xviii Preface solutions and scientists are well versed in understanding phenomena from small families of exact solutions. Short wavelength asymptotics can be used to great advantage in many disparate domains. They explain and extend the basic rules of linear geomet- ric optics. They explain the dispersion and diffraction of linear electromag- netic waves. There are nonlinear optical effects, generation of harmonics, rotation of the axis of elliptical polarization, and self-focusing, which are also well described. Geometric optics has many applications within the subject of partial differential equations. They play a key role in the problem of solvability of linear equations via results on propagation of singularities as presented in §5.5. They are used in deriving necessary conditions, for example for hy- poellipticity and hyperbolicity. They are used by Ralston to prove necessity in the conjecture of Lax and Phillips on local decay. Via propagation of singularities they also play a central role in the proof of sufficiency. Propa- gation of singularities plays a central role in problems of observability and controlability (see §5.6). The microlocal elliptic regularity theorem and the propagation of singularities for symmetric hyperbolic operators of constant multiplicity is treated in this book. These are the two basic results of linear microlocal analysis. These notes are not a systematic introduction to that subject, but they present an important part en passant. Chapters 9 and 10 are devoted to the phenomenon of resonance whereby waves with distinct phases interact nonlinearly. They are preparatory for Chapter 11. That chapter constructs a family of solutions of the compress- ible 2d Euler equations exhibiting three incoming wave packets interacting to generate an infinite number of oscillatory wave packets whose velocities are dense in the unit circle. Because of the central role played by rays and characteristic hypersur- faces, the analysis of conormal waves is closely related to geometric optics. The reader is referred to the treatment of progressing waves in [Lax, 2006] and to [Beals, 1989] for this material. Acknowledgments. I have been studying hyperbolic partial differential equations for more that forty years. During that period, I have had the pleasure and privilege to work for extended periods with (in order of ap- pearance) M. Taylor, M. Reed, C. Bardos, G. etivier, G. Lebeau, J.-L. Joly, and L. Halpern. I thank them all for the things that they have taught me and the good times spent together. My work in geometric optics is mostly joint with J.-L. Joly and G. etivier. This collaboration is the motivation and central theme of the book. I gratefully acknowledge my indebtedness to them.
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