Geometric optics and the wave equation on manifolds with corners

2006 ◽  
pp. 315-333 ◽  
Author(s):  
András Vasy
Keyword(s):  
Wave Equation ◽  
Geometric Optics ◽  
Manifolds With Corners

scholarly journals GEOMETRIC-OPTICS FOR NONLINEAR CONCENTRATING WAVES IN FOCUSING AND NON-FOCUSING TWO GEOMETRIES

2004 ◽  
Vol 06 (01) ◽  
pp. 1-23 ◽  
Author(s):  
SLIM IBRAHIM
Keyword(s):  
Wave Equation ◽  
Geometrical Optics ◽  
Variable Coefficients ◽  
Geometric Optics ◽  
Scattering Operator ◽  
Wave Operators

With the methods used in [1] and [4], we prove that in the absence of focus, nonlinear geometrical optics of the critical wave equation with variable coefficients, is reduced to linear geometrical optics combined with wave operators for the critical wave equation with coefficients fixed on concentrating points. On the odd-dimensional spheres, we prove that passing through a focus is generated by a modified scattering operator.

DIFFRACTION BY EDGES

2008 ◽  
Vol 22 (23) ◽  
pp. 2287-2328 ◽  
Author(s):  
ANDRÁS VASY
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Boundary Conditions ◽  
A Priori ◽  
Differential Forms ◽  
Natural Boundary ◽  
Propagation Of Singularities ◽  
Natural Boundary Conditions ◽  
Manifolds With Corners

In these expository notes we explain the role of geometric optics in wave propagation on domains or manifolds with corners or edges. Both the propagation of singularities, which describes where solutions of the wave equation may be singular, and the diffractive improvement under non-focusing hypotheses, which states that in certain places the diffracted wave is more regular than a priori expected, is described. In addition, the wave equation on differential forms with natural boundary conditions, which in particular includes a formulation of Maxwell's equations, is studied.

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