Geometric optics expansions for hyperbolic corner problems, I: Self-interaction phenomenon

Geometric Optics Expansions for Hyperbolic Corner Problems II: From Weak Stability to Violent Instability

In this article we are interested in the rigorous construction of geometric optics expansions for weakly well-posed 4 hyperbolic corner problems. More precisely we focus on the case where selfinteracting phases occur and where one of them is 5 exactly the phase where the uniform Kreiss-Lopatinskii condition fails. We show that the associated WKB expansion suffers 6 arbitrarily many amplificatio...

Geometric Optics Expansions with Amplification for Hyperbolic Boundary Value Problems: Linear Problems

We compute and justify rigorous geometric optics expansions for linear hyperbolic boundary value problems that do not satisfy the uniform Lopatinskii condition. We exhibit an amplification phenomenon for the reflection of small high frequency oscillations at the boundary. Our analysis has two important consequences for such hyperbolic boundary value problems. Firstly, we make precise the optima...

Hyperbolic Partial Differential Equations and Geometric Optics

§1.1. The method of characteristics §1.2. Examples of propagation of singularities using progressing waves §1.3. Group velocity and the method of nonstationary phase §1.4. Fourier synthesis and rectilinear propagation §1.5. A cautionary example in geometric optics §1.6. The law of reflection §1.6.1. The method of images §1.6.2. The plane wave derivation §1.6.3. Reflected high frequency wave pac...

2 . Gibbs Phenomenon for Wavelet Sampling Expansions

We deal with the maximum Gibbs ripple of the sampling wavelet series of a discontinuous .function f at a point t ~ R, .for all possible values o.['a satisfying f (t) = ee.f (t 0) + (1 cO.f (t + 0). For the Shannon wavelet series, we make a complete description of all ripples, .for any ot in [0,1]. We show that Meyer sampling series exhibit Gibbs Phenomenon.lor ce < 0 .12495 and ct > 0.306853. W...

REFRACTION PROBLEMS IN GEOMETRIC OPTICS April 22, 2013