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.

VL - 3 SN - 2523-3688 UR - https://doi.org/10.1007/s42286-020-00036-8 IS - 1 JO - Water Waves ER - TY - JOUR T1 - Traveling Quasi-periodic Water Waves with Constant Vorticity Y1 - 2021 A1 - Massimiliano Berti A1 - Luca Franzoi A1 - Alberto Maspero AB -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.

VL - 240 SN - 1432-0673 UR - https://doi.org/10.1007/s00205-021-01607-w IS - 1 JO - Archive for Rational Mechanics and Analysis ER - TY - RPRT T1 - Almost global existence of solutions for capillarity-gravity water waves equations with periodic spatial boundary conditions Y1 - 2017 A1 - Massimiliano Berti A1 - Jean-Marc Delort AB - 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. UR - http://preprints.sissa.it/handle/1963/35285 U1 - 35590 U2 - Mathematics ER - TY - RPRT T1 - Time quasi-periodic gravity water waves in finite depth Y1 - 2017 A1 - P Baldi A1 - Massimiliano Berti A1 - Emanuele Haus A1 - Riccardo Montalto AB - We prove the existence and the linear stability of Cantor families of small amplitude time quasi-periodic standing water wave solutions - namely periodic and even in the space variable x - of a bi-dimensional ocean with finite depth under the action of pure gravity. Such a result holds for all the values of the depth parameter in a Borel set of asymptotically full measure. This is a small divisor problem. The main difficulties are the quasi-linear nature of the gravity water waves equations and the fact that the linear frequencies grow just in a sublinear way at infinity. We overcome these problems by first reducing the linearized operators obtained at each approximate quasi-periodic solution along the Nash-Moser iteration to constant coefficients up to smoothing operators, using pseudo-differential changes of variables that are quasi-periodic in time. Then we apply a KAM reducibility scheme which requires very weak Melnikov non-resonance conditions (losing derivatives both in time and space), which we are able to verify for most values of the depth parameter using degenerate KAM theory arguments. UR - http://preprints.sissa.it/handle/1963/35296 U1 - 35602 U2 - Mathematics ER - TY - RPRT T1 - Large KAM tori for perturbations of the dNLS equation Y1 - 2016 A1 - Massimiliano Berti A1 - Thomas Kappeler A1 - Riccardo Montalto AB - We prove that small, semi-linear Hamiltonian perturbations of the defocusing nonlinear Schr\"odinger (dNLS) equation on the circle have an abundance of invariant tori of any size and (finite) dimension which support quasi-periodic solutions. When compared with previous results the novelty consists in considering perturbations which do not satisfy any symmetry condition (they may depend on x in an arbitrary way) and need not be analytic. The main difficulty is posed by pairs of almost resonant dNLS frequencies. The proof is based on the integrability of the dNLS equation, in particular the fact that the nonlinear part of the Birkhoff coordinates is one smoothing. We implement a Newton-Nash-Moser iteration scheme to construct the invariant tori. The key point is the reduction of linearized operators, coming up in the iteration scheme, to 2×2 block diagonal ones with constant coefficients together with sharp asymptotic estimates of their eigenvalues. UR - http://preprints.sissa.it/handle/1963/35284 U1 - 35589 U2 - Mathematics ER - TY - CONF T1 - Ship Sinkage and Trim Predictions Based on a CAD Interfaced Fully Nonlinear Potential Model T2 - The 26th International Ocean and Polar Engineering Conference Y1 - 2016 A1 - Andrea Mola A1 - Luca Heltai A1 - Antonio DeSimone A1 - Massimiliano Berti JF - The 26th International Ocean and Polar Engineering Conference PB - International Society of Offshore and Polar Engineers VL - 3 ER - TY - JOUR T1 - An Abstract Nash–Moser Theorem and Quasi-Periodic Solutions for NLW and NLS on Compact Lie Groups and Homogeneous Manifolds Y1 - 2014 A1 - Massimiliano Berti A1 - Livia Corsi A1 - Michela Procesi AB - 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. PB - Springer UR - http://urania.sissa.it/xmlui/handle/1963/34651 U1 - 34858 U2 - Mathematics ER - TY - JOUR T1 - KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation JF - Mathematische Annalen Y1 - 2014 A1 - P Baldi A1 - Massimiliano Berti A1 - Riccardo Montalto AB - We prove the existence of small amplitude quasi-periodic solutions for quasi-linear and fully nonlinear forced perturbations of the linear Airy equation. For Hamiltonian or reversible nonlinearities we also prove their linear stability. The key analysis concerns the reducibility of the linearized operator at an approximate solution, which provides a sharp asymptotic expansion of its eigenvalues. For quasi-linear perturbations this cannot be directly obtained by a KAM iteration. Hence we first perform a regularization procedure, which conjugates the linearized operator to an operator with constant coefficients plus a bounded remainder. These transformations are obtained by changes of variables induced by diffeomorphisms of the torus and pseudo-differential operators. At this point we implement a Nash-Moser iteration (with second order Melnikov non-resonance conditions) which completes the reduction to constant coefficients. © 2014 Springer-Verlag Berlin Heidelberg. N1 - cited By (since 1996)0; Article in Press ER - TY - JOUR T1 - KAM for quasi-linear KdV JF - C. R. Math. Acad. Sci. Paris Y1 - 2014 A1 - P Baldi A1 - Massimiliano Berti A1 - Riccardo Montalto AB -We prove the existence and stability of Cantor families of quasi-periodic, small-amplitude solutions of quasi-linear autonomous Hamiltonian perturbations of KdV.

