01306nas a2200145   4500008004100000022001300041245007600054210006900130300001200199490000800211520079500219100002401014700001701038856010501055       2011                        eng d  a0010361600aBranching of Cantor Manifolds of Elliptic Tori and Applications to PDEs0 aBranching of Cantor Manifolds of Elliptic Tori and Applications   a741-7960 v3053 aWe 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.1 aBerti, Massimiliano1 aBiasco, Luca  uhttps://www.math.sissa.it/publication/branching-cantor-manifolds-elliptic-tori-and-applications-pdes00957nas a2200109   4500008004300000245006000043210005600103520060900159100001900768700002400787856003600811       2006                        en_Ud  00aA Birkhoff-Lewis-Type Theorem for Some Hamiltonian PDEs0 aBirkhoffLewisType Theorem for Some Hamiltonian PDEs3 aIn this paper we give an extension of the Birkhoff--Lewis theorem to some semilinear PDEs. Accordingly we prove existence of infinitely many periodic orbits with large period accumulating at the origin. Such periodic orbits bifurcate from resonant finite dimensional invariant tori of the fourth order normal form of the system. Besides standard nonresonance and nondegeneracy assumptions, our main result is obtained assuming a regularizing property of the nonlinearity. We apply our main theorem to a semilinear beam equation and to a nonlinear Schr\\\\\\\"odinger equation with smoothing nonlinearity.1 aBambusi, Dario1 aBerti, Massimiliano  uhttp://hdl.handle.net/1963/215900778nas a2200109   4500008004300000245007400043210006900117520040200186100002400588700002000612856003600632       2004                        en_Ud  00aBifurcation of free vibrations for completely resonant wave equations0 aBifurcation of free vibrations for completely resonant wave equa3 aWe prove existence of small amplitude, 2 pi/omega -periodic in time solutions of completely resonant nonlinear wave equations with Dirichlet boundary conditions for any frequency omega belonging to a Cantor-like set of positive measure and for a generic set of nonlinearities. The proof relies on a suitable Lyapunov-Schmidt decomposition and a variant of the Nash-Moser Implicit Function Theorem.1 aBerti, Massimiliano1 aBolle, Philippe  uhttp://hdl.handle.net/1963/2245