P.1. How this book came to be, and its peculiarities
This book presents an introduction to hyperbolic partial differential equa-
tions. A major subtheme is linear and nonlinear geometric optics. The two
central results of linear microlocal analysis are derived from geometric op-
tics. The treatment of nonlinear geometric optics gives an introduction to
methods developed within the last twenty years, including a rethinking of
the linear case.
Much of the material has grown out of courses that I have taught. The
crucial step was a series of lectures on nonlinear geometric optics at the
Institute for Advanced Study/Park City Mathematics Institute in July 1995.
The Park City notes were prepared with the assistance of Markus Keel
and appear in [Rauch, 1998]. They presented a straight line path to some
theorems in nonlinear geometric optics. Graduate courses at the University
of Michigan in 1993 and 2008 were important. Much of the material was
refined in invited minicourses:
Ecole Normale Sup´ erieure de Cachan, 1997;
• Nordic Conference on Conservation Laws at the Mittag-Leffler Institute
and KTH in Stockholm, December 1997 (Chapters 9–11);
• Centro di Ricerca Matematica Ennio De Giorgi, Pisa, February 2004;
• Universit´ e de Provence, Marseille, March 2004 (§3.4, 5.4, Appendix 2.I);
• Universit` a di Pisa, February–May 2005, March–April 2006 (Chapter 3,
§6.7, 6.8), March–April 2007 (Chapters 9–11);
• Universit´ e de Paris Nord, February 2006–2010 (§1.4–1.7).