Nearly integrable infinite-dimensional Hamiltonian systems / Sergej B. Kuksin

Author(s): Kuksin, Sergej B., 1955-
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Subjects: Hamiltonian systems
Schrödinger equation
Formats: Print
Material Type: Books
Language: English
Audience: Unspecified
Published: Berlin ; New York : Springer-Verlag, c1993
Series: Lecture notes in mathematics 1556
Lecture notes in mathematics (Springer-Verlag) 1556
LC Classification: Q, QA
Physical Description: xxvii, 101 p. ; 24 cm
Table of Contents: Introduction
1. Finite-dimensional situation
2. Infinite dimensional systems (the problem and the result)
3. Applications
4. Remarks on averaging theorems
5. Remarks on nearly integrable symplectomorphisms
6. Notations
Pt. 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 9
1.4. Toroidal phase space 11
1.5. A version of the former constructions 12
Pt. 2. Statement of the main theorem and its consequences 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 44
Pt. 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 of notations 94
Bibliography 96
Index 101
Notes: LCCN: 93037172
ISBN: 0387571612
Includes bibliographical references and index
OCLC Number: 28927866
ISBN/ISSN: 0387571612