P.2. A bird’s eye view of hyperbolic equations xvii needs to be understood. This poses serious analytic, computational, and engineering challenges. A second way to bring nonlinear effects to the fore is to increase the amplitude of disturbances. It was only with the advent of the laser that suﬃciently intense optical fields were produced so that nonlinear effects are routinely observed. The conclusion is that for nonlinearity to be important, either the fields or the propagation distances must be large. For the latter, dissipative losses must be small. The ray description as a simplification of the Maxwell equations is anal- ogous to the fact that classical mechanics gives a good approximation to so- lutions of the Schr¨ odinger equation of quantum mechanics. The associated ideas are called the quasiclassical approximation. The methods developed for hyperbolic equations also work for this important problem in quantum me- chanics. A brief treatment is presented in §5.2.2. The role of rays in optics is played by the paths of classical mechanics. There is an important difference in the two cases. The Schr¨ odinger equation has a small parameter, Planck’s constant. The quasiclassical approximation is an approximation valid for small Planck’s constant. The mathematical theory involves the limit as this constant tends to zero. The Maxwell equations apparently have a small parameter too, the inverse of the speed of light. One might guess that rays occur in a theory where this speed tends to infinity. This is not the case. For the Maxwell equations in a vacuum the small parameter that appears is the wavelength which is introduced via the initial data. It is not in the equa- tion. The equations describing the dispersion of light when it interacts with matter do have a small parameter, the inverse of the resonant frequencies of the material, and the analysis involves data tuned to this frequency just as the quasiclassical limit involves data tuned to Planck’s constant. Dispersion is one of my favorite topics. Interested readers are referred to the articles [Donnat and Rauch, 1997] (both) and [Rauch, 2007]. Short wavelength phenomena cannot simply be studied by numerical simulations. If one were to discretize a cubic meter of space with mesh size 10−5 cm so as to have five mesh points per wavelength, there would be 1021 data points in each time slice. Since this is nearly as large as the number of atoms per cubic centimeter, there is no chance for the memory of a computer to be suﬃcient to store enough data, let alone make calculations. Such brute force approaches are doomed to fail. A more intelligent approach would be to use radical local mesh refinement so that the fine mesh was used only when needed. Still, this falls far outside the bounds of present computing power. Asymptotic analysis offers an alternative approach that is not only powerful but is mathematically elegant. In the scientific literature it is also embraced because the resulting equations sometimes have exact

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