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31
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.
Periodic solutions for a class of nonlinear partial differential equations in higher dimension
, 2008
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Quasiperiodic solutions of completely resonant forced wave equations, preprint Sissa
, 2005
"... Abstract. We prove existence of quasiperiodic solutions with two frequencies of completely resonant, periodically forced nonlinear wave equations with periodic spatial boundary conditions. We consider both the cases the forcing frequency is: (Case A) a rational number and (Case B) an irrational num ..."
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Cited by 7 (2 self)
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Abstract. We prove existence of quasiperiodic solutions with two frequencies of completely resonant, periodically forced nonlinear wave equations with periodic spatial boundary conditions. We consider both the cases the forcing frequency is: (Case A) a rational number and (Case B) an irrational number.
A BirkhoffLewis type theorem for the nonlinear wave equation
, 2009
"... We give an extension of the celebrated Birkhoff–Lewis theorem to the nonlinear wave equation. Accordingly we find infinitely many periodic orbits with longer and longer minimal periods accumulating at the origin, which is an elliptic equilibrium of the associated infinite dimensional Hamiltonian sys ..."
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Cited by 6 (3 self)
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We give an extension of the celebrated Birkhoff–Lewis theorem to the nonlinear wave equation. Accordingly we find infinitely many periodic orbits with longer and longer minimal periods accumulating at the origin, which is an elliptic equilibrium of the associated infinite dimensional Hamiltonian system.
DIOPHANTINE CONDITIONS IN WELLPOSEDNESS THEORY OF COUPLED KDVTYPE SYSTEMS: LOCAL THEORY
, 2009
"... We consider the local wellposedness problem of a oneparameter family of coupled KdVtype systems both in the periodic and nonperiodic setting. In particular, we show that certain resonances occur, closely depending on the value of a coupling parameter α when α = 1. In the periodic setting, we u ..."
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Cited by 5 (2 self)
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We consider the local wellposedness problem of a oneparameter family of coupled KdVtype systems both in the periodic and nonperiodic setting. In particular, we show that certain resonances occur, closely depending on the value of a coupling parameter α when α = 1. In the periodic setting, we use the Diophantine conditions to characterize the resonances, and establish sharp local wellposedness of the system in H s (Tλ), s ≥ s ∗, where s ∗ = s ∗ (α) ∈ ( 1,1] is determined by the Diophantine characterization of certain 2 constants derived from the coupling parameter α. We also present a sharp local (and global) result in L 2 (R). In the appendix, we briefly discuss the local wellposedness result in H −1 2 (Tλ) for α = 1 without the mean 0 assumption, by introducing the vectorvalued X s,b spaces.
QUASIPERIODIC SOLUTIONS OF THE SCHRÖDINGER EQUATION WITH ARBITRARY ALGEBRAIC NONLINEARITIES
, 907
"... Abstract. We present a geometric formulation of existence of time quasiperiodic solutions. As an application, we prove the existence of quasiperiodic solutions of b frequencies, b ≤ d + 2, in arbitrary dimension d and for arbitrary non integrable algebraic nonlinearity p. This reflects the conserv ..."
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Cited by 4 (0 self)
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Abstract. We present a geometric formulation of existence of time quasiperiodic solutions. As an application, we prove the existence of quasiperiodic solutions of b frequencies, b ≤ d + 2, in arbitrary dimension d and for arbitrary non integrable algebraic nonlinearity p. This reflects the conservation of d momenta, energy and L 2 norm. In 1d, we prove the existence of quasiperiodic solutions with arbitrary b and for arbitrary p, solving a problem that started Hamiltonian PDE. Contents