KP or mKP: noncommutative mathematics of Lagrangian, Hamiltonian and integrable systems
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, RI
American Math. Soc.
2000
|
| Series: | Mathematical surveys and monographs
78 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVIII, 600 S. |
| ISBN: | 0821814001 |
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MARC
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| 100 | 1 | |a Kupershmidt, Boris A. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a KP or mKP |b noncommutative mathematics of Lagrangian, Hamiltonian and integrable systems |c Boris A. Kupershmidt |
| 264 | 1 | |a Providence, RI |b American Math. Soc. |c 2000 | |
| 300 | |a XVIII, 600 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Mathematical surveys and monographs |v 78 | |
| 650 | 4 | |a Géométrie différentielle non commutative | |
| 650 | 7 | |a Sistemas dinâmicos |2 larpcal | |
| 650 | 7 | |a Sistemas hamiltonianos |2 larpcal | |
| 650 | 4 | |a Systèmes hamiltoniens | |
| 650 | 4 | |a Hamiltonian systems | |
| 650 | 4 | |a Noncommutative differential geometry | |
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| 689 | 0 | |5 DE-604 | |
| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |z 978-1-4704-1305-7 |
| 830 | 0 | |a Mathematical surveys and monographs |v 78 |w (DE-604)BV000018014 |9 78 | |
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Record in the Search Index
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|---|---|
| adam_text | CONTENTS
PREFACE
ACKNOWLEDGMENTS
PART A. CONTINUOUS SPACE TIME 1
CHAPTER 1. The KP Hierarchy 2
§1.1. The Basic Equations And Their First Properties 2
§1.2. Hamiltonian Formalism For The KP Hierarchy 8
§1.3. Quaternionic KP Hierarchy 13
§1.4. The KP Hierarchy With Values In Finite Dimensional
Associative Algebras 17
§1.5. Dressing Motions 22
CHAPTER 2. The MKP Hierarchy 24
§2.1. Construction Of The Basic Equations And The
Commutativity Of The Flows In The MKP Hierarchy 24
§2.2. The Hamiltonian Formalism For The MKP Hierarchy 26
§2.3. The MKP Hierarchy With Values In Finite Dimensional
Associative Algebras 30
§2.4. The Equations Of Dispersive Water Waves 34
§2.5. The Burgers Hierarchy 36
§2.6. The Korteweg De Vries Hierarchy 39
§2.7. The MKP Hierarchy Dressed Up 44
CHAPTER 3. Between KP And MKP 47
§3.1. The Miura Map In The Language Of Lax Representations 47
§3.2. The Miura Map In The Language Of Wilson Equations 52
§3.3. The Miura Map From MKP To KP Is Hamiltonian 54
§3.4. From DWW To KP 67
§3.5. From Nonlinear Schrodinger To KP 69
§3.6. From Derivative NLS To MKP 76
vii
viii B. A. KUPERSHMIDT
§3.7. Between DNLS And NLS 81
§3.8. Water Form Of Nonlinear Schrodinger 83
§3.9. The Real Miura Map Between The KdV And MkdV
Hierarchies 88
§3.10. KP Factorized, Or MKP11 111
§3.11. P.S.: The Fully Nonabelian Miura Map Between The KP
And Potential MKP Hierarchies Revisited And Found
Perfectly Hamiltonian 116
CHAPTER 4. Noncommutative Lagrangian Formalism 126
§4.1. Motivation Extracted From A Lobotomy Of The KdV
Equation 126
§4.2. Variational Derivatives And Related Notions 129
§4.3. Transformation Formula For The Variational Derivatives 134
§4.4. The Variational Complex 137
§4.5. The Residue Formula 142
§4.6. The Legendre Transform 144
§4.7. Localizations 148
CHAPTER 5. Noncommutative Hamiltonian Formalism 150
§5.1. The Main Result Of The Hamiltonian Formalism 150
§5.2. Hamiltonian Maps 155
§5.3. Linear And Afflne Hamiltonian Operators, Lie Algebras
And Two Cocycles 157
§5.4. The Local Global Principle 162