PB - Elsevier VL - 352 UR - http://urania.sissa.it/xmlui/handle/1963/35067 IS - 7-8 U1 - 35302 U2 - Mathematics U4 - 1 ER - TY - JOUR T1 - KAM for Reversible Derivative Wave Equations JF - Arch. Ration. Mech. Anal. Y1 - 2014 A1 - Massimiliano Berti A1 - Luca Biasco A1 - Michela Procesi AB -We prove the existence of Cantor families of small amplitude, analytic, linearly stable quasi-periodic solutions of reversible derivative wave equations.

PB - Springer VL - 212 UR - http://urania.sissa.it/xmlui/handle/1963/34646 IS - 3 U1 - 34850 U2 - Mathematics ER - TY - CONF T1 - Potential Model for Ship Hydrodynamics Simulations Directly Interfaced with CAD Data Structures T2 - The 24th International Ocean and Polar Engineering Conference Y1 - 2014 A1 - Andrea Mola A1 - Luca Heltai A1 - Antonio DeSimone A1 - Massimiliano Berti JF - The 24th International Ocean and Polar Engineering Conference PB - International Society of Offshore and Polar Engineers VL - 4 ER - TY - JOUR T1 - Existence and stability of quasi-periodic solutions for derivative wave equations JF - Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica e Applicazioni Y1 - 2013 A1 - Massimiliano Berti A1 - Luca Biasco A1 - Michela Procesi KW - Constant coefficients KW - Dynamical systems KW - Existence and stability KW - Infinite dimensional KW - KAM for PDEs KW - Linearized equations KW - Lyapunov exponent KW - Lyapunov methods KW - Quasi-periodic solution KW - Small divisors KW - Wave equations AB - 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*. VL - 24 N1 - cited By (since 1996)0 ER - TY - JOUR T1 - KAM theory for the Hamiltonian derivative wave equation JF - Annales Scientifiques de l'Ecole Normale Superieure Y1 - 2013 A1 - Massimiliano Berti A1 - Luca Biasco A1 - Michela Procesi AB -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. © 2013 Société Mathématique de France.

