@article {FEOLA2019932,
	title = {Reducibility of first order linear operators on tori via Moser{\textquoteright}s theorem},
	journal = {Journal of Functional Analysis},
	volume = {276},
	number = {3},
	year = {2019},
	pages = {932 - 970},
	abstract = {<p>In this paper we prove reducibility of a class of first order, quasi-linear, quasi-periodic time dependent PDEs on the torus∂tu+ζ.∂xu+a(ωt,x).∂xu=0,x∈Td,ζ∈Rd,ω∈Rν. As a consequence we deduce a stability result on the associated Cauchy problem in Sobolev spaces. By the identification between first order operators and vector fields this problem can be formulated as the problem of finding a change of coordinates which conjugates a weakly perturbed constant vector field on Tν+d to a constant diophantine flow. For this purpose we generalize Moser{\textquoteright}s straightening theorem: considering smooth perturbations we prove that the corresponding straightening torus diffeomorphism is smooth, under the assumption that the perturbation is small only in some given Sobolev norm and that the initial frequency belongs to some Cantor-like set. In view of applications in KAM theory for PDEs we provide also tame estimates on the change of variables.</p>},
	keywords = {Hyperbolic PDEs, KAM theory, Nash{\textendash}Moser, Reducibility},
	issn = {0022-1236},
	doi = {https://doi.org/10.1016/j.jfa.2018.10.009},
	url = {http://www.sciencedirect.com/science/article/pii/S0022123618303793},
	author = {Roberto Feola and Filippo Giuliani and Riccardo Montalto and Michela Procesi}
}
@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}
}
@mastersthesis {2014,
	title = {KAM for quasi-linear and fully nonlinear perturbations of Airy and KdV equations},
	year = {2014},
	school = {SISSA},
	url = {http://urania.sissa.it/xmlui/handle/1963/7476},
	author = {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}
}
