We 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.

}, isbn = {2523-3688}, url = {https://doi.org/10.1007/s42286-020-00036-8}, author = {Massimiliano Berti and Roberto Feola and Luca Franzoi} } @article {2021, title = {Traveling Quasi-periodic Water Waves with Constant Vorticity}, volume = {240}, year = {2021}, month = {2021/04/01}, pages = {99 - 202}, abstract = {We 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.

}, isbn = {1432-0673}, url = {https://doi.org/10.1007/s00205-021-01607-w}, author = {Massimiliano Berti and Luca Franzoi and Alberto Maspero} } @article {2017, title = {Almost global existence of solutions for capillarity-gravity water waves equations with periodic spatial boundary conditions}, number = {arXiv;1702.04674}, year = {2017}, abstract = {The 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.}, url = {http://preprints.sissa.it/handle/1963/35285}, author = {Massimiliano Berti and Jean-Marc Delort} } @article {2017, title = {Time quasi-periodic gravity water waves in finite depth}, number = {arXiv;1708.01517}, year = {2017}, abstract = {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.}, url = {http://preprints.sissa.it/handle/1963/35296}, author = {P Baldi and Massimiliano Berti and Emanuele Haus and Riccardo Montalto} } @article {2016, title = {Large KAM tori for perturbations of the dNLS equation}, number = {arXiv;1603.09252}, year = {2016}, abstract = {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{\texttimes}2 block diagonal ones with constant coefficients together with sharp asymptotic estimates of their eigenvalues.}, url = {http://preprints.sissa.it/handle/1963/35284}, author = {Massimiliano Berti and Thomas Kappeler and Riccardo Montalto} } @conference {mola2016ship, title = {Ship Sinkage and Trim Predictions Based on a CAD Interfaced Fully Nonlinear Potential Model}, booktitle = {The 26th International Ocean and Polar Engineering Conference}, volume = {3}, year = {2016}, pages = {511{\textendash}518}, publisher = {International Society of Offshore and Polar Engineers}, organization = {International Society of Offshore and Polar Engineers}, author = {Andrea Mola and Luca Heltai and Antonio DeSimone and Massimiliano Berti} } @article {2014, title = {An Abstract Nash{\textendash}Moser Theorem and Quasi-Periodic Solutions for NLW and NLS on Compact Lie Groups and Homogeneous Manifolds}, number = {Communications in mathematical physics;volume 334; issue 3; pages 1413-1454;}, year = {2014}, publisher = {Springer}, abstract = {We 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.}, doi = {10.1007/s00220-014-2128-4}, url = {http://urania.sissa.it/xmlui/handle/1963/34651}, author = {Massimiliano Berti and Livia Corsi and Michela Procesi} } @article {Baldi20141, title = {KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation}, journal = {Mathematische Annalen}, year = {2014}, note = {cited By (since 1996)0; Article in Press}, pages = {1-66}, abstract = {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. {\textcopyright} 2014 Springer-Verlag Berlin Heidelberg.}, issn = {00255831}, doi = {10.1007/s00208-013-1001-7}, author = {P Baldi and Massimiliano Berti and Riccardo Montalto} } @article {2014, title = {KAM for quasi-linear KdV}, journal = {C. R. Math. Acad. Sci. Paris}, volume = {352}, number = {Comptes Rendus Mathematique;volume 352; issue 7-8; pages 603-607;}, year = {2014}, pages = {603-607}, publisher = {Elsevier}, abstract = {We prove the existence and stability of Cantor families of quasi-periodic, small-amplitude solutions of quasi-linear autonomous Hamiltonian perturbations of KdV.

}, doi = {10.1016/j.crma.2014.04.012}, url = {http://urania.sissa.it/xmlui/handle/1963/35067}, author = {P Baldi and Massimiliano Berti and Riccardo Montalto} } @article {2014, title = {KAM for Reversible Derivative Wave Equations}, journal = {Arch. Ration. Mech. Anal.}, volume = {212}, number = {Archive for rational mechanics and analysis;volume 212; issue 3; pages 905-955;}, year = {2014}, pages = {905-955}, publisher = {Springer}, abstract = {We prove the existence of Cantor families of small amplitude, analytic, linearly stable quasi-periodic solutions of reversible derivative wave equations.