VL - 46 N1 - cited By (since 1996)4 ER - TY - JOUR T1 - A note on KAM theory for quasi-linear and fully nonlinear forced KdV JF - Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 24 (2013), no. 3: 437–450 Y1 - 2013 A1 - P Baldi A1 - Massimiliano Berti A1 - Riccardo Montalto KW - KAM for PDEs AB - We present the recent results in [3] concerning quasi-periodic solutions for quasi-linear and fully nonlinear forced perturbations of KdV equations. For Hamiltonian or reversible nonlinearities the solutions are linearly stable. The proofs are based on a combination of di erent ideas and techniques: (i) a Nash-Moser iterative scheme in Sobolev scales. (ii) A regularization procedure, which conjugates the linearized operator to a di erential operator with constant coe cients plus a bounded remainder. These transformations are obtained by changes of variables induced by di eomorphisms of the torus and pseudo-di erential operators. (iii) A reducibility KAM scheme, which completes the reduction to constant coe cients of the linearized operator, providing a sharp asymptotic expansion of the perturbed eigenvalues. PB - European Mathematical Society U1 - 7268 U2 - Mathematics U4 - 1 U5 - MAT/05 ANALISI MATEMATICA ER - TY - JOUR T1 - Quasi-periodic solutions with Sobolev regularity of NLS on Td with a multiplicative potential JF - Journal of the European Mathematical Society Y1 - 2013 A1 - Massimiliano Berti A1 - Philippe Bolle AB - We 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. VL - 15 N1 - cited By (since 1996)5 ER - TY - JOUR T1 - Sobolev quasi-periodic solutions of multidimensional wave equations with a multiplicative potential JF - Nonlinearity Y1 - 2012 A1 - Massimiliano Berti A1 - Philippe Bolle AB - 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 'separation properties' of the small divisors assuming weaker non-resonance conditions than in [11]. © 2012 IOP Publishing Ltd. VL - 25 N1 - cited By (since 1996)3 ER - TY - JOUR T1 - Branching of Cantor Manifolds of Elliptic Tori and Applications to PDEs JF - Communications in Mathematical Physics Y1 - 2011 A1 - Massimiliano Berti A1 - Luca Biasco AB - 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. © 2011 Springer-Verlag. VL - 305 N1 - cited By (since 1996)8 ER - TY - JOUR T1 - Degenerate KAM theory for partial differential equations JF - Journal of Differential Equations Y1 - 2011 A1 - Dario Bambusi A1 - Massimiliano Berti A1 - Elena Magistrelli AB - 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 Rüssmann type and an asymptotic behavior. An application to nonlinear wave equations is given. © 2010 Elsevier Inc. VL - 250 N1 - cited By (since 1996)3 ER - TY - JOUR T1 - Nonlinear wave and Schrödinger equations on compact Lie groups and homogeneous spaces JF - Duke Mathematical Journal Y1 - 2011 A1 - Massimiliano Berti A1 - Michela Procesi AB - 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¨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. VL - 159 IS - 3 ER - TY - JOUR T1 - An abstract Nash-Moser theorem with parameters and applications to PDEs JF - Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis Y1 - 2010 A1 - Massimiliano Berti A1 - Philippe Bolle A1 - Michela Procesi KW - Abstracting KW - Aircraft engines KW - Finite dimensional KW - Hamiltonian PDEs KW - Implicit function theorem KW - Invariant tori KW - Iterative schemes KW - Linearized operators KW - Mathematical operators KW - Moser theorem KW - Non-Linearity KW - Nonlinear equations KW - Nonlinear wave equation KW - Periodic solution KW - Point of interest KW - Resonance phenomena KW - Small divisors KW - Sobolev KW - Wave equations AB - 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. © 2009 Elsevier Masson SAS. All rights reserved. VL - 27 N1 - cited By (since 1996)9 ER - TY - JOUR T1 - Sobolev periodic solutions of nonlinear wave equations in higher spatial dimensions JF - Archive for Rational Mechanics and Analysis Y1 - 2010 A1 - Massimiliano Berti A1 - Philippe Bolle AB - 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". © Springer-Verlag (2009). VL - 195 N1 - cited By (since 1996)6 ER - TY - JOUR T1 - Cantor families of periodic solutions for completely resonant wave equations JF - Frontiers of Mathematics in China Y1 - 2008 A1 - Massimiliano Berti A1 - Philippe Bolle AB - 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. © 2008 Higher Education Press. VL - 3 N1 - cited By (since 1996)0 ER - TY - JOUR T1 - Cantor families of periodic solutions for wave equations via a variational principle JF - Advances in Mathematics Y1 - 2008 A1 - Massimiliano Berti A1 - Philippe