@article {2017,
	title = {Almost global existence of solutions for capillarity-gravity water waves equations with periodic spatial boundary conditions},
	number = {arXiv;1702.04674},
	year = {2017},
	abstract = {The goal of this monograph is to prove that any solution of the Cauchy problem for the capillarity-gravity water waves equations, in one space dimension, with periodic, even in space, initial data of small size ϵ, is almost globally defined in time on Sobolev spaces, i.e. it exists on a time interval of length of magnitude ϵ-N for any N, as soon as the initial data are smooth enough, and the gravity-capillarity parameters are taken outside an exceptional subset of zero measure. In contrast to the many results known for these equations on the real line, with decaying Cauchy data, one cannot make use of dispersive properties of the linear flow. Instead, our method is based on a normal forms procedure, in order to eliminate those contributions to the Sobolev energy that are of lower degree of homogeneity in the solution. Since the water waves equations are a quasi-linear system, usual normal forms approaches would face the well known problem of losses of derivatives in the unbounded transformations. In this monograph, to overcome such a difficulty, after a paralinearization of the capillarity-gravity water waves equations, necessary to obtain energy estimates, and thus local existence of the solutions, we first perform several paradifferential reductions of the equations to obtain a diagonal system with constant coefficients symbols, up to smoothing remainders. Then we may start with a normal form procedure where the small divisors are compensated by the previous paradifferential regularization.The reversible structure of the water waves equations, and the fact that we look for solutions even in x, guarantees a key cancellation which prevents the growth of the Sobolev norms of the solutions.},
	url = {http://preprints.sissa.it/handle/1963/35285},
	author = {Massimiliano Berti and Jean-Marc Delort}
}
@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}
}
@article {2002,
	title = {Arnold diffusion: a functional analysis approach},
	journal = {Pr. Inst. Mat. Nats. Akad. Nauk Ukr. Mat. Zastos., 43, Part 1, 2, Natsional. Akad. Nauk Ukra{\"\i}ni, Inst. Mat., Kiev, 2002},
	year = {2002},
	publisher = {Natsional. Akad. Nauk Ukra{\"\i}ni},
	abstract = {We present, in the context of nearly integrable Hamiltonian systems, a functional analysis
approach to study the {\textquotedblleft}splitting of the whiskers{\textquotedblright} and the {\textquotedblleft}shadowing problem{\textquotedblright} developed
in collaboration with P. Bolle in the recent papers [1] and [2] . This method is applied to the
problem of Arnold diffusion for nearly integrable partially isochronous systems improving
known results.},
	author = {Massimiliano Berti}
}
@article {2000,
	title = {Arnold{\textquoteright}s Diffusion in nearly integrable isochronous Hamiltonian systems},
	number = {SISSA;98/00/M},
	year = {2000},
	publisher = {SISSA Library},
	url = {http://hdl.handle.net/1963/1554},
	author = {Massimiliano Berti and Philippe Bolle}
}
