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21
Birkhoff Normal Form For Some Nonlinear PDEs
 COMMUN. MATH. PHYS
, 2003
"... We consider the problem of extending to PDEs Birkhoff normal form theorem on Hamiltonian systems close to nonresonant elliptic equilibria. As a model problem we take the nonlinear wave equation with Dirichlet boundary conditions on [0; ]; g is an analytic skewsymmetric function which vanishes for u ..."
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Cited by 46 (5 self)
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We consider the problem of extending to PDEs Birkhoff normal form theorem on Hamiltonian systems close to nonresonant elliptic equilibria. As a model problem we take the nonlinear wave equation with Dirichlet boundary conditions on [0; ]; g is an analytic skewsymmetric function which vanishes for u = 0 and is periodic with period 2 in the x variable. We prove, under a nonresonance condition which is fulfilled for most g's, that for any integer M there exists a canonical transformation that puts the Hamiltonian in Birkhoff normal form up to a reminder of order M . The canonical transformation is well defined in a neighbourhood of the origin of a Sobolev type phase space of sufficiently high order. Some dynamical consequences are obtained. The technique of proof is applicable to quite general equations in one space dimension.
Almost global existence for Hamiltonian semilinear KleinGordon equations with small Cauchy data on Zoll manifolds
, 2005
"... This paper is devoted to the proof of almost global existence results for KleinGordon equations on Zoll manifolds (e.g. spheres of arbitrary dimension) with Hamiltonian nonlinearities, when the Cauchy data are smooth and small. The proof relies on Birkhoff normal form methods and on the specific d ..."
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Cited by 29 (11 self)
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This paper is devoted to the proof of almost global existence results for KleinGordon equations on Zoll manifolds (e.g. spheres of arbitrary dimension) with Hamiltonian nonlinearities, when the Cauchy data are smooth and small. The proof relies on Birkhoff normal form methods and on the specific distribution of eigenvalues of the laplacian perturbed by a potential on Zoll manifolds.
Families Of Periodic Solutions Of Resonant PDEs
 J. Nonlinear Sci
, 2001
"... . We study small oscillations of a semilinear partial dierential equation about a completely resonant equilibrium point. We use averaging methods to construct a suitable functional in the unit ball of the conguration space. We prove that to each nondegenerate critical point of such a functional ther ..."
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Cited by 23 (1 self)
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. We study small oscillations of a semilinear partial dierential equation about a completely resonant equilibrium point. We use averaging methods to construct a suitable functional in the unit ball of the conguration space. We prove that to each nondegenerate critical point of such a functional there corresponds a family of small amplitude periodic orbits of the system. The proof is based on Lyapunov{Schmidt decomposition. Our technique establishes a rst direct relation between averaging techniques and Lyapunov{Schmidt decomposition. An application to the construction of small oscillations of the nonlinear string equation u tt u xx u 3 = 0 (and of its perturbations) is also done. 1. Introduction In this paper we study existence of families of small amplitude periodic solutions in some semilinear partial dierential equation of the form u + Au = f(u) ; u 2 ` 2 s (1) where A is a positive selfadjoint operator with pure point spectrum that we assume to be completely resonant, ...
On metastability in FPU
 Comm. Math. Phys
"... We present an analytical study of the Fermi–Pasta–Ulam (FPU) α– model with periodic boundary conditions. We analyze the dynamics corresponding to initial data with some low frequency Fourier modes excited. We show that, correspondignly, a pair of KdV equations constitute the resonant normal form of ..."
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Cited by 18 (8 self)
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We present an analytical study of the Fermi–Pasta–Ulam (FPU) α– model with periodic boundary conditions. We analyze the dynamics corresponding to initial data with some low frequency Fourier modes excited. We show that, correspondignly, a pair of KdV equations constitute the resonant normal form of the system. We also use such a normal form in order to prove the existence of a metastability phenomenon. More precisely, we show that the time average of the modal energy spectrum rapidly attains a well defined distribution corresponding to a packet of low frequencies modes. Subsequently, the distribution remains unchanged up to the time scales of validity of our approximation. The phenomenon is controlled by the specific energy. 1
The Classical Kam Theory At The Dawn Of The TwentyFirst Century
 MOSCOW MATH. J
, 2003
"... We survey several recent achievements in the KAM theory. The achievements chosen pertain to Hamiltonian systems only and are closely connected with the content of Kolmogorov's original theorem of 1954. They include the weak nondegeneracy conditions, Gevrey smoothness of families of perturbed in ..."