}, doi = {10.1007/s00205-014-0726-0}, url = {http://urania.sissa.it/xmlui/handle/1963/34646}, author = {Massimiliano Berti and Luca Biasco and Michela Procesi} } @conference {mola2014ship, title = {Potential Model for Ship Hydrodynamics Simulations Directly Interfaced with CAD Data Structures}, booktitle = {The 24th International Ocean and Polar Engineering Conference}, volume = {4}, year = {2014}, pages = {815{\textendash}822}, publisher = {International Society of Offshore and Polar Engineers}, organization = {International Society of Offshore and Polar Engineers}, author = {Andrea Mola and Luca Heltai and Antonio DeSimone and Massimiliano Berti} } @article {Berti2013199, title = {Existence and stability of quasi-periodic solutions for derivative wave equations}, journal = {Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica e Applicazioni}, volume = {24}, number = {2}, year = {2013}, note = {cited By (since 1996)0}, pages = {199-214}, abstract = {In 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*.}, keywords = {Constant coefficients, Dynamical systems, Existence and stability, Infinite dimensional, KAM for PDEs, Linearized equations, Lyapunov exponent, Lyapunov methods, Quasi-periodic solution, Small divisors, Wave equations}, issn = {11206330}, doi = {10.4171/RLM/652}, author = {Massimiliano Berti and Luca Biasco and Michela Procesi} } @article {Berti2013301, title = {KAM theory for the Hamiltonian derivative wave equation}, journal = {Annales Scientifiques de l{\textquoteright}Ecole Normale Superieure}, volume = {46}, number = {2}, year = {2013}, note = {cited By (since 1996)4}, pages = {301-373}, abstract = {We 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. {\textcopyright} 2013 Soci{\'e}t{\'e} Math{\'e}matique de France.

}, issn = {00129593}, author = {Massimiliano Berti and Luca Biasco and Michela Procesi} } @article {2013, title = {A note on KAM theory for quasi-linear and fully nonlinear forced KdV}, journal = {Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 24 (2013), no. 3: 437{\textendash}450}, year = {2013}, publisher = {European Mathematical Society}, abstract = {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.}, keywords = {KAM for PDEs}, doi = {10.4171/RLM/660}, author = {P Baldi and Massimiliano Berti and Riccardo Montalto} } @article {Berti2013229, title = {Quasi-periodic solutions with Sobolev regularity of NLS on Td with a multiplicative potential}, journal = {Journal of the European Mathematical Society}, volume = {15}, number = {1}, year = {2013}, note = {cited By (since 1996)5}, pages = {229-286}, abstract = {We prove the existence of quasi-periodic solutions for Schr{\"o}dinger 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$\infty$ then the solutions are C$\infty$. 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. {\textcopyright} European Mathematical Society 2013.}, issn = {14359855}, doi = {10.4171/JEMS/361}, author = {Massimiliano Berti and Philippe Bolle} } @article {Berti20122579, title = {Sobolev quasi-periodic solutions of multidimensional wave equations with a multiplicative potential}, journal = {Nonlinearity}, volume = {25}, number = {9}, year = {2012}, note = {cited By (since 1996)3}, pages = {2579-2613}, abstract = {We 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 {\textquoteright}separation properties{\textquoteright} of the small divisors assuming weaker non-resonance conditions than in [11]. {\textcopyright} 2012 IOP Publishing Ltd.}, issn = {09517715}, doi = {10.1088/0951-7715/25/9/2579}, author = {Massimiliano Berti and Philippe Bolle} } @article {Berti2011741, title = {Branching of Cantor Manifolds of Elliptic Tori and Applications to PDEs}, journal = {Communications in Mathematical Physics}, volume = {305}, number = {3}, year = {2011}, note = {cited By (since 1996)8}, pages = {741-796}, abstract = {We 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. {\textcopyright} 2011 Springer-Verlag.}, issn = {00103616}, doi = {10.1007/s00220-011-1264-3}, author = {Massimiliano Berti and Luca Biasco} } @article {Bambusi20113379, title = {Degenerate KAM theory for partial differential equations}, journal = {Journal of Differential Equations}, volume = {250}, number = {8}, year = {2011}, note = {cited By (since 1996)3}, pages = {3379-3397}, abstract = {This 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 R{\"u}ssmann type and an asymptotic behavior. An application to nonlinear wave equations is given. {\textcopyright} 2010 Elsevier Inc.