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19
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)
Quasiperiodic motions in dynamical systems. Review of a renormalisation group approach
, 2009
"... Power series expansions naturally arise whenever solutions of ordinary differential equations are studied in the regime of perturbation theory. In the case of quasiperiodic solutions the issue of convergence of the series is plagued of the socalled small divisor problem. In this paper we review a ..."
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Cited by 11 (6 self)
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Power series expansions naturally arise whenever solutions of ordinary differential equations are studied in the regime of perturbation theory. In the case of quasiperiodic solutions the issue of convergence of the series is plagued of the socalled small divisor problem. In this paper we review a method recently introduced to deal with such a problem, based on renormalisation group ideas and multiscale techniques. Applications to both quasiintegrable Hamiltonian systems (KAM theory) and nonHamiltonian dissipative systems are discussed. The method is also suited to situations in which the perturbation series diverges and a resummation procedure can be envisaged, leading to a solution which is not analytic in the perturbation parameter: we consider explicitly examples of solutions which are only C ∞ in the perturbation parameter, or even defined on a Cantor set.
Periodic solutions for a class of nonlinear partial differential equations in higher dimension
, 2008
"... ..."
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.
An abstract NashMoser Theorem with parameters and applications to PDEs
"... Abstract. We prove an abstract NashMoser 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 schem ..."
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Cited by 6 (2 self)
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Abstract. We prove an abstract NashMoser 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 a low regularity. 1
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.
Conservation of resonant periodic solutions for the onedimensional nonlinear Schrödinger equation
 Comm. Math. Phys
, 2006
"... Abstract. We consider the onedimensional nonlinear Schrödinger equation with Dirichlet boundary conditions in the fully resonant case (absence of the mass term). We investigate conservation of small amplitude periodic solutions for a large measure set of frequencies. In particular we show that ..."
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Cited by 5 (4 self)
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Abstract. We consider the onedimensional nonlinear Schrödinger equation with Dirichlet boundary conditions in the fully resonant case (absence of the mass term). We investigate conservation of small amplitude periodic solutions for a large measure set of frequencies. In particular we show that there are infinitely many periodic solutions which continue the linear ones involving an arbitrary number of resonant modes, provided the corresponding frequencies are large enough, say greater than a certain threshold value depending on the number of resonant modes. If the frequencies of the latter are close enough to such a threshold, then they can not be too distant from each other. Hence we can interpret such solutions as perturbations of wave packets with large wave number. 1. Introduction and
Periodic solutions for the Schrödinger equation with non local smoothing nonlinearities in higher dimension
, 2007
"... We consider the nonlinear Schrödinger equation in higher dimension with Dirichlet boundary conditions and with a nonlocal smoothing nonlinearity. We prove the existence of small amplitude periodic solutions. In the fully resonant case we find solutions which at leading order are wave packets, in th ..."
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Cited by 5 (2 self)
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We consider the nonlinear Schrödinger equation in higher dimension with Dirichlet boundary conditions and with a nonlocal smoothing nonlinearity. We prove the existence of small amplitude periodic solutions. In the fully resonant case we find solutions which at leading order are wave packets, in the sense that they continue linear solutions with an arbitrarily large number of resonant modes. The main difficulty in the proof consists in solving a “small divisor problem” which we do by using a renormalisation group approach.