KP or mKP : noncommutative mathematics of Lagrangian, Hamiltonian, and integrable systems / Boris A. Kupershmidt.

Format:

Book

Author/Creator:

Kupershmidt, Boris A., 1946-

Series:

Mathematical surveys and monographs ; no. 78.

Mathematical surveys and monographs, 0076-5376 ; v. 78

Language:

English

Subjects (All):

Hamiltonian systems.

Noncommutative differential geometry.

Physical Description:

xviii, 600 pages ; 27 cm.

Place of Publication:

Providence, R.I. : American Mathematical Society, [2000]

Contents:

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

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 MKP[superscript II] 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 Affine 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. M[superscript 2]KP 177

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

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

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

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 H[subscript t] = HH[subscript x]H 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 Sympletic 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

Appendix A1. 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 [delta] 541

A3.3. The Image Of The Operator [characters not reproducible] + [epsilon]ad[subscript u] 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. GL[subscript m+1]-Generalities And A GL[subscript 2]-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:

Includes bibliographical references (pages 591-595) and index.

ISBN:

0821814001

OCLC:

42921336