}, issn = {00220396}, doi = {10.1016/j.jde.2010.11.002}, author = {Dario Bambusi and Massimiliano Berti and Elena Magistrelli} } @article {2011, title = {Nonlinear wave and Schr{\"o}dinger equations on compact Lie groups and homogeneous spaces}, journal = {Duke Mathematical Journal}, volume = {159}, year = {2011}, month = {2011}, chapter = {479}, abstract = {We 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{\textasciidieresis}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.}, issn = {0012-7094}, doi = {10.1215/00127094-1433403}, author = {Massimiliano Berti and Michela Procesi} } @article {Berti2010377, title = {An abstract Nash-Moser theorem with parameters and applications to PDEs}, journal = {Annales de l{\textquoteright}Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis}, volume = {27}, number = {1}, year = {2010}, note = {cited By (since 1996)9}, pages = {377-399}, abstract = {We 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. {\textcopyright} 2009 Elsevier Masson SAS. All rights reserved.}, keywords = {Abstracting, Aircraft engines, Finite dimensional, Hamiltonian PDEs, Implicit function theorem, Invariant tori, Iterative schemes, Linearized operators, Mathematical operators, Moser theorem, Non-Linearity, Nonlinear equations, Nonlinear wave equation, Periodic solution, Point of interest, Resonance phenomena, Small divisors, Sobolev, Wave equations}, issn = {02941449}, doi = {10.1016/j.anihpc.2009.11.010}, author = {Massimiliano Berti and Philippe Bolle and Michela Procesi} } @article {Berti2009609, title = {Sobolev periodic solutions of nonlinear wave equations in higher spatial dimensions}, journal = {Archive for Rational Mechanics and Analysis}, volume = {195}, number = {2}, year = {2010}, note = {cited By (since 1996)6}, pages = {609-642}, abstract = {We 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". {\textcopyright} Springer-Verlag (2009).}, issn = {00039527}, doi = {10.1007/s00205-008-0211-8}, author = {Massimiliano Berti and Philippe Bolle} } @article {Berti2008151, title = {Cantor families of periodic solutions for completely resonant wave equations}, journal = {Frontiers of Mathematics in China}, volume = {3}, number = {2}, year = {2008}, note = {cited By (since 1996)0}, pages = {151-165}, abstract = {We 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. {\textcopyright} 2008 Higher Education Press.}, issn = {16733452}, doi = {10.1007/s11464-008-0011-3}, author = {Massimiliano Berti and Philippe Bolle} } @article {Berti20081671, title = {Cantor families of periodic solutions for wave equations via a variational principle}, journal = {Advances in Mathematics}, volume = {217}, number = {4}, year = {2008}, note = {cited By (since 1996)6}, pages = {1671-1727}, abstract = {We 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. {\textcopyright} 2007 Elsevier Inc. All rights reserved.}, issn = {00018708}, doi = {10.1016/j.aim.2007.11.004}, author = {Massimiliano Berti and Philippe Bolle} } @article {Berti2008247, title = {Cantor families of periodic solutions of wave equations with C k nonlinearities}, journal = {Nonlinear Differential Equations and Applications}, volume = {15}, number = {1-2}, year = {2008}, note = {cited By (since 1996)10}, pages = {247-276}, abstract = {We 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. {\textcopyright} 2008 Birkhaueser.}, issn = {10219722}, doi = {10.1007/s00030-007-7025-5}, author = {Massimiliano Berti and Philippe Bolle} } @article {2008, title = {Forced Vibrations of a Nonhomogeneous String}, journal = {SIAM J. Math. Anal. 40 (2008) 382-412}, number = {SISSA;36/2006/M}, year = {2008}, abstract = {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.}, doi = {10.1137/060665038}, url = {http://hdl.handle.net/1963/2643}, author = {P Baldi and Massimiliano Berti} } @article {Berti2008601, title = {On periodic elliptic equations with gradient dependence}, journal = {Communications on Pure and Applied Analysis}, volume = {7}, number = {3}, year = {2008}, note = {cited By (since 1996)1}, pages = {601-615}, abstract = {We 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.}, issn = {15340392}, author = {Massimiliano Berti and Matzeu, M and Enrico Valdinoci} } @article {Berti2008391, title = {Variational methods for Hamiltonian PDEs}, journal = {NATO Science for Peace and Security Series B: Physics and Biophysics}, year = {2008}, note = {cited By (since 1996)0}, pages = {391-420}, abstract = {We 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. {\textcopyright} 2008 Springer Science + Business Media B.V.