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.