§5.5. Hamiltonian Analog Of A Homomorphism Of Lie Algebras 166
CHAPTER 6. MKP = M+KP 167
§6.1. KP, MKP, KdV, And Other Equations as Noncommutative
Hamiltonian Systems 167
§6.2. Inverting The Noninvertible Miura Map Between The MKP
And KP Hierarchies 174
§6.3. M2KP 177
CONTENTS ix
§6.4. Clebsch Representations 183
§6.5. The Kontsevich Type Formula 189
§6.6. The Third Hamiltonian Structure Of The MKP Hierarchy 192
CHAPTER 7. The Quasirelativistic KP Hierarchy 194
§7.1. Construction Of Basic Equations And The Commutativity
Of The Flows 194
§7.2. Hamiltonian Formalism For The Quasirelativistic Flows 201
§7.3. Quasirelativistic NLS Hierarchy 209
CHAPTER 8. The Second Construction Of Integrals Of The KP Hierarchy 212
§8.1. The Wilson Formulae 212
§8.2. The Cherednik Flaschka Formulae 214
§8.3. The Inversion Formula 219
PART B. DISCRETE SPACE CONTINUOUS TIME 221
CHAPTER 9. KP, Then MKP 222
§9.1. Evolutions Of KP Type 222
§9.2. The Dressing Scene 227
§9.3. Evolutions Of MKP Type 228
§9.4. The Modified Dressing Scene 231
§9.5. KP From MKP 232
§9.6. The Miura Map In The Dressing Spaces 235
§9.7. The Classical Limit 237
§9.8. The Quasiclassical Limit 238
§9.9. KP Factorizations And The Modified Toda Lattice 242
CHAPTER 10. The Noncommutative Differential Difference Calculus 246
§10.1. The Variational Language 246
§10.2. The Natural Properties Of Variational Derivatives 252
§10.3. The Variational Complex 255
x B. A. KUPERSHMIDT
§10.4. The Residue Formula 260
CHAPTER 11. The Noncommutative Hamiltonian Formalism Over
Differential Difference Rings 263
§11.1. The Main Result Of The Hamiltonian Formalism 263
§11.2. Hamiltonian Maps 265
§11.3. Affine Hamiltonian Operators And Two Cocycles On
Lie Algebras, Etc. 266
CHAPTER 12. Hamiltonian Formalism For Discrete Integrable Systems
Of KP And MKP Types 267
§12.1. The KP Type Systems 267
§12.2. The MKP Type Systems 270
§12.3. The Miura Map From KP To MKP Is Hamiltonian 272
§12.4. Gap Specializations And The Second Hamiltonian Structure 279
§12.5. The Kontsevich Type Formula 281
§12.6. The Third Hamiltonian Structure Of The MKP Hierarchy 282
CHAPTER 13. The Gibbons Forms 291
§13.1. The Gibbons Form Of The KP Hierarchy 291
§13.2. The Gibbons Forms Of The MKP Hierarchy 295
§13.3. The Miura Map Between The Gibbons Forms Of The KP
and MKP Hierarchies 298
§13.4. The Fourth Gibbons Form Of The MKP Hierarchy 303
§13.5. The Fully Bilinear Form Of The KP Hierarchy 306
§13.6. The Fifth Gibbons Form Of The MKP Hierarchy 307
§13.7. The Gibbons Form Of The KP Hierarchy In The
G Coordinates 312
§13.8. The Potential MKP Hierarchy In The G Coordinates As A
Nonholonomic Dynamical Hierarchy, And The Associated
Miura Map 315
§13.9. The Gibbons Form Under The Gap Specialization 318
CHAPTER 14. The Hydrodynamical Representation 321
CONTENTS xi
§14.1. Motivation 321
§14.2. Hamiltonian Approach In The KP Case 324
§14.3. Algebraic Treatment Of The KP Case 328
§14.4. The Hydrodynamical Form Of The MKP Hierarchy 332
§14.5. The Hydrodynamical Miura Map 336
§14.6. The Hydrodynamical Form Of The KP Hierarchy
In The G Coordinates 341
§14.7. The Hydrodynamical Form Of The MKP Hierarchy
In The G Coordinates 345
§14.8. Noncommutative Lattice Analogs Of The Inviscid
Burgers Hierarchy 352
§14.9. The Dressing Form Of The Hydrodynamical Representations 354
CHAPTER 15. Relativistic Toda Lattice And Related Systems 357
§15.1. Quasirelativistic Ansatz And Its First Properties 357
§15.2. At The Edge Of The Universe 361
§15.3. Hamiltonian Formalism For The Quasirelativistic KP
Hierarchy 363
§15.4. Quasirelativistic Gibbons Form 366