Bolle AB - 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. © 2007 Elsevier Inc. All rights reserved. VL - 217 N1 - cited By (since 1996)6 ER - TY - JOUR T1 - Cantor families of periodic solutions of wave equations with C k nonlinearities JF - Nonlinear Differential Equations and Applications Y1 - 2008 A1 - Massimiliano Berti A1 - Philippe Bolle AB - 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. © 2008 Birkhaueser. VL - 15 N1 - cited By (since 1996)10 ER - TY - JOUR T1 - Forced Vibrations of a Nonhomogeneous String JF - SIAM J. Math. Anal. 40 (2008) 382-412 Y1 - 2008 A1 - P Baldi A1 - Massimiliano Berti AB - We prove existence of vibrations of a nonhomogeneous string under a nonlinear time periodic forcing term in the case in which the forcing frequency avoids resonances with the vibration modes of the string (nonresonant case). The proof relies on a Lyapunov-Schmidt reduction and a Nash-Moser iteration scheme. UR - http://hdl.handle.net/1963/2643 U1 - 1480 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - On periodic elliptic equations with gradient dependence JF - Communications on Pure and Applied Analysis Y1 - 2008 A1 - Massimiliano Berti A1 - Matzeu, M A1 - Enrico Valdinoci AB - 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. VL - 7 N1 - cited By (since 1996)1 ER - TY - JOUR T1 - Variational methods for Hamiltonian PDEs JF - NATO Science for Peace and Security Series B: Physics and Biophysics Y1 - 2008 A1 - Massimiliano Berti AB - 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. © 2008 Springer Science + Business Media B.V. SN - 9781402069628 N1 - cited By (since 1996)0 ER - TY - JOUR T1 - A Birkhoff-Lewis-Type Theorem for Some Hamiltonian PDEs JF - SIAM J. Math. Anal. 37 (2006) 83-102 Y1 - 2006 A1 - Dario Bambusi A1 - Massimiliano Berti AB - 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. UR - http://hdl.handle.net/1963/2159 U1 - 2085 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Cantor families of periodic solutions for completely resonant nonlinear wave equations JF - Duke Math. J. 134 (2006) 359-419 Y1 - 2006 A1 - Massimiliano Berti A1 - Philippe Bolle AB - 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. UR - http://hdl.handle.net/1963/2161 U1 - 2083 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Forced vibrations of wave equations with non-monotone nonlinearities JF - Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006) 439-474 Y1 - 2006 A1 - Massimiliano Berti A1 - Luca Biasco AB - 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. UR - http://hdl.handle.net/1963/2160 U1 - 2084 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Periodic solutions of nonlinear wave equations for asymptotically full measure sets of frequencies JF - Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica e Applicazioni Y1 - 2006 A1 - P Baldi A1 - Massimiliano Berti AB - We prove existence and multiplicity of small amplitude periodic solutions of completely resonant nonlinear wave equations with Dirichlet boundary conditions for asymptotically full measure sets of frequencies, extending the results of [7] to new types of nonlinearities. VL - 17 N1 - cited By (since 1996)5 ER - TY - JOUR T1 - Quasi-periodic solutions of completely resonant forced wave equations JF - Comm. Partial Differential Equations 31 (2006) 959 - 985 Y1 - 2006 A1 - Massimiliano Berti A1 - Michela Procesi AB - 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. UR - http://hdl.handle.net/1963/2234 U1 - 2010 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Periodic solutions of nonlinear wave equations with non-monotone forcing terms JF - Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 16 (2005), no. 2, 117-124 Y1 - 2005 A1 - Massimiliano Berti A1 - Luca Biasco PB - Accademia Nazionale dei Lincei UR - http://hdl.handle.net/1963/4581 U1 - 4349 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER - TY - JOUR T1 - Quasi-periodic oscillations for wave equations under periodic forcing JF - Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 16 (2005), no. 2, 109-116 Y1 - 2005 A1 - Massimiliano Berti A1 - Michela Procesi PB - Accademia Nazionale dei Lincei UR - http://hdl.handle.net/1963/4583 U1 - 4350 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER - TY - JOUR T1 - Bifurcation of free vibrations for completely resonant wave equations JF - Boll. Unione Mat. Ital. Sez. B 7 (2004) 519-528 Y1 - 2004 A1 - Massimiliano Berti A1 - Philippe Bolle AB - 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. UR - http://hdl.handle.net/1963/2245 U1 - 1999 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Multiplicity