Quasi-periodic solutions of nonlinear wave equations in the D-dimensional torus
By: Massimiliano Berti, Philippe Bolle
Contributors: Bolle, Philippe, author
Series: EMS Monographs in Mathematics; EMS monographs in mathematics
Publisher: European Mathematical Society
Published: 2020
Place of publication: Berlin
Language: eng
Number of pages: 358
ISBN-13: 9783037192115
Subjects: Nonlinear wave equations; Numerical solutions; Hamiltonian systems
Description: "Many partial differential equations (PDEs) arising in physics, such as the nonlinear wave equation and the Schr̲dinger 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." - publisher
Notes: Includes bibliographical references and index.
Table of contents: title: KAM for PDEs and strategy of proof; title: Hamiltonian formulation; title: Functional setting; title: Multiscale analysis; title: Nash-Moser theorem; title: Linearized operator at an approximate solution; title: Splitting of low-high normal subspaces up to O (E4); title: Approximate right inverse in normal directions; title: Splitting between low-high normal subspaces; title: Construction of approximate right inverse; title: Proof of the Nash-Moser theorem; title: Genericity of the assumptions