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Zbl 0694.70001
Toda, Morikazu
Theory of nonlinear lattices. 2nd enlarged edition. (English)
[B] Springer Series in Solid-State Sciences, 20. Berlin etc.: Springer-Verlag. x, 225 p. DM 60.00 (1989). ISBN 3-540-18327-2

The book should be called more appropriately ``The Toda Lattice''. The bulk of the book is devoted to the solutions of the ``classical'' Toda lattice (the periodic and the infinite versions). In my opinion, the scope of the book is too narrow. \par The presentation is big on explicit formulas and rather small on the conceptual side, which makes the book hard to read for a mathematician (or anybody else for that matter) not very familiar with the subject. On the other hand, the book is not addressed to an expert either. It is the second English edition [see the review of the first 1981 ed. in Zbl 0465.70014], of the Japanese original which appeared in 1978 (Zbl 0502.70013). The numerous developments in the integrable Hamiltonian systems that took place in the past decade are left out. As the author says in the preface, ``a new chapter has been added to describe several important recent achievements''. Unfortunately, the addition (chapter 6: Recent advances in the theory of nonlinear lattices) falls short of this goal. It barely touches on a few subjects related to the Toda lattice and does not do justice to any of them. Suffices to say that chapter 6 is 18 pages long and consists of 9 subsections each devoted to a separate topic. The topics vary from the KdV equation to the ``Integrability'' to the ``Bethe Ansatz'' to the ``Numerical Results''. The book would have been better without it.
[E.Gutkin]

MSC 2000:: *70-02 Research monographs (mechanics of particles and systems)
70Jxx Linear vibration theory (general mechanics)
82B05 Classical equilibrium statistical mechanics (general)

Keywords: Toda Lattice; integrable Hamiltonian systems; nonlinear lattices; KdV equation; Bethe Ansatz

Citations: Zbl 0465.70014; Zbl 0502.70013

Cited in: Zbl 0830.58015

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	Scientific prize winners of the ICM 2010
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	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

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