, in the framework of nonlinear diffractive optics, the author in [8] derived a Nonlinear Schrödinger equation (NLS) as a model for propagation of waves in 1d PhC. This was extended by [4] in a multi-dimensional setting but still for homogeneous PhC.The present analysis relies on the spectral properties of the waveguide which are precisely given in the next sections. In particular we provide a resolution of identity of a class of transverse operators modeling the situation. It is mainly based on A. Fig...