Lie algebraic methods in integrable systems

By A. Roy Chowdhury

Reviewed by A.P.Fordy

Chapman and Hall, 2000,
Research Notes in Mathematics 415,
Hardback: ISBN 1-584880-37-6,

Lie algebras play an important role in the theory of integrable systems, most obviously in the Lax or zero curvature formulation. Each such formulation is associated with a particular Lie algebra and, depending upon the representation, this can be interpreted as linear spectral problem, Bäcklund transformation or Hamiltonian structure. Lie algebras play a crucial role in several other aspects of integrable systems, such as symmetries and r-matrices. The idea of the book is therefore very interesting, bringing together many Lie algebraic topics under one banner. Other available books concentrate only on particular aspects.

Unfortunately, the book is not as interesting as it promises. The first question to ask is, For whom is the book written?'' The Readership'', listed on the back cover, is Mathematicians and theoretical physicists working on nonlinear integrable systems, chaos and dynamical systems.'' It is my opinion that people working in chaos or general dynamical systems will get very little out of this book, so let us turn to Mathematicians and theoretical physicists working on nonlinear integrable systems.'' I presume experts are excluded, since they have no need of such a book, so could I give this book to one of my post-graduate students and expect him/her to learn anything?

My answer is, No,'' since no chapter of this book is even remotely close to being self contained. The author has not written a consistent list of pre-requisites'' when presenting the material. The brief glimpse of some of the basic feature of soliton theory seen from a traditional viewpoint'' (page 3) clearly assumes the reader to be familiar with standard soliton theory, but to be just in need of a quick reminder. Chapter 2 collects basic Lie algebra theory in concise form.'' This is a big subject and it is excusable for the author to collect some basic facts'' to be used in the integrable systems context. However, I'm afraid that the author has not presented a very logical and coherent picture. For instance, the exponential map is introduced on page 18, while the formal definition of a Lie group is left for another 12 pages. Kac--Moody algebras are defined on page 48 in the basic survey and again on page 141, in the chapter on co-adjoint orbits, as if never seen before!

A very bad feature from the student point of view is the inconsistency of notation. For instance, in chapter 3, on the Wahlquist--Estabrook prolongation method, the primary KdV variables (field variable and its derivatives) are (q,u,p,r,...) on one page, (u,z,p,...) two pages later, (q,u,p,z,...) on the next, and so on. On one page we have upper indices on F,G and on the next, lower indices (these are the same objects, not some tensor having its indices raised and lowered. The book is also full of misprints, such as u=qx,p=ux=qxxx. If such mistakes occurred in just a few places, they would be excusable, but almost every page is littered with them!

Each topic is presented as scattered results from the original papers, with no real understanding or synthesis. I would not recommend this book, either as a place to gain an understanding or as a source of useful information. Contrary to a statement on the back of the book, this is {\em not} a user-friendly handbook.''

A listing of books reviewed in UK Nonlinear News is available.

UK Nonlinear News thanks Chapman and Hall for providing a review copy of this book.

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