§15.5. Hydrodynamical Forms Of The Quasirelativistic
KP Hierarchy 367
§15.6. A Deformation Of The MKP Hierarchy 368
PART C. DISCRETE SPACE TIME 376
CHAPTER 16. The Idea Of Lax Representations And Its Discrete Time
Analog 377
CHAPTER 17. Systems Of The KP Type 383
§17.1. The KP Hierarchy 383
§17.2. The Gibbons Form And Its Symplectic Properties 388
§17.3. The Hydrodynamical Form 393
§17.4. The KP Hierarchy In The G Coordinates 401
§17.5. The Gibbons Form In The G Coordinates 413
xii B. A. KUPERSHMIDT
§17.6. The Hydrodynamical Form In The G Coordinates 416
§17.7. The Factorized KP And The Modified Toda Lattice 423
CHAPTER 18. Systems Of The MKP Type 432
§18.1. The MKP Hierarchy 432
§18.2. The KP To MKP Miura Map 436
§18.3. The First Two Gibbons Forms 441
§18.4. The Third Gibbons Form And The Associated Miura Map 444
§18.5. The Fourth Gibbons Form 446
§18.6. The Hydrodynamical Representation And The
Associated Miura Map 449
§18.7. Space Time Discretizations Of The Equation Ht = HHXH
Form A Family Of Hamiltonian Maps 455
§18.8. The MKP Hierarchy In The G Coordinates 458
§18.9. The Gibbons Form In The G Coordinates And Its
Symplectic Properties 466
CHAPTER 19. The Toda Lattice, The Relativistic Toda Lattice,
And Related Systems 470
§19.1. The Problem Of Discrete Dressing 470
§19.2. The Negative Evolution Of The Toda Lattice 473
§19.3. The Relativistic Toda Lattice 476
§19.4. The Shadow Relativistic Toda Lattice 481
§19.5. The Negative Evolution Of The Modified Toda Lattice 485
§19.6. The Negative Evolution Of The Volterra System 490
§19.7. The Positive Evolution Of The Volterra System 493
§19.8. The Volterra System From The Toda Lattice Point Of View 496
§19.9. Generalized Volterra Systems 499
§19.10. The Gap KP Hierarchy 503
§19.11. Time Discretization As A Factorization 506
§19.12. The Problem Of Discrete Dressing Resolved 513
CONTENTS xiii
PART D. APPENDICES 518
APPENDIX Al. Complexification Of Hamiltonian Systems 519
APPENDIX A2. Asymptotic Expansions Of Hamiltonian Systems 523
§A2.1. Motivation From An Example: The KdV Equation 523
§A2.2. Vector Fields, Differential Forms, Variational Derivatives 527
§A2.3. Hamiltonian Structures 530
APPENDIX A3. Variational Calculus Over Noncommutative Rings 531
§A3.1. The Basic Objects 532
§A3.2. The Image And Kernel Of The Variational Operator 5 541
§A3.3. The Image Of The Operator d + eadu 547
APPENDIX A4. Hamiltonian Correspondencies 550
§A4.1. From Geometry To Algebra 550
§A4.2. The Infinite Dimensional Case 556
§A4.3. Closed 1 Forms As Lagrangian Submanifolds,
The Variational Version 561
§A4.4. Generating Functions Of Symplectic Maps
And Their Generalizations 562
APPENDIX A5. Covariant Aspects Of The Hamiltonian Formalism 566
§A5.1. GLm+i Generalities And A GL2 Example:
The KdV Equation 566
§A5.2. Infinitesimal Geometric Perturbations 571
APPENDIX A6. Noncommutative Solitions 577
APPENDIX A7. The Noncommutative KP Equation 581
APPENDIX A8. A List Of Scalar Equations 582
APPENDIX A9. Open Problems And Conjectures 586
NOTES AND COMMENTS 589
BIBLIOGRAPHY 591
INDEX/NOTATIONS 596
|
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| author | Kupershmidt, Boris A. |
| author_facet | Kupershmidt, Boris A. |
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| dewey-ones | 516 - Geometry |
| dewey-raw | 516.3/6 |
| dewey-search | 516.3/6 |
| dewey-sort | 3516.3 16 |
| dewey-tens | 510 - Mathematics |
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| id | DE-604.BV013324546 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:43:50Z |
| institution | BVB |
