Hamiltonian Methods in the Theory of Solitons (Google eBook)

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Springer Science & Business Media, Aug 10, 2007 - Science - 594 pages
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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

Introduction
1
References
7
The Nonlinear Schrödinger Equation NS Model
9
Zero Curvature Representation
11
2 Zero Curvature Condition
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
References
573
Conclusion
577
List of Symbols
579
Index
584
Copyright

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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.

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About the author (2007)

Ludwig D. Faddeev was born in Leningrad, USSR in 1934. He graduated from the Leningrad State University in 1956 and received his Ph.D. from there in 1959. Since 1959 he has been affiliated with the Leningrad branch of Steklov Mathematical Institute and was its Director from 1976 to 2000. Currently Faddeev is Director of the Euler International Mathematical Institute in St. Petersburg, Russia, and Academician-Secretary of the Mathematics Division of the Russian Academy of Sciences. He was President of the International Mathematical Union during1986-1990.

Faddeev’s principal interests and contributions cover the large area of mathematical physics. They include, in chronological order, quantum scattering theory, spectral theory of automorphic functions, quantization of Yang-Mills theories, Hamiltonian methods in classical and quantum integrable systems, quantum groups and quantum integrable systems on a lattice. Faddeev’s work laid a mathematical foundation for functional methods in quantum gauge theories. A great deal of his work was directed towards development of Hamiltonian methods in classical and quantum field theories.

 

 

Leon A. Takhtajan was born in Erevan, Republic of Armenia of the USSR, in 1950. He was schooled in Leningrad, graduated from the Leningrad State University in 1973, and received his Ph.D. from the Leningrad branch of Steklov Mathematical Institute in 1975, with which he was affiliated during1973-1998. Since 1992 he has been Professor of Mathematics at Stony Brook University, USA.

Takhtajan’s principal interests and contributions are in the area of mathematical physics. They include classical and quantum integrable systems, quantum groups, Weil-Petersson geometry of moduli spaces of Riemann surfaces and moduli spaces of vector bundles, and application of quantum methods to algebraic and complex analysis. His work, together with L.D. Faddeev and E.K. Sklyanin, led to the development of the quantum inverse scattering method from which the theory of quantum groups was born.

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