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. M´ 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. M´ etivier. This collaboration is the motivation
and central theme of the book. I gratefully acknowledge my indebtedness