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Algebraic Structures In Integrability: Foreword By Victor Kac [Hardback]

(Landau Ins For Theoretical Physics, Russia)
  • Formāts: Hardback, 348 pages
  • Izdošanas datums: 13-Jul-2020
  • Izdevniecība: World Scientific Publishing Co Pte Ltd
  • ISBN-10: 9811219648
  • ISBN-13: 9789811219641
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Algebraic Structures In Integrability: Foreword By Victor KacPalielināt
  • Formāts: Hardback, 348 pages
  • Izdošanas datums: 13-Jul-2020
  • Izdevniecība: World Scientific Publishing Co Pte Ltd
  • ISBN-10: 9811219648
  • ISBN-13: 9789811219641
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9789811219641.html
Keywords:
"Relationships of the theory of integrable systems with various branches of mathematics are extremely deep and diverse. On the other hand, the most fundamental exactly integrable systems often have applications in theoretical physics. Therefore, many mathematicians and physicists are interested in integrable models. The book is intelligible to graduate and PhD students and can serve as an introduction to separate sections of the theory of classical integrable systems for scientists with algebraic inclinations. For the young, the book can serve as a starting point in the study of various aspects of integrability, while professional algebraists will be able to use some examples of algebraic structures, which appear in the theory of integrable systems, for wide-ranging generalizations. The statements are formulated in the simplest possible form. However, some ways of generalization are indicated. In the proofs, only essential points are mentioned, while for technical details, references are provided. The focus is on carefully selected examples. In addition, the book proposes many unsolved problems of various levels of complexity. A deeper understanding of every chapter of the book may require the study of more rigorous and specialized literature"--
Foreword xiii
Preface xv
1 Introduction
1(26)
1.1 List of basic notations
1(3)
1.1.1 Constants, vectors and matrices
1(1)
1.1.2 Derivations and differential operators
1(1)
1.1.3 Differential algebra
2(1)
1.1.4 Algebra
3(1)
1.2 Lax pairs
4(4)
1.2.1 Case of ODE
4(2)
1.2.2 Lax pairs for partial differential equations
6(1)
1.2.3 Matrix Riemann--Hilbert problem and dressing procedure
7(1)
1.3 Hamiltonian structures
8(4)
1.3.1 Hamiltonian form of evolution equations
10(2)
1.4 Infinitesimal symmetries
12(4)
1.4.1 Naive symmetry test
14(2)
1.5 First integrals and local conservation laws
16(2)
1.6 Transformations
18(9)
1.6.1 Point and contact transformations
18(3)
1.6.2 Differential substitutions of Miura type
21(6)
Part 1 Lax representations for integrable systems
27(54)
2 Lax pairs and decomposition of Lie algebras
29(52)
2.0.1 Definitions of symmetries and conservation laws
29(2)
2.1 Scalar Lax pairs for evolution equations
31(14)
2.1.1 Pseudo-differential series
31(2)
2.1.2 Korteweg---de Vries hierarchy
33(8)
2.1.3 Gelfand--Dikii hierarchy and its generalizations
41(4)
2.2 Matrix Lax pairs
45(7)
2.2.1 The NLS hierarchy
45(4)
2.2.2 Generalizations
49(3)
2.3 Decompositions of loop algebras and Lax pairs
52(19)
2.3.1 Factoring subalgebras for Q = SP3
56(5)
2.3.2 Integrable top-like systems
61(1)
2.3.3 Classical SO3 spinning tops
62(3)
2.3.4 Generalization of Euler and Steklov--Lyapunov models to son-case
65(1)
2.3.5 Factoring subalgebras for Kac--Moody algebras
66(2)
2.3.6 Integrable systems of Landau--Lifschitz type
68(2)
2.3.7 Integrable hyperbolic models of chiral type
70(1)
2.4 Finite-dimensional factorization method, reductions and nonassociative algebras
71(10)
2.4.1 Factorization method
71(2)
2.4.2 Reductions
73(4)
2.4.3 Generalized factorization method
77(4)
Part 2 Algebraic structures in theory of bi-Hamiltonian systems
81(42)
3 Bi-Hamiltonian formalism
83(40)
3.0.1 Shift argument method
84(2)
3.0.2 Bi-Hamiltonian form for KdV equation
86(1)
3.1 Bi-Hamiltonian formalism and pairs of compatible algebras
86(23)
3.1.1 Compatible Lie algebras. Examples and applications
89(5)
3.1.2 Compatible Lie brackets associated with θ-functions
94(3)
3.1.3 Associative algebras compatible with Matm
97(12)
3.2 Polynomial forms of Calogero--Moser elliptic systems
109(14)
3.2.1 Calogero--Moser Hamiltonians
109(3)
3.2.2 Quasi-exactly solvable differential operators
112(4)
3.2.3 Commutative subalgebras in U(glN+1) and Calogero--Moser quantum Hamiltonians
116(2)
3.2.4 Bi-Hamiltonian origin of the classical elliptic Calogero-Moser model
118(5)
Part 3 Symmetry approach to integrability
123(184)
4 Basic concepts of symmetry approach
125(52)
4.1 Description of some classification results
125(9)
4.1.1 Hyperbolic equations
125(1)
4.1.2 Evolution equations
126(5)
4.1.3 Systems of two equations
