Dirac structures and integrability of nonlinear evolution equations:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Chichester u.a.
Wiley
1993
|
| Series: | Nonlinear science : theory and applications
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | VII, 176 S. |
| ISBN: | 0471938939 |
Staff View
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| 100 | 1 | |a Dorfman, Irene |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Dirac structures and integrability of nonlinear evolution equations |c Irene Dorfman |
| 264 | 1 | |a Chichester u.a. |b Wiley |c 1993 | |
| 300 | |a VII, 176 S. | ||
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| 490 | 0 | |a Nonlinear science : theory and applications | |
| 650 | 4 | |a Mathematische Physik | |
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Record in the Search Index
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| adam_text | Contents
Preface ix
Acknowledgements xi
1 Introduction 1
2 Algebraic theory of Dirac structures 8
2.1 Graded spaces, complexes, cohomologies 8
2.2 Complexes over Lie algebras 11
2.3 Lie derivative; symmetries and invariants; reduction
procedure 15
2.4 Dirac structures 16
2.5 Symplectic operators 20
2.6 Hamiltonian operators 22
2.7 Lie algebra structures in the space of 1 forms 25
2.8 The Schouten bracket 27
2.9 The classical Yang Baxter equation and its
Belavin Drinfeld solutions 29
2.10 Notes 31
3 Nijenhuis operators and pairs of Dirac structures 33
3.1 Nijenhuis operators and deformations of Lie algebras 33
3.2 Properties of Nijenhuis operators; symmetry generation 36
3.3 Conjugate to a Nijenhuis operator; Nijenhuis torsion 37
3.4 Hierarchies of 2 forms generated by a Nijenhuis operator;
regular structures 39
3.5 Hamiltonian pairs and associated Nijenhuis operators 42
3.6 Nijenhuis relations; pairs of Dirac structures 45
3.7 Lenard scheme of integrability for Dirac structures 48
vi Contents
3.8 Applications; Lenard scheme for Hamiltonian and
symplectic pairs 51
3.9 Conditions guaranteeing symplecticity of a pair 54
3.10 Notes 55
4 The complex of formal variational calculus 57
4.1 Construction of the complex 57
4.2 Invariant operations expressed in terms of Frechet derivatives 61
4.3 Exactness problem; dependence on the choice of the basic ring 62
4.4 Cohomology group H°(Q,d) 65
4.5 The equation df/du — g in the rings of polynomials and
smooth functions 68
4.6 General procedure of solving the equation df/Su = g 69
4.7 Notes 73
5 Local Hamiltonian operators and evolution equations related to
them 75
5.1 Matrix differential operators 75
5.2 Hamiltonian conditions in an explicit form 77
5.3 First order Hamiltonian operators in the one variable case 80
5.4 Third order Hamiltonian operators 82
5.5 The family of equations of Korteweg de Vries and Harry
Dym type 85
5.6 Infinite dimensional Kirillov Kostant structures, KdV
equation and coupled nonlinear wave equation 89
5.7 Structure functions; shift of the argument and deformations
of Kirillov Kostant structures 96
5.8 The Virasoro algebra and two Hamiltonian structures of the
KdV equation; generalizations to multi variable case 100
5.9 Kirillov Kostant type hydrodynamic structures; Benney s
moment equations 102
5.10 Dubrovin Novikov type Hamiltonian structures 104
5.11 The Adler Gelfand Dikii method of constructing
Hamiltonian pairs 106
5.12 Some other local Hamiltonian operators 111
5.13 Notes 113
6 Local symplectic operators and evolution equations related
to them 115
6.1 Symplecticity conditions in an explicit form 115
6.2 One variable case: first and third order symplectic operators 117
6.3 A pair of local symplectic operators for the
Krichever Novikov equation 120
Contents vii
6.4 Lenard scheme for the Krichever Novikov equation 123
6.5 Two distinct Lenard schemes for the potential KdV equation 125
6.6 Dirac structures related to Liouville, sine Gordon, modified
KdV and nonlinear Schrodinger equations 131
6.7 Many variable case with linear dependence on uf 137
6.8 Upper bounds for the level of symplectic and Hamiltonian
operators 142
6.9 Symplectic operators of differential geometric type 146
6.10 Notes 147
7 t Scheme of integrability 148
7.1 An alternative scheme to generate commuting symmetries 148
7.2 T Scheme for the KdV, Benjamin Ono and
Kadomtsev Petviashvili equations 151
7.3 T Scheme in Hamiltonian framework; symmetries of Dirac
structures 154
7.4 Examples of r Schemes with conserved Dirac structures 157
7.5 Lie derivatives in constructing Hamiltonian pairs 159
7.6 Simultaneous action of the Lenard scheme and the t scheme 161
7.7 Notes 165
References 166
Index 173
|
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| id | DE-604.BV008325899 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T17:18:19Z |
| institution | BVB |
| isbn | 0471938939 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-005499272 |
| oclc_num | 27107910 |
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| physical | VII, 176 S. |
| publishDate | 1993 |
| publishDateSearch | 1993 |
| publishDateSort | 1993 |
| publisher | Wiley |
| record_format | marc |
| series2 | Nonlinear science : theory and applications |
| spelling | Dorfman, Irene Verfasser aut Dirac structures and integrability of nonlinear evolution equations Irene Dorfman Chichester u.a. Wiley 1993 VII, 176 S. txt rdacontent n rdamedia nc rdacarrier Nonlinear science : theory and applications Mathematische Physik Evolution equations, Nonlinear Hamiltonian systems Mathematical physics Hamiltonsches System (DE-588)4139943-2 gnd rswk-swf Nichtlineare Evolutionsgleichung (DE-588)4221363-0 gnd rswk-swf Nichtlineare Evolutionsgleichung (DE-588)4221363-0 s Hamiltonsches System (DE-588)4139943-2 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005499272&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Dorfman, Irene Dirac structures and integrability of nonlinear evolution equations Mathematische Physik Evolution equations, Nonlinear Hamiltonian systems Mathematical physics Hamiltonsches System (DE-588)4139943-2 gnd Nichtlineare Evolutionsgleichung (DE-588)4221363-0 gnd |
| subject_GND | (DE-588)4139943-2 (DE-588)4221363-0 |
| title | Dirac structures and integrability of nonlinear evolution equations |
| title_auth | Dirac structures and integrability of nonlinear evolution equations |
| title_exact_search | Dirac structures and integrability of nonlinear evolution equations |
| title_full | Dirac structures and integrability of nonlinear evolution equations Irene Dorfman |
| title_fullStr | Dirac structures and integrability of nonlinear evolution equations Irene Dorfman |
| title_full_unstemmed | Dirac structures and integrability of nonlinear evolution equations Irene Dorfman |
| title_short | Dirac structures and integrability of nonlinear evolution equations |
| title_sort | dirac structures and integrability of nonlinear evolution equations |
| topic | Mathematische Physik Evolution equations, Nonlinear Hamiltonian systems Mathematical physics Hamiltonsches System (DE-588)4139943-2 gnd Nichtlineare Evolutionsgleichung (DE-588)4221363-0 gnd |
| topic_facet | Mathematische Physik Evolution equations, Nonlinear Hamiltonian systems Mathematical physics Hamiltonsches System Nichtlineare Evolutionsgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005499272&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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