Results 1  10
of
11
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 ..."
Abstract

Cited by 15 (5 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 11 (6 self)
 Add to MetaCart
(Show Context)
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.
Nonlinear wave and Schrödinger equations on compact Lie groups and homogeneous spaces
"... Abstract. We prove the existence of Cantor families of small amplitude periodic solutions for wave and Schrödinger equations on compact Lie groups and homogeneous spaces with merely differentiable nonlinearities. The NLS equation on homogeneous spaces arises as a mean field approximation of conden ..."
Abstract

Cited by 9 (3 self)
 Add to MetaCart
(Show Context)
Abstract. We prove the existence of Cantor families of small amplitude periodic solutions for wave and Schrödinger equations on compact Lie groups and homogeneous spaces with merely differentiable nonlinearities. The NLS equation on homogeneous spaces arises as a mean field approximation of condensates of manybody lattice problems. The highly degenerate eigenvalues of the Laplace Beltrami operator give NashMoser implicit function theorem. We provide a new algebraic framework to prove the key tame estimates for the inverse linearized operators along Banach scales of Sobolev functions. We exploit properties of the eigenvalues and eigenfunctions of the Laplace Beltrami operator on Lie groups. 1
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 ..."
Abstract

Cited by 6 (2 self)
 Add to MetaCart
(Show Context)
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
KAM for quasilinear and fully nonlinear forced KdV
"... We prove the existence of quasiperiodic, small amplitude, solutions for quasilinear and fully nonlinear forced perturbations of KdV equations. For Hamiltonian or reversible nonlinearities we also obtain the linear stability of the solutions. The proofs are based on a combination of different ideas ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
We prove the existence of quasiperiodic, small amplitude, solutions for quasilinear and fully nonlinear forced perturbations of KdV equations. For Hamiltonian or reversible nonlinearities we also obtain the linear stability of the solutions. The proofs are based on a combination of different ideas and techniques: (i) a NashMoser iterative scheme in Sobolev scales. (ii) A regularization procedure, which conjugates the linearized operator to a differential operator with constant coefficients plus a bounded remainder. These transformations are obtained by changes of variables induced by diffeomorphisms of the torus and pseudodifferential operators. (iii) A reducibility KAM scheme, which completes the reduction to constant
Periodic solutions of nonlinear Schrödinger equations: a paradifferential approach
, 2011
"... This paper is devoted to the construction of periodic solutions of nonlinear Schrödinger equations on the torus, for a large set of frequencies. Usual proofs of such results rely on the use of Nash–Moser methods. Our approach avoids this, exploiting the possibility of reducing, through paradifferent ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
This paper is devoted to the construction of periodic solutions of nonlinear Schrödinger equations on the torus, for a large set of frequencies. Usual proofs of such results rely on the use of Nash–Moser methods. Our approach avoids this, exploiting the possibility of reducing, through paradifferential conjugation, the equation under study to an equivalent form for which periodic solutions may be constructed by a classical iteration scheme.
Digital Object Identifier Mathematical Physics Branching of Cantor Manifolds of Elliptic Tori and Applications to PDEs
, 2011
"... Abstract: We consider infinite dimensional Hamiltonian systems. We prove the existence of "Cantor manifolds" of elliptic toriof any finite higher dimensionaccumulating on a given elliptic KAM torus. Then, close to an elliptic equilibrium, we show the existence of Cantor manifolds of ell ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract: We consider infinite dimensional Hamiltonian systems. We prove the existence of "Cantor manifolds" of elliptic toriof any finite higher dimensionaccumulating 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 answer to a conjecture of Bourgain, proving the existence of invariant elliptic tori with tangential frequency along a preassigned direction. The proofs are based on an improved KAM theorem. Its main advantages are an explicit characterization of the Cantor set of parameters and weaker smallness conditions on the perturbation. We apply these results to the nonlinear wave equation.
Stabilità dell’integrabilità Hamiltoniana:
"... Hamiltoniani la cui funzione di Hamilton è “vicina ” a quella di un sistema analiticamente integrabile. Un sistema a ℓ gradi di libertà e con Hamiltoniana H(p,q) analitica sullo spazio delle fasi è analiticamente integrabile su una regione W dello spazio delle fasi se (1) i punti di W sono rappresen ..."
Abstract
 Add to MetaCart
(Show Context)
Hamiltoniani la cui funzione di Hamilton è “vicina ” a quella di un sistema analiticamente integrabile. Un sistema a ℓ gradi di libertà e con Hamiltoniana H(p,q) analitica sullo spazio delle fasi è analiticamente integrabile su una regione W dello spazio delle fasi se (1) i punti di W sono rappresentabili via ℓ coordinate A = (A1,...,Aℓ) ∈ R ℓ, detteazioni,eℓ angoliα = (α1,...,αℓ) ∈ [0,2π] ℓ nelsensocheipunti(p,q) ∈ W possono essere rappresentati, a mezzo di una trasformazione analitica 1 di coordinate p = P ( A,α),q = Q ( A,α) con ( A,α) ∈ U × T ℓ ove U ⊂ R ℓ