Nonlinear Dynamical Systems of Mathematical Physics: Spectral and Symplectic Integrability Analysis

Contents......Page 12
Preface......Page 8
1.1.1 Invariant measure......Page 22
1.1.2 The Liouville condition......Page 23
1.1.3 The Poincar e theorem......Page 24
1.1.4 The Birkho  Khinchin theorem......Page 25
1.1.5 The Birkho  Khinchin theorem for discrete dynamical systems......Page 26
1.2.1 Poisson brackets......Page 27
1.2.2 The Liouville theorem and Hamilton Jacobi method......Page 28
1.2.3 Dirac reduction: Symplectic and Poissonian structures on submanifolds......Page 32
2.1 The Poisson structures and Lie group actions on manifolds: Introduction......Page 36
2.2 Lie group actions on Poisson manifolds and the orbit struc- ture......Page 37
2.3 The canonical reduction method on symplectic spaces and related geometric structures on principal fiber bundles......Page 39
2.4 The form of reduced symplectic structures on cotangent spaces to Lie group manifolds and associated canonical connections......Page 42
2.6 The geometric structure of non-abelian Yang   Mills gauge field equations via the reduction method......Page 44
2.7.1 The quantization scheme, observables and Poisson manifolds......Page 48
2.7.2 The Hopf and quantum algebras......Page 51
2.7.3 Integrable flows related to Hopf algebras and their Poissonian representations......Page 53
2.7.4 Casimir elements and their special properties......Page 54
2.7.5 Poisson co-algebras and their realizations......Page 55
2.7.6 Casimir elements and the Heisenberg   Weil algebra related structures......Page 57
2.7.7 The Heisenberg   Weil co-algebra structure and related integrable flows......Page 61
3.1 Introduction......Page 64
3.2 Preliminaries......Page 68
3.3 Integral submanifold embedding problem for an abelian Lie algebra of invariants......Page 73
3.4 Integral submanifold embedding problem for a nonabelian Lie algebra of invariants......Page 87
3.5 Examples......Page 93
3.6 Existence problem for a global set of invariants......Page 96
3.7.1 The Henon   Heiles system......Page 97
3.7.2 A truncated four-dimensional Fokker   Planck Hamiltonian system......Page 100
4.2 Implectic operators and dynamical systems......Page 104
4.3 Symmetry properties and recursion operators......Page 110
4.4 B  cklund transformations......Page 111
4.5 Properties of solutions of some infinite sequences of dy- namical systems......Page 113
4.6 Integro-differential systems......Page 119
5.1.1 Generalized eigenvalue problem......Page 122
5.1.2 Properties of the spectral problem......Page 123
5.1.3 Analysis of a generating function for conservation laws......Page 124
5.2.1 Gradient-holonomic properties of the generating functional of conservation laws......Page 125
5.2.2 Involutivity of conservation laws......Page 128
5.3.1 The monodromy matrix and the Lax representation......Page 129
5.3.2 The gradient-holonomic method for constructing conservation laws......Page 130
5.3.3 Construction of compatible implectic operators......Page 132
5.3.4 Reconstruction of the Lax operator algorithm......Page 135
5.3.5 Asymptotic construction of recursive and implectic operators for Lax integrable dynamical systems......Page 136
5.3.6 A small parameter method for constructing recursion and implectic operators......Page 138
5.4.1 Preliminaries......Page 149
5.4.2 Hierarchies of symmetries and related Hamiltonian structures......Page 152
5.4.3 A Lie-algebraic algorithm for investigating integrability......Page 154
6.1 A non-isospectrally Lax integrable KdV dynamical system......Page 158
6.1.1 A non-isospectrally integrable nonlinear nonautonomous Schr  odinger dynamical system......Page 160
6.1.2 Lagrangian and Hamiltonian analysis of dynamical systems on functional manifolds: The Poisson  Dirac reduction......Page 162
6.1.3 Remarks......Page 168
