We consider the Euler–Korteweg system with space periodic boundary conditions $$ x \in {\mathbb {T}}^d$$. We prove a local in time existence result of classical solutions for irrotational velocity fields requiring natural minimal regularity assumptions on the initial data.

1 aBerti, Massimiliano1 aMaspero, Alberto1 aMurgante, Federico uhttps://doi.org/10.1007/s10884-020-09927-301345nas a2200169 4500008004100000020001400041245006600055210006500121260001500186300001300201490000600214520084700220100002401067700001901091700001801110856004701128 2021 eng d a2523-368800aQuadratic Life Span of Periodic Gravity-capillary Water Waves0 aQuadratic Life Span of Periodic Gravitycapillary Water Waves c2021/04/01 a85 - 1150 v33 aWe consider the gravity-capillary water waves equations for a bi-dimensional fluid with a periodic one-dimensional free surface. We prove a rigorous reduction of this system to Birkhoff normal form up to cubic degree. Due to the possible presence of three-wave resonances for general values of gravity, surface tension, and depth, such normal form may be not trivial and exhibit a chaotic dynamics (Wilton ripples). Nevertheless, we prove that for all the values of gravity, surface tension, and depth, initial data that are of size $$ \varepsilon $$in a sufficiently smooth Sobolev space leads to a solution that remains in an $$ \varepsilon $$-ball of the same Sobolev space up times of order $$ \varepsilon ^{-2}$$. We exploit that the three-wave resonances are finitely many, and the Hamiltonian nature of the Birkhoff normal form.

1 aBerti, Massimiliano1 aFeola, Roberto1 aFranzoi, Luca uhttps://doi.org/10.1007/s42286-020-00036-801072nas a2200169 4500008004100000020001400041245006500055210006400120260001500184300001300199490000800212520057200220100002400792700001800816700002100834856004700855 2021 eng d a1432-067300aTraveling Quasi-periodic Water Waves with Constant Vorticity0 aTraveling Quasiperiodic Water Waves with Constant Vorticity c2021/04/01 a99 - 2020 v2403 aWe prove the first bifurcation result of time quasi-periodic traveling wave solutions for space periodic water waves with vorticity. In particular, we prove the existence of small amplitude time quasi-periodic solutions of the gravity-capillary water waves equations with constant vorticity, for a bidimensional fluid over a flat bottom delimited by a space-periodic free interface. These quasi-periodic solutions exist for all the values of depth, gravity and vorticity, and restrict the surface tension to a Borel set of asymptotically full Lebesgue measure.

1 aBerti, Massimiliano1 aFranzoi, Luca1 aMaspero, Alberto uhttps://doi.org/10.1007/s00205-021-01607-w02169nas a2200109 4500008004100000245012900041210006900170520172600239100002401965700002201989856004802011 2017 en d00aAlmost global existence of solutions for capillarity-gravity water waves equations with periodic spatial boundary conditions0 aAlmost global existence of solutions for capillaritygravity wate3 aThe goal of this monograph is to prove that any solution of the Cauchy problem for the capillarity-gravity water waves equations, in one space dimension, with periodic, even in space, initial data of small size ϵ, is almost globally defined in time on Sobolev spaces, i.e. it exists on a time interval of length of magnitude ϵ−N for any N, as soon as the initial data are smooth enough, and the gravity-capillarity parameters are taken outside an exceptional subset of zero measure. In contrast to the many results known for these equations on the real line, with decaying Cauchy data, one cannot make use of dispersive properties of the linear flow. Instead, our method is based on a normal forms procedure, in order to eliminate those contributions to the Sobolev energy that are of lower degree of homogeneity in the solution. Since the water waves equations are a quasi-linear system, usual normal forms approaches would face the well known problem of losses of derivatives in the unbounded transformations. In this monograph, to overcome such a difficulty, after a paralinearization of the capillarity-gravity water waves equations, necessary to obtain energy estimates, and thus local existence of the solutions, we first perform several paradifferential reductions of the equations to obtain a diagonal system with constant coefficients symbols, up to smoothing remainders. Then we may start with a normal form procedure where the small divisors are compensated by the previous paradifferential regularization.The reversible structure of the water waves equations, and the fact that we look for solutions even in x, guarantees a key cancellation which prevents the growth of the Sobolev norms of the solutions.1 aBerti, Massimiliano1 aDelort, Jean-Marc uhttp://preprints.sissa.it/handle/1963/3528501540nas 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/1963/3529601362nas a2200121 4500008004100000245005800041210005800099520096700157100002401124700002101148700002301169856004801192 2016 en d00aLarge KAM tori for perturbations of the dNLS equation0 aLarge KAM tori for perturbations of the dNLS equation3 aWe 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.1 aBerti, Massimiliano1 