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**Hamiltonian methods in the theory of solitons.
(Гамил’тонов подход в теории солитонов.)**
*(Russian)*
Zbl 0632.58003

Moskva: “Nauka”. Glavnaya Redaktsiya Fiziko-Matematicheskoj Literatury. 528 p. R. 3.00 (GOE 87 A 25546) (1986).

The book gives the exposition of the inverse scattering method and its application to soliton theory. It deals with the classical part of the subject only; the authors are planning to write a second volume devoted to quantum aspects. The main characteristic feature of this book is the consistent Hamiltonian approach to the theory. The nonlinear Schrödinger equation (not the KdV equation as usual) is considered as a main example; the investigation of this equation forms the first part of the book. The second part is devoted to such fundamental models as the sine-Gordon equation, Heisenberg equation, Toda lattice, etc., the classification of integrable models and the methods for constructing their solutions.

The following list of chapters gives more details as to the 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.

The following list of chapters gives more details as to the 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.

Reviewer: Yu.E.Glikhlikh

### MSC:

37-02 | Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory |

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.) |

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 |

37K30 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures |

37K60 | Lattice dynamics; integrable lattice equations |

81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |

81U40 | Inverse scattering problems in quantum theory |

58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |

35Q51 | Soliton equations |

35Q55 | NLS equations (nonlinear Schrödinger equations) |

35Q58 | Other completely integrable PDE (MSC2000) |