KdV & KAM
In this text the authors consider the Korteweg-de Vries (KdV) equation (ut = - uxxx + 6uux) with periodic boundary conditions. Derived to describe long surface waves in a narrow and shallow channel, this equation in fact models waves in hom
- PDF / 20,185,314 Bytes
- 284 Pages / 439.37 x 666.142 pts Page_size
- 49 Downloads / 205 Views
A Series of Modern Surveys in Mathematics
Editorial Board S. Feferman, Stanford M. Gromov, Bures-sur-Yvette J. Jost, Leipzig J. Kollar, Princeton H.W. Lenstra, Jr., Berkeley P.-L. Lions, Paris M. Rapoport, Koln J.Tits, Paris D. B. Zagier, Bonn G. M. Ziegler, Berlin Managing Editor R. Remmert, Miinst~r
Volume 45
Springer-Verlag Berlin Heidelberg GmbH
Thomas Kappeler Jurgen Poschel
KdV&KAM
,
Springer
Thomas Kappeler Institut fiir Mathematik Universităt Ziirich Winterthurerstr. 190 8057 Ztirich, Switzerland e-mail: [email protected] ]tirgen Poschel Fakultăt Mathematik und Physik Universităt Stuttgart PfaffenwaIdring 57 70569 Stuttgart, Germany e-mail: [email protected]
Cataloging-in-Publication Data applied for Bibliographic information published by Oie Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at .
Mathematics Subject Classification (2000): 34B 30 , 35B20 , 37KlO, 37K35, 37J4° ISSN 0071-1136 ISBN 978-3-642-05694-9 ISBN 978-3-662-08054-2 (eBook) DOI 10.1007/978-3-662-08054-2 This work is subject to copyright. AII rights are reserved, whether the whole or part of the materi-
al is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfIlms ar in any other ways, 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 aIways be obtained from Springer-Verlag Berlin Heidelberg GmbH • Violations are liable for prosecution under the German Copy- right Law. © Springer-Verlag Berlin Heidelberg 2003 Originally published by Springer-Verlag Berlin Heidelberg New York in 2003 Softcover reprint of the harocover 1 st edition 2003
Typeset by the authors using a Springer TEX macro package. 44/3142LK - 5 43210 Printed on acid-free paper
In memory of JURGEN MOSER
teacher mentor friend
Preface
This book is concerned with two aspects of the theory of integrable partial differential equations. The first aspect is a normal form theory for such equations, which we exemplify by the periodic Korteweg de Vries equation - undoubtedly one of the most important nonlinear, integrable pdes. This makes for the' KdV' part of the title of the book. The second aspect is a theory for Hamiltonian perturbations of such pdes. Its prototype is the so called KAM theory, developed for finite dimensional systems by Kolmogorov, Arnold and Moser. This makes for the 'KAM' part of the title of the book. To be more specific, our starting point is the periodic KdV equation considered as an infinite dimensional, integrable Hamiltonian system admitting a complete set of independent integrals in involution. We show that this leads to a single, global, real analytic system of Birkhoff coordinates - the cartesian version of action-angle coordinates -, such that the KdV Hamiltonian becomes a function of
Data Loading...