6.2.2 The algebraic structure of the Lax integrable dynamical system......Page 169
6.2.3 The periodic problem and canonical variational relationships......Page 174
6.2.4 An integrable nonlinear dynamical system of Ito......Page 179
6.2.5 The Benney  Kaup dynamical system......Page 184
6.2.6 Integrability analysis of the inverse Korteweg  de Vries equation (inv KdV)......Page 186
6.2.7 Integrability analysis of the inverse nonlinear Benney  Kaup system......Page 196
6.3.1 Introduction......Page 206
6.3.2 Lagrangian analysis......Page 207
6.3.3 Gradient-holonomic analysis......Page 209
6.3.4 Lax form and finite-dimensional reductions......Page 213
6.4.1 Differential-geometric integrability analysis......Page 216
6.4.2 Bi-Hamiltonian structure and Lax representation......Page 218
6.5.1 Introduction......Page 222
6.5.2 The generalized Riemann hydrodynamical equation for N = 2: Conservation laws, bi-Hamiltonian structure and Lax representation......Page 224
6.5.3 The generalized Riemann hydrodynamic equation for N = 3: Conservation laws, bi-Hamiltonian structure and Lax representation......Page 230
6.5.4 The hierarchies of conservation laws and their analysis......Page 237
6.5.5 Generalized Riemann hydrodynamic equation for N = 4: Conservation laws, bi-Hamiltonian structure and Lax representation......Page 239
6.6.1 Introduction......Page 244
6.6.2.1 Differential-algebraic preliminaries......Page 245
6.6.2.2 The generalized Riemann hydrodynamic equation: the case N = 3......Page 246
6.6.2.3 Solution set analysis of the functional-differential equation D  + rD   = 1......Page 249
6.6.2.4 The generalized Riemann hydrodynamical equation for N = 4......Page 252
6.6.3 Differential-algebraic analysis of the Lax integrability of the KdV dynamical system......Page 255
6.7.1 Introduction......Page 258
6.7.2 Symmetry properties......Page 260
6.7.3 Dirac  Fock  Podolsky problem analysis......Page 261
6.7.4 Symplectic reduction......Page 263
6.8.1 Introduction......Page 265
6.8.2 Symplectic and symmetry analysis......Page 266
6.8.3 Incompressible superfluids: Symmetry analysis and conservation laws......Page 271
6.9.1 Introduction......Page 275
6.9.2 General differential-geometric analysis......Page 277
6.9.3 Lie-algebraic analysis of the case n = 2......Page 279
6.9.4 Generalized spectral problem......Page 280
6.9.5 Novikov   Marchenko commutator equation......Page 283
6.9.6 Representation of the holonomy Lie algebra sl(2)......Page 285
6.9.7 Algebraic-geometric properties of the integrable Riccati equations: The case n = 2......Page 287
6.9.8 Jacobi inversion and Abel transformation......Page 289
6.9.9 Convergence of Abelian integrals......Page 292
6.9.10 Analytical expressions for exact solutions......Page 294
6.9.12 A final remark......Page 298
7.1 Introduction: Diff     -actions......Page 300
7.2 Lie-algebraic structure of the A-action......Page 302
7.3 Casimir functionals and reduction problem......Page 303
7.4 Associated momentum map and versal deformations of the Diff    action......Page 306
8.1 Short introduction......Page 310
8.2 Lie-algebra of Lax integrable (2+1)-dimensional dynamical systems......Page 311
8.2.1 The Poisson bracket on the extended phase space......Page 314
8.2.2 Hierarchies of additional symmetries......Page 317
8.2.3 Closing remarks......Page 321
9.1 Problem setting......Page 324
9.2 Factorization properties......Page 325
9.3 Hamiltonian analysis......Page 326
9.4 Tensor products of Poisson structures and source like factorized operator dynamical systems......Page 328
9.5 Remarks......Page 329
10.1 Spectral operators and generalized eigenfunctions expansions......Page 330
10.2 Semilinear forms, generalized kernels and congruence of operators......Page 332
10.3 Congruent kernel operators and related Delsarte transmutation maps......Page 335
