The book was written to present a proof of the following KAM theorem: most
space-periodic finite-gap solutions of a Lax-integrable Hamiltonian partial
differential equation (PDE) persist under a small Hamiltonian perturbation of the equation as
time-quasiperiodic solutions of the perturbed equation. In order to prove the theorem
we develop a theory of Hamiltonian PDEs (Chapter 1) and give short presentations of
abstract Lax-integrable equations (Chapter 2) as well as of classical Lax-integrable PDEs
(Chapters 3 and 4). Next, in Chapters 5-7 we develop normal forms for Lax-integrable
PDEs in the vicinity of manifolds, formed by the finite-gap solutions. Finally, we prove
the main theorem applying an abstract KAM theorem (Chapters 8 and 10 of Part II) to
equations, written in the normal form. Our presentation is rather complete; the only
nontrivial result which is given without a proof is the celebrated Its-Matveev theta formula
for finite-gap solutions of a Lax-integrable PDE. The above-mentioned normal form
results, and the abstract KAM theorem, are important effective tools to study non-linear
PDEs, apart from the persistence of finite-gap solutions (e.g. see Kuksin 1993, Bobenko
and Kuksin 1995a, and Kuksin and Poschel 1996 for some other KAM results).
Author(s): Sergei B. Kuksin
Series: OLS Mathematics and Its Applications 19
Publisher: Oxford University Press
Year: 2000
Language: English
Pages: 226
Kuksin,S.B. Analysis of Hamiltonian PDEs OLS Mathematics and Its Applications vol.19 ......Page 4
Copyright ......Page 5
Preface ......Page 6
Contents ......Page 8
Notation xi ......Page 12
I UNPERTURBED EQUATIONS ......Page 14
1.1 Differentiable and analytic maps 3 ......Page 16
1.2 Scales of Hilbert spaces and interpolation 5 ......Page 18
1.3 Differential forms 10 ......Page 23
1.4 Symplectic structures and Hamiltonian equations 14 ......Page 27
1.5 Symplectic transformations 19 ......Page 32
1.6 A Darboux lemma 26 ......Page 39
Appendix 1. Time-quasiperiodic solutions 27 ......Page 40
Appendix 2. Hilbert matrices and the Schur criterion 28 ......Page 41
2 Integrable subsystems of Hamiltonian equations and Lax-integrable equations 30 ......Page 43
2.1 Three examples 31 ......Page 44
2.2 Integrable subsystems 34 ......Page 47
2.3 Lax-integrable equations 37 ......Page 50
3.1 Finite-gap manifolds 40 ......Page 53
3.2 The Its-Matveev theta formulas 47 ......Page 60
3.3 Small-gap solutions 52 ......Page 65
3.4 Higher equations from the KdV hierarchy 58 ......Page 71
Appendix 3. On the Its-Matveev formulas 59 ......Page 72
Appendix 4. On the vectors V and W 61 ......Page 74
Appendix 5. A small-gap limit for theta functions 63 ......Page 76
Appendix 6. A Non-degeneracy Lemma 65 ......Page 78
4.1 The L, A pair 70 ......Page 83
4.2 Theta formulas 74 ......Page 87
4.3 Even periodic and odd periodic solutions 77 ......Page 90
4.4 Local structure of finite-gap manifolds 80 ......Page 93
4.5 Proof of Lemma 4.4 82 ......Page 95
Appendix 7. On the algebraic functions of infinite-dimensional arguments 86 ......Page 99
5.1 The linearized equation 87 ......Page 100
5.2 Floquet solutions 88 ......Page 101
5.3 Complete systems of Floquet solutions 92 ......Page 105
5.4 Lower-dimensional invariant tori in finite-dimensional systems and Floquet’s theorem 102 ......Page 115
6.1 Abstract setting 104 ......Page 117
6.2 Linearized KdV equation 105 ......Page 118
6.3 Higher KdV equations 112 ......Page 125
6.4 Linearized Sine-Gordon equation 113 ......Page 126
7.1 A normal form theorem 119 ......Page 132
7.2 Proof of Lemma 7.3 125 ......Page 138
7.3 Examples 128 ......Page 141
II PERTURBED EQUATIONS ......Page 144
8.1 The Main Theorem and related results 133 ......Page 146
8.2 Reduction to a parameter-depending case 136 ......Page 149
8.3 A KAM theorem for parameter-depending equations 138 ......Page 151
8.4 Completion of the proof of the Main Theorem 139 ......Page 152
8.5 Around the Main Theorem 141 ......Page 154
Appendix 8. Lipschitz analysis and Hausdorff measure 143 ......Page 156
9.1 Perturbed KdV equation 145 ......Page 158
9.2 Higher KdV equations 147 ......Page 160
9.3 Time-quasiperiodic perturbations of Lax-integrable equations 148 ......Page 161
9.4 Perturbed SG equation 151 ......Page 164
9.5 KAM persistence of lower-dimensional invariant tori of non-linear finite-dimensional systems 153 ......Page 166
10.1 Preliminary reductions 154 ......Page 167
10.2 Proof of the theorem 155 ......Page 168
10.3 Proof of Lemma 10.3 (estimation of the small divisors) 171 ......Page 184
Appendix 9. Some inequalities for Fourier series 174 ......Page 187
Appendix 10. On the Craig-Wayne-Bourgain KAM scheme 176 ......Page 189
11 Linearized equations 179 ......Page 192
12 First-order linear differential equations on the ft-torus 184 ......Page 197
A.2 Theorems A and B 192 ......Page 205
A.3 Sketch of the proof 195 ......Page 208
A.5 Proof of theorem B 196 ......Page 209
References 206 ......Page 219
Index 211 ......Page 224
cover......Page 1
