02929nam a22003615a 4500001001100000003001200011005001700023006001900040007001500059008004100074020001800115024002100133040001400154072001600168084002200184100003400206245014700240260008200387300003400469336002600503337002600529338003600555347002400591490005300615506006600668520157500734650004502309650004602354650004002400700003002440856003202470856006502502258-200908CH-001817-320200908234501.0a fot ||| 0|cr nn mmmmamaa200908e20201009sz fot ||| 0|eng d a978303719711070a10.4171/2112doi ach0018173 7aPBK2bicssc a37-xxa35-xx2msc1 aBerti, Massimiliano,eauthor.10aQuasi-Periodic Solutions of Nonlinear Wave Equations on the $d$-Dimensional Torush[electronic resource] /cMassimiliano Berti, Philippe Bolle3 aZuerich, Switzerland :bEuropean Mathematical Society Publishing House,c2020 a1 online resource (374 pages) atextbtxt2rdacontent acomputerbc2rdamedia aonline resourcebcr2rdacarrier atext filebPDF2rda0 aEMS Monographs in Mathematics (EMM) ;x2523-51921 aRestricted to subscribers:uhttps://www.ems-ph.org/ebooks.php aMany partial differential equations (PDEs) arising in physics, such as the nonlinear wave equation and the SchrĂ¶dinger 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 Nashâ€“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.07aCalculus & mathematical analysis2bicssc07aDynamical systems and ergodic theory2msc07aPartial differential equations2msc1 aBolle, Philippe,eauthor.40uhttps://doi.org/10.4171/211423cover imageuhttps://www.ems-ph.org/img/books/berti_mini.jpg