Symmetries and recursion operators for classical and supersymmetric differential equations:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Dordrecht [u.a.]
Kluwer Acad. Publ.
2000
|
| Series: | Mathematics and its applications
507 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVI, 384 S. Ill. |
| ISBN: | 0792363159 |
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| 100 | 1 | |a Krasil'ščik, Iosif S. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Symmetries and recursion operators for classical and supersymmetric differential equations |c by I. S. Krasil'shchik and P. H. M. Kersten |
| 264 | 1 | |a Dordrecht [u.a.] |b Kluwer Acad. Publ. |c 2000 | |
| 300 | |a XVI, 384 S. |b Ill. | ||
| 336 | |b txt |2 rdacontent | ||
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| 490 | 1 | |a Mathematics and its applications |v 507 | |
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Record in the Search Index
| _version_ | 1804128124574629888 |
|---|---|
| adam_text | Contents
Preface xi
Chapter 1. Classical symmetries 1
1. Jet spaces 1
1.1. Finite jets 1
1.2. Nonlinear differential operators 5
1.3. Infinite jets 7
2. Nonlinear PDE 12
2.1: Equations and solutions 12
2.2. The Cartan distributions 16
2.3. Symmetries 21
2.4. Prolongations 28
3. Symmetries of the Burgers equation 30
4. Symmetries of the nonlinear diffusion equation 34
4.1. Case 1: p = 0, k = 0 35
4.2. Case 2: p = 0, k j 0, q = 1 35
4.3. Case 3: p = 0, k ^ 0, q ^ 1 36
4.4. Case 4: p = 4/5, fc = 0 36
4.5. Case 5: p £ 4/5, p £ 0, k = 0 36
4.6. Case 6: p = 4/5, k ^ 0, q = 1 36
4.7. Case 7: p ^ 0, p ^ 4/5, k^0,q = l 37
4.8. Case 8: p^0, p^ 4/5, q = p+l 37
4.9. Case 9: p + 0, p ^ 4/5, q ^ 1, 9 ^ p + 1 37
5. The nonlinear Dirac equations 37
5.1. Case 1: c = 0, A 1 =0 39
5.2. Case 2: e = 0, A 1 ^0 43
5.3. Case 3: e ± 0, A 1 =0 43
5.4. Case 4: e ^ 0, A 1 ^0 43
6. Symmetries of the self dual SU(2) Yang Mills equations 43
6.1. Self dual SU(2) Yang Mills equations 43
6.2. Classical symmetries of self dual Yang Mills equations 46
6.3. Instanton solutions 49
6.4. Classical symmetries for static gauge fields 51
6.5. Monopole solution 52
Chapter 2. Higher symmetries and conservation laws 57
1. Basic structures 57
vi CONTENTS
1.1. Calculus 57
1.2. Cartan distribution 59
1.3. Cartan connection 61
1.4. C differential operators 63
2. Higher symmetries and conservation laws 67
2.1. Symmetries 67
2.2. Conservation laws 72
3. The Burgers equation 80
3.1. Defining equations 80
3.2. Higher order terms 81
3.3. Estimating Jacobi brackets 82
3.4. Low order symmetries 83
3.5. Action of low order symmetries 83
3.6. Final description 83
4. The Hilbert Cartan equation 84
4.1. Classical symmetries 85
4.2. Higher symmetries 87
4.3. Special cases 91
5. The classical Boussinesq equation 93
Chapter 3. Nonlocal theory 99
1. Coverings 99
2. Nonlocal symmetries and shadows 103
3. Reconstruction theorems 105
4. Nonlocal symmetries of the Burgers equation 109
5. Nonlocal symmetries of the KDV equation 111
6. Symmetries of the massive Thirring model 115
6.1. Higher symmetries 116
6.2. Nonlocal symmetries 120
6.2.1. Construction of nonlocal symmetries 121
6.2.2. Action of nonlocal symmetries 124
7. Symmetries of the Federbush model 129
7.1. Classical symmetries 129
7.2. First and second order higher symmetries 130
7.3. Recursion symmetries 135
7.4. Discrete symmetries 138
7.5. Towards infinite number of hierarchies of symmetries 138
7.5.1. Construction of Y+(2,0) and y+(2,0) 139
7.5.2. Hamiltonian structures 140
7.5.3. The infinity of the hierarchies 144
7.6. Nonlocal symmetries 146
