Nonlinear Oscillations of Hamiltonian PDEs

Massimiliano Berti (auth.)0817646809, 9780817646806, 9780817646813

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.

Table of contents :
Front Matter….Pages I-XII
Finite Dimension….Pages 1-27
Infinite Dimension….Pages 29-57
A Tutorial in Nash–Moser Theory….Pages 59-71
Application to the Nonlinear Wave Equation….Pages 73-109
Forced Vibrations….Pages 111-137
Back Matter….Pages 139-180