Nonlinear Oscillations of Hamiltonian Pdes

by Berti, Massimiliano

• ISBN13: 9780817646806
• ISBN10: 0817646809
• Format: Hardcover
• Copyright: 2007-11-01
• Publisher: Birkhauser

Summary

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 bifurcation results, the analogous theory for nonlinear wave equations is developed. The theory and applications of the Nash'Moser theorem to a class of nonlinear wave equations is also discussed together with other basic notions of Hamiltonian PDEs and number theory. The main examples of Hamiltonian PDEs presented include: the nonlinear wave equation, the nonlinear Schrödinger equation, beam equations, and the Euler equations of hydrodynamics. This text serves as an introduction to research in this fascinating and rapidly growing field. Graduate students and researchers interested in variational techniques and nonlinear analysis applied to Hamiltonian PDEs will find inspiration in the book.

Table of Contents

Finite Dimension	p. 1
The Lyapunov Center Theorem	p. 4
The Weinstein-Moser and Fadell-Rabinowitz Resonant Center Theorems	p. 7
The Variational Lyapunov-Schmidt Reduction	p. 13
Solution of the Range Equation	p. 14
Solution of the Bifurcation Equation	p. 15
Proof of the Weinstein-Moser Theorem	p. 16
Proof of the Fadell-Rabinowitz Theorem	p. 20
Infinite Dimension	p. 29
The Lyapunov Center Theorem for PDEs	p. 31
Completely Resonant Wave Equations	p. 33
The Case p Odd	p. 35
The Variational Lyapunov-Schmidt Reduction	p. 36
The Range Equation	p. 37
The Bifurcation Equation	p. 41
The Mountain Pass Argument	p. 43
The Case p Even	p. 48
Multiplicity	p. 53
The Small-Divisor Problem	p. 54
A Tutorial in Nash-Moser Theory	p. 59
Introduction	p. 59
An Analytic Nash-Moser Theorem	p. 60
A Differentiable Nash-Moser Theorem	p. 66
Application to the Nonlinear Wave Equation	p. 73
The Zeroth-Order Bifurcation Equation	p. 75
The Finite-Dimensional Reduction	p. 77
Solution of the (Q2)-Equation	p. 77
Solution of the Range Equation	p. 80
The Nash-Moser Scheme	p. 83
Solution of the (Q1)-Equation	p. 94
The Linearized Operator	p. 98
Decomposition of L[subscript n]	p. 98
Step 1: Inversion of D	p. 100
Step 2: Inversion of L[subscript n]	p. 104
Forced Vibrations	p. 111
The Forcing Frequency [omega isin] Q	p. 111
The Variational Lyapunov-Schmidt Reduction	p. 113
The Range Equation	p. 115
The Bifurcation Equation	p. 117
Monotone f	p. 119
Step 1: the L[superscript infinity] Estimate	p. 119
Step 2: the H[superscript 1] Estimate	p. 122
Nonmonotone f	p. 124
Step 1: the L[superscript 2k] Estimate	p. 129
Step 2: the L[superscript infinity] Estimate	p. 130
Step 3: The H[superscript 1] Estimate	p. 132
The "Maximum Principle"	p. 133
Hamiltonian PDEs	p. 139
The Nonlinear Schrodinger Equation	p. 139
The Beam Equation	p. 139
The KdV Equation	p. 140
The Euler Equations of Hydrodynamics	p. 140
Critical Point Theory	p. 145
Preliminaries	p. 145
Minima	p. 146
The Minimax Idea	p. 147
The Mountain Pass Theorem	p. 149
Free Vibrations of Nonlinear Wave Equations: A Global Result	p. 155
Approximation of Irrationals by Rationals	p. 161
Continued Fractions	p. 163
The Banach Algebra Property of X[subscript sigma s]	p. 167
Solutions	p. 169
References	p. 171
List of Symbols	p. 177
Index	p. 179
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