aKappeler, Thomas1 aMontalto, Riccardo uhttp://preprints.sissa.it/handle/1963/3528400651nas a2200157 4500008004100000245009600041210006900137260005800206300001400264490000600278100001700284700001700301700002200318700002400340856012900364 2016 eng d00aShip Sinkage and Trim Predictions Based on a CAD Interfaced Fully Nonlinear Potential Model0 aShip Sinkage and Trim Predictions Based on a CAD Interfaced Full bInternational Society of Offshore and Polar Engineers a511–5180 v31 aMola, Andrea1 aHeltai, Luca1 aDeSimone, Antonio1 aBerti, Massimiliano uhttps://www.math.sissa.it/publication/ship-sinkage-and-trim-predictions-based-cad-interfaced-fully-nonlinear-potential-model01482nas a2200133 4500008004100000245013000041210007100171260001300242520098000255100002401235700001701259700002101276856005101297 2014 en d00aAn Abstract Nash–Moser Theorem and Quasi-Periodic Solutions for NLW and NLS on Compact Lie Groups and Homogeneous Manifolds0 aAbstract Nash–Moser Theorem and QuasiPeriodic Solutions for NLW bSpringer3 aWe prove an abstract implicit function theorem with parameters for smooth operators defined on scales of sequence spaces, modeled for the search of quasi-periodic solutions of PDEs. The tame estimates required for the inverse linearised operators at each step of the iterative scheme are deduced via a multiscale inductive argument. The Cantor-like set of parameters where the solution exists is defined in a non inductive way. This formulation completely decouples the iterative scheme from the measure theoretical analysis of the parameters where the small divisors non-resonance conditions are verified. As an application, we deduce the existence of quasi-periodic solutions for forced NLW and NLS equations on any compact Lie group or manifold which is homogeneous with respect to a compact Lie group, extending previous results valid only for tori. A basic tool of harmonic analysis is the highest weight theory for the irreducible representations of compact Lie groups.1 aBerti, Massimiliano1 aCorsi, Livia1 aProcesi, Michela uhttp://urania.sissa.it/xmlui/handle/1963/3465101519nas 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 aWe 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/1963/3506700608nas a2200157 4500008004100000245004900041210004900090260001300139300001200152490000800164520016500172100002400337700001700361700002100378856005100399 2014 en d00aKAM for Reversible Derivative Wave Equations0 aKAM for Reversible Derivative Wave Equations bSpringer a905-9550 v2123 aWe prove the existence of Cantor families of small amplitude, analytic, linearly stable quasi-periodic solutions of reversible derivative wave equations.

1 aBerti, Massimiliano1 aBiasco, Luca1 aProcesi, Michela uhttp://urania.sissa.it/xmlui/handle/1963/3464600655nas a2200157 4500008004100000245010000041210006900141260005800210300001400268490000600282100001700288700001700305700002200322700002400344856012900368 2014 eng d00aPotential Model for Ship Hydrodynamics Simulations Directly Interfaced with CAD Data Structures0 aPotential Model for Ship Hydrodynamics Simulations Directly Inte bInternational Society of Offshore and Polar Engineers a815–8220 v41 aMola, Andrea1 aHeltai, Luca1 aDeSimone, Antonio1 aBerti, Massimiliano uhttps://www.math.sissa.it/publication/potential-model-ship-hydrodynamics-simulations-directly-interfaced-cad-data-structures01347nas a2200289 4500008004100000022001300041245008600054210006900140300001200209490000700221520039800228653002600626653002200652653002800674653002500702653001700727653002500744653002200769653002100791653002800812653001900840653001900859100002400878700001700902700002100919856011700940 2013 eng d a1120633000aExistence and stability of quasi-periodic solutions for derivative wave equations0 aExistence and stability of quasiperiodic solutions for derivativ a199-2140 v243 aIn this note we present the new KAM result in [3] which proves the existence of Cantor families of small amplitude, analytic, quasi-periodic solutions of derivative wave equations, with zero Lyapunov exponents and whose linearized equation is reducible to constant coefficients. In turn, this result is derived by an abstract KAM theorem for infinite dimensional reversible dynamical systems*.10aConstant coefficients10aDynamical systems10aExistence and stability10aInfinite dimensional10aKAM for PDEs10aLinearized equations10aLyapunov exponent10aLyapunov methods10aQuasi-periodic solution10aSmall divisors10aWave equations1 aBerti, Massimiliano1 aBiasco, Luca1 aProcesi, Michela uhttps://www.math.sissa.it/publication/existence-and-stability-quasi-periodic-solutions-derivative-wave-equations00759nas a2200157 4500008004100000022001300041245006000054210006000114300001200174490000700186520025600193100002400449700001700473700002100490856009000511 2013 eng d a0012959300aKAM theory for the Hamiltonian derivative wave equation0 aKAM theory for the Hamiltonian derivative wave equation a301-3730 v463 aWe prove an infinite dimensional KAM theorem which implies the existence of Can- tor families of small-amplitude, reducible, elliptic, analytic, invariant tori of Hamiltonian derivative wave equations. © 2013 Société Mathématique de France.