}, isbn = {9781402069628}, issn = {18746500}, doi = {10.1007/978-1-4020-6964-2-16}, author = {Massimiliano Berti} } @article {2006, title = {A Birkhoff-Lewis-Type Theorem for Some Hamiltonian PDEs}, journal = {SIAM J. Math. Anal. 37 (2006) 83-102}, number = {arXiv.org;math/0310182v1}, year = {2006}, abstract = {In 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.}, doi = {10.1137/S0036141003436107}, url = {http://hdl.handle.net/1963/2159}, author = {Dario Bambusi and Massimiliano Berti} } @article {2006, title = {Cantor families of periodic solutions for completely resonant nonlinear wave equations}, journal = {Duke Math. J. 134 (2006) 359-419}, number = {arXiv.org;math/0410618v1}, year = {2006}, abstract = {We 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.}, doi = {10.1215/S0012-7094-06-13424-5}, url = {http://hdl.handle.net/1963/2161}, author = {Massimiliano Berti and Philippe Bolle} } @article {2006, title = {Forced vibrations of wave equations with non-monotone nonlinearities}, journal = {Ann. Inst. H. Poincar{\'e} Anal. Non Lin{\'e}aire 23 (2006) 439-474}, number = {arXiv.org;math/0410619v1}, year = {2006}, abstract = {We 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.}, doi = {10.1016/j.anihpc.2005.05.004}, url = {http://hdl.handle.net/1963/2160}, author = {Massimiliano Berti and Luca Biasco} } @article {Baldi2006257, title = {Periodic solutions of nonlinear wave equations for asymptotically full measure sets of frequencies}, journal = {Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica e Applicazioni}, volume = {17}, number = {3}, year = {2006}, note = {cited By (since 1996)5}, pages = {257-277}, abstract = {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.}, issn = {11206330}, author = {P Baldi and Massimiliano Berti} } @article {2006, title = {Quasi-periodic solutions of completely resonant forced wave equations}, journal = {Comm. Partial Differential Equations 31 (2006) 959 - 985}, number = {SISSA;106/2004/M}, year = {2006}, abstract = {We 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.}, doi = {10.1080/03605300500358129}, url = {http://hdl.handle.net/1963/2234}, author = {Massimiliano Berti and Michela Procesi} } @article {2005, title = {Periodic solutions of nonlinear wave equations with non-monotone forcing terms}, journal = {Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 16 (2005), no. 2, 117-124}, year = {2005}, publisher = {Accademia Nazionale dei Lincei}, url = {http://hdl.handle.net/1963/4581}, author = {Massimiliano Berti and Luca Biasco} } @article {2005, title = {Quasi-periodic oscillations for wave equations under periodic forcing}, journal = {Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 16 (2005), no. 2, 109-116}, year = {2005}, publisher = {Accademia Nazionale dei Lincei}, url = {http://hdl.handle.net/1963/4583}, author = {Massimiliano Berti and Michela Procesi} } @article {2004, title = {Bifurcation of free vibrations for completely resonant wave equations}, journal = {Boll. Unione Mat. Ital. Sez. B 7 (2004) 519-528}, number = {SISSA;27/2004/M}, year = {2004}, abstract = {We 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.}, url = {http://hdl.handle.net/1963/2245}, author = {Massimiliano Berti and Philippe Bolle} } @article {2004, title = {Multiplicity of periodic solutions of nonlinear wave equations}, journal = {Nonlinear Anal. 56 (2004) 1011-1046}, year = {2004}, publisher = {Elsevier}, doi = {10.1016/j.na.2003.11.001}, url = {http://hdl.handle.net/1963/2974}, author = {Massimiliano Berti and Philippe Bolle} } @article {2004, title = {Periodic orbits close to elliptic tori and applications to the three-body problem}, journal = {Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 3 (2004) 87-138}, number = {SISSA;28/2003/M}, year = {2004}, publisher = {Scuola Normale Superiore di Pisa}, abstract = {We prove, under suitable non-resonance and non-degeneracy {\textquoteleft}{\textquoteleft}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 {\textquoteleft}{\textquoteleft}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.)