Un peu d'optique diffractive non-lineaire a phases courbes

Abstract : We present in this work some new results concerning nonlinear diffractive optics. First, we give three scales asymptotics of smooth oscillating solutions of hyperbolic systems. This is a WKB multiphase method, with nonplanar phases: the faster scale describes oscillations, the intermediate one describes diffractive effects, transversally to the propagation. This propagation, corresponding to the slower scale, follows the rays of geometric optics. We use nonplanar phases so as to treat the case of variable coefficients systems ; coherence assumptions and small divisors properties are needed, and we show their genericity. We give examples of interactions of diffracted waves, especially for nonlinear acoustics. Moreover, we consider several functional cases where diffraction takes place: periodic, weakly decreasing, or shock profiles, with respect to the ``intermediate variable''. These different behaviours allow us to study perturbations of the oscillating phases, as well as separation of light and shadow. In each case, we look at the influence of rectification effects. Finally, we describe oscillations reflecting near a diffractive point (where rays meet the boundary tangently), for a semilinear dissipative Klein-Gordon equation. Interactions and the ``shadow zone'' are emphasized via a $H^1$-asymptotics.
Document type :
Mathematics [math]. Université Rennes 1, 2000. French

Contributor : Eric Dumas <>
Submitted on : Monday, March 10, 2003 - 1:15:51 PM
Last modification on : Friday, March 27, 2015 - 9:59:52 AM


  • HAL Id : tel-00002525, version 1



Eric Dumas. Un peu d'optique diffractive non-lineaire a phases courbes. Mathematics [math]. Université Rennes 1, 2000. French. <tel-00002525>




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