Solitons: differential equations, symmetries and infinite dimensional algebras. Transl. from the Japanese by Miles Reid.(English)Zbl 0986.37068
Cambridge Tracts in Mathematics. 135. Cambridge: Cambridge University Press. ix, 108 p. (2000).
This is a translation of a Japanese text written in 1992. In the words of the authors, “this book has concentrated mainly on the results on soliton equations and their symmetries of the Kyoto school surrounding Mikio Sato, work that reached its peak in the years around 1981.” It gives an introduction to some of the algebraic structure underlying the KP hierarchy, culminating in the relation between Plücker relations and the Hirota formalism. Prerequisites are kept to a minimum, but considerable background is necessary to put these developments in context. Each of the ten chapters concludes with a few exercises, with answers included at the end of the tract. For a balanced perspective, the newcomer to the field should also read one of the classical introductions to the actual physical applications of solitons, and to the problems in nonlinear PDE raised by them.
Reviewer: Satyanad Kichenassamy (Reims)
MSC:
| 37K40 | Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems |
| 37-02 | Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory |
| 35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 35Q51 | Soliton equations |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 37K30 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures |
| 17B69 | Vertex operators; vertex operator algebras and related structures |