131(3)
4.2 Necessary integrability conditions
134(23)
4.2.1 Evolutionary vector fields, recursion operator and variational derivative
134(1)
4.2.2 Formal symmetries
135(3)
4.2.3 Conservation laws
138(1)
4.2.4 Formal symplectic operator
139(2)
4.2.5 Canonical densities and integrability conditions
141(4)
4.2.6 Invariance of integrability conditions with respect to changes of variables
145(2)
4.2.7 Classification of integrable equations of KdV type
147(3)
4.2.8 Integrable equations of Harry--Dym type
150(1)
4.2.9 Nonlocal variables, evolution equations with constraints and inversion of differential substitutions
151(6)
4.3 Weakly nonlocal recursion and Hamiltonian operators
157(10)
4.3.1 Weakly nonlocal recursion operators
158(4)
4.3.2 Weakly nonlocal Hamiltonian operators
162(1)
4.3.3 Recursion operators for Krichever--Novikov equation
163(2)
4.3.4 Weakly nonlocal Hamiltonian operators for Krichever--Novikov equation
165(2)
4.4 Integrable nonevolution equations
167(10)
4.4.1 Formal symmetry and symplectic operator
168(2)
4.4.2 Examples
170(1)
4.4.3 Integrability conditions
171(3)
4.4.4 Weakly nonlocal recursion operators
174(2)
4.4.5 Discussion
176(1)
5 Integrable hyperbolic equations of Liouville type
177(26)
5.1 Generalized x and y-integrals
178(3)
5.2 Laplace invariants for linear hyperbolic operator
181(2)
5.3 Nonlinear hyperbolic equations of Liouville type
183(4)
5.4 Differential substitutions and equations of Liouville type
187(3)
5.4.1 Differential substitutions of the first order
189(1)
5.5 Pre-Hamiltonian operators
190(6)
5.5.1 Examples of pre-Hamiltonian operators related to Liouville type equations
192(4)
5.6 Integrable multi-component Liouville-type hyperbolic systems
196(7)
6 Integrable nonabelian equations
203(48)
6.1 ODE on free associative algebras
203(26)
6.1.1 Equations with matrix variables
203(5)
6.1.2 Systems of differential equations on free associative algebra
208(1)
6.1.3 Quadratic homogeneous nonabelian systems
209(1)
6.1.4 Two-component nonabelian systems
210(6)
6.1.5 Integrable scalar quadratic homogeneous systems
216(5)
6.1.6 Nonabelization of integrable homogeneous scalar systems
221(5)
6.1.7 Integrable inhomogeneous nonabelian systems
226(3)
6.2 Nonabelian Hamiltonian formalism and Poisson brackets defined on traces of matrices
229(16)
6.2.1 Trace Poisson brackets
229(8)
6.2.2 Nonabelian Poisson brackets on free associative algebras
237(6)
6.2.3 Double Poisson brackets on free associative algebras
243(2)
6.3 Evolution equations on free associative algebras
245(6)
6.3.1 Matrix integrable equations
245(2)
6.3.2 Nonabelian evolution equations
247(4)
7 Integrable evolution systems and nonassociative algebras
251(29)
7.1 Nonassociative algebraic structures related to integrability
251(3)
7.1.1 Left-symmetric algebras
252(1)
7.1.2 Jordan algebras
252(1)
7.1.3 Triple Jordan systems
253(1)
7.2 Jordan KdV systems
254(4)
7.3 Left-symmetric algebras and Burgers type systems
258(1)
7.4 Integrable equations associated with triple Jordan systems
259(6)
7.4.1 Systems of mKdV type
259(4)
7.4.2 Systems of NLS type
263(2)
7.4.3 Systems of derivative NLS type
265(1)
7.5 Integrable systems corresponding to new algebraic structures
265(4)
7.5.1 Equations of potential mKdV type
265(1)
7.5.2 Systems of Olver--Sokolov type
266(1)
7.5.3 Systems of mKdV type with two algebraic operations
267(2)
7.6 Rational integrable systems
269(4)
7.6.1 Inverse element as a solution of a system of differential equations
269(2)
7.6.2 Several classes of integrable rational Jordan models
271(2)
7.7 Deformations of nonassociative algebras and integrable systems of geometric type
273(7)
7.7.1 Geometric description of deformations
274(1)
7.7.2 Algebraic description of deformations
274(2)
7.7.3 Evolution equations of geometric type
276(4)
8 Integrable vector evolution equations
280(17)
8.1 Examples of integrable polynomial vector systems
280(2)
8.2 Symmetry approach to classification of integrable vector equations
282(15)
8.2.1 Canonical densities
283(2)
8.2.2 Euler operator and Frechet derivative
285(1)
8.2.3 Vector isotropic equations of KdV type
286(3)
8.2.4 Vector equations of geometric type
289(2)
8.2.5 Vector auto-Backlund transformations
291(1)
8.2.6 Integrable equations on the sphere
292(1)
8.2.7 Equations with two scalar products on the sphere
293(2)
8.2.8 Anisotropic equations with constant vector
295(2)
9 Appendices
297(10)
9.1 Appendix
1. Hyperbolic equations with third order integrable symmetries
297(1)
9.2 Appendix
2. Scalar hyperbolic equations of Liouville type
298(2)
9.3 Appendix
3. Integrable evolution equations
300(4)
9.3.1 Third order equations
301(1)
9.3.2 Fifth order equations
302(2)
9.4 Appendix
4. Quasilinear systems of two equations of second order
304(3)
Bibliography 307(18)
Index 325