1 aBerti, Massimiliano1 aBiasco, Luca1 aProcesi, Michela uhttps://www.math.sissa.it/publication/kam-theory-hamiltonian-derivative-wave-equation01376nas 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-kdv01345nas a2200145 4500008004100000022001300041245009800054210006900152300001200221490000700233520079700240100002401037700002001061856011801081 2013 eng d a1435985500aQuasi-periodic solutions with Sobolev regularity of NLS on Td with a multiplicative potential0 aQuasiperiodic solutions with Sobolev regularity of NLS on Td wit a229-2860 v153 aWe prove the existence of quasi-periodic solutions for Schrödinger equations with a multiplicative potential on Td , d ≥ 1, finitely differentiable nonlinearities, and tangential frequencies constrained along a pre-assigned direction. The solutions have only Sobolev regularity both in time and space. If the nonlinearity and the potential are C∞ then the solutions are C∞. The proofs are based on an improved Nash-Moser iterative scheme, which assumes the weakest tame estimates for the inverse linearized operators ("Green functions") along scales of Sobolev spaces. The key off-diagonal decay estimates of the Green functions are proved via a new multiscale inductive analysis. The main novelty concerns the measure and "complexity" estimates. © European Mathematical Society 2013.1 aBerti, Massimiliano1 aBolle, Philippe uhttps://www.math.sissa.it/publication/quasi-periodic-solutions-sobolev-regularity-nls-td-multiplicative-potential01296nas a2200145 4500008004100000022001300041245010400054210006900158300001400227490000700241520073600248100002400984700002001008856012201028 2012 eng d a0951771500aSobolev quasi-periodic solutions of multidimensional wave equations with a multiplicative potential0 aSobolev quasiperiodic solutions of multidimensional wave equatio a2579-26130 v253 aWe prove the existence of quasi-periodic solutions for wave equations with a multiplicative potential on T d , d ≥ 1, and finitely differentiable nonlinearities, quasi-periodically forced in time. The only external parameter is the length of the frequency vector. The solutions have Sobolev regularity both in time and space. The proof is based on a Nash-Moser iterative scheme as in [5]. The key tame estimates for the inverse linearized operators are obtained by a multiscale inductive argument, which is more difficult than for NLS due to the dispersion relation of the wave equation. We prove the 'separation properties' of the small divisors assuming weaker non-resonance conditions than in [11]. © 2012 IOP Publishing Ltd.1 aBerti, Massimiliano1 aBolle, Philippe uhttps://www.math.sissa.it/publication/sobolev-quasi-periodic-solutions-multidimensional-wave-equations-multiplicative01306nas a2200145 4500008004100000022001300041245007600054210006900130300001200199490000800211520079500219100002401014700001701038856010501055 2011 eng d a0010361600aBranching of Cantor Manifolds of Elliptic Tori and Applications to PDEs0 aBranching of Cantor Manifolds of Elliptic Tori and Applications a741-7960 v3053 aWe consider infinite dimensional Hamiltonian systems. We prove the existence of "Cantor manifolds" of elliptic tori-of any finite higher dimension-accumulating on a given elliptic KAM torus. Then, close to an elliptic equilibrium, we show the existence of Cantor manifolds of elliptic tori which are "branching" points of other Cantor manifolds of higher dimensional tori. We also answer to a conjecture of Bourgain, proving the existence of invariant elliptic tori with tangential frequency along a pre-assigned direction. The proofs are based on an improved KAM theorem. Its main advantages are an explicit characterization of the Cantor set of parameters and weaker smallness conditions on the perturbation. We apply these results to the nonlinear wave equation. © 2011 Springer-Verlag.1 aBerti, Massimiliano1 aBiasco, Luca uhttps://www.math.sissa.it/publication/branching-cantor-manifolds-elliptic-tori-and-applications-pdes00919nas a2200157 4500008004100000022001300041245006100054210006100115300001400176490000800190520040200198100001900600700002400619700002300643856009500666 2011 eng d a0022039600aDegenerate KAM theory for partial differential equations0 aDegenerate KAM theory for partial differential equations a3379-33970 v2503 aThis paper deals with degenerate KAM theory for lower dimensional elliptic tori of infinite dimensional Hamiltonian systems, depending on