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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.
Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation
, 2008
"... We consider the cubic defocusing nonlinear Schrödinger equation on the two dimensional torus. We exhibit smooth solutions for which the support of the conserved energy moves to higher Fourier modes. This behavior is quantified by the growth of higher Sobolev norms: given any δ ≪ 1, K ≫ 1, s> 1, ..."
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Cited by 38 (3 self)
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We consider the cubic defocusing nonlinear Schrödinger equation on the two dimensional torus. We exhibit smooth solutions for which the support of the conserved energy moves to higher Fourier modes. This behavior is quantified by the growth of higher Sobolev norms: given any δ ≪ 1, K ≫ 1, s> 1, we construct smooth initial data u0 with ‖u0‖H s < δ, so that the corresponding time evolution u satisfies ‖u(T)‖Hs> K at some time T. This growth occurs despite the Hamiltonian’s bound on ‖u(t)‖H ˙ 1 and despite the conservation of the quantity ‖u(t)‖L2. The proof contains two arguments which may be of interest beyond the particular result described above. The first is a construction of the solution’s frequency support that simplifies the system of ODE’s describing each Fourier mode’s evolution. The second is a construction of solutions to these simpler systems of ODE’s which begin near one invariant manifold and ricochet from arbitrarily small neighborhoods of an arbitrarily large number of other invariant manifolds. The techniques used here are related to but are distinct from those traditionally used
Cantor families of periodic solutions for completely resonant nonlinear wave equations.
 Duke Math. J.
, 2006
"... Abstract We prove the existence of small amplitude, (2π/ω) ..."
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Cited by 31 (9 self)
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Abstract We prove the existence of small amplitude, (2π/ω)
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, ...
A Property Of Exponential Stability In Nonlinear Wave Equations Near The Fundamental Linear Mode
 PHYSICA D
, 1998
"... We consider the nonlinear wave equation u tt c 2 u xx = (u) ; u(t; 0) = u(t; ) = 0 ; (0:1) where is an analytic function satisfying (0) = 0 and 0 (0) 6= 0. For each value of the harmonic energy E (i.e. energy when = 0), we consider the closed curve E which is the phase space trajectory of t ..."
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Cited by 21 (11 self)
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We consider the nonlinear wave equation u tt c 2 u xx = (u) ; u(t; 0) = u(t; ) = 0 ; (0:1) where is an analytic function satisfying (0) = 0 and 0 (0) 6= 0. For each value of the harmonic energy E (i.e. energy when = 0), we consider the closed curve E which is the phase space trajectory of the linear mode with lowest frequency. If the perturbation parameter and the harmonic energy E are small enough, then, near E we construct, by means of a suitable \simplied equation", a closed curve = E; with the following property: provided the distance between the initial datum and such a curve is small enough, the corresponding solution remains close to E; up to times exponentially long with 1=; one cannot expect the curves E; to be trajectories of solutions of (0.1). Such curves depend smoothly on the parameters E and . The above result is deduced from a general theorem on generic hamiltonian perturbations of completely resonant linear systems.
Degenerate elliptic resonances
, 2004
"... Quasiperiodic motions on invariant tori of an integrable system of dimension smaller than half the phase space dimension may continue to exists after small perturbations. The parametric equations of the invariant tori can often be computed as formal power series in the perturbation parameter and c ..."
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Cited by 20 (17 self)
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Quasiperiodic motions on invariant tori of an integrable system of dimension smaller than half the phase space dimension may continue to exists after small perturbations. The parametric equations of the invariant tori can often be computed as formal power series in the perturbation parameter and can be given a meaning via resummations. Here we prove that, for a class of elliptic tori, a resummation algorithm can be devised and proved to be convergent, thus extending to such lowerdimensional invariant tori the methods employed to prove convergence of the Lindstedt series either for the maximal (i.e. KAM) tori or for the hyperbolic lowerdimensional invariant tori.
Branching of Cantor manifolds of elliptic tori . . .
"... We consider infinite dimensional Hamiltonian systems. First we prove the existence of “Cantor manifolds” of elliptic toriof 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 wh ..."
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Cited by 15 (5 self)
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We consider infinite dimensional Hamiltonian systems. First we prove the existence of “Cantor manifolds” of elliptic toriof 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 provide a positive answer to a conjecture of Bourgain [8] proving the existence of invariant elliptic KAM tori with tangential frequency constrained to a fixed Diophantine direction. These results are obtained under the natural nonresonance and nondegeneracy conditions. As applications we prove the existence of new kinds of quasi periodic solutions of the one dimensional nonlinear wave equation. The proofs are based on averaging normal forms and a sharp KAM theorem, whose advantages are an explicit characterisation of the Cantor set of parameters, quite convenient for measure estimates, and weaker smallness conditions on the perturbation.