8. Backlund transformations and recursion operators 149
Chapter 4. Brackets 155
1. Differential calculus over commutative algebras 155
1.1. Linear differential operators 155
CONTENTS vii
1.2. Jets 159
1.3. Derivations 160
1.4. Forms 164
1.5. Smooth algebras 168
2. Frolicher Nijenhuis bracket 171
2.1. Calculus in form valued derivations 171
2.2. Algebras with flat connections and cohomology 176
3. Structure of symmetry algebras 181
3.1. Recursion operators and structure of symmetry algebras 182
3.2. Concluding remarks 184
Chapter 5. Deformations and recursion operators 187
1. C cohomologies of partial differential equations 187
2. Spectral sequences and graded evolutionary derivations 196
3. C cohomologies of evolution equations 208
4. From deformations to recursion operators 217
5. Deformations of the Burgers equation 221
6. Deformations of the KdV equation 227
7. Deformations of the nonlinear Schrodinger equation 231
8. Deformations of the classical Boussinesq equation 233
9. Symmetries and recursion for the Sym equation 235
9.1. Symmetries 235
9.2. Conservation laws and nonlocal symmetries 239
9.3. Recursion operator for symmetries 241
Chapter 6. Super and graded theories 243
1. Graded calculus 243
1.1. Graded polyderivations and forms 243
1.2. Wedge products 245
1.3. Contractions and graded Richardson Nijenhuis bracket 246
1.4. De Rham complex and Lie derivatives 248
1.5. Graded Frolicher Nijenhuis bracket 249
2. Graded extensions 251
2.1. General construction 251
2.2. Connections 252
2.3. Graded extensions of differential equations 253
2.4. The structural element and C cohomologies 253
2.5. Vertical subtheory 255
2.6. Symmetries and deformations 256
2.7. Recursion operators 257
2.8. Commutativity theorem 260
3. Nonlocal theory and the case of evolution equations 261
3.1. The GDE(M) category 262
3.2. Local representation 262
3.3. Evolution equations 264
3.4. Nonlocal setting and shadows 265
viii CONTENTS
3.5. The functors K and T 267
3.6. Reconstructing shadows 268
4. The Kupershmidt super KdV equation 270
4.1. Higher symmetries 271
4.2. A nonlocal symmetry 273
5. The Kupershmidt super mKdV equation 275
5.1. Higher symmetries 276
5.2. A nonlocal symmetry 278
6. Supersymmetric KdV equation 280
6.1. Higher symmetries 281
6.2. Nonlocal symmetries and conserved quantities 282
7. Supersymmetric mKdV equation 290
8. Supersymmetric extensions of the NLS 293
8.1. Construction of supersymmetric extensions 293
8.2. Symmetries and conserved quantities 297
8.2.1. Case A 297
8.2.2. Case B 303
9. Concluding remarks 307
Chapter 7. Deformations of supersymmetric equations 309
1. Supersymmetric KdV equation 309
1.1. Nonlocal variables 309
1.2. Symmetries 310
1.3. Deformations 312
1.4. Passing from deformations to classical recursion operators 313
2. Supersymmetric extensions of the NLS equation 315
2.1. Case A 316
2.2. Case B 318
3. Supersymmetric Boussinesq equation 320
3.1. Construction of supersymmetric extensions 320
3.2. Construction of conserved quantities and nonlocal variables 321
3.3. Symmetries 322
3.4. Deformation and recursion operator 323
4. Supersymmetric extensions of the KdV equation, N = 2 324
4.1. Case a = 2 325
4.1.1. Conservation laws 326
4.1.2. Higher and nonlocal symmetries 328
4.1.3. Recursion operator 330
4.2. Case a = 4 331
4.2.1. Conservation laws 331
4.2.2. Higher and nonlocal symmetries 334
4.2.3. Recursion operator 335
4.3. Case a = 1 337
4.3.1. Conservation laws 337
4.3.2. Higher and nonlocal symmetries 341
CONTENTS ix
4.3.3. Recursion operator 347
Chapter 8. Symbolic computations in differential geometry 349
