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

PB - Springer VL - 212 UR - http://urania.sissa.it/xmlui/handle/1963/34646 IS - 3 U1 - 34850 U2 - Mathematics ER - TY - JOUR T1 - Existence and stability of quasi-periodic solutions for derivative wave equations JF - Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica e Applicazioni Y1 - 2013 A1 - Massimiliano Berti A1 - Luca Biasco A1 - Michela Procesi KW - Constant coefficients KW - Dynamical systems KW - Existence and stability KW - Infinite dimensional KW - KAM for PDEs KW - Linearized equations KW - Lyapunov exponent KW - Lyapunov methods KW - Quasi-periodic solution KW - Small divisors KW - Wave equations AB - 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*. VL - 24 N1 - cited By (since 1996)0 ER - TY - JOUR T1 - KAM theory for the Hamiltonian derivative wave equation JF - Annales Scientifiques de l'Ecole Normale Superieure Y1 - 2013 A1 - Massimiliano Berti A1 - Luca Biasco A1 - Michela Procesi AB -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. © 2013 Société Mathématique de France.

VL - 46 N1 - cited By (since 1996)4 ER - TY - JOUR T1 - Branching of Cantor Manifolds of Elliptic Tori and Applications to PDEs JF - Communications in Mathematical Physics Y1 - 2011 A1 - Massimiliano Berti A1 - Luca Biasco AB - 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. © 2011 Springer-Verlag. VL - 305 N1 - cited By (since 1996)8 ER - TY - JOUR T1 - Forced vibrations of wave equations with non-monotone nonlinearities JF - Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006) 439-474 Y1 - 2006 A1 - Massimiliano Berti A1 - Luca Biasco AB - 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. UR - http://hdl.handle.net/1963/2160 U1 - 2084 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Periodic solutions of nonlinear wave equations with non-monotone forcing terms JF - Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 16 (2005), no. 2, 117-124 Y1 - 2005 A1 - Massimiliano Berti A1 - Luca Biasco PB - Accademia Nazionale dei Lincei UR - http://hdl.handle.net/1963/4581 U1 - 4349 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER - TY - JOUR T1 - Periodic orbits close to elliptic tori and applications to the three-body problem JF - Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 3 (2004) 87-138 Y1 - 2004 A1 - Massimiliano Berti A1 - Luca Biasco A1 - Enrico Valdinoci AB - We prove, under suitable non-resonance and non-degeneracy ``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 ``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.) PB - Scuola Normale Superiore di Pisa UR - http://hdl.handle.net/1963/2985 U1 - 1348 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Drift in phase space: a new variational mechanism with optimal diffusion time JF - J. Math. Pures Appl. 82 (2003) 613-664 Y1 - 2003 A1 - Massimiliano Berti A1 - Luca Biasco A1 - Philippe Bolle AB - 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 ``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. PB - Elsevier UR - http://hdl.handle.net/1963/3020 U1 - 1313 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - An optimal fast-diffusion variational method for non isochronous system Y1 - 2002 A1 - Luca Biasco A1 - Massimiliano Berti A1 - Philippe Bolle PB - SISSA Library UR - http://hdl.handle.net/1963/1579 U1 - 2539 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Optimal stability and instability results for a class of nearly integrable Hamiltonian systems JF - Atti.Accad.Naz.Lincei Cl.Sci.Fis.Mat.Natur.Rend.Lincei (9) Mat.Appl.13(2002),no.2,77-84 Y1 - 2002 A1 - Massimiliano Berti A1 - Luca Biasco A1 - Philippe Bolle PB - SISSA Library UR - http://hdl.handle.net/1963/1596 U1 - 2522 U2 - Mathematics U3 - Functional Analysis and Applications ER -