简介
本书是天元基金影印数学丛书之一，介绍了可积偏微分方程理论的两个方面（KdV＆KAM）。第一个方面是可积偏微分方程的正规形式理论，以很重要的非线性可积偏微分方程——周期的Korteweg de Vries方程为例来阐述这个正规形式理论，这构成了书的“KdV”部分。第二个方面是可积偏微分方程的哈密尔顿摄动理论，它的原始模型是由Kolmogorov，Arnold和Moser发展起来的针对有限维系统的理论，这构成了书的“KAM”部分。本书内容包括：Classical Background，Birkhoff Coordinates，Perturbed KdV Equations，The KAM Proof，Kuksin’s Lemma，Background Material，Psi-Functions and Frequencies，Birkhoff Normal Forms，Some Technicalities。本书不仅是为可积偏微分方程理论和哈密尔顿摄动理论的专家所写，也是为整体分析和KAM理论领域的研究工作者和研究生所写。为了使读者更容易读懂本书，作者增加了描述有限维哈密尔顿系统的一章，所省略的证明都是可以在熟知的教科书中找到。本书可作为高等学校数学类专业研究生的教学用书，也可供研究院（所）从事KdV方程和KAM理论研究和学习的研究生或研究人员使用。
目录
Chapter Ⅰ The Beginning
1 Overview
Chapter Ⅱ Classical Background
2 Hamiltonian Formalism
3 Liouviile Integrable Systems
4 Birkhoff Integrable Systems
5 KAMTheory
Chapter Ⅲ Birkhoff Coordinates
6 Background and Results
7 Actions
8 Angles
9 Cartesian Coordinates
10 Orthogonality Relations
11 The Diffeomorphism Property
12 The Symplectomorphism Property
Chapter Ⅳ Perturbed KdV Equations
13 The Main Theorems
14 Birkhoff Normal Form
15 Global Coordinates and Frequencies
16 The KAM Theorem
17 Proof of the Main Theorems
Chapter Ⅴ The KAM Proof
18 Set Up and Summary of Main Results
19 The Linearized Equation
20 TheKAMStep
21 Iteration and Convergence
22 The Excluded Set of Parameters
Chapter Ⅵ Kuksin's Lemma
23 Kuksin's Lemma
Chapter Ⅶ Background Material
A Analyticity
B Spectra
C KdV Hierarchy
Chapter Ⅷ Psi-Functions and Frequencies
D Construction of the Psi-Functions
E A Trace Formula
F Frequencies
Chapter Ⅸ Birkhoff Normal Forms
G Two Results on Birkhoff Normal Forms
H Birkhoff Normal Form of Order 6
I Kramer's Lemma
J Nondegeneracy of the Second KdV Hamiltonian
Chapter Ⅹ Some Technicalities
K Symplectic Formalism
L Infinite Products
M Auxiliary Results
References
Index
Notations
1 Overview
Chapter Ⅱ Classical Background
2 Hamiltonian Formalism
3 Liouviile Integrable Systems
4 Birkhoff Integrable Systems
5 KAMTheory
Chapter Ⅲ Birkhoff Coordinates
6 Background and Results
7 Actions
8 Angles
9 Cartesian Coordinates
10 Orthogonality Relations
11 The Diffeomorphism Property
12 The Symplectomorphism Property
Chapter Ⅳ Perturbed KdV Equations
13 The Main Theorems
14 Birkhoff Normal Form
15 Global Coordinates and Frequencies
16 The KAM Theorem
17 Proof of the Main Theorems
Chapter Ⅴ The KAM Proof
18 Set Up and Summary of Main Results
19 The Linearized Equation
20 TheKAMStep
21 Iteration and Convergence
22 The Excluded Set of Parameters
Chapter Ⅵ Kuksin's Lemma
23 Kuksin's Lemma
Chapter Ⅶ Background Material
A Analyticity
B Spectra
C KdV Hierarchy
Chapter Ⅷ Psi-Functions and Frequencies
D Construction of the Psi-Functions
E A Trace Formula
F Frequencies
Chapter Ⅸ Birkhoff Normal Forms
G Two Results on Birkhoff Normal Forms
H Birkhoff Normal Form of Order 6
I Kramer's Lemma
J Nondegeneracy of the Second KdV Hamiltonian
Chapter Ⅹ Some Technicalities
K Symplectic Formalism
L Infinite Products
M Auxiliary Results
References
Index
Notations
- 名称
- 类型
- 大小
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