| isbn | 0821814001 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009086963 |
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| owner | DE-355 DE-BY-UBR DE-703 DE-83 |
| owner_facet | DE-355 DE-BY-UBR DE-703 DE-83 |
| physical | XVIII, 600 S. |
| publishDate | 2000 |
| publishDateSearch | 2000 |
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| publisher | American Math. Soc. |
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| series | Mathematical surveys and monographs |
| series2 | Mathematical surveys and monographs |
| spelling | Kupershmidt, Boris A. Verfasser aut KP or mKP noncommutative mathematics of Lagrangian, Hamiltonian and integrable systems Boris A. Kupershmidt Providence, RI American Math. Soc. 2000 XVIII, 600 S. txt rdacontent n rdamedia nc rdacarrier Mathematical surveys and monographs 78 Géométrie différentielle non commutative Sistemas dinâmicos larpcal Sistemas hamiltonianos larpcal Systèmes hamiltoniens Hamiltonian systems Noncommutative differential geometry Nichtkommutative Differentialgeometrie (DE-588)4311174-9 gnd rswk-swf Hamiltonsches System (DE-588)4139943-2 gnd rswk-swf Hamiltonsches System (DE-588)4139943-2 s Nichtkommutative Differentialgeometrie (DE-588)4311174-9 s DE-604 Erscheint auch als Online-Ausgabe 978-1-4704-1305-7 Mathematical surveys and monographs 78 (DE-604)BV000018014 78 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009086963&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kupershmidt, Boris A. KP or mKP noncommutative mathematics of Lagrangian, Hamiltonian and integrable systems Mathematical surveys and monographs Géométrie différentielle non commutative Sistemas dinâmicos larpcal Sistemas hamiltonianos larpcal Systèmes hamiltoniens Hamiltonian systems Noncommutative differential geometry Nichtkommutative Differentialgeometrie (DE-588)4311174-9 gnd Hamiltonsches System (DE-588)4139943-2 gnd |
| subject_GND | (DE-588)4311174-9 (DE-588)4139943-2 |
| title | KP or mKP noncommutative mathematics of Lagrangian, Hamiltonian and integrable systems |
| title_auth | KP or mKP noncommutative mathematics of Lagrangian, Hamiltonian and integrable systems |
| title_exact_search | KP or mKP noncommutative mathematics of Lagrangian, Hamiltonian and integrable systems |
| title_full | KP or mKP noncommutative mathematics of Lagrangian, Hamiltonian and integrable systems Boris A. Kupershmidt |
| title_fullStr | KP or mKP noncommutative mathematics of Lagrangian, Hamiltonian and integrable systems Boris A. Kupershmidt |
| title_full_unstemmed | KP or mKP noncommutative mathematics of Lagrangian, Hamiltonian and integrable systems Boris A. Kupershmidt |
| title_short | KP or mKP |
| title_sort | kp or mkp noncommutative mathematics of lagrangian hamiltonian and integrable systems |
| title_sub | noncommutative mathematics of Lagrangian, Hamiltonian and integrable systems |
| topic | Géométrie différentielle non commutative Sistemas dinâmicos larpcal Sistemas hamiltonianos larpcal Systèmes hamiltoniens Hamiltonian systems Noncommutative differential geometry Nichtkommutative Differentialgeometrie (DE-588)4311174-9 gnd Hamiltonsches System (DE-588)4139943-2 gnd |
| topic_facet | Géométrie différentielle non commutative Sistemas dinâmicos Sistemas hamiltonianos Systèmes hamiltoniens Hamiltonian systems Noncommutative differential geometry Nichtkommutative Differentialgeometrie Hamiltonsches System |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009086963&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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