Nonlinear Geometric Optics

Nonlinear Geometric Optics
Jeffrey Rauch, University of Michigan
October 19, 2009

Abstract:  Geometric optics is a family of methods to construct approximate solutions to partial differential equations.  The solutions have a small parameter often representing a wavelength.The approximations are accurate in the short wavelength limit. In this limit, direct numerical simulation is not possible.  The history is long and rich. In this talk we recall some of the  history and basic ideas of the subject and recent advances,  notably for nonlinear hyperbolic equations. For the compressible Euler equations there are solutions with three incoming wave trains and with outgoing waves with velocities dense in the unit sphere.  My work in this area is with J.L. Joly and G. Metivier.