Soliton Equations and Hamiltonian Systems.
Title:
Soliton Equations and Hamiltonian Systems.
Author:
Dickey, L. A.
ISBN:
9789812794512
Personal Author:
Edition:
2nd ed.
Physical Description:
1 online resource (421 pages)
Series:
Advanced Series in Mathematical Physics ; v.26
Advanced Series in Mathematical Physics
Contents:
Contents -- Preface to the Second Edition -- Introduction to the First Edition -- Chapter 1 Integrable Systems Generated by Linear Differential nth Order Operators -- 1.1 Differential Algebra A -- 1.2 Space of Functionals A -- 1.3 Ring of Pseudodifferential Operators -- 1.4 Lax Pairs. GD Hierarchies of Equations -- 1.5 First Integrals (Constants of Motion) -- 1.6 Compatibility of the Equations of a Hierarchy -- 1.7 Soliton Solutions -- 1.8 Resolvent. Adler Mapping -- Chapter 2 Hamiltonian Structures -- 2.1 Finite-Dimensional Case -- 2.2 Hamilton Mapping -- 2.3 Variational Principles -- 2.4 Symplectic Form on an Orbit of the Coadjoint Representation of a Lie Group -- 2.5 Purely Algebraic Treatment of the Hamiltonian Structure -- 2.6 Examples -- Chapter 3 Hamiltonian Structure of the GD Hierarchies -- 3.1 Lie Algebra V Dual Space Q1 and Module Q0 -- 3.2 Proof of Theorem 3.1.2 -- 3.3 Poisson Bracket -- 3.4 Reduction to the Submanifold Un-1 = 0 -- 3.5 Variational Derivative of the Resolvent -- 3.6 Hamiltonians of the GD Hierarchies -- 3.7 Theory of the KdV-Hierarchy (n = 2) Independent of the General Case -- Chapter 4 Modified KdV and GD. The Kupershmidt-Wilson Theorem -- 4 1 Miura Transformation. The Kupershmidt-Wilson Theorem -- 4.2 Modified KdV Equation. Backlund Transformations -- 4.3 More on Modified GD Equations -- Chapter 5 The KP Hierarchy -- 5.1 Definition of the KP Hierarchy -- 5.2 Reduction of the KP Hierarchy to GD -- 5.3 First Integrals and Soliton Solutions -- 5.4 Hamiltonian Structure -- 5.5 Resolvent -- 5.6 Hamiltonians of the KP Hierarchy -- Chapter 6 Baker Function T-Function -- 6.1 Dressing -- 6.2 Baker Function -- 6.3 Shift Operator and T-Function -- 6.4 Resolvent and Baker Function. Fay Identities -- 6.5 Vertex Operators -- 6.6 T-Function and Fock Representation.
6A Appendix. List of Useful Formulas for the Faa di Bruno Polynomials -- Chapter 7 Additional Symmetries String Equation -- 7.1 Additional Symmetries -- 7.2 Generating Function for Additional Symmetries -- 7.3 String Equation -- Chapter 8 Grassmannian. Algebraic-Geometrical Krichever Solutions -- 8.1 Infinite-Dimensional Grassmannian -- 8.2 Modified Definition of the Grassmannian T-Function -- 8.3 Algebraic-Geometrical Solutions of Krichever -- 8A Appendix. Abel Mapping and the 0-Function -- Chapter 9 Matrix First-Order Operator AKNS-D Hierarchy -- 9.1 Hierarchy of Equations Generated by a First-Order Matrix Differential Operator -- 9.2 Hamiltonian Structure -- 9.3 Hamiltonians of the AKNS-D Hierarchy -- 9.4 GD Hierarchies as Reductions of the Matrix Hierarchies (Drinfeld-Sokolov Reduction) -- 9A Appendix. Extension of the Algebra A to an Algebra Closed with Respect to the Indefinite Integration -- Chapter 10 Generalization of the AKNS-D Hierarchy: Single-Pole and Multi-Pole Matrix Hierarchies -- 10.1 Single-Pole Matrix Hierarchy -- 10.2 Single-Pole Hierarchy. Presentation not Depending on a Distinguished Operator 1 -- 10.3 Multi-Pole (General Zakharov-Shabat) Hierarchy -- 10.4 Example: Principal Chiral Field Equation -- 10.5 Grassmannian -- Chapter 11 Isomonodromic Deformations and the Most General Matrix Hierarchy -- 11.1 Isomonodromic Deformations -- 11.2 General Matrix Hierarchy -- Chapter 12 Tau Functions of Matrix Hierarchies -- 12.1 Segal-Wilson's T-Function for AKNS-D -- 12.2 Tau Functions for More General Matrix Hierarchies -- Chapter 13 KP Modified KP Constrained KP Discrete KP and q-KP -- 13.1 Modified GD (Cont'd) -- 13.2 Modified KP and Constrained KP -- 13.3 Discrete KP -- 13.4 q-KP.
