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 - 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 - 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 - 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 - 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 - 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 - 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 -