Volume: 78; 2000; 600 pp; Hardcover
MSC: Primary 37; 46; Secondary 35; 49; 58
Print ISBN: 978-0-8218-1400-0
Product Code: SURV/78
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Electronic ISBN: 978-1-4704-1305-7
Product Code: SURV/78.E
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KP or mKP: Noncommutative Mathematics of Lagrangian, Hamiltonian, and Integrable Systems
Share this pageBoris A. Kupershmidt
This book develops a theory that can be viewed as a noncommutative counterpart of the following topics: dynamical systems in general and integrable systems in particular; Hamiltonian formalism; variational calculus, both in continuous space and discrete. The text is self-contained and includes a large number of exercises. Many different specific models are analyzed extensively and motivations for the new notions are provided.
Readership
Graduate students, research mathematicians, and physicists interested in global analysis and analysis on manifolds.
Reviews & Endorsements
This remarkable book … is written so as to be completely self-contained with most arguments presented in great detail … a remarkable account of a substantial body of work … likely to remain a significant work of reference as the field continues to develop.
-- Mathematical Reviews
Table of Contents
Table of Contents
KP or mKP: Noncommutative Mathematics of Lagrangian, Hamiltonian, and Integrable Systems
- Contents vii8 free
- Preface xv16 free
- Acknowledgments xix20 free
- Part A. Continuous Space Time 122 free
- Chapter 1. The KP Hierarchy 223
- Chapter 2. The MKP Hierarchy 2445
- §2.1. Construction Of The Basic Equations And The Commutativity Of The Flows In The MKP Hierarchy 2445
- §2.2. The Hamiltonian Formalism For The MKP Hierarchy 2647
- §2.3. The MKP Hierarchy With Values In Finite-Dimensional Associative Algebras 3051
- §2.4. The Equations Of Dispersive Water Waves 3455
- §2.5. The Burgers Hierarchy 3657
- §2.6. The Korteweg-De Vries Hierarchy 3960
- §2.7. The MKP Hierarchy Dressed Up 4465
- Chapter 3. Between KP And MKP 4768
- §3.1. The Miura Map In The Language Of Lax Representations 4768
- §3.2. The Miura Map In The Language Of Wilson Equations 5273
- §3.3. The Miura Map From MKP To KP Is Hamiltonian 5475
- §3.4. From DWW To KP 6788
- §3.5. From Nonlinear Schrodinger To KP 6990
- §3.6. From Derivative NLS To MKP 7697
- §3.7. Between DNLS And NLS 81102
- §3.8. Water Form Of Nonlinear Schrodinger 83104
- §3.9. The Real Miura Map Between The KdV And MkdV Hierarchies 88109
- §3.10. KP Factorized, Or MKP[sup(II)] 111132
- §3.11. P. S.: The Fully Nonabelian Miura Map Between The KPAnd Potential MKP Hierarchies Revisited And Found Perfectly Hamiltonian 116137
- Chapter 4. Noncommutative Lagrangian Formalism 126147
- §4.1. Motivation Extracted From A Lobotomy Of The KdV Equation 126147
- §4.2. Variational Derivatives And Related Notions 129150
- §4.3. Transformation Formula For The Variational Derivatives 134155
- §4.4. The Variational Complex 137158
- §4.5. The Residue Formula 142163
- §4.6. The Legendre Transform 144165
- §4.7. Localizations 148169
- Chapter 5. Noncommutative Hamiltonian Formalism 150171
- Chapter 6. MKP = M+ KP 167188
- §6.1. KP, MKP, KdV, And Other Equations as Noncommutative Hamiltonian Systems 167188
- §6.2. Inverting The Noninvertible Miura Map Between The MKP And KP Hierarchies 174195
- §6.3. M2KP 177198
- §6.4. Clebsch Representations 183204
- §6.5. The Kontsevich-Type Formula 189210
- §6.6. The Third Hamiltonian Structure Of The MKP Hierarchy 192213
- Chapter 7. The Quasirelativistic KP Hierarchy 194215
- Chapter 8. The Second Construction Of Integrals Of The KP Hierarchy 212233
- Part B. Discrete Space-Continuous Time 221242
- Chapter 9. KP, Then MKP 222243
- §9.1. Evolutions Of KP Type 222243
- §9.2. The Dressing Scene 227248
- §9.3. Evolutions Of MKP Type 228249
- §9.4. The Modified Dressing Scene 231252
- §9.5. KP From MKP 232253
- §9.6. The Miura Map In The Dressing Spaces 235256
- §9.7. The Classical Limit 237258
- §9.8. The Quasiclassical Limit 238259
- §9.9. KP Factorizations And The Modified Toda Lattice 242263
- Chapter 10. The Noncommutative Differential-Difference Calculus 246267
- Chapter 11. The Noncommutative Hamiltonian Formalism Over Differential-Difference Rings 263284
- Chapter 12. Hamiltonian Formalism For Discrete Integrable Systems Of KP And MKP Types 267288
- §12.1. The KP-Type Systems 267288
- §12.2. The MKP-Type Systems 270291
