Niveau: Supérieur, Doctorat, Bac+8
About nonlinear geometric optics E. Dumas Institut Fourier, UMR 5582 (CNRS-UJF) 100 rue des Mathematiques Domaine Universitaire BP 74, 38402 Saint Martin d'Heres - France email: Abstract We give an idea of the evolution of mathematical nonlinear geomet- ric optics from its foundation by Lax in 1957, and present applications in various fields of mathematics and physics. 1 Introduction Geometric optics goes back at least to the XVIIth Century, with Fermat, Snell and Descartes, who described the “paths” (rays) followed by the light. Nowadays, Physics tells us that we may reasonably replace the waves from Quantum Mechanics with classical particles, in the semi-classical approx- imation (when considering Planck's constant ~, or the wavelength, as in- finitely small). The mathematical transcription of these problems consists in studying the asymptotic behavior of solutions to partial differential equations where different scales (represented by small parameters) are present, often in a high frequency oscillatory context. We present the first historical results of the field, and then review some extensions and applications of the method. We shall see how geometric optics applies to Maxwell's equations (from optics, ferromagnetism, . . . ), to the wave or Klein-Gordon equation, to fluid dynamics and plasma physics, to general hyperbolic systems and conservation laws, as well as to nonlinear Schrodinger equations, among others.
- phase-amplitude represen- tation
- profile equations
- optics goes
- geometric optics
- weakly nonlinear
- semi-classical approx- imation
- equations takes