In this paper we prove reducibility of a class of first order, quasi-linear, quasi-periodic time dependent PDEs on the torus∂tu+ζ⋅∂xu+a(ωt,x)⋅∂xu=0,x∈Td,ζ∈Rd,ω∈Rν. As a consequence we deduce a stability result on the associated Cauchy problem in Sobolev spaces. By the identification between first order operators and vector fields this problem can be formulated as the problem of finding a change of coordinates which conjugates a weakly perturbed constant vector field on Tν+d to a constant diophantine flow. For this purpose we generalize Moser's straightening theorem: considering smooth perturbations we prove that the corresponding straightening torus diffeomorphism is smooth, under the assumption that the perturbation is small only in some given Sobolev norm and that the initial frequency belongs to some Cantor-like set. In view of applications in KAM theory for PDEs we provide also tame estimates on the change of variables.

VL - 276 UR - http://www.sciencedirect.com/science/article/pii/S0022123618303793 ER - TY - RPRT T1 - Time quasi-periodic gravity water waves in finite depth Y1 - 2017 A1 - P Baldi A1 - Massimiliano Berti A1 - Emanuele Haus A1 - Riccardo Montalto AB - We 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. UR - http://preprints.sissa.it/handle/1963/35296 U1 - 35602 U2 - Mathematics ER - TY - RPRT T1 - Large KAM tori for perturbations of the dNLS equation Y1 - 2016 A1 - Massimiliano Berti A1 - Thomas Kappeler A1 - Riccardo Montalto AB - We prove that small, semi-linear Hamiltonian perturbations of the defocusing nonlinear Schr\"odinger (dNLS) equation on the circle have an abundance of invariant tori of any size and (finite) dimension which support quasi-periodic solutions. When compared with previous results the novelty consists in considering perturbations which do not satisfy any symmetry condition (they may depend on x in an arbitrary way) and need not be analytic. The main difficulty is posed by pairs of almost resonant dNLS frequencies. The proof is based on the integrability of the dNLS equation, in particular the fact that the nonlinear part of the Birkhoff coordinates is one smoothing. We implement a Newton-Nash-Moser iteration scheme to construct the invariant tori. The key point is the reduction of linearized operators, coming up in the iteration scheme, to 2×2 block diagonal ones with constant coefficients together with sharp asymptotic estimates of their eigenvalues. UR - http://preprints.sissa.it/handle/1963/35284 U1 - 35589 U2 - Mathematics ER - TY - JOUR T1 - KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation JF - Mathematische Annalen Y1 - 2014 A1 - P Baldi A1 - Massimiliano Berti A1 - Riccardo Montalto AB - We 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. N1 - cited By (since 1996)0; Article in Press ER - TY - THES T1 - KAM for quasi-linear and fully nonlinear perturbations of Airy and KdV equations Y1 - 2014 A1 - Riccardo Montalto PB - SISSA UR - http://urania.sissa.it/xmlui/handle/1963/7476 U1 - 7571 U2 - Mathematics U4 - 1 U5 - MAT/05 ER - TY - JOUR T1 - KAM for quasi-linear KdV JF - C. R. Math. Acad. Sci. Paris Y1 - 2014 A1 - P Baldi A1 - Massimiliano Berti A1 - Riccardo Montalto AB -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 -