Solitons: Differential Equations, Symmetries and Infinite Dimensional Algebras [Brossura]

There are various approaches to classical integrable models. Each comes from a different school, and each has a certain flavour. Examples include: The Hamiltonian approach of the Faddeev school, the algebraic geometric approach of the Novikov school, and the analytic approaches of the Princeton, NYU and Clarkson schools.

This book is devoted to an introduction to the free fermion/representation theoretic approach of M. Sato and the Kyoto school.

It is a short (basically 100 pages + solutions, etc) and introductory book that, on first glance, gives the impression of being highly readable.

This is indeed the case: the book *is* highly readable, but a *full* appreciation of the underlying theory requires filling in the details, and there are quite a few details to fill in, and a superficial reading of the prose will not get the reader very far.

Sato's theory, and its subsequent development by his collaborators and students, is so rich, and has so many different aspects to it, from analysis to quantum field theory to geometry, that there is indeed so much to think about, so many ends to tie.

The book contains three parts:

1. Integrable nonlinear PDE's, their Lax formulation, using pseudo differential operators, their reformulation as bilinear (Hirota) differential equations, and the concept of tau functions.

2. Rewriting the theory in part 1 in the language of free fermions, Clifford algebra, infinite dimensional Lie algebra, and rewriting tau functions as fermion expectation values.

3. The geometric interpretation of the theory of parts 1 and 2 in terms of orbits of highest weight vectors under the action of Lie algebras, tau functions as points on Sato's (infinite dimensional) Grassmannian of the fermion Hilbert space, Hirota's bilinear equations as Plucker relations of Sato's infinite dimensional Grassmannian.

The book does not contain all elements of the Kyoto approach, and should be supplemented by reading further review papers. In particular,

Solitons and Infinite Dimensional Lie Algebras,
by M Jimbo and T Miwa,
Publ. RIMS, Kyoto Univ, vol 19 (1983) 943--1001

is highly recommended for complementing the book, but should be read only after completing parts 1 and 2 of the book, as it requires more mathematical maturity.

The book is suitable to be used in the classroom (it contains very doable exercises and outlines of almost all solutions), but an overview of the subject will be acquired only upon a very detailed reading.