Many partial differential equations (PDEs) arising in physics, such as the nonlinear wave equation and the Schrödinger equation, can be viewed as infinite-dimensional Hamiltonian systems. In the last thirty years, several existence results of time quasi-periodic solutions have been proved adopting a "dynamical systems" point of view. Most of them deal with equations in one space dimension, whereas for multidimensional PDEs a satisfactory picture is still under construction.
An updated introduction to the now rich subject of KAM theory for PDEs is provided in the first part of this research monograph. We then focus on the nonlinear wave equation, endowed with periodic boundary conditions. The main result of the monograph proves the bifurcation of small amplitude finite-dimensional invariant tori for this equation, in any space dimension. This is a difficult small divisor problem due to complex resonance phenomena between the normal mode frequencies of oscillations. The proof requires various mathematical methods, ranging from Nash–Moser and KAM theory to reduction techniques in Hamiltonian dynamics and multiscale analysis for quasi-periodic linear operators, which are presented in a systematic and self-contained way. Some of the techniques introduced in this monograph have deep connections with those used in Anderson localization theory.
This book will be useful to researchers who are interested in small divisor problems, particularly in the setting of Hamiltonian PDEs, and who wish to get acquainted with recent developments in the field.
Keywords: Infinite-dimensional Hamiltonian systems, nonlinear wave equation, KAM for PDEs, quasi-periodic solutions and invariant tori, small divisors, Nash–Moser theory, multiscale analysis
Author(s): Massimiliano Berti, Philippe Bolle
Series: EMS Monographs in Mathematics
Publisher: European Mathematical Society
Year: 2021
Introduction
Main result and historical context
Statement of the main results
Basic notation
KAM for PDEs and strategy of proof
The Newton–Nash–Moser algorithm
The reducibility approach to KAM for PDEs
Transformation laws and reducibility
Perturbative reducibility
Reducibility results
KAM for 1-dimensional NLW and NLS equationswith Dirichlet boundary conditions
KAM for 1-dimensional NLW and NLS equationswith periodic boundary conditions
Space multidimensional PDEs
1-dimensional quasi- and fully nonlinear PDEs, water waves
The multiscale approach to KAM for PDEs
Time-periodic case
Quasiperiodic case
The multiscale analysis of Chapter 5
Outline of the proof of Theorem 1.2.1
Hamiltonian formulation
Hamiltonian form of NLW equation
Action-angle and normal variables
Admissible Diophantine directions
Functional setting
Phase space and basis
Linear operators and matrix representation
Decay norms
Off-diagonal decay of - + V(x)
Interpolation inequalities
Multiscale Analysis
Multiscale proposition
Matrix representation
Multiscale step
Separation properties of bad sites
Definition of the sets (; , Xr, )
Right inverse of [Lr, ]N2N for N < N02
Inverse of Lr, , N for N N02
The set (; 1, Xr, ) is good at any scale
Inverse of Lr, , N for N N02
Measure estimates
Preliminaries
Measure estimate of
Measure estimate of G0N, 12 for N N02
Measure estimate of G0Nk, 12 for k 1
Stability of the L2 good parameters under variation of Xr,
Conclusion: proof of (5.1.17) and (5.1.18)
Nash–Moser theorem
Statement
Shifted tangential frequencies up to O(4 )
First approximate solution
Linearized operator at an approximate solution
Symplectic approximate decoupling
Proof of Proposition 7.1.1
Proof of Lemma 7.1.2
Splitting of low-high normal subspaces up to O(4)
Choice of M
Homological equations
Averaging step
Approximate right inverse in normal directions
Split admissible operators
Approximate right inverse
Splitting between low-high normal subspaces
Splitting step and corollary
The linearized homological equation
Solution of homological equations: proof of Lemma 10.2.2
Splitting step: proof of Proposition 10.1.1
Construction of approximate right inverse
Splitting of low-high normal subspaces
Approximate right inverse of LD
Approximate right inverse of L = LD +
Approximate right inverse of - J (A0 + )
Proof of the Nash–Moser Theorem
Approximate right inverse of L
Nash–Moser iteration
C solutions
Genericity of the assumptions
Genericity of nonresonance and nondegeneracy conditions
Hamiltonian and reversible PDEs
Hamiltonian and reversible vector fields
Nonlinear wave and Klein–Gordon equations
Nonlinear Schrödinger equation
Perturbed KdV equations
Multiscale Step
Matrices with off-diagonal decay
Multiscale step proposition
Normal form close to an isotropic torus
Symplectic coordinates near an invariant torus
Symplectic coordinates near an approximately invariant torus
Bibliography
Index
