%0 Journal Article
%J Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica e Applicazioni
%D 2013
%T Existence and stability of quasi-periodic solutions for derivative wave equations
%A Massimiliano Berti
%A Luca Biasco
%A Michela Procesi
%K Constant coefficients
%K Dynamical systems
%K Existence and stability
%K Infinite dimensional
%K KAM for PDEs
%K Linearized equations
%K Lyapunov exponent
%K Lyapunov methods
%K Quasi-periodic solution
%K Small divisors
%K Wave equations
%X In this note we present the new KAM result in [3] which proves the existence of Cantor families of small amplitude, analytic, quasi-periodic solutions of derivative wave equations, with zero Lyapunov exponents and whose linearized equation is reducible to constant coefficients. In turn, this result is derived by an abstract KAM theorem for infinite dimensional reversible dynamical systems*.
%B Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica e Applicazioni
%V 24
%P 199-214
%G eng
%R 10.4171/RLM/652
%0 Journal Article
%J Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis
%D 2010
%T An abstract Nash-Moser theorem with parameters and applications to PDEs
%A Massimiliano Berti
%A Philippe Bolle
%A Michela Procesi
%K Abstracting
%K Aircraft engines
%K Finite dimensional
%K Hamiltonian PDEs
%K Implicit function theorem
%K Invariant tori
%K Iterative schemes
%K Linearized operators
%K Mathematical operators
%K Moser theorem
%K Non-Linearity
%K Nonlinear equations
%K Nonlinear wave equation
%K Periodic solution
%K Point of interest
%K Resonance phenomena
%K Small divisors
%K Sobolev
%K Wave equations
%X We prove an abstract Nash-Moser implicit function theorem with parameters which covers the applications to the existence of finite dimensional, differentiable, invariant tori of Hamiltonian PDEs with merely differentiable nonlinearities. The main new feature of the abstract iterative scheme is that the linearized operators, in a neighborhood of the expected solution, are invertible, and satisfy the "tame" estimates, only for proper subsets of the parameters. As an application we show the existence of periodic solutions of nonlinear wave equations on Riemannian Zoll manifolds. A point of interest is that, in presence of possibly very large "clusters of small divisors", due to resonance phenomena, it is more natural to expect solutions with only Sobolev regularity. © 2009 Elsevier Masson SAS. All rights reserved.
%B Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis
%V 27
%P 377-399
%G eng
%R 10.1016/j.anihpc.2009.11.010