@article {2014, title = {An Abstract Nash{\textendash}Moser Theorem and Quasi-Periodic Solutions for NLW and NLS on Compact Lie Groups and Homogeneous Manifolds}, number = {Communications in mathematical physics;volume 334; issue 3; pages 1413-1454;}, year = {2014}, publisher = {Springer}, abstract = {We prove an abstract implicit function theorem with parameters for smooth operators defined on scales of sequence spaces, modeled for the search of quasi-periodic solutions of PDEs. The tame estimates required for the inverse linearised operators at each step of the iterative scheme are deduced via a multiscale inductive argument. The Cantor-like set of parameters where the solution exists is defined in a non inductive way. This formulation completely decouples the iterative scheme from the measure theoretical analysis of the parameters where the small divisors non-resonance conditions are verified. As an application, we deduce the existence of quasi-periodic solutions for forced NLW and NLS equations on any compact Lie group or manifold which is homogeneous with respect to a compact Lie group, extending previous results valid only for tori. A basic tool of harmonic analysis is the highest weight theory for the irreducible representations of compact Lie groups.}, doi = {10.1007/s00220-014-2128-4}, url = {http://urania.sissa.it/xmlui/handle/1963/34651}, author = {Massimiliano Berti and Livia Corsi and Michela Procesi} } @article {Berti2010377, title = {An abstract Nash-Moser theorem with parameters and applications to PDEs}, journal = {Annales de l{\textquoteright}Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis}, volume = {27}, number = {1}, year = {2010}, note = {cited By (since 1996)9}, pages = {377-399}, abstract = {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. {\textcopyright} 2009 Elsevier Masson SAS. All rights reserved.}, keywords = {Abstracting, Aircraft engines, Finite dimensional, Hamiltonian PDEs, Implicit function theorem, Invariant tori, Iterative schemes, Linearized operators, Mathematical operators, Moser theorem, Non-Linearity, Nonlinear equations, Nonlinear wave equation, Periodic solution, Point of interest, Resonance phenomena, Small divisors, Sobolev, Wave equations}, issn = {02941449}, doi = {10.1016/j.anihpc.2009.11.010}, author = {Massimiliano Berti and Philippe Bolle and Michela Procesi} }