Quasi-periodic solutions of nonlinear wave equations in the D-dimensional torus / Massimiliano Berti, Philippe Bolle.

Format:

Book

Author/Creator:

Berti, Massimiliano, author.

Bolle, Philippe, author.

Series:

EMS monographs in mathematics

EMS Monographs in Mathematics

Language:

English

Subjects (All):

Nonlinear wave equations--Numerical solutions.

Nonlinear wave equations.

Hamiltonian systems.

Physical Description:

xv, 358 pages ; 24 cm.

Place of Publication:

Berlin : European Mathematical Society, [2020]

Summary:

"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." - publisher

Contents:

KAM for PDEs and strategy of proof

Hamiltonian formulation

Functional setting

Multiscale analysis

Nash-Moser theorem

Linearized operator at an approximate solution

Splitting of low-high normal subspaces up to O (E4)

Approximate right inverse in normal directions

Splitting between low-high normal subspaces

Construction of approximate right inverse

Proof of the Nash-Moser theorem

Genericity of the assumptions

Notes:

Includes bibliographical references and index.

ISBN:

9783037192115

3037192119

OCLC:

1199124577