Nearly Integrable Infinite-Dimensional Hamiltonian Systems
The book is devoted to partial differential equations of Hamiltonian form, close to integrable equations. For such equations a KAM-like theorem is proved, stating that solutions of the unperturbed equation that are quasiperiodic in time mostly persist in
- PDF / 8,790,428 Bytes
- 128 Pages / 468 x 684 pts Page_size
- 28 Downloads / 220 Views
1556
Sergej B. Kuksin
Nearly Integrable Infinite-Dimensional Hamiltonian Systems
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest
Author Sergej B. Kuksin Institute for Information Transmission Problems Ermolovoy St. 19 101 447 Moscow, Russia
Mathematics Subject Classification (1991): 34G20, 35Q55, 58F07, 58F27, 58F39 ISBN 3-540-57161-2 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-57161-2 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer- Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1993 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210 - Printed on acid-free paper
Contents Introduction
Vll
1. Finite-dimensional situation
Vlll
2. Infinite dimensional systems (the problem and the result)
xi
3. Applications
xvi
4. Remarks on averaging theorems
xxv
5. Remarks on nearly integrable symplectomorphisms
xxv
6. Notations
xxvii
Part 1. Symplectic structures and Hamiltonian systems in scales of Hilbert spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1. Symplectic Hilbert scales and Hamiltonian equations
1
1.2. Canonical transformations
6
1.3. Local solvability of Hamiltonian equations.
.
.. . .. . . . .
1.4. Toroidal phase space.........
. . .. . .
1.5. A version of the former constructions Part 2. Statement of the main theorem and its consequences
9 11 12 13
2.1. Statement of the main theorem
14
2.2. Reformulation of Theorem 1.1.
18
2.3. Nonlinear Schrodinger equation
22
2.4. Schrodinger equation with random potential
28
2.5. Nonlinear Schrodinger equation on the real line
33
2.6. Nonlinear string equation
35
2.7. On non-commuting operators J, A and partially hyperbolic invariant tori
39
Appendix. On superposition operator in Sobolev spaces
v
44
Part 3. Proof of the main theorem
45
3.1. Preliminary transformations
45
3.2. Proof of Theorem 1.1
53
3.3. Proof of Lemma 2.2 (solving the homological equations)
67
3.4. Proof of Lemma 3.1 (estimation of the small divisors)
73
3.5. Proof of Lemma 2.3 (estimation of the change of variables)
78
3.6. Proof of Refinement 2
82
3.7. On reducibility of variational equations
84
3.8. Proof of Theorem 1.2
85
Appendix A. Interpolation theorem..................................... 91 Appendix B. Some estimates for Fourier series
92
Appendix C. Lipschitz homeomorphisms of Borel sets
92
Appendix D. Cauchy estimate
93
List
Data Loading...
