Hardcover ISBN:  9783037192115 
Product Code:  EMSMONO/11 
List Price:  $75.00 
AMS Member Price:  $60.00 
Hardcover ISBN:  9783037192115 
Product Code:  EMSMONO/11 
List Price:  $75.00 
AMS Member Price:  $60.00 

Book DetailsEMS Monographs in MathematicsVolume: 11; 2020; 374 ppMSC: Primary 37; 35
Many partial differential equations (PDEs) arising in physics, such as the nonlinear wave equation and the Schrödinger equation, can be viewed as infinitedimensional Hamiltonian systems. In the last thirty years, several existence results of time quasiperiodic 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 finitedimensional 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 quasiperiodic linear operators, which are presented in a systematic and selfcontained 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.
ReadershipResearchers interested in partial differential equations and dynamical systems.

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Many partial differential equations (PDEs) arising in physics, such as the nonlinear wave equation and the Schrödinger equation, can be viewed as infinitedimensional Hamiltonian systems. In the last thirty years, several existence results of time quasiperiodic 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 finitedimensional 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 quasiperiodic linear operators, which are presented in a systematic and selfcontained 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.
Researchers interested in partial differential equations and dynamical systems.