one parameter only. We assume that the linear frequencies are analytic functions of the parameter, satisfy a weak non-degeneracy condition of Rüssmann type and an asymptotic behavior. An application to nonlinear wave equations is given. © 2010 Elsevier Inc.1 aBambusi, Dario1 aBerti, Massimiliano1 aMagistrelli, Elena uhttps://www.math.sissa.it/publication/degenerate-kam-theory-partial-differential-equations01454nas a2200145 4500008004100000022001400041245009100055210007000146260000900216490000800225520090000233100002401133700002101157856013001178 2011 eng d a0012-709400aNonlinear wave and Schrödinger equations on compact Lie groups and homogeneous spaces0 aNonlinear wave and Schrödinger equations on compact Lie groups a c20110 v1593 aWe develop linear and nonlinear harmonic analysis on compact Lie groups and homogeneous spaces relevant for the theory of evolutionary Hamiltonian PDEs. A basic tool is the theory of the highest weight for irreducible representations of compact Lie groups. This theory provides an accurate description of the eigenvalues of the Laplace-Beltrami operator as well as the multiplication rules of its eigenfunctions. As an application, we prove the existence of Cantor families of small amplitude time-periodic solutions for wave and Schr¨odinger equations with differentiable nonlinearities. We apply an abstract Nash-Moser implicit function theorem to overcome the small divisors problem produced by the degenerate eigenvalues of the Laplace operator. We provide a new algebraic framework to prove the key tame estimates for the inverse linearized operators on Banach scales of Sobolev functions.1 aBerti, Massimiliano1 aProcesi, Michela uhttps://www.math.sissa.it/publication/nonlinear-wave-and-schr%C3%B6dinger-equations-compact-lie-groups-and-homogeneous-spaces02010nas a2200385 4500008004100000022001300041245007600054210006900130300001200199490000700211520082800218653001601046653002101062653002301083653002101106653003001127653001901157653002201176653002501198653002701223653001801250653001801268653002401286653002801310653002201338653002201360653002401382653001901406653001201425653001901437100002401456700002001480700002101500856010301521 2010 eng d a0294144900aAn abstract Nash-Moser theorem with parameters and applications to PDEs0 aabstract NashMoser theorem with parameters and applications to P a377-3990 v273 aWe prove an abstract Nash-Moser implicit function theorem with parameters which covers the applications to the existence of finite dimensional, differentiable, invariant tori of Hamiltonian PDEs with merely differentiable nonlinearities. The main new feature of the abstract iterative scheme is that the linearized operators, in a neighborhood of the expected solution, are invertible, and satisfy the "tame" estimates, only for proper subsets of the parameters. As an application we show the existence of periodic solutions of nonlinear wave equations on Riemannian Zoll manifolds. A point of interest is that, in presence of possibly very large "clusters of small divisors", due to resonance phenomena, it is more natural to expect solutions with only Sobolev regularity. © 2009 Elsevier Masson SAS. All rights reserved.10aAbstracting10aAircraft engines10aFinite dimensional10aHamiltonian PDEs10aImplicit function theorem10aInvariant tori10aIterative schemes10aLinearized operators10aMathematical operators10aMoser theorem10aNon-Linearity10aNonlinear equations10aNonlinear wave equation10aPeriodic solution10aPoint of interest10aResonance phenomena10aSmall divisors10aSobolev10aWave equations1 aBerti, Massimiliano1 aBolle, Philippe1 aProcesi, Michela uhttps://www.math.sissa.it/publication/abstract-nash-moser-theorem-parameters-and-applications-pdes01245nas a2200145 4500008004100000022001300041245008800054210006900142300001200211490000800223520070400231100002400935700002000959856012000979 2010 eng d a0003952700aSobolev periodic solutions of nonlinear wave equations in higher spatial dimensions0 aSobolev periodic solutions of nonlinear wave equations in higher a609-6420 v1953 aWe prove the existence of Cantor families of periodic solutions for nonlinear wave equations in higher spatial dimensions with periodic boundary conditions. We study both forced and autonomous PDEs. In the latter case our theorems generalize previous results of Bourgain to more general nonlinearities of class C k and assuming weaker non-resonance conditions. Our solutions have