We prove the existence of Cantor families of small amplitude, analytic, linearly stable quasi-periodic solutions of reversible derivative wave equations.

}, doi = {10.1007/s00205-014-0726-0}, url = {http://urania.sissa.it/xmlui/handle/1963/34646}, author = {Massimiliano Berti and Luca Biasco and Michela Procesi} } @article {Berti2013199, title = {Existence and stability of quasi-periodic solutions for derivative wave equations}, journal = {Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica e Applicazioni}, volume = {24}, number = {2}, year = {2013}, note = {cited By (since 1996)0}, pages = {199-214}, abstract = {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*.}, keywords = {Constant coefficients, Dynamical systems, Existence and stability, Infinite dimensional, KAM for PDEs, Linearized equations, Lyapunov exponent, Lyapunov methods, Quasi-periodic solution, Small divisors, Wave equations}, issn = {11206330}, doi = {10.4171/RLM/652}, author = {Massimiliano Berti and Luca Biasco and Michela Procesi} } @article {Berti2013301, title = {KAM theory for the Hamiltonian derivative wave equation}, journal = {Annales Scientifiques de l{\textquoteright}Ecole Normale Superieure}, volume = {46}, number = {2}, year = {2013}, note = {cited By (since 1996)4}, pages = {301-373}, abstract = {We prove an infinite dimensional KAM theorem which implies the existence of Can- tor families of small-amplitude, reducible, elliptic, analytic, invariant tori of Hamiltonian derivative wave equations. {\textcopyright} 2013 Soci{\'e}t{\'e} Math{\'e}matique de France.

}, issn = {00129593}, author = {Massimiliano Berti and Luca Biasco and Michela Procesi} } @article {Berti2011741, title = {Branching of Cantor Manifolds of Elliptic Tori and Applications to PDEs}, journal = {Communications in Mathematical Physics}, volume = {305}, number = {3}, year = {2011}, note = {cited By (since 1996)8}, pages = {741-796}, abstract = {We consider infinite dimensional Hamiltonian systems. We prove the existence of "Cantor manifolds" of elliptic tori-of any finite higher dimension-accumulating on a given elliptic KAM torus. Then, close to an elliptic equilibrium, we show the existence of Cantor manifolds of elliptic tori which are "branching" points of other Cantor manifolds of higher dimensional tori. We also answer to a conjecture of Bourgain, proving the existence of invariant elliptic tori with tangential frequency along a pre-assigned direction. The proofs are based on an improved KAM theorem. Its main advantages are an explicit characterization of the Cantor set of parameters and weaker smallness conditions on the perturbation. We apply these results to the nonlinear wave equation. {\textcopyright} 2011 Springer-Verlag.}, issn = {00103616}, doi = {10.1007/s00220-011-1264-3}, author = {Massimiliano Berti and Luca Biasco} } @article {2006, title = {Forced vibrations of wave equations with non-monotone nonlinearities}, journal = {Ann. Inst. H. Poincar{\'e} Anal. Non Lin{\'e}aire 23 (2006) 439-474}, number = {arXiv.org;math/0410619v1}, year = {2006}, abstract = {We prove existence and regularity of periodic in time solutions of completely resonant nonlinear forced wave equations with Dirichlet boundary conditions for a large class of non-monotone forcing terms. Our approach is based on a variational Lyapunov-Schmidt reduction. It turns out that the infinite dimensional bifurcation equation exhibits an intrinsic lack of compactness. We solve it via a minimization argument and a-priori estimate methods inspired to regularity theory of Rabinowitz.}, doi = {10.1016/j.anihpc.2005.05.004}, url = {http://hdl.handle.net/1963/2160}, author = {Massimiliano Berti and Luca Biasco} } @article {2005, title = {Periodic solutions of nonlinear wave equations with non-monotone forcing terms}, journal = {Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 16 (2005), no. 2, 117-124}, year = {2005}, publisher = {Accademia Nazionale dei Lincei}, url = {http://hdl.handle.net/1963/4581}, author = {Massimiliano Berti and Luca Biasco} } @article {2004, title = {Periodic orbits close to elliptic tori and applications to the three-body problem}, journal = {Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 3 (2004) 87-138}, number = {SISSA;28/2003/M}, year = {2004}, publisher = {Scuola Normale Superiore di Pisa}, abstract = {We prove, under suitable non-resonance and non-degeneracy {\textquoteleft}{\textquoteleft}twist\\\'\\\' conditions, a Birkhoff-Lewis type result showing the existence of infinitely many periodic solutions, with larger and larger minimal period, accumulating onto elliptic invariant tori (of Hamiltonian systems). We prove the applicability of this result to the spatial planetary three-body problem in the small eccentricity-inclination regime. Furthermore, we find other periodic orbits under some restrictions on the period and the masses of the {\textquoteleft}{\textquoteleft}planets\\\'\\\'. The proofs are based on averaging theory, KAM theory and variational methods. (Supported by M.U.R.S.T. Variational Methods and Nonlinear Differential Equations.)}, url = {http://hdl.handle.net/1963/2985}, author = {Massimiliano Berti and Luca Biasco and Enrico Valdinoci} } @article {2003, title = {Drift in phase space: a new variational mechanism with optimal diffusion time}, journal = {J. Math. Pures Appl. 82 (2003) 613-664}, number = {arXiv.org;math/0205307v1}, year = {2003}, publisher = {Elsevier}, abstract = {We consider non-isochronous, nearly integrable, a-priori unstable Hamiltonian systems with a (trigonometric polynomial) $O(\\\\mu)$-perturbation which does not preserve the unperturbed tori. We prove the existence of Arnold diffusion with diffusion time $ T_d = O((1/ \\\\mu) \\\\log (1/ \\\\mu))$ by a variational method which does not require the existence of {\textquoteleft}{\textquoteleft}transition chains of tori\\\'\\\' provided by KAM theory. We also prove that our estimate of the diffusion time $T_d $ is optimal as a consequence of a general stability result derived from classical perturbation theory.}, doi = {10.1016/S0021-7824(03)00032-1}, url = {http://hdl.handle.net/1963/3020}, author = {Massimiliano Berti and Luca Biasco and Philippe Bolle} } @article {2002, title = {An optimal fast-diffusion variational method for non isochronous system}, number = {SISSA;8/2002/M}, year = {2002}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/1579}, author = {Luca Biasco and Massimiliano Berti and Philippe Bolle} } @article {2002, title = {Optimal stability and instability results for a class of nearly integrable Hamiltonian systems}, journal = {Atti.Accad.Naz.Lincei Cl.Sci.Fis.Mat.Natur.Rend.Lincei (9) Mat.Appl.13(2002),no.2,77-84}, number = {SISSA;25/2002/M}, year = {2002}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/1596}, author = {Massimiliano Berti and Luca Biasco and Philippe Bolle} }