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

Description

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 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