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Hamiltonian methods in the theory of solitons. Transl. from the Russian by A. G. Reyman. Reprint of the 1987 original. (English) Zbl 1111.37001

Classics in Mathematics. Berlin: Springer (ISBN 978-3-540-69843-2/pbk). xiv, 592 p. (2007).
This is a reprint of the original 1987 edition (see Zbl 0632.58004). The Russian original was reviewed in Zbl 0632.58003.
This fantastic book definitely deserves to be included in Springer’s Classics in Mathematics series as it represents a milestone in dynamical systems literature.
For convenience of the readers who are not aware of this book yet, we give the table of contents:
Part I. Nonlinear Schrödinger equation. Chapter 1. Representation of zero curvature. Chapter 2. Riemann problem. Chapter 3. Hamiltonian formulation.
Part II. General theory of integrable evolution equations. Chapter 1. Main examples and their general properties. Chapter 2. Fundamental continuous models. Chapter 3. Fundamental models on a lattice. Chapter 4. Lie-algebraic approach to the classification and investigation of integrable models.

MSC:

37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
37K60 Lattice dynamics; integrable lattice 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.)
37K30 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures
81U40 Inverse scattering problems in quantum theory
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
35Q51 Soliton equations
35Q55 NLS equations (nonlinear Schrödinger equations)
35Q58 Other completely integrable PDE (MSC2000)
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