Quasi-Periodic Solutions of Nonlinear Wave Equations on the d - Dimensional Torus

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