Sobolev regularity both in time and space. The proofs are based on a differentiable Nash-Moser iteration scheme, where it is sufficient to get estimates of interpolation-type for the inverse linearized operators. Our approach works also in presence of very large "clusters of small divisors". © Springer-Verlag (2009).1 aBerti, Massimiliano1 aBolle, Philippe uhttps://www.math.sissa.it/publication/sobolev-periodic-solutions-nonlinear-wave-equations-higher-spatial-dimensions00787nas a2200145 4500008004100000022001300041245008100054210006900135300001200204490000600216520026300222100002400485700002000509856011200529 2008 eng d a1673345200aCantor families of periodic solutions for completely resonant wave equations0 aCantor families of periodic solutions for completely resonant wa a151-1650 v33 aWe present recent existence results of Cantor families of small amplitude periodic solutions for completely resonant nonlinear wave equations. The proofs rely on the Nash-Moser implicit function theory and variational methods. © 2008 Higher Education Press.1 aBerti, Massimiliano1 aBolle, Philippe uhttps://www.math.sissa.it/publication/cantor-families-periodic-solutions-completely-resonant-wave-equations01284nas a2200145 4500008004100000022001300041245008900054210006900143300001400212490000800226520074600234100002400980700002001004856011401024 2008 eng d a0001870800aCantor families of periodic solutions for wave equations via a variational principle0 aCantor families of periodic solutions for wave equations via a v a1671-17270 v2173 aWe prove existence of small amplitude periodic solutions of completely resonant wave equations with frequencies in a Cantor set of asymptotically full measure, via a variational principle. A Lyapunov-Schmidt decomposition reduces the problem to a finite dimensional bifurcation equation-variational in nature-defined on a Cantor set of non-resonant parameters. The Cantor gaps are due to "small divisors" phenomena. To solve the bifurcation equation we develop a suitable variational method. In particular, we do not require the typical "Arnold non-degeneracy condition" of the known theory on the nonlinear terms. As a consequence our existence results hold for new generic sets of nonlinearities. © 2007 Elsevier Inc. All rights reserved.1 aBerti, Massimiliano1 aBolle, Philippe uhttps://www.math.sissa.it/publication/cantor-families-periodic-solutions-wave-equations-variational-principle00827nas a2200145 4500008004100000022001300041245008400054210006900138300001200207490000700219520030000226100002400526700002000550856011100570 2008 eng d a1021972200aCantor families of periodic solutions of wave equations with C k nonlinearities0 aCantor families of periodic solutions of wave equations with C k a247-2760 v153 aWe prove bifurcation of Cantor families of periodic solutions for wave equations with nonlinearities of class C k . It requires a modified Nash-Moser iteration scheme with interpolation estimates for the inverse of the linearized operators and for the composition operators. © 2008 Birkhaueser.1 aBerti, Massimiliano1 aBolle, Philippe uhttps://www.math.sissa.it/publication/cantor-families-periodic-solutions-wave-equations-c-k-nonlinearities00637nas 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/1963/264300666nas a2200157 4500008004100000022001300041245006000054210005700114300001200171490000600183520016900189100002400358700001400382700002200396856009000418 2008 eng d a1534039200aOn periodic elliptic equations with gradient dependence0 aperiodic elliptic equations with gradient dependence a601-6150 v73 aWe construct entire solutions of Δu = f(x, u, ∇u) which are superpositions of odd, periodic functions and linear ones, with prescribed integer or rational slope.1 aBerti, Massimiliano1 aMatzeu, M1 aValdinoci, Enrico uhttps://www.math.sissa.it/publication/periodic-elliptic-equations-gradient-dependence01563nas a2200133 4500008004100000020001800041022001300059245004500072210004500117300001200162520115200174100002401326856007901350 2008 eng d a9781402069628 a1874650000aVariational methods for Hamiltonian PDEs0 aVariational methods for Hamiltonian PDEs a391-4203 aWe present recent existence results of periodic solutions for completely resonant nonlinear wave equations in which both "small divisor" difficulties and infinite dimensional bifurcation phenomena occur. These results can be seen as generalizations of the classical finite-dimensional