10.4 Differential-geometric structure of the Lagrangian identity and related Delsarte transmutation operators......Page 346
10.5 The general differential-geometric and topological structure of Delsarte transmutation operators: A generalized de Rham   Hodge theory......Page 352
10.6 A special case: Relations with Lax systems......Page 364
10.7 Geometric and spectral theory aspects of Delsarte   Darboux binary transformations......Page 366
10.8 The spectral structure of Delsarte   Darboux transmutation operators in multi-dimensions......Page 371
10.9 Delsarte   Darboux transmutation operators for special multi-dimensional expressions and their applications......Page 380
11.1 Differential-geometric problem setting......Page 394
11.2 The differential invariants......Page 395
12.1 Introduction......Page 400
12.2 Mathematical preliminaries: Fock space and realizations......Page 403
12.3 The Fock space embedding method, nonlinear dynamical systems and their complete linearization......Page 413
12.5.1 Introduction......Page 416
12.5.2 Loop Grassmann manifolds......Page 422
12.5.3 Symplectic structures on loop Grassmann manifolds and Casimir invariants......Page 424
12.5.4 An intrinsic loop Grassmannian structure and dual momentum maps......Page 426
12.5.5 Holonomy group structure of the quantum computing medium......Page 431
12.5.6 Holonomic quantum computations: Examples......Page 435
13.1 Introductory setting......Page 440
13.2 Classical relativistic electrodynamics revisited......Page 445
13.3.1 Motion of a point particle in a vacuum - an alternative electrodynamic model......Page 448
13.3.2 Motion of two interacting charge systems in a vacuum   an alternative electrodynamic model......Page 449
13.3.3 A moving charged point particle formulation dual to the classical alternative electrodynamic model......Page 451
13.4 Vacuum field electrodynamics: Hamiltonian analysis......Page 453
13.5.1 The problem setting......Page 456
13.5.2 Free point particle electrodynamics model and its quantization......Page 457
13.5.3 Classical charged point particle electrodynamics model and its quantization......Page 459
13.5.4 Modified charged point particle electrodynamics model and its quantization......Page 460
13.6 Some relevant observations......Page 462
13.7.1 The classical relativistic electrodynamics back- grounds: A charged point particle analysis......Page 464
13.7.2 Least action principle analysis......Page 467
13.8.1 A free charged point particle in a vacuum......Page 473
13.8.2 Charged point particle electrodynamics......Page 475
13.9.1 A new hadronic string model: Least action formulation......Page 478
13.9.2 Lagrangian and Hamiltonian analysis......Page 479
13.9.4 Maxwell\'s electromagnetism theory from the vacuum field theory perspective......Page 483
14.1 General setting......Page 488
14.2.1 Exterior forms of degree 2 and their canonical representation......Page 492
14.2.2 Locally trivial fiber bundles and their structures......Page 494
14.2.3 Subbundles and factor bundles......Page 500
14.2.4 Manifolds. Tangent and cotangent bundles......Page 501
14.2.5 The rank theorem for a differential map and its corollaries......Page 503
14.2.6 Vector fields......Page 504
14.2.7 Differential forms......Page 505
14.2.8 Differential systems......Page 509
14.2.9 The class of an ideal. Darboux\'s theorem......Page 511
14.2.10 Dynamical systems on symplectic manifolds. Complete integrability and ergodicity......Page 517
14.3.1 General setting......Page 522
14.3.2 The Maurer  Cartan one-form construction......Page 527
14.3.3 Cartan  Frobenius integrability of ideals in a Grassmann algebra......Page 530
14.3.4 The differential-geometric structure of a class of integrable ideals in a Grassmann algebra......Page 532
14.3.5 Example: Burgers\' dynamical system and its integrability......Page 534
Bibliography......Page 538
Index......Page 560