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

PB - Elsevier VL - 352 UR - http://urania.sissa.it/xmlui/handle/1963/35067 IS - 7-8 U1 - 35302 U2 - Mathematics U4 - 1 ER - TY - JOUR T1 - A note on KAM theory for quasi-linear and fully nonlinear forced KdV JF - Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 24 (2013), no. 3: 437–450 Y1 - 2013 A1 - P Baldi A1 - Massimiliano Berti A1 - Riccardo Montalto KW - KAM for PDEs AB - We 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. PB - European Mathematical Society U1 - 7268 U2 - Mathematics U4 - 1 U5 - MAT/05 ANALISI MATEMATICA ER - TY - JOUR T1 - Forced Vibrations of a Nonhomogeneous String JF - SIAM J. Math. Anal. 40 (2008) 382-412 Y1 - 2008 A1 - P Baldi A1 - Massimiliano Berti AB - We 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. UR - http://hdl.handle.net/1963/2643 U1 - 1480 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Periodic solutions of nonlinear wave equations for asymptotically full measure sets of frequencies JF - Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica e Applicazioni Y1 - 2006 A1 - P Baldi A1 - Massimiliano Berti AB - We 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. VL - 17 N1 - cited By (since 1996)5 ER -