Nonlinear Oscillations of Hamiltonian PDEs

  • Massimiliano Berti

Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 74)

Table of contents

  1. Front Matter
    Pages I-XII
  2. Massimiliano Berti
    Pages 1-27
  3. Massimiliano Berti
    Pages 29-57
  4. Massimiliano Berti
    Pages 59-71
  5. Massimiliano Berti
    Pages 73-109
  6. Massimiliano Berti
    Pages 111-137
  7. Back Matter
    Pages 139-180

About this book

Introduction

Many partial differential equations (PDEs) that arise in physics can be viewed as infinite-dimensional Hamiltonian systems. This monograph presents recent existence results of nonlinear oscillations of Hamiltonian PDEs, particularly of periodic solutions for completely resonant nonlinear wave equations.

After introducing the reader to classical finite-dimensional dynamical system theory, including the Weinstein–Moser and Fadell–Rabinowitz resonant center theorems, the author develops the analogous theory for completely resonant nonlinear wave equations. Within this theory, both problems of small divisors and infinite bifurcation phenomena occur, requiring the use of Nash–Moser theory as well as minimax variational methods. These techniques are presented in a self-contained manner together with other basic notions of Hamiltonian PDEs and number theory.

This text serves as an introduction to research in this fascinating and rapidly growing field. Graduate students and researchers interested in nonlinear variational techniques as well in small divisors problems applied to Hamiltonian PDEs will find inspiration in the book.

Keywords

Dynamical system Nash-Moser theorem bifurcations critical point theory number theory partial differential equation wave equation

Authors and affiliations

  • Massimiliano Berti
    • 1
  1. 1.Dipartimento di Matematica e Applicazioni ‘R. Caccioppoli’Università Federico II Napoli via CintiaMonte S. AngeloItaly

Bibliographic information

  • DOI https://doi.org/10.1007/978-0-8176-4681-3
  • Copyright Information Birkhäuser Boston 2007
  • Publisher Name Birkhäuser Boston
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-0-8176-4680-6
  • Online ISBN 978-0-8176-4681-3
  • About this book