The interested reader has access to a vast literature on hyperbolic Partial Differential Equations. The ubiquitous presence of hyperbolic PDEs (from pure Mathematics to applied Engineering) has made these equations the centre of many important references. Probably because of this fact, those books covering more than the fundamentals of the theory are almost forced to concentrate efforts to specialized topics of the field. The book under review has chosen short wavelength asymptotics as the main theme of its exposition. The author, Prof. J. Rauch is a renowned researcher on PDEs with interesting contributions on non-linear microlocal analysis, control of waves and non-linear Geometric Optics. He is also the author of a monograph on PDEs of which this second book can be considered a sequel for deeper and specialized study.

The book contains some general facts about PDEs (characteristics, the Cauchy problem, dispersion,…) but it rapidly focus its attention to linear and non-linear geometric optics, that is, the realm of solutions of hyperbolic PDEs with short wavelength in the sense of the Fourier expansion. In this situation, the existence of solutions, interference, microlocal analysys, resonance and other interesting issues are tackled. It is important to note that short wave length solutions are not only important in Optics, but also in other relevant situations as Fluids. These other situations are also contemplated in the book which, in addition, is illustrated with many examples and exercises that help the reading of this dense work. The result is a book certainly intended for researchers and graduate students interested in the topic.

Reviewer:

Marco Castrillon Lopez