resonant center theorems of Weinstein-Moser and Fadell-Rabinowitz. The proofs are based on variational bifurcation theory: after a Lyapunov-Schmidt reduction, the small divisor problem in the range equation is overcome with a Nash-Moser implicit function theorem for a Cantor set of non-resonant parameters. Next, the infinite dimensional bifurcation equation, variational in nature, possesses minimax mountain-pass critical points. The big difficulty is to ensure that they are not in the "Cantor gaps". This is proved under weak non-degeneracy conditions. Finally, we also discuss the existence of forced vibrations with rational frequency. This problem requires variational methods of a completely different nature, such as constrained minimization and a priori estimates derivable from variational inequalities. © 2008 Springer Science + Business Media B.V.1 aBerti, Massimiliano uhttps://www.math.sissa.it/publication/variational-methods-hamiltonian-pdes00957nas a2200109 4500008004300000245006000043210005600103520060900159100001900768700002400787856003600811 2006 en_Ud 00aA Birkhoff-Lewis-Type Theorem for Some Hamiltonian PDEs0 aBirkhoffLewisType Theorem for Some Hamiltonian PDEs3 aIn this paper we give an extension of the Birkhoff--Lewis theorem to some semilinear PDEs. Accordingly we prove existence of infinitely many periodic orbits with large period accumulating at the origin. Such periodic orbits bifurcate from resonant finite dimensional invariant tori of the fourth order normal form of the system. Besides standard nonresonance and nondegeneracy assumptions, our main result is obtained assuming a regularizing property of the nonlinearity. We apply our main theorem to a semilinear beam equation and to a nonlinear Schr\\\\\\\"odinger equation with smoothing nonlinearity.1 aBambusi, Dario1 aBerti, Massimiliano uhttp://hdl.handle.net/1963/215901190nas a2200109 4500008004300000245009100043210006900134520079700203100002401000700002001024856003601044 2006 en_Ud 00aCantor families of periodic solutions for completely resonant nonlinear wave equations0 aCantor families of periodic solutions for completely resonant no3 aWe prove the existence of small amplitude, $2\\\\pi \\\\slash \\\\om$-periodic in time solutions of completely resonant nonlinear wave equations with Dirichlet boundary conditions, for any frequency $ \\\\om $ belonging to a Cantor-like set of positive measure and for a new set of nonlinearities. The proof relies on a suitable Lyapunov-Schmidt decomposition and a variant of the Nash-Moser Implicit Function Theorem. In spite of the complete resonance of the equation we show that we can still reduce the problem to a {\\\\it finite} dimensional bifurcation equation. Moreover, a new simple approach for the inversion of the linearized operators required by the Nash-Moser scheme is developed. It allows to deal also with nonlinearities which are not odd and with finite spatial regularity.1 aBerti, Massimiliano1 aBolle, Philippe uhttp://hdl.handle.net/1963/216100868nas a2200109 4500008004300000245007300043210006900116520049600185100002400681700001700705856003600722 2006 en_Ud 00aForced vibrations of wave equations with non-monotone nonlinearities0 aForced vibrations of wave equations with nonmonotone nonlinearit3 aWe prove existence and regularity of periodic in time solutions of completely resonant nonlinear forced wave equations with Dirichlet boundary conditions for a large class of non-monotone forcing terms. Our approach is based on a variational Lyapunov-Schmidt reduction. It turns out that the infinite dimensional bifurcation equation exhibits an intrinsic lack of compactness. We solve it via a minimization argument and a-priori estimate methods inspired to regularity theory of Rabinowitz.1 aBerti, Massimiliano1 aBiasco, Luca uhttp://hdl.handle.net/1963/216000834nas 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-000676nas a2200109 4500008004300000245007400043210006900117520029900186100002400485700002100509856003600530 2006 en_Ud 00aQuasi-periodic solutions of completely resonant forced wave equations0 aQuasiperiodic solutions of completely resonant forced wave equat3 aWe prove existence of quasi-periodic solutions with two frequencies of completely resonant, periodically forced nonlinear wave equations with periodic spatial boundary conditions. We consider both the cases the forcing frequency is: (Case A) a rational number and (Case B) an