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Soliton equations and Hamiltonian systems. (English) Zbl 0753.35075
Advanced Series in Mathematical Physics. 12. Singapore: World Scientific. ix, 310 p. (1991).
The book of L. A. Dickey presents one more point of view on the mathematical theory of solitons or, in other words, on the theory of nonlinear partial differential equations. Repeating the author’s words one can say: “So, I. M. Gelfand and the author picture this science to themselves” and to readers. The series of joint papers of I. M. Gelfand and L. A. Dickey in the middle of seventies was an important step in the development of the mathematical theory of nonlinear integrable equations. The algebraic approach, mathematical power and the spirit of Gelfand’s school are fully represented in the referred book.
The algebraic point of view is a key element of the whole exposition. Differential algebra, ring of pseudodifferential operators, resolvent, algebraic formulation of Hamiltonian structures, KdV and KP hierarchies, Grassmannian, \(\Sigma\)-function, matrix hierarchies, stationary equations are basic notions and tools of the approach discussed by the author.
The book is basically organized in three parts. In this first part the KdV and Gelfand-Dickey equations and their Hamiltonian structures are considered (chapters 2-4). The KP-hierarchy is Hamiltonian structure, Baker function, \(\bar\Sigma\)-function, Grassmannian symmetries are discussed in chapters 5-8. Matrix linear problems and associated integrable equations are treated in chapters 9, 10, 16. Chapters 11, 12, 13, 14-15 are devoted to the stationary equations. Multi-time Lagrangian and Hamiltonian formalism is considered in chapter 17.
As a whole the book presents a very good exposition of the important part of the soliton theory.

35Q53 KdV equations (Korteweg-de Vries equations)
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35Q51 Soliton equations
35Q58 Other completely integrable PDE (MSC2000)