# Hamiltonian Methods in the Theory of Solitons (Google eBook)

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

### 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.

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.