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Table of contents

  1. Front Matter
    Pages I-XIII
  2. Thomas Kappeler, Jürgen Pöschel
    Pages 1-17
  3. Thomas Kappeler, Jürgen Pöschel
    Pages 19-49
  4. Thomas Kappeler, Jürgen Pöschel
    Pages 51-109
  5. Thomas Kappeler, Jürgen Pöschel
    Pages 111-143
  6. Thomas Kappeler, Jürgen Pöschel
    Pages 145-175
  7. Thomas Kappeler, Jürgen Pöschel
    Pages 177-186
  8. Thomas Kappeler, Jürgen Pöschel
    Pages 187-210
  9. Thomas Kappeler, Jürgen Pöschel
    Pages 211-231
  10. Thomas Kappeler, Jürgen Pöschel
    Pages 233-256
  11. Thomas Kappeler, Jürgen Pöschel
    Pages 257-266
  12. Back Matter
    Pages 267-280

About this book

Introduction

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 homogeneous, weakly nonlinear and weakly dispersive media in general.

Viewing the KdV equation as an infinite dimensional, and in fact integrable Hamiltonian system, we first construct action-angle coordinates which turn out to be globally defined. They make evident that all solutions of the periodic KdV equation are periodic, quasi-periodic or almost-periodic in time. Also, their construction leads to some new results along the way.

Subsequently, these coordinates allow us to apply a general KAM theorem for a class of integrable Hamiltonian pde's, proving that large families of periodic and quasi-periodic solutions persist under sufficiently small Hamiltonian perturbations.

The pertinent nondegeneracy conditions are verified by calculating the first few Birkhoff normal form terms -- an essentially elementary calculation.

Keywords

Calculation Finite Integrable Systems KAM Theory KdV Equation Perturbation Theory equation function proof theorem

Authors and affiliations

  • Thomas Kappeler
    • 1
  • Jürgen Pöschel
    • 2
  1. 1.Institut für MathematikUniversität ZürichZürichSwitzerland
  2. 2.Fakultät Mathematik und PhysikUniversität StuttgartStuttgartGermany

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-662-08054-2
  • Copyright Information Springer-Verlag Berlin Heidelberg 2003
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-642-05694-9
  • Online ISBN 978-3-662-08054-2
  • Series Print ISSN 0071-1136
  • Buy this book on publisher's site