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Birkhoff Normal Form for Some Nonlinear PDEs
Abstract: 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 waveExpand
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Birkhoff normal form for partial differential equations with tame modulus
We prove an abstract Birkhoff normal form theorem for Hamiltonian Partial Differential Equations. The theorem applies to semilinear equations with nonlinearity satisfying a property that we call ofExpand
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On dispersion of small energy solutions of the nonlinear Klein Gordon equation with a potential
In this paper we study small amplitude solutions of nonlinear Klein Gordon equations with a potential. Under suitable smoothness and decay assumptions on the potential and a genericity assumption onExpand
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A Birkhoff normal form theorem for some semilinear PDEs
In these lectures we present an extension of Birkhoff normal form theorem to some Hamiltonian PDEs. The theorem applies to semilinear equations with non- linearity of a suitable class. We present anExpand
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Nekhoroshev theorem for small amplitude solutions in nonlinear Schrödinger equations
Abstract. We prove a Nekhoroshev type result [26,27] for the nonlinear Schrödinger equation \begin{eqnarray} iu_t=-u_{xx}-mu-u \varphi (|u|^2) , \end{eqnarray} with vanishing or periodic boundaryExpand
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Reducibility of the Quantum Harmonic Oscillator in $d$-dimensions with Polynomial Time Dependent Perturbation
We prove a reducibility result for a quantum harmonic oscillator in arbitrary dimensions with arbitrary frequencies perturbed by a linear operator which is a polynomial of degree two in $x_j$, $-iExpand
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Families of Periodic Solutions of Resonant PDEs
TLDR
We construct some families of small amplitude periodic solutions close to a completely resonant equilibrium point of a semilinear reversible partial differential equation that converge to 2π/n when the amplitude tends to zero. Expand
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Almost global existence for Hamiltonian semilinear Klein‐Gordon equations with small Cauchy data on Zoll manifolds
This paper is devoted to the proof of almost global existence results for Klein-Gordon equations on Zoll manifolds (e.g., spheres of arbitrary dimension) with Hamiltonian nonlinearities, when theExpand
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On long time stability in Hamiltonian perturbations of non-resonant linear PDEs
We consider small Hamiltonian perturbations of a system of infinitely many harmonic oscillators. We assume that the frequencies j, j 1, fulfil j ~ jd with d 1, and a suitable non-resonance conditionExpand
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Time Quasi-Periodic Unbounded Perturbations¶of Schrödinger Operators and KAM Methods
Abstract: We eliminate by KAM methods the time dependence in a class of linear differential equations in ℓ2 subject to an unbounded, quasi-periodic forcing. This entails the pure-point nature of theExpand
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