Analysis of Hamiltonian PDEs

by Sergei B. Kuksin

Overview

For the last 20-30 years, interest among mathematicians and physicists in infinite-dimensional Hamiltonian systems and Hamiltonian partial differential equations has been growing strongly, and many papers and a number of books have been written on integrable Hamiltonian PDEs. During the last decade though, the interest has shifted steadily towards non-integrable Hamiltonian PDEs. Here, not algebra but analysis and symplectic geometry are the appropriate analysing tools. The present book is the first one to use this approach to Hamiltonian PDEs and present a complete proof of the "KAM for PDEs" theorem. It will be an invaluable source of information for postgraduate mathematics and physics students and researchers.

Product Details

ISBN-13:	9780198503958
Publisher:	Oxford University Press
Publication date:	11/09/2000
Series:	Oxford Lecture Series in Mathematics and Its Applications , #19
Pages:	224
Product dimensions:	6.10(w) x 9.00(h) x 0.70(d)

About the Author

Heriot-Watt University

Table of Contents

PrefaceNotationsI. Unperturbed equations1. Some analysis in Hilbert spaces and scales2. Integrable subsystems and Lax-integrable equations3. Finite-gap manifolds for the KdV equation and theta-formulas4. Sine-Gordon equation5. Linearised equations and their Floquet solutions6. Linearised Lax-integrable equations7. Normal formsII. Perturbed equations1. A KAM theorem for perturbed nonlinear equations2. Examples3. Proof of KAM-theorem on parameter-depending equations4. Linearised equations5. First-order linear differential equations on n-torusAddendum: The theorem of A.N. KolmogorovIndexBibliography

Editorial Reviews

"The aim of this book is to present the following form of the proof of Kolmogorov-Arnold-Moser (KAM) theorem: most of the space-periodic finite-gap solutions of a Lax-integrable Hamiltonian partial differential equations (PDE) persist under a small perturbation of the equation as timequasiperiodic solutions of the perturbed equation. This theorem provides an important tool for an effective study of PDEs."—EMS

"The book was written to present a complete proof of the following infinite-dimensional KAM theorem: most space-periodic finite-gap solutions of a Lax-integrable partial differential equation persist under a small Hamiltonian perturbation of the equation as time-periodic solutions of the perturbed equation. . .The book provides a very useful source of information for both integrable and non-integrable differential equations."—MATH

"This is the first monograph where KAM-theory for PDEs is discussed systematically; most journal publications on the subject deal with particular examples rather than with general settings. The author succeeds in presenting a harmonic combination of general theory with nontrivial examples such as KdV (including KdV hierarchy) and sine-Gordon equations ... the book is carefully written ..."—Mathematical Reviews

From the Publisher

Kuksin (mathematics, Heriot-Watt U., Edinburgh, Scotland and Steklov Mathematical Institute, Moscow, Russia) focuses on the use of analysis and symplectic geometry to analyze Hamiltonian PDEs. He develops a theory of Hamiltonian PDEs, offers a short presentation of abstract Lax-integrable equations and classical Lax-integrable PDEs, and develops normal forms for Lax-integrable PDEs in the vicinity of manifolds, formed by the finite-gap solutions. Finally, he proves the main KAM theorem applying an abstract KAM theorem to equations, written in the normal form. Of likely interest to postgraduate mathematics and physics students and researchers with some knowledge of basic symplectic geometry, non-linear PDEs, Sobolev spaces, and interpolation. Annotation c. Book News, Inc., Portland, OR (booknews.com)

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