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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 ..."
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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.
Persistence of Diophantine flows for quadratic nearlyintegrable Hamiltonians under slowly decaying aperiodic time dependence
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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 ..."
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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.
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"... Abstract. We prove, by applying a KAM algorithm, existence of large families of stable and unstable quasi periodic solutions for the NLS in any number of independent frequencies. The main tools are the existence of a nondegenerate integrable normal form proved in [18] and [20] and a suitable genera ..."
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Abstract. We prove, by applying a KAM algorithm, existence of large families of stable and unstable quasi periodic solutions for the NLS in any number of independent frequencies. The main tools are the existence of a nondegenerate integrable normal form proved in [18] and [20] and a suitable generalization of the quasiTöplitz functions introduced in [24]