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Multi-Hamiltonian theory of dynamical systems. (English) Zbl 0912.58017
Texts and Monographs in Physics. Berlin: Springer. x, 350 p. (1998).
The theory of solitons started off as a number of interesting, though rather special, coincidences. However, in the last 20 years it has blossomed to become a very broad mathematical body. It has applications to the study of a wide range of phenomena and is deeply related to several fields of mathematics and physics. The rapid development of the theory was the effort of several groups in different continents. One of the aspects of the theory that up to now has not been covered extensively in the form of a text book is precisely the multi-Hamiltonian approach. In general terms, for the vast majority of soliton equations, behind the complete integrability lies the existence of two (or more) compatible Poisson structures, i.e., two Poisson brackets whose linear combination is also a Poisson bracket. This remark, which can be traced back to the work of F. Magri [J. Math. Phys. 19, No. 5, 1156-1162 (1978; Zbl 0383.35065)], has been explored in a very fruitful way in the work of a number of authors.
The concept of bi-Hamiltonian vector fields, when conjoined with the idea of recursion operators and hereditary symmetries, forms the substratum for the approach in the book under review. The field has received contributions from the work of many researchers, and the author of the book has himself contributed with a number of research articles. The viewpoint of the book is an algebraic one. The main objects, such as symplectic structures, Poisson brackets, recursion operators, and the like, are considered without much reference to a specific function space where they belong. This allows one to focus on their interplay in diverse situations of complete integrability, the goal being the development of a theory suitable to infinite dimensional systems both in the case of continuous as well as discrete dynamical systems. It is remarkable that such infinite dimensional theory brings a refreshing view to some finite dimensional systems, in particular to classical issues such as Liouville integrability and separability.
The structure of the book is basically the following. After a short overview in the Introduction, Chapter 2 contains a review of the necessary tools of tensor calculus. This makes the book pretty much self-contained. Chapter 3 deals with Lie Algebras, Lie brackets, Hamiltonian vector fields, bi-Hamiltonian structures, general tensor invariants, and, finally, with the Virasoro algebra. Chapter 4 concerns the Lax representation and isospectral deformations. It also deals with nonisospectral deformations, which are related to the so-called master-symmetries. Chapter 5 is devoted to the study of the \(N\)-soliton solutions from an algebraic point of view, the point being that the general theory developed in the book gives a complete set of conserved quantities (the \(N\)-soliton manifold) as well as the vector fields forming the basis of the tangent bundle. Chapter 6 concerns the construction of finite-dimensional completely integrable systems by restricting infinite-dimensional ones to invariant manifolds. It also studies the concept of separability and deals with topics such as Ostrogradsky generalized coordinates, Newton coordinates, bi-Hamiltonian chains, Poisson-Nijenhuis manifolds and Calogero-Moser systems. Chapter 7 focusses on systematic ways of constructing multi-Hamiltonian systems. The discussion includes techniques such as the Gelfand-Dikii approach, the Adler-Kostant-Symes theorem, as well as the R-matrix approach. Chapter 8 dwells on the algebraic theory of \((2+1)\) dimensional field systems. This topic of intense recent interest is roughly divided into two parts in this chapter, the first one being the Sato approach and the second one being non-commutative operator valued field variables.
The book, which is accessible to a large audience including graduate students in mathematics or physics, could be used as a textbook on an advanced course on the algebraic aspects of soliton theory. It should also be useful to researchers who plan to apply such techniques to other fields. Among the strong points, one can mention the unique aspect of emphasizing algebraic aspects of integrability. It deals with recent research topics in a very solid and balanced way. It is well documented, with a good number of helpful proofs and a large up-to-date bibliography. Obviously a work of the scope of this book would have to stay away from a few topics in order to remain within reasonable size. Some of these missing topics are Riemann-Hilbert problems, the usual direct/inverse scattering approach, the bispectral problem, and string theory, to cite a few. However, many of these topics are already covered in a number of other classical soliton texts such as M. J. Ablowitz and H. Segur, ‘Solitons and the inverse scattering transform’ (1981; Zbl 0472.35002), F. Calogero and A. Degasperis, ‘Spectral transform and solitons. I’ (1982; Zbl 0501.35072), and S. Novikov, S. V. Manakov, L. P. Pitaevskij and V. E. Zakharov, ‘Theory of solitons.’ (1984; Zbl 0598.35002). In conclusion, the work under review will prove to be useful as a reference, a text-book, as well as a research tool.

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.)
37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
35Q51 Soliton equations
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems