P.1. How this book came to be, and its peculiarities xiii diffractive nonlinear geometric optics in contrast have reached a mature state. Readers of this book should be in a position to readily attack the papers describing that material. The methods of geometric optics are presented as a way to understand the qualitative behavior of partial differential equations. Many examples proper to the theory of partial differential equations are discussed in the text, notably the basic results of microlocal analysis. In addition two long examples, stabilization of waves in §5.6 and dense oscillations for inviscid compressible fluid flow in Chapter 11 are presented. There are many impor- tant examples in science and technology. Readers are encouraged to study some of them by consulting the literature. In the scientific literature there will not be theorems. The results of this book turn many seemingly ad hoc approximate methods into rigorous asymptotic analyses. Only a few of the many important hyperbolic systems arising in appli- cations are discussed. I recommend the books [Courant, 1962], [Benzoni- Gavage and Serre, 2007], and [M´ etivier, 2009]. The asymptotic expansions of geometric optics explain the physical theory, also called geometric optics, describing the rectilinear propagation, reflection, and refraction of light rays. A brief discussion of the latter ideas is presented in the introductory chap- ter that groups together elementary examples that could be, but are usually not, part of a partial differential equations course. The WKB expansions of geometric optics also play a crucial role in understanding the connection of classical and quantum mechanics. That example, though not hyperbolic, is presented in §5.2.2. The theory of hyperbolic mixed initial boundary value problems, a sub- ject with many interesting applications and many diﬃcult challenges, is not discussed. Nor is the geometric optics approach to shocks. I have omitted several areas where there are already good sources for example, the books [Smoller, 1983], [Serre, 1999], [Serre, 2000], [Dafermos, 2010], [Majda, 1984], [Bressan, 2000] on conservation laws, and the books [H¨ ormander, 1997] and [Taylor, 1991] on the use of pseudodifferential tech- niques in nonlinear problems. Other books on hyperbolic partial differential equations include [Hadamard, 1953], [Leray, 1953], [Mizohata, 1965], and [Benzoni-Gavage and Serre, 2007]. Lax’s 1963 Stanford notes occupy a spe- cial place for me. I took a course from him in the late 1960s that corre- sponded to the enlarged version [Lax, 2006]. When I approached him to ask if he’d be my thesis director he asked what interested me. I indicated two subjects from the course, mixed initial boundary value problems and the section on waves and rays. The first became the topic of my thesis, and the second is the subject of this book and at the core of much of my research. I

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