- §12.3. The Miura Map From KP To MKP Is Hamiltonian 272293
- §12.4. Gap Specializations And The Second Hamiltonian Structure 279300
- §12.5. The Kontsevich-Type Formula 281302
- §12.6. The Third Hamiltonian Structure Of The MKP Hierarchy 282303
- Chapter 13. The Gibbons Forms 291312
- §13.1. The Gibbons Form Of The KP Hierarchy 291312
- §13.2. The Gibbons Forms Of The MKP Hierarchy 295316
- §13.3. The Miura Map Between The Gibbons Forms Of The KP and MKP Hierarchies 298319
- §13.4. The Fourth Gibbons Form Of The MKP Hierarchy 303324
- §13.5. The Fully Bilinear Form Of The KP Hierarchy 306327
- §13.6. The Fifth Gibbons Form Of The MKP Hierarchy 307328
- §13.7. The Gibbons Form Of The KP Hierarchy In The G-Coordinates 312333
- §13.8. The Potential MKP Hierarchy In The G-Coordinates As ANonholonomic Dynamical Hierarchy, And The Associated Miura Map 315336
- §13.9. The Gibbons Form Under The Gap Specialization 318339
- Chapter 14. The Hydrodynamical Representation 321342
- §14.1. Motivation 321342
- §14.2. Hamiltonian Approach In The KP Case 324345
- §14.3. Algebraic Treatment Of The KP Case 328349
- §14.4. The Hydrodynamical Form Of The MKP Hierarchy 332353
- §14.5. The Hydrodynamical Miura Map 336357
- §14.6. The Hydrodynamical Form Of The KP Hierarchy In The G-Coordinates 341362
- §14.7. The Hydrodynamical Form Of The MKP Hierarchy In The G-Coordinates 345366
- §14.8. Noncommutative Lattice Analogs Of The Inviscid Burgers Hierarchy 352373
- §14.9. The Dressing Form Of The Hydrodynamical Representations 354375
- Chapter 15. Relativistic Toda Lattice And Related Systems 357378
- §15.1. Quasirelativistic Ansatz And Its First Properties 357378
- §15.2. At The Edge Of The Universe 361382
- §15.3. Hamiltonian Formalism For The Quasirelativistic KP Hierarchy 363384
- §15.4. Quasirelativistic Gibbons Form 366387
- §15.5. Hydrodynamical Forms Of The Quasirelativistic KP Hierarchy 367388
- §15.6. A Deformation Of The MKP Hierarchy 368389
- Part C. Discrete Space Time 376397
- Chapter 16. The Idea Of Lax Representations And Its Discrete-Time Analog 377398
- Chapter 17. Systems Of The KP Type 383404
- §17.1. The KP Hierarchy 383404
- §17.2. The Gibbons Form And Its Symplectic Properties 388409
- §17.3. The Hydrodynamical Form 393414
- §17.4. The KP Hierarchy In The G-Coordinates 401422
- §17.5. The Gibbons Form In The G-Coordinates 413434
- §17.6. The Hydrodynamical Form In The G-Coordinates 416437
- §17.7. The Factorized KP And The Modified Toda Lattice 423444
- Chapter 18. Systems Of The MKP Type 432453
- §18.1. The MKP Hierarchy 432453
- §18.2. The KP To MKP Miura Map 436457
- §18.3. The First Two Gibbons Forms 441462
- §18.4. The Third Gibbons Form And The Associated Miura Map 444465
- §18.5. The Fourth Gibbons Form 446467
- §18.6. The Hydrodynamical Representation And The Associated Miura Map 449470
- §18.7. Space-Time Discretizations Of The Equation H[sub(t)] = HH[sub(x)]H Form A Family Of Hamiltonian Maps 455476
- §18.8. The MKP Hierarchy In The G-Coordinates 458479
- §18.9. The Gibbons Form In The G-Coordinates And Its Symplectic Properties 466487
- Chapter 19. The Toda Lattice, The Relativistic Toda Lattice, And Related Systems 470491
- §19.1. The Problem Of Discrete Dressing 470491
- §19.2. The Negative Evolution Of The Toda Lattice 473494
- §19.3. The Relativistic Toda Lattice 476497
- §19.4. The Shadow Relativistic Toda Lattice 481502
- §19.5. The Negative Evolution Of The Modified Toda Lattice 485506
- §19.6. The Negative Evolution Of The Volterra System 490511
- §19.7. The Positive Evolution Of The Volterra System 493514
- §19.8. The Volterra System From The Toda Lattice Point Of View 496517
- §19.9. Generalized Volterra Systems 499520
- §19.10. The Gap-KP Hierarchy 503524
- §19.11. Time-Discretization As A Factorization 506527
- §19.12. The Problem Of Discrete Dressing Resolved 513534
- Part D. Appendices 518539
- Appendix A1. Complexification Of Hamiltonian Systems 519540
- Appendix A2. Asymptotic Expansions Of Hamiltonian Systems 523544
- Appendix A3. Variational Calculus Over Noncommutative Rings 531552
- Appendix A4. Hamiltonian Correspondencies 550571
- Appendix A5. Covariant Aspects Of The Hamiltonian Formalism 566587
- Appendix A6. Noncommutative Solitions 577598
- Appendix A7. The Noncommutative KP Equation 581602
- Appendix A8. A List Of Scalar Equations 582603
- Appendix A9. Open Problems And Conjectures 586607
- Notes And Comments 589610
- Bibliography 591612
- Index/Notations 596617