01345nas 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-multiplicative02010nas 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-nonlinearities01190nas 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/1963/216100778nas 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/1963/297401027nas 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/1963/164800795nas 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/1963/159600394nas 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