Analysis of Hamiltonian PDEs:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Oxford [u.a.]
Oxford Univ. Press
2000
|
| Edition: | 1. publ. |
| Series: | Oxford lecture series in mathematics and its applications
19 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XII, 212 S. |
| ISBN: | 0198503954 |
Staff View
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| 100 | 1 | |a Kuksin, Sergej B. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Analysis of Hamiltonian PDEs |c Sergei B. Kuksin |
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Record in the Search Index
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| adam_text | Contents
Notation xi
I UNPERTURBED EQUATIONS
1 Some analysis in Hilbert spaces and scales 3
1.1 Differentiable and analytic maps 3
1.2 Scales of Hilbert spaces and interpolation 5
1.3 Differential forms 10
1.4 Symplectic structures and Hamiltonian equations 14
1.5 Symplectic transformations 19
1.6 A Darboux lemma 26
Appendix 1. Time quasiperiodic solutions 27
Appendix 2. Hilbert matrices and the Schur criterion 28
2 Integrable subsystems of Hamiltonian equations and
Lax integrable equations 30
2.1 Three examples 31
2.2 Integrable subsystems 34
2.3 Lax integrable equations 37
3 Finite gap manifolds for the KdV equation and theta formulas 40
3.1 Finite gap manifolds 40
3.2 The Its Matveev theta formulas 47
3.3 Small gap solutions 52
3.4 Higher equations from the KdV hierarchy 58
Appendix 3. On the Its Matveev formulas 59
Appendix 4. On the vectors V and W 61
Appendix 5. A small gap limit for theta functions 63
Appendix 6. A Non degeneracy Lemma 65
4 The Sine Gordon equation 70
4.1 The L, A pair 70
4.2 Theta formulas 74
4.3 Even periodic and odd periodic solutions 77
4.4 Local structure of finite gap manifolds 80
viii Contents
4.5 Proof of Lemma 4.4 82
Appendix 7. On the algebraic functions of infinite dimensional
arguments 86
5 Linearized equations and their Floquet solutions 87
5.1 The linearized equation 87
5.2 Floquet solutions 88
5.3 Complete systems of Floquet solutions 92
5.4 Lower dimensional invariant tori in finite dimensional systems
and Floquet s theorem 102
6 Linearized Lax integrable equations 104
6.1 Abstract setting 104
6.2 Linearized KdV equation 105
6.3 Higher KdV equations 112
6.4 Linearized Sine Gordon equation 113
7 The normal form 119
7.1 A normal form theorem 119
7.2 Proof of Lemma 7.3 125
7.3 Examples 128
II PERTURBED EQUATIONS
8 A KAM theorem for perturbed non linear equations 133
8.1 The Main Theorem and related results 133
8.2 Reduction to a parameter depending case 136
8.3 A KAM theorem for parameter depending equations 138
8.4 Completion of the proof of the Main Theorem 139
8.5 Around the Main Theorem 141
Appendix 8. Lipschitz analysis and Hausdorff measure 143
9 Examples 145
9.1 Perturbed KdV equation 145
9.2 Higher KdV equations 147
9.3 Time quasiperiodic perturbations of Lax integrable equations 148
9.4 Perturbed SG equation 151
9.5 KAM persistence of lower dimensional invariant tori of
non linear finite dimensional systems 153
10 Proof of Theorem 8.3 on parameter depending equations 154
10.1 Preliminary reductions 154
10.2 Proof of the theorem 155
10.3 Proof of Lemma 10.3 (estimation of the small divisors) 171
Contents ix
Appendix 9. Some inequalities for Fourier series 174
Appendix 10. On the Craig Wayne Bourgain KAM scheme 176
11 Linearized equations 179
12 First order linear differential equations on the n torus 184
Addendum. The theorem of A.N. Kolmogorov 192
A.I Introduction 192
A.2 Theorems A and B 192
A.3 Sketch of the proof 195
A.4 Reformulation of the theorem s assertion 196
A.5 Proof of theorem B 196
References 206
Index 211
|
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| spelling | Kuksin, Sergej B. Verfasser aut Analysis of Hamiltonian PDEs Sergei B. Kuksin 1. publ. Oxford [u.a.] Oxford Univ. Press 2000 XII, 212 S. txt rdacontent n rdamedia nc rdacarrier Oxford lecture series in mathematics and its applications 19 Differential equations, Partial Hamiltonian operator Hamilton-Gleichungen (DE-588)4289066-4 gnd rswk-swf Hamilton-Gleichungen (DE-588)4289066-4 s DE-604 Oxford lecture series in mathematics and its applications 19 (DE-604)BV009910017 19 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009154365&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kuksin, Sergej B. Analysis of Hamiltonian PDEs Oxford lecture series in mathematics and its applications Differential equations, Partial Hamiltonian operator Hamilton-Gleichungen (DE-588)4289066-4 gnd |
| subject_GND | (DE-588)4289066-4 |
| title | Analysis of Hamiltonian PDEs |
| title_auth | Analysis of Hamiltonian PDEs |
| title_exact_search | Analysis of Hamiltonian PDEs |
| title_full | Analysis of Hamiltonian PDEs Sergei B. Kuksin |
| title_fullStr | Analysis of Hamiltonian PDEs Sergei B. Kuksin |
| title_full_unstemmed | Analysis of Hamiltonian PDEs Sergei B. Kuksin |
| title_short | Analysis of Hamiltonian PDEs |
| title_sort | analysis of hamiltonian pdes |
| topic | Differential equations, Partial Hamiltonian operator Hamilton-Gleichungen (DE-588)4289066-4 gnd |
| topic_facet | Differential equations, Partial Hamiltonian operator Hamilton-Gleichungen |
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