PREFACE

the nonlinear Schrödinger (NLS) equation, the beam equation, the higher-dimensional membrane equation, the water-waves equations, i.e., the Euler equations of hydrodynamics describing the evolution of an incompressible irrotational fluid under the action of gravity and surface tension, as well as its approximate models like the Korteweg de Vries (KdV), Boussinesq, Benjamin–Ono, and Kadomtsev–Petviashvili (KP) equations, among many others. We refer to [102] for a general introduction to Hamiltonian PDEs. In this monograph we shall adopt a “dynamical systems” point of view regarding the NLW equation (0.0.2) equipped with periodic boundary conditions x 2 T WD .R=2 Z/ as an infinite-dimensional Hamiltonian system, and we shall prove the existence of Cantor families of finite-dimensional invariant tori filled by quasiperiodic solutions of (0.0.2). The first results in this direction were obtained by Bourgain [42]. The search for invariant sets for the flow is an essential change of paradigm in the study of hyperbolic equations with respect to the more traditional pursuit of the initial value problem. This perspective has allowed the discovery of many new results, inspired by finite-dimensional Hamiltonian systems, for Hamiltonian PDEs. When the space variable x belongs to a compact manifold, say x 2 Œ0; with Dirichlet boundary conditions or x 2 T (periodic boundary conditions), the dynamics of a Hamiltonian PDE (0.0.1), like (0.0.2), is expected to have a “recurrent” behavior in time, with many periodic and quasiperiodic solutions, i.e., solutions (defined for all times) of the form