Geometric Asymptotics for Nonlinear PDE, I V. P. Maslov and G. A. Omelyanov [Translations of Mathematical Monographs Vol. 202] American Mathematical Society (2001) 285 pp., price USD 99.- ISBN: 0821821091.
Reviewer: R. Ghrist, University of Illinois, Urbana-Champaign, USA.

Level: intermediate.

This monograph covers a particular class of methods for nonlinear wave solutions to PDEs. The type of wave solutions considered are those which have fast variation over a small spatial scale and slow variation elsewhere. Examples of such include ``smoothed'' shock waves and soliton or breather-type solutions with spatially concentrated waves. The variety of equations which exhibit such solutions is both wide and well-studied [K-dV, K-P, Burgers, Boussinesq, etc.].

There are well-known methods for asymptotics of wave solutions. The key ideas in this book begin with the WKB method, inspired by ideas in geometric optics. Briefly, one develops rays for the linear asymptotics on which there lies a Hamiltonian dynamics structure [the eikonal equation]. These methods can be globalized by adding geometric structure to the bicharacteristics: this is an example of what the authors mean by geometric asymptotics.

The text is organized around examples and classes of equations. Beginning with one-dimensional waves in the Korteweg-de Vries equation, the authors present their methods and contrast them with more classical (e.g., multiple scales) techniques. This quickly extends to numerous other one-dimensional equations. The second chapter extends the methods to two-dimensional waves; again, numerous examples are used to explain the methods.

Unfortunately, there does not appear to be a central or general presentation of the methods developed. The reader who is looking to this book as a reference will likely be disappointed; however, the reader who wants to learn how to use the techniques `by hand' in specific equations will be quite satisfied.

Later chapters continue the theme into more complex examples, including Toda lattices and breathers. The final chapter is a detailed treatment of vortices which has a slightly different character than the remainder of the text.