# Quasi-Periodic Solutions of Nonlinear Wave Equations on the \(d\)-Dimensional Torus

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*Massimiliano Berti; Philippe Bolle*

A publication of the European Mathematical Society

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. The authors 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.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

#### Readership

Researchers interested in partial differential equations and dynamical systems.