of periodic solutions of nonlinear wave equations JF - Nonlinear Anal. 56 (2004) 1011-1046 Y1 - 2004 A1 - Massimiliano Berti A1 - Philippe Bolle PB - Elsevier UR - http://hdl.handle.net/1963/2974 U1 - 1359 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Periodic orbits close to elliptic tori and applications to the three-body problem JF - Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 3 (2004) 87-138 Y1 - 2004 A1 - Massimiliano Berti A1 - Luca Biasco A1 - Enrico Valdinoci AB - We 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.) PB - Scuola Normale Superiore di Pisa UR - http://hdl.handle.net/1963/2985 U1 - 1348 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Soluzioni periodiche di PDEs Hamiltoniane JF - Bollettino dell\\\'Unione Matematica Italiana Serie 8 7-B (2004), p. 647-661 Y1 - 2004 A1 - Massimiliano Berti PB - Unione Matematica Italiana UR - http://hdl.handle.net/1963/4582 U1 - 4351 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER - TY - JOUR T1 - Drift in phase space: a new variational mechanism with optimal diffusion time JF - J. Math. Pures Appl. 82 (2003) 613-664 Y1 - 2003 A1 - Massimiliano Berti A1 - Luca Biasco A1 - Philippe Bolle AB - 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 ``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. PB - Elsevier UR - http://hdl.handle.net/1963/3020 U1 - 1313 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Periodic solutions of nonlinear wave equations with general nonlinearities JF - Comm.Math.Phys. 243 (2003) no.2, 315 Y1 - 2003 A1 - Massimiliano Berti A1 - Philippe Bolle PB - SISSA Library UR - http://hdl.handle.net/1963/1648 U1 - 2470 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Arnold diffusion: a functional analysis approach JF - Pr. Inst. Mat. Nats. Akad. Nauk Ukr. Mat. Zastos., 43, Part 1, 2, Natsīonal. Akad. Nauk Ukraïni, Īnst. Mat., Kiev, 2002 Y1 - 2002 A1 - Massimiliano Berti AB - We 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. PB - Natsīonal. Akad. Nauk Ukraïni U1 - 7269 U2 - Mathematics U4 - 1 U5 - MAT/05 ANALISI MATEMATICA ER - TY - JOUR T1 - Chaotic dynamics for perturbations of infinite-dimensional Hamiltonian systems JF - Nonlinear Anal. 48 (2002) 481-504 Y1 - 2002 A1 - Massimiliano Berti A1 - Carlo Carminati PB - Elsevier UR - http://hdl.handle.net/1963/1279 U1 - 3176 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Fast Arnold diffusion in systems with three time scales JF - Discrete Contin. Dyn. Syst. 8 (2002) 795-811 Y1 - 2002 A1 - Massimiliano Berti A1 - Philippe Bolle AB - 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. PB - American Institute of Mathematical Sciences UR - http://hdl.handle.net/1963/3058 U1 - 1275 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - A functional analysis approach to Arnold diffusion JF - Ann. Inst. H. Poincare Anal. Non Lineaire 19 (2002) 395-450 Y1 - 2002 A1 - Massimiliano Berti A1 - Philippe Bolle AB - 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. PB - Elsevier UR - http://hdl.handle.net/1963/3151 U1 - 1182 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - An optimal fast-diffusion variational method for non isochronous system Y1 - 2002 A1 - Luca Biasco A1 - Massimiliano Berti A1 - Philippe Bolle PB - SISSA Library UR - http://hdl.handle.net/1963/1579 U1 - 2539 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Optimal stability and instability results for a class of nearly integrable Hamiltonian systems JF - Atti.Accad.Naz.Lincei Cl.Sci.Fis.Mat.Natur.Rend.Lincei (9) Mat.Appl.13(2002),no.2,77-84 Y1 - 2002 A1 - Massimiliano Berti A1 - Luca Biasco A1 - Philippe Bolle PB - SISSA Library UR - http://hdl.handle.net/1963/1596 U1 - 2522 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Non-compactness and multiplicity results for the Yamabe problem on Sn JF - J. Funct. Anal. 180 (2001) 210-241 Y1 - 2001 A1 - Massimiliano Berti A1 - Andrea Malchiodi PB - Elsevier UR - http://hdl.handle.net/1963/1345 U1 - 3110 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Arnold's Diffusion in nearly integrable isochronous Hamiltonian systems Y1 - 2000 A1 - Massimiliano Berti A1 - Philippe Bolle PB - SISSA Library UR - http://hdl.handle.net/1963/1554 U1 - 2564 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Diffusion time and splitting of separatrices for nearly integrable JF - Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl., 2000, 11, 235 Y1 - 2000 A1 - Massimiliano Berti A1 - Philippe Bolle PB - SISSA Library UR - http://hdl.handle.net/1963/1547 U1 - 2571 U2 - Mathematics U3 - Functional Analysis and Applications ER -