irrational number.1 aBerti, Massimiliano1 aProcesi, Michela uhttp://hdl.handle.net/1963/223400415nas a2200109 4500008004100000245008300041210006900124260003500193100002400228700001700252856003600269 2005 en d00aPeriodic solutions of nonlinear wave equations with non-monotone forcing terms0 aPeriodic solutions of nonlinear wave equations with nonmonotone bAccademia Nazionale dei Lincei1 aBerti, Massimiliano1 aBiasco, Luca uhttp://hdl.handle.net/1963/458100410nas a2200109 4500008004100000245007400041210006900115260003500184100002400219700002100243856003600264 2005 en d00aQuasi-periodic oscillations for wave equations under periodic forcing0 aQuasiperiodic oscillations for wave equations under periodic for bAccademia Nazionale dei Lincei1 aBerti, Massimiliano1 aProcesi, Michela uhttp://hdl.handle.net/1963/458300778nas a2200109 4500008004300000245007400043210006900117520040200186100002400588700002000612856003600632 2004 en_Ud 00aBifurcation of free vibrations for completely resonant wave equations0 aBifurcation of free vibrations for completely resonant wave equa3 aWe prove existence of small amplitude, 2 pi/omega -periodic in time solutions of completely resonant nonlinear wave equations with Dirichlet boundary conditions for any frequency omega belonging to a Cantor-like set of positive measure and for a generic set of nonlinearities. The proof relies on a suitable Lyapunov-Schmidt decomposition and a variant of the Nash-Moser Implicit Function Theorem.1 aBerti, Massimiliano1 aBolle, Philippe uhttp://hdl.handle.net/1963/224500380nas a2200109 4500008004300000245006700043210006700110260001300177100002400190700002000214856003600234 2004 en_Ud 00aMultiplicity of periodic solutions of nonlinear wave equations0 aMultiplicity of periodic solutions of nonlinear wave equations bElsevier1 aBerti, Massimiliano1 aBolle, Philippe uhttp://hdl.handle.net/1963/297401171nas a2200133 4500008004300000245008600043210006900129260003700198520070300235100002400938700001700962700002200979856003601001 2004 en_Ud 00aPeriodic orbits close to elliptic tori and applications to the three-body problem0 aPeriodic orbits close to elliptic tori and applications to the t bScuola Normale Superiore di Pisa3 aWe prove, under suitable non-resonance and non-degeneracy ``twist\\\'\\\' conditions, a Birkhoff-Lewis type result showing the existence of infinitely many periodic solutions, with larger and larger minimal period, accumulating onto elliptic invariant tori (of Hamiltonian systems). We prove the applicability of this result to the spatial planetary three-body problem in the small eccentricity-inclination regime. Furthermore, we find other periodic orbits under some restrictions on the period and the masses of the ``planets\\\'\\\'. The proofs are based on averaging theory, KAM theory and variational methods. (Supported by M.U.R.S.T. Variational Methods and Nonlinear Differential Equations.)1 aBerti, Massimiliano1 aBiasco, Luca1 aValdinoci, Enrico uhttp://hdl.handle.net/1963/298500322nas a2200097 4500008004100000245004600041210004600087260003100133100002400164856003600188 2004 en d00aSoluzioni periodiche di PDEs Hamiltoniane0 aSoluzioni periodiche di PDEs Hamiltoniane bUnione Matematica Italiana1 aBerti, Massimiliano uhttp://hdl.handle.net/1963/458201027nas a2200133 4500008004300000245008200043210006900125260001300194520058900207100002400796700001700820700002000837856003600857 2003 en_Ud 00aDrift in phase space: a new variational mechanism with optimal diffusion time0 aDrift in phase space a new variational mechanism with optimal di bElsevier3 aWe consider non-isochronous, nearly integrable, a-priori unstable Hamiltonian systems with a (trigonometric polynomial) $O(\\\\mu)$-perturbation which does not preserve the unperturbed tori. We prove the existence of Arnold diffusion with diffusion time $ T_d = O((1/ \\\\mu) \\\\log (1/ \\\\mu))$ by a variational method which does not require the existence of ``transition chains of tori\\\'\\\' provided by KAM theory. We also prove that our estimate of the diffusion time $T_d $ is optimal as a consequence of a general stability result derived from classical perturbation theory.1 aBerti, Massimiliano1 aBiasco, Luca1 aBolle, Philippe uhttp://hdl.handle.net/1963/302000397nas a2200109 4500008004100000245007900041210006900120260001800189100002400207700002000231856003600251 2003 en d00aPeriodic solutions of nonlinear wave equations with general