## Hamiltonian Methods in the Theory of Solitons (Google eBook)This book presents the foundations of the inverse scattering method and its applications to the theory of solitons in such a form as we understand it in Leningrad. The concept of solitonwas introduced by Kruskal and Zabusky in 1965. A soliton (a solitary wave) is a localized particle-like solution of a nonlinear equation which describes excitations of finite energy and exhibits several characteristic features: propagation does not destroy the profile of a solitary wave; the interaction of several solitary waves amounts to their elastic scat tering, so that their total number and shape are preserved. Occasionally, the concept of the soliton is treated in a more general sense as a localized solu tion of finite energy. At present this concept is widely spread due to its universality and the abundance of applications in the analysis of various processes in nonlinear media. The inverse scattering method which is the mathematical basis of soliton theory has developed into a powerful tool of mathematical physics for studying nonlinear partial differential equations, almost as vigoraus as the Fourier transform. The book is based on the Hamiltonian interpretation of the method, hence the title. Methods of differential geometry and Hamiltonian formal ism in particular are very popular in modern mathematical physics. It is precisely the general Hamiltonian formalism that presents the inverse scat tering method in its most elegant form. Moreover, the Hamiltonian formal ism provides a link between classical and quantum mechanics. |

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### Contents

1 | |

7 | |

9 | |

11 | |

20 | |

3 Properties of the Monodromy Matrix in the QuasiPeriodic Case | 26 |

4 Local Integrals of the Motion | 33 |

5 The Monodromy Matrix in the Rapidly Decreasing Case | 39 |

Basic Examples and Their General Properties | 280 |

2 Examples of Lattice Models | 292 |

3 Zero Curvature Representation as a Method for Constructing Integrable Equations | 305 |

4 Gauge Equivalence of the NS Model 𝑥 1 and the HM Model | 315 |

5 Hamiltonian Formulation of the Chiral Field Equations and Related Models | 321 |

6 The Riemann Problem as a Method for Constructing Solutions of Integrable Equations | 333 |

7 A Scheme for Constructing the General Solution of the Zero Curvature Equation Concluding Remarks on Integrable Equations | 339 |

8 Notes and References | 345 |

6 Analytic Properties of Transition Coefficients | 46 |

7 The Dynamics of Transition Coefficients | 51 |

8 The Case of Finite Density Jost Solutions | 55 |

9 The Case of Finite Density Transition Coefficients | 62 |

10 The Case of Finite Density Time Dynamics and Integrals of the Motion | 72 |

11 Notes and References | 78 |

References | 80 |

The Riemann Problem | 81 |

2 The Rapidly Decreasing Case Analysis of the Riemann Problem | 89 |

3 Application of the Inverse Scattering Problem to the NS Model | 108 |

4 Relationship Between the Riemann Problem Method and the GelfandLevitanMarchenko Integral Equations Formulation | 114 |

5 The Rapidly Decreasing Case Soliton Solutions | 126 |

6 Solution of the Inverse Problem in the Case of Finite Density The Riemann Problem Method | 137 |

7 Solution of the Inverse Problem in the Case of Finite Density The GelfandLevitanMarchenko Formulation | 146 |

8 Soliton Solutions in the Case of Finite Density | 165 |

9 Notes and References | 177 |

References | 182 |

The Hamiltonian Formulation | 186 |

2 Poisson Commutativity of the Motion Integrals in the QuasiPeriodic Case | 194 |

3 Derivation of the Zero Curvature Representation from the Fundamental Poisson Brackets | 199 |

4 Integrals of the Motion in the Rapidly Decreasing Case and in the Case of Finite Density | 205 |

5 The 𝚲Operator and a Hierarchy of Poisson Structures | 210 |

6 Poisson Brackets of Transition Coefficients in the Rapidly Decreasing Case | 222 |

7 ActionAngle Variables in the Rapidly Decreasing Case | 229 |

8 Soliton Dynamics from the Hamiltonian Point of View | 241 |

9 Complete Integrability in the Case of Finite Density | 249 |

10 Notes and References | 267 |

References | 274 |

General Theory of Integrable Evolution Equations | 279 |

References | 350 |

Fundamental Continuous Models | 356 |

2 The Inverse Problem for the HM Model | 370 |

3 Hamiltonian Formulation of the HM Model | 384 |

4 The Auxiliary Linear Problem for the SG Model | 393 |

5 The Inverse Problem for the SG Model | 407 |

6 Hamiltonian Formulation of the SG Model | 431 |

7 The SG Model in LightCone Coordinates | 446 |

8 The LandauLifshitz Equation as a Universal Integrable Model with TwoDimensional Auxiliary Space | 457 |

9 Notes and References | 463 |

References | 467 |

Fundamental Models on the Lattice | 471 |

2 The Auxiliary Linear Problem for the Toda Model in the Rapidly Decreasing Case | 475 |

3 The Inverse Problem and Soliton Dynamics for the Toda Model in the Rapidly Decreasing Case | 489 |

4 Complete Integrability of the Toda Model in the Rapidly Decreasing Case | 499 |

5 The Lattice LL Model as a Universal Integrable System with TwoDimensional Auxiliary Space | 508 |

6 Notes and References | 519 |

References | 521 |

LieAlgebraic Approach to the Classification and Analysis of Integrable Models | 523 |

2 Trigonometric and Elliptic rMatrices and the Related Fundamental Poisson Brackets | 533 |

3 Fundamental Poisson Brackets on the Lattice | 540 |

4 Geometric Interpretation of the Zero Curvature Representation and the Riemann Problem Method | 543 |

5 The General Scheme as Illustrated with the NS Model | 558 |

6 Notes and References | 566 |

573 | |

Conclusion | 577 |

579 | |

584 | |

### Common terms and phrases

action-angle variables Anal analytic continuation asymptotic behaviour asymptotic expansion auxiliary linear problem boundary conditions canonical completely integrable consider continuous spectrum coordinates corresponding defined diagonal differential equation discrete spectrum Dokl dynamics English transl equa equations of motion expression Faddeev follows formulae fundamental Poisson brackets given half-plane HM model integral equations integral representations inverse problem inverse scattering method involution Jacobi identity Jost solutions kernel Korteweg-de Vries Korteweg-de Vries equation lattice Lie algebra Lie groups Lie-Poisson brackets limit Math monodromy matrix nonlinear equations NS equation NS model operator parameters phase space Phys Poisson brackets Poisson structure Priloz properties quantum quasi-periodic r-matrix rapidly decreasing reduced monodromy matrix relation respect Riemann problem right hand side Russian satisfy Schrodinger SG model Teor theorem tion transformation transition coefficients transition matrix vanishes variational derivatives vector Zakharov zero curvature condition zero curvature representation

### Popular passages

Page 18 - ... the value of the integral does not depend on the choice of the curve.

Page 12 - We shall assume that the reader is familiar with the basic concepts of formal language theory and computational complexity as contained, for example, In [9"].

Page 573 - Integrable equations on Lie algebras arising in problems of mathematical physics. Izv. Akad. Nauk SSSR, Ser. Mat. 48, No.

Page 573 - NY 1964 [D 1983] Drinfeld, VG: Hamiltonian structures on Lie groups, Lie bialgebras and the geometrical meaning of classical Yang-Baxter equations. Dokl. Akad. Nauk SSSR 268, 285-287 (1983) [Russian]; English transl.

Page 7 - In: Les Houches, Session XXXIX, 1982, Recent Advances in Field Theory and Statistical Mechanics, Zuber, J.-B., Stora, R.