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Quasi-Periodic Solutions of Nonlinea...

Berti, Massimiliano,

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

Record Type:	Electronic resources : Monograph/item
Title/Author:	Quasi-Periodic Solutions of Nonlinear Wave Equations on the d-Dimensional Torus/ Massimiliano Berti, Philippe Bolle
Author:	Berti, Massimiliano,
other author:	Bolle, Philippe,
Published:	Zuerich, Switzerland :European Mathematical Society Publishing House, : 2020,
Description:	1 online resource (374 pages)
Subject:	Calculus & mathematical analysis -
Online resource:	https://doi.org/10.4171/211
Online resource:	https://www.ems-ph.org/img/books/berti_mini.jpg
ISBN:	9783037197110

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

Quasi-Periodic Solutions of Nonlinear Wave Equations on the d-Dimensional Torus[electronic resource] /Massimiliano Berti, Philippe Bolle - Zuerich, Switzerland :European Mathematical Society Publishing House,2020 - 1 online resource (374 pages) - EMS Monographs in Mathematics (EMM) ;2523-5192.

Restricted to subscribers:https://www.ems-ph.org/ebooks.php

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.

ISBN: 9783037197110

Standard No.: 10.4171/211doiSubjects--Topical Terms:

3480907
Calculus & mathematical analysis

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