WAVE TRANSMISSION IN DISPERSIVE MEDIA

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.

Download Full-text

High-frequency nonlinear optics: from the nonlinear Schrödinger approximation to ultrashort-pulses equations

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s030821050900002x ◽

2011 ◽

Vol 141 (2) ◽

pp. 253-286 ◽

Cited By ~ 4

Author(s):

David Lannes

Keyword(s):

High Frequency ◽

Hyperbolic Systems ◽

Ultrashort Pulses ◽

Frequency Dispersion ◽

Dispersive Media ◽

Nls Equation ◽

Short Pulses ◽

Nonlinear Schrödinger ◽

Striking Fact ◽

Large Time Scale

After a brief introduction and physical motivation, we show how the nonlinear Schrödinger (NLS) equation can be derived from a general class of nonlinear hyperbolic systems. Its purpose is to describe the behaviour of high-frequency oscillating wave packets over a large time-scale that requires us to take into account diffractive effects. We then show that the NLS approximation fails for short pulses and propose some alternative models, including a modified Schrödinger equation with improved frequency dispersion. It turns out that these models have better properties and are quite accurate for short pulses. For ultrashort pulses, however, they must also be abandoned for more complex approaches. We give the main steps for such an analysis and explain one striking fact about ultrashort pulses: their dynamics in dispersive media is linear.

Download Full-text

Diffractive wave transmission in dispersive media

Journal of Differential Equations ◽

10.1016/j.jde.2007.08.003 ◽

2008 ◽

Vol 244 (4) ◽

pp. 972-1010 ◽

Cited By ~ 2

Author(s):

Vincent Lescarret

Keyword(s):

Wave Transmission ◽

Dispersive Media

Download Full-text

WKB EXPANSIONS FOR HYPERBOLIC BOUNDARY VALUE PROBLEMS IN A STRIP: SELFINTERACTION MEETS STRONG WELL-POSEDNESS

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s1474748018000506 ◽

2018 ◽

Vol 19 (5) ◽

pp. 1629-1675

Author(s):

Antoine Benoit

Keyword(s):

Boundary Value Problems ◽

Hyperbolic Systems ◽

Boundary Value ◽

Initial Boundary ◽

Well Posedness ◽

Characteristic Variety ◽

Strip Geometry ◽

Initial Boundary Value ◽

Hyperbolic Boundary ◽

Wkb Expansion

In this article we are interested in the rigorous construction of WKB expansions for hyperbolic boundary value problems in the strip $\mathbb{R}^{d-1}\times [0,1]$. In this geometry, a new inversibility condition has to be imposed to construct the WKB expansion. This new condition is due to selfinteraction phenomenon which naturally appear when several boundary conditions are imposed. More precisely, by selfinteraction we mean that some rays can regenerated themselves after some rebounds against the sides of the strip. This phenomenon is not new and has already been studied in Benoit (Geometric optics expansions for hyperbolic corner problems, I: self-interaction phenomenon, Anal. PDE9(6) (2016), 1359–1418), Sarason and Smoller (Geometrical optics and the corner problem, Arch. Rat. Mech. Anal.56 (1974/75), 34–69) in the corner geometry. In this framework the existence of such selfinteracting rays is linked to specific geometries of the characteristic variety and may seem to be somewhat anecdotal. However for the strip geometry such rays become generic. The new inversibility condition, used to construct the WKB expansion, is a microlocalized version of the one characterizing the uniform in time strong well-posedness (Benoit, Lower exponential strong well-posedness of hyperbolic boundary value problems in a strip (preprint)). It is interesting to point here that such a situation already occurs in the half space geometry (Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math.23 (1970), 277–298).

Download Full-text