1. Super (graded) calculus 350
2. Classical differential geometry 355
3. Overdetermined systems of PDE 356
3.1. General case 357
3.2. The Burgers equation 360
3.3. Polynomial and graded cases 371
Bibliography 373
Index 379
|
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| id | DE-604.BV013349382 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:44:15Z |
| institution | BVB |
| isbn | 0792363159 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009103885 |
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| owner_facet | DE-703 |
| physical | XVI, 384 S. Ill. |
| publishDate | 2000 |
| publishDateSearch | 2000 |
| publishDateSort | 2000 |
| publisher | Kluwer Acad. Publ. |
| record_format | marc |
| series | Mathematics and its applications |
| series2 | Mathematics and its applications |
| spelling | Krasil'ščik, Iosif S. Verfasser aut Symmetries and recursion operators for classical and supersymmetric differential equations by I. S. Krasil'shchik and P. H. M. Kersten Dordrecht [u.a.] Kluwer Acad. Publ. 2000 XVI, 384 S. Ill. txt rdacontent n rdamedia nc rdacarrier Mathematics and its applications 507 Rekursionsoperator (DE-588)4436515-9 gnd rswk-swf Symmetrie (DE-588)4058724-1 gnd rswk-swf Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 gnd rswk-swf Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 s Rekursionsoperator (DE-588)4436515-9 s DE-604 Symmetrie (DE-588)4058724-1 s Kersten, Paul H. M. Verfasser aut Mathematics and its applications 507 (DE-604)BV008163334 507 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009103885&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Krasil'ščik, Iosif S. Kersten, Paul H. M. Symmetries and recursion operators for classical and supersymmetric differential equations Mathematics and its applications Rekursionsoperator (DE-588)4436515-9 gnd Symmetrie (DE-588)4058724-1 gnd Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 gnd |
| subject_GND | (DE-588)4436515-9 (DE-588)4058724-1 (DE-588)4128900-6 |
| title | Symmetries and recursion operators for classical and supersymmetric differential equations |
| title_auth | Symmetries and recursion operators for classical and supersymmetric differential equations |
| title_exact_search | Symmetries and recursion operators for classical and supersymmetric differential equations |
| title_full | Symmetries and recursion operators for classical and supersymmetric differential equations by I. S. Krasil'shchik and P. H. M. Kersten |
| title_fullStr | Symmetries and recursion operators for classical and supersymmetric differential equations by I. S. Krasil'shchik and P. H. M. Kersten |
| title_full_unstemmed | Symmetries and recursion operators for classical and supersymmetric differential equations by I. S. Krasil'shchik and P. H. M. Kersten |
| title_short | Symmetries and recursion operators for classical and supersymmetric differential equations |
| title_sort | symmetries and recursion operators for classical and supersymmetric differential equations |
| topic | Rekursionsoperator (DE-588)4436515-9 gnd Symmetrie (DE-588)4058724-1 gnd Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 gnd |
| topic_facet | Rekursionsoperator Symmetrie Nichtlineare partielle Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009103885&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV008163334 |
| work_keys_str_mv | AT krasilscikiosifs symmetriesandrecursionoperatorsforclassicalandsupersymmetricdifferentialequations AT kerstenpaulhm symmetriesandrecursionoperatorsforclassicalandsupersymmetricdifferentialequations |