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.

%B Journal of Functional Analysis %V 276 %P 932 - 970 %G eng %U http://www.sciencedirect.com/science/article/pii/S0022123618303793 %R https://doi.org/10.1016/j.jfa.2018.10.009 %0 Report %D 2017 %T Time quasi-periodic gravity water waves in finite depth %A P Baldi %A Massimiliano Berti %A Emanuele Haus %A Riccardo Montalto %X 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. %G en %U http://preprints.sissa.it/handle/1963/35296 %1 35602 %2 Mathematics %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2017-09-27T15:46:10Z No. of bitstreams: 1 1708.01517.pdf: 1909445 bytes, checksum: 71fc741593666cacf41187c00b092503 (MD5) %0 Report %D 2016 %T Large KAM tori for perturbations of the dNLS equation %A Massimiliano Berti %A Thomas Kappeler %A Riccardo Montalto %X 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. %G en %U http://preprints.sissa.it/handle/1963/35284 %1 35589 %2 Mathematics %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2017-05-30T09:09:35Z No. of bitstreams: 1 1603.09252.pdf: 1306610 bytes, checksum: 1e34137dcb21eb2e2a8e93d0c2d009a3 (MD5) %0 Journal Article %J Mathematische Annalen %D 2014 %T KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation %A P Baldi %A Massimiliano Berti %A Riccardo Montalto %X 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. %B Mathematische Annalen %P 1-66 %G eng %R 10.1007/s00208-013-1001-7 %0 Thesis %D 2014 %T KAM for quasi-linear and fully nonlinear perturbations of Airy and KdV equations %A Riccardo Montalto %I SISSA %G en %U http://urania.sissa.it/xmlui/handle/1963/7476 %1 7571 %2 Mathematics %4 1 %# MAT/05 %$ Submitted by rmontalt@sissa.it (rmontalt@sissa.it) on 2014-10-27T10:15:33Z No. of bitstreams: 1 Tesi-Phd-Montalto-Riccardo.pdf: 1843377 bytes, checksum: b4e504e6ad1860c3d3580beb4e24efff (MD5) %0 Journal Article %J C. R. Math. Acad. Sci. Paris %D 2014 %T KAM for quasi-linear KdV %A P Baldi %A Massimiliano Berti %A Riccardo Montalto %XWe prove the existence and stability of Cantor families of quasi-periodic, small-amplitude solutions of quasi-linear autonomous Hamiltonian perturbations of KdV.

%B C. R. Math. Acad. Sci. Paris %I Elsevier %V 352 %P 603-607 %G en %U http://urania.sissa.it/xmlui/handle/1963/35067 %N 7-8 %1 35302 %2 Mathematics %4 1 %$ Approved for entry into archive by Maria Pia Calandra (calapia@sissa.it) on 2015-11-30T15:30:40Z (GMT) No. of bitstreams: 0 %R 10.1016/j.crma.2014.04.012 %0 Journal Article %J Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 24 (2013), no. 3: 437–450 %D 2013 %T A note on KAM theory for quasi-linear and fully nonlinear forced KdV %A P Baldi %A Massimiliano Berti %A Riccardo Montalto %K KAM for PDEs %X 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. %B Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 24 (2013), no. 3: 437–450 %I European Mathematical Society %G en %1 7268 %2 Mathematics %4 1 %# MAT/05 ANALISI MATEMATICA %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2013-12-09T08:16:04Z No. of bitstreams: 1 Baldi-Berti-Montalto-Lincei-Note-KAM-forced-KdV.pdf: 322654 bytes, checksum: ddae153eff7ad17e2be36cb3ba1af9bf (MD5) %R 10.4171/RLM/660