01540nas a2200133   4500008004100000245006000041210005900101520111900160100001301279700002401292700001901316700002301335856004801358       2017                        en  d00aTime quasi-periodic gravity water waves in finite depth0 aTime quasiperiodic gravity water waves in finite depth3 aWe prove the existence and the linear stability of Cantor families of small amplitude time quasi-periodic standing water wave solutions - namely periodic and even in the space variable x - of a bi-dimensional ocean with finite depth under the action of pure gravity. Such a result holds for all the values of the depth parameter in a Borel set of asymptotically full measure. This is a small divisor problem. The main difficulties are the quasi-linear nature of the gravity water waves equations and the fact that the linear frequencies grow just in a sublinear way at infinity. We overcome these problems by first reducing the linearized operators obtained at each approximate quasi-periodic solution along the Nash-Moser iteration to constant coefficients up to smoothing operators, using pseudo-differential changes of variables that are quasi-periodic in time. Then we apply a KAM reducibility scheme which requires very weak Melnikov non-resonance conditions (losing derivatives both in time and space), which we are able to verify for most values of the depth parameter using degenerate KAM theory arguments.1 aBaldi, P1 aBerti, Massimiliano1 aHaus, Emanuele1 aMontalto, Riccardo  uhttp://preprints.sissa.it/handle/1963/3529601519nas a2200145   4500008004100000022001300041245008300054210006900137300000900206520098400215100001301199700002401212700002301236856011401259       2014                        eng d  a0025583100aKAM for quasi-linear and fully nonlinear forced perturbations of Airy equation0 aKAM for quasilinear and fully nonlinear forced perturbations of   a1-663 aWe 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. © 2014 Springer-Verlag Berlin Heidelberg.1 aBaldi, P1 aBerti, Massimiliano1 aMontalto, Riccardo  uhttps://www.math.sissa.it/publication/kam-quasi-linear-and-fully-nonlinear-forced-perturbations-airy-equation00573nas a2200157   4500008004100000245002900041210002800070260001300098300001200111490000800123520017300131100001300304700002400317700002300341856005100364       2014                        en  d00aKAM for quasi-linear KdV0 aKAM for quasilinear KdV  bElsevier  a603-6070 v3523 a<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>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