}, url = {http://hdl.handle.net/1963/2985}, author = {Massimiliano Berti and Luca Biasco and Enrico Valdinoci} } @article {2004, title = {Soluzioni periodiche di PDEs Hamiltoniane}, journal = {Bollettino dell\\\'Unione Matematica Italiana Serie 8 7-B (2004), p. 647-661}, year = {2004}, publisher = {Unione Matematica Italiana}, url = {http://hdl.handle.net/1963/4582}, author = {Massimiliano Berti} } @article {2003, title = {Drift in phase space: a new variational mechanism with optimal diffusion time}, journal = {J. Math. Pures Appl. 82 (2003) 613-664}, number = {arXiv.org;math/0205307v1}, year = {2003}, publisher = {Elsevier}, abstract = {We 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 {\textquoteleft}{\textquoteleft}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.}, doi = {10.1016/S0021-7824(03)00032-1}, url = {http://hdl.handle.net/1963/3020}, author = {Massimiliano Berti and Luca Biasco and Philippe Bolle} } @article {2003, title = {Periodic solutions of nonlinear wave equations with general nonlinearities}, journal = {Comm.Math.Phys. 243 (2003) no.2, 315}, number = {SISSA;78/2002/M}, year = {2003}, publisher = {SISSA Library}, doi = {10.1007/s00220-003-0972-8}, url = {http://hdl.handle.net/1963/1648}, author = {Massimiliano Berti and Philippe Bolle} } @article {2002, title = {Arnold diffusion: a functional analysis approach}, journal = {Pr. Inst. Mat. Nats. Akad. Nauk Ukr. Mat. Zastos., 43, Part 1, 2, Natsional. Akad. Nauk Ukra{\"\i}ni, Inst. Mat., Kiev, 2002}, year = {2002}, publisher = {Natsional. Akad. Nauk Ukra{\"\i}ni}, abstract = {We present, in the context of nearly integrable Hamiltonian systems, a functional analysis approach to study the {\textquotedblleft}splitting of the whiskers{\textquotedblright} and the {\textquotedblleft}shadowing problem{\textquotedblright} 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.}, author = {Massimiliano Berti} } @article {2002, title = {Chaotic dynamics for perturbations of infinite-dimensional Hamiltonian systems}, journal = {Nonlinear Anal. 48 (2002) 481-504}, number = {SISSA;65/99/M}, year = {2002}, publisher = {Elsevier}, doi = {10.1016/S0362-546X(00)00200-5}, url = {http://hdl.handle.net/1963/1279}, author = {Massimiliano Berti and Carlo Carminati} } @article {2002, title = {Fast Arnold diffusion in systems with three time scales}, journal = {Discrete Contin. Dyn. Syst. 8 (2002) 795-811}, number = {SISSA;21/2001/M}, year = {2002}, publisher = {American Institute of Mathematical Sciences}, abstract = {We 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.}, url = {http://hdl.handle.net/1963/3058}, author = {Massimiliano Berti and Philippe Bolle} } @article {2002, title = {A functional analysis approach to Arnold diffusion}, journal = {Ann. Inst. H. Poincare Anal. Non Lineaire 19 (2002) 395-450}, year = {2002}, publisher = {Elsevier}, abstract = {We 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.}, doi = {10.1016/S0294-1449(01)00084-1}, url = {http://hdl.handle.net/1963/3151}, author = {Massimiliano Berti and Philippe Bolle} } @article {2002, title = {An optimal fast-diffusion variational method for non isochronous system}, number = {SISSA;8/2002/M}, year = {2002}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/1579}, author = {Luca Biasco and Massimiliano Berti and Philippe Bolle} } @article {2002, title = {Optimal stability and instability results for a class of nearly integrable Hamiltonian systems}, journal = {Atti.Accad.Naz.Lincei Cl.Sci.Fis.Mat.Natur.Rend.Lincei (9) Mat.Appl.13(2002),no.2,77-84}, number = {SISSA;25/2002/M}, year = {2002}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/1596}, author = {Massimiliano Berti and Luca Biasco and Philippe Bolle} } @article {2001, title = {Non-compactness and multiplicity results for the Yamabe problem on Sn}, journal = {J. Funct. Anal. 180 (2001) 210-241}, number = {SISSA;130/99/M}, year = {2001}, publisher = {Elsevier}, doi = {10.1006/jfan.2000.3699}, url = {http://hdl.handle.net/1963/1345}, author = {Massimiliano Berti and Andrea Malchiodi} } @article {2000, title = {Arnold{\textquoteright}s Diffusion in nearly integrable isochronous Hamiltonian systems}, number = {SISSA;98/00/M}, year = {2000}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/1554}, author = {Massimiliano Berti and Philippe Bolle} } @article {2000, title = {Diffusion time and splitting of separatrices for nearly integrable}, journal = {Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl., 2000, 11, 235}, number = {SISSA;90/00/M}, year = {2000}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/1547}, author = {Massimiliano Berti and Philippe Bolle} }