@article {2017,
	title = {Time quasi-periodic gravity water waves in finite depth},
	number = {arXiv;1708.01517},
	year = {2017},
	abstract = {We prove the existence and the linear stability of Cantor families of small amplitude time quasi-periodic standing water wave solutions - namely periodic and even in the space variable x - of a bi-dimensional ocean with finite depth under the action of pure gravity. Such a result holds for all the values of the depth parameter in a Borel set of asymptotically full measure. This is a small divisor problem. The main difficulties are the quasi-linear nature of the gravity water waves equations and the fact that the linear frequencies grow just in a sublinear way at infinity. We overcome these problems by first reducing the linearized operators obtained at each approximate quasi-periodic solution along the Nash-Moser iteration to constant coefficients up to smoothing operators, using pseudo-differential changes of variables that are quasi-periodic in time. Then we apply a KAM reducibility scheme which requires very weak Melnikov non-resonance conditions (losing derivatives both in time and space), which we are able to verify for most values of the depth parameter using degenerate KAM theory arguments.},
	url = {http://preprints.sissa.it/handle/1963/35296},
	author = {P Baldi and Massimiliano Berti and Emanuele Haus and Riccardo Montalto}
}
@article {Baldi20141,
	title = {KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation},
	journal = {Mathematische Annalen},
	year = {2014},
	note = {cited By (since 1996)0; Article in Press},
	pages = {1-66},
	abstract = {We prove the existence of small amplitude quasi-periodic solutions for quasi-linear and fully nonlinear forced perturbations of the linear Airy equation. For Hamiltonian or reversible nonlinearities we also prove their linear stability. The key analysis concerns the reducibility of the linearized operator at an approximate solution, which provides a sharp asymptotic expansion of its eigenvalues. For quasi-linear perturbations this cannot be directly obtained by a KAM iteration. Hence we first perform a regularization procedure, which conjugates the linearized operator to an operator with constant coefficients plus a bounded remainder. These transformations are obtained by changes of variables induced by diffeomorphisms of the torus and pseudo-differential operators. At this point we implement a Nash-Moser iteration (with second order Melnikov non-resonance conditions) which completes the reduction to constant coefficients. {\textcopyright} 2014 Springer-Verlag Berlin Heidelberg.},
	issn = {00255831},
	doi = {10.1007/s00208-013-1001-7},
	author = {P Baldi and Massimiliano Berti and Riccardo Montalto}
}
@article {2014,
	title = {KAM for quasi-linear KdV},
	journal = {C. R. Math. Acad. Sci. Paris},
	volume = {352},
	number = {Comptes Rendus Mathematique;volume 352; issue 7-8; pages 603-607;},
	year = {2014},
	pages = {603-607},
	publisher = {Elsevier},
	abstract = {<p>We prove the existence and stability of Cantor families of quasi-periodic, small-amplitude solutions of quasi-linear autonomous Hamiltonian perturbations of KdV.</p>},
	doi = {10.1016/j.crma.2014.04.012},
	url = {http://urania.sissa.it/xmlui/handle/1963/35067},
	author = {P Baldi and Massimiliano Berti and Riccardo Montalto}
}
@article {2013,
	title = {A note on KAM theory for quasi-linear and fully nonlinear forced KdV},
	journal = {Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 24 (2013), no. 3: 437{\textendash}450},
	year = {2013},
	publisher = {European Mathematical Society},
	abstract = {We present the recent results in [3] concerning quasi-periodic solutions
for quasi-linear and fully nonlinear forced perturbations of KdV equations.
For Hamiltonian or reversible nonlinearities the solutions are linearly stable.
The proofs are based on a combination of di erent ideas and techniques:
(i) a Nash-Moser iterative scheme in Sobolev scales. (ii) A regularization
procedure, which conjugates the linearized operator to a di erential operator
with constant coe cients plus a bounded remainder. These transformations
are obtained by changes of variables induced by di eomorphisms of the torus
and pseudo-di erential operators. (iii) A reducibility KAM scheme, which
completes the reduction to constant coe cients of the linearized operator,
providing a sharp asymptotic expansion of the perturbed eigenvalues.},
	keywords = {KAM for PDEs},
	doi = {10.4171/RLM/660},
	author = {P Baldi and Massimiliano Berti and Riccardo Montalto}
}
@article {2008,
	title = {Forced Vibrations of a Nonhomogeneous String},
	journal = {SIAM J. Math. Anal. 40 (2008) 382-412},
	number = {SISSA;36/2006/M},
	year = {2008},
	abstract = {We prove existence of vibrations of a nonhomogeneous string under a nonlinear time periodic forcing term in the case in which the forcing frequency avoids resonances with the vibration modes of the string (nonresonant case). The proof relies on a Lyapunov-Schmidt reduction and a Nash-Moser iteration scheme.},
	doi = {10.1137/060665038},
	url = {http://hdl.handle.net/1963/2643},
	author = {P Baldi and Massimiliano Berti}
}
@article {Baldi2006257,
	title = {Periodic solutions of nonlinear wave equations for asymptotically full measure sets of frequencies},
	journal = {Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica e Applicazioni},
	volume = {17},
	number = {3},
	year = {2006},
	note = {cited By (since 1996)5},
	pages = {257-277},
	abstract = {We prove existence and multiplicity of small amplitude periodic solutions of completely resonant nonlinear wave equations with Dirichlet boundary conditions for asymptotically full measure sets of frequencies, extending the results of [7] to new types of nonlinearities.},
	issn = {11206330},
	author = {P Baldi and Massimiliano Berti}
}