nonlinearities0 aPeriodic solutions of nonlinear wave equations with general nonl bSISSA Library1 aBerti, Massimiliano1 aBolle, Philippe uhttp://hdl.handle.net/1963/164800800nas a2200109 4500008004100000245005300041210005200094260003600146520039600182100002400578856008800602 2002 en d00aArnold diffusion: a functional analysis approach0 aArnold diffusion a functional analysis approach bNatsīonal. Akad. Nauk Ukraïni3 aWe present, in the context of nearly integrable Hamiltonian systems, a functional analysis approach to study the “splitting of the whiskers” and the “shadowing problem” developed in collaboration with P. Bolle in the recent papers [1] and [2] . This method is applied to the problem of Arnold diffusion for nearly integrable partially isochronous systems improving known results.1 aBerti, Massimiliano uhttps://www.math.sissa.it/publication/arnold-diffusion-functional-analysis-approach00397nas a2200109 4500008004100000245008300041210006900124260001300193100002400206700002100230856003600251 2002 en d00aChaotic dynamics for perturbations of infinite-dimensional Hamiltonian systems0 aChaotic dynamics for perturbations of infinitedimensional Hamilt bElsevier1 aBerti, Massimiliano1 aCarminati, Carlo uhttp://hdl.handle.net/1963/127900795nas a2200121 4500008004300000245006000043210006000103260004800163520038200211100002400593700002000617856003600637 2002 en_Ud 00aFast Arnold diffusion in systems with three time scales0 aFast Arnold diffusion in systems with three time scales bAmerican Institute of Mathematical Sciences3 aWe consider the problem of Arnold Diffusion for nearly integrable partially isochronous Hamiltonian systems with three time scales. By means of a careful shadowing analysis, based on a variational technique, we prove that, along special directions, Arnold diffusion takes place with fast (polynomial) speed, even though the \\\"splitting determinant\\\" is exponentially small.1 aBerti, Massimiliano1 aBolle, Philippe uhttp://hdl.handle.net/1963/305800697nas a2200121 4500008004300000245005500043210005300098260001300151520033100164100002400495700002000519856003600539 2002 en_Ud 00aA functional analysis approach to Arnold diffusion0 afunctional analysis approach to Arnold diffusion bElsevier3 aWe discuss in the context of nearly integrable Hamiltonian systems a functional analysis approach to the \\\"splitting of separatrices\\\" and to the \\\"shadowing problem\\\". As an application we apply our method to the problem of Arnold Diffusion for nearly integrable partially isochronous systems improving known results.1 aBerti, Massimiliano1 aBolle, Philippe uhttp://hdl.handle.net/1963/315100423nas a2200121 4500008004100000245007600041210006900117260001800186100001700204700002400221700002000245856003600265 2002 en d00aAn optimal fast-diffusion variational method for non isochronous system0 aoptimal fastdiffusion variational method for non isochronous sys bSISSA Library1 aBiasco, Luca1 aBerti, Massimiliano1 aBolle, Philippe uhttp://hdl.handle.net/1963/157900446nas a2200121 4500008004100000245009900041210006900140260001800209100002400227700001700251700002000268856003600288 2002 en d00aOptimal stability and instability results for a class of nearly integrable Hamiltonian systems0 aOptimal stability and instability results for a class of nearly bSISSA Library1 aBerti, Massimiliano1 aBiasco, Luca1 aBolle, Philippe uhttp://hdl.handle.net/1963/159600389nas a2200109 4500008004100000245007400041210006900115260001300184100002400197700002200221856003600243 2001 en d00aNon-compactness and multiplicity results for the Yamabe problem on Sn0 aNoncompactness and multiplicity results for the Yamabe problem o bElsevier1 aBerti, Massimiliano1 aMalchiodi, Andrea uhttp://hdl.handle.net/1963/134500394nas a2200109 4500008004100000245007600041210006900117260001800186100002400204700002000228856003600248 2000 en d00aArnold's Diffusion in nearly integrable isochronous Hamiltonian systems0 aArnolds Diffusion in nearly integrable isochronous Hamiltonian s bSISSA Library1 aBerti, Massimiliano1 aBolle, Philippe uhttp://hdl.handle.net/1963/155400389nas a2200109 4500008004100000245007100041210006900112260001800181100002400199700002000223856003600243 2000 en d00aDiffusion time and splitting of separatrices for nearly integrable0 aDiffusion time and splitting of separatrices for nearly integrab bSISSA Library1 aBerti, Massimiliano1 aBolle, Philippe uhttp://hdl.handle.net/1963/1547