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Tables of Contents for Analysis of Hamiltonian Pdes
Chapter/Section Title
Page #
Page Count
Notation
xi
I. Unperturbed Equations
Some analysis in Hilbert spaces and scales
3
27
Differentiable and analytic maps
3
2
Scales of Hilbert spaces and interpolation
5
5
Differential forms
10
4
Symplectic structures and Hamiltonian equations
14
5
Symplectic transformations
19
7
A Darboux lemma
26
4
Time-quasiperiodic solutions
27
1
Hilbert matrices and the Schur criterion
28
2
Integrable subsystems of Hamiltonian equations and Lax-integrable equations
30
10
Three examples
31
3
Integrable subsystems
34
3
Lax-integrable equations
37
3
Finite-gap manifolds for the KdV equation and theta formulas
40
30
Finite-gap manifolds
40
7
The Its-Matveev theta formulas
47
5
Small-gap solutions
52
6
Higher equations from the KdV hierarchy
58
12
On the Its-Matveev formulas
59
2
On the vectors V and W
61
2
A small-gap limit for theta functions
63
2
A Non-degeneracy Lemma
65
5
The Sine-Gordon equation
70
17
The L, A pair
70
4
Theta formulas
74
3
Even Periodic and odd periodic solutions
77
3
Local structure of finite-gap manifolds
80
2
Proof of Lemma 4.4
82
5
On the algebraic functions of infinite-dimensional arguments
86
1
Linearized equations and their Floquet solutions
87
17
The linearized equation
87
1
Floquet solutions
88
4
Complete systems of Floquet solutions
92
10
Lower-dimensional invariant tori in finite-dimensional systems and Floquet's theorem
102
2
Linearized Lax-integrable equations
104
15
Abstract setting
104
1
Linearized KdV equation
105
7
Higher KdV equations
112
1
Linearized Sine-Gordon equation
113
6
The normal form
119
14
A normal form theorem
119
6
Proof of Lemma 7.3
125
3
Examples
128
5
II PERTURBED EQUATIONS
A KAM theorem for perturbed non-linear equations
133
12
The Main Theorem and related results
133
3
Reduction to a parameter-depending case
136
2
A KAM theorem for parameter-depending equations
138
1
Completion of the proof the Main Theorem
139
2
Around the Main Theorem
141
4
Lipschitz analysis and Hausdorff measure
143
2
Examples
145
9
Perturbed KdV equation
145
2
Higher KdV equations
147
1
Time-quasiperiodic perturbations of Lax-integrable equations
148
3
Perturbed SG equation
151
2
KAM persistence of lower-dimensional invariant tori of non-linear finite-dimensional systems
153
1
Proof of Theorem 8.3 on parameter-depending equations
154
25
Preliminary reductions
154
1
Proof of the theorem
155
16
Proof of Lemma 10.3 (estimation of the small divisors)
171
8
Some inequalities for Fourier series
174
2
On the Craig-Wayne-Bourgain KAM scheme
176
3
Linearized equations
179
5
First-order linear differential equations on the n-torus
184
8
Addendum. The theorem of A.N. Kolmogorov
192
14
A.1 Introduction
192
1
A.2 Theorems A and B
192
3
A.3 Sketch of the proof
195
1
A.4 Reformulation of the theorem's assertion
196
1
A.5 Proof of theorem B
196
10
References
206
5
Index
211