Chapter 14 Another Chain of KP Hierarchies and Integrals Over Matrix Varieties -- 14.1 Introduction. More About the Modified KP -- 14.2 Stabilizing Chain -- 14.3 Solutions to the Chain -- 14.4 Solutions in the Form of Series in Schur Polynomials. Stabilization -- 14.5 From the Stabilizing Chain to the Kontsevich Integral -- Chapter 15 Transformational Properties of a Differential Operator under Diffeomorphisms and Classical W-Algebras -- 15.1 Tensors with Respect to Diffeomorphisms and the AGD-Algebra -- 15.2 Another Construction of Primary Fields -- Chapter 16 Further Restrictions of the KP -- Stationary Equations -- 16.1 The Ring of Functions on the Phase Space of the Equation -- 16.2 Characteristics of the First Integrals -- 16.3 Hamiltonian Structure -- 16.4 Stationary Equations of the KdV Hierarchy ([GD79]) -- 16.5 Integration after Liouville -- 16.6 Return to the Original Variables -- Chapter 17 Stationary Equations of the Matrix Hierarchy -- 17.1 First Integrals -- 17.2 Hamiltonian Structure of Stationary Equations -- 17.3 Action-Angle Variables -- 17A Appendix. Genus of the Riemann Surfaces and the Newton Diagram -- Chapter 18 Stationary Equations of the Matrix Hierarchy (Cont'd) -- 18.1 Baker Function. Return to Original Variables -- 18.2 Rotation of the n-Dimensional Rigid Body -- Chapter 19 Field Lagrangian and Hamiltonian Formalism -- 19.1 Introduction -- 19.2 Variational Bi-Complex -- 19.3 Exactness of the Bi-Complex -- 19.4 Variational Derivative -- 19.5 Lagrangian-Hamiltonian Formalism -- 19.6 Variational Bi-Complex of a Differential Equation. First Integrals -- 19.7 Poisson Bracket -- 19.8 Relationship with the Single-Time Formalism -- Chapter 20 Further Examples and Applications -- 20.1 KP-Hierarchy.
20.2 The Zakharov-Shabat Equation with Rational Dependence on the Spectral Parameter -- 20.3 Principal Chiral Field -- 20.4 Lagrangians of the nth Reduced KP (GD) Hierarchy -- Bibliography -- Index.
Abstract:
The theory of soliton equations and integrable systems has developed rapidly during the last 30 years with numerous applications in mechanics and physics. For a long time, books in this field have not been written but the flood of papers was overwhelming: many hundreds, maybe thousands of them. All this output followed one single work by Gardner, Green, Kruskal, and Mizura on the Korteweg-de Vries equation (KdV), which had seemed to be merely an unassuming equation of mathematical physics describing waves in shallow water. Besides its obvious practical use, this theory is attractive also because it satisfies the aesthetic need in a beautiful formula which is so inherent to mathematics. The second edition is up-to-date and differs from the first one considerably. One third of the book (five chapters) is completely new and the rest is refreshed and edited. Contents: Integrable Systems Generated by Linear Differential n th Order Operators; Hamiltonian Structures; Hamiltonian Structure of the GD Hierarchies; Modified KdV and GD. The Kupershmidt-Wilson Theorem; The KP Hierarchy; Baker Function, τ-Function; Additional Symmetries, String Equation; Grassmannian. Algebraic-Geometrical Krichever Solutions; Matrix First-Order Operator, AKNS-D Hierarchy; Generalization of the AKNS-D Hierarchy: Single-Pole and Multi-Pole Matrix Hierarchies; Isomonodromic Deformations and the Most General Matrix Hierarchy; Tau Functions of Matrix Hierarchies; KP, Modified KP, Constrained KP, Discrete KP, and q -KP; Another Chain of KP Hierarchies and Integrals Over Matrix Varieties; Transformational Properties of a Differential Operator under Diffeomorphisms and Classical W -Algebras; Further Restrictions of the KP, Stationary Equations; Stationary Equations of the Matrix Hierarchy; Field Lagrangian and Hamiltonian Formalism; Further Examples and Applications. Readership:
Applied mathematicians and mathematical physicists.
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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