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Cited by 15 (0 self)
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We survey several recent achievements in the KAM theory. The achievements chosen pertain to Hamiltonian systems only and are closely connected with the content of Kolmogorov's original theorem of 1954. They include the weak nondegeneracy conditions, Gevrey smoothness of families of perturbed invariant tori, the "exponential condensation" of perturbed tori, destruction mechanisms of the resonant unperturbed tori, the excitation of the elliptic normal modes of the unperturbed tori, and "atropic" invariant tori (i.e., tori that are neither isotropic nor coisotropic). The exposition is informal and nontechnical, and, as a rule, the methods of proofs are not discussed.
On dispersion of small energy solutions of the nonlinear Klein Gordon equation with a potential
, 2009
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PERTURBATIONS OF VECTOR FIELDS ON TORI: RESONANT NORMAL FORMS AND DIOPHANTINE PHENOMENA
, 2002
"... This paper concerns perturbations of smooth vector fields on Tn (constant if n 3) with zerothorder C ∞ and Gevrey Gσ, σ 1, pseudodifferential operators. Simultaneous resonance is introduced and simultaneous resonant normal forms are exhibited (via conjugation with an elliptic pseudodifferentia ..."
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Cited by 7 (2 self)
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This paper concerns perturbations of smooth vector fields on Tn (constant if n 3) with zerothorder C ∞ and Gevrey Gσ, σ 1, pseudodifferential operators. Simultaneous resonance is introduced and simultaneous resonant normal forms are exhibited (via conjugation with an elliptic pseudodifferential operator) under optimal simultaneous Diophantine conditions outside the resonances. In the C ∞ category the results are complete, while in the Gevrey category the effect of the loss of the Gevrey regularity of the conjugating operators due to Diophantine conditions is encountered. The normal forms are used to study global hypoellipticity in C ∞ and Gevrey Gσ. Finally, the exceptional sets associated with the simultaneous Diophantine conditions are studied. A generalized Hausdorff dimension is used to give precise estimates of the ‘size ’ of different exceptional sets, including some inhomogeneous examples.
Averaging Theorem For Quasilinear Hamiltonian PDEs In Arbitrary Space Dimensions
 Ann. Henri Poincar
, 2002
"... We study the dynamics of quasilinear Hamiltonian wave equations with Dirichlet boundary conditions in an ndimensional parallepided. We prove an averaging theorem according to which the solution corresponding to an arbitrary small amplitude smooth initial datum remains arbitrarily close to a finite ..."
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Cited by 4 (2 self)
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We study the dynamics of quasilinear Hamiltonian wave equations with Dirichlet boundary conditions in an ndimensional parallepided. We prove an averaging theorem according to which the solution corresponding to an arbitrary small amplitude smooth initial datum remains arbitrarily close to a finite dimensional torus up to very long times. We expect the result to be valid for a very general class of quasilinear Hamiltonian equations.
A TUTORIAL ON KAM THEORY
"... This is a tutorial on some of the main ideas in KAM theory. The goal is to present the background and to explain and compare somewhat informally some of the main methods of proof. It is an expanded version of the lectures given by the author in the Summer Research Institute on Smooth Ergodic Theory ..."
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Cited by 1 (0 self)
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This is a tutorial on some of the main ideas in KAM theory. The goal is to present the background and to explain and compare somewhat informally some of the main methods of proof. It is an expanded version of the lectures given by the author in the Summer Research Institute on Smooth Ergodic Theory Seattle, 1999. The style is pedagogical and expository and it only aims to be an introduction to the primary literature. It does not aim to be a systematic survey nor to present full proofs.