We prove the existence and stability of Cantor families of quasi-periodic, small-amplitude solutions of quasi-linear autonomous Hamiltonian perturbations of KdV.

1 aBaldi, P1 aBerti, Massimiliano1 aMontalto, Riccardo uhttp://urania.sissa.it/xmlui/handle/1963/3506701376nas a2200145 4500008004100000245007300041210006900114260003400183520083400217653001701051100001301068700002401081700002301105856010201128 2013 en d00aA note on KAM theory for quasi-linear and fully nonlinear forced KdV0 anote on KAM theory for quasilinear and fully nonlinear forced Kd bEuropean Mathematical Society3 aWe 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.10aKAM for PDEs1 aBaldi, P1 aBerti, Massimiliano1 aMontalto, Riccardo uhttps://www.math.sissa.it/publication/note-kam-theory-quasi-linear-and-fully-nonlinear-forced-kdv00637nas a2200109 4500008004300000245004900043210004900092520031300141100001300454700002400467856003600491 2008 en_Ud 00aForced Vibrations of a Nonhomogeneous String0 aForced Vibrations of a Nonhomogeneous String3 aWe 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.1 aBaldi, P1 aBerti, Massimiliano uhttp://hdl.handle.net/1963/264300834nas a2200145 4500008004100000022001300041245010300054210006900157300001200226490000700238520027500245100001300520700002400533856013100557 2006 eng d a1120633000aPeriodic solutions of nonlinear wave equations for asymptotically full measure sets of frequencies0 aPeriodic solutions of nonlinear wave equations for asymptoticall a257-2770 v173 aWe 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.1 aBaldi, P1 aBerti, Massimiliano uhttps://www.math.sissa.it/publication/periodic-solutions-nonlinear-wave-equations-asymptotically-full-measure-sets-frequenci-0