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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.
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 ..."
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Cited by 9 (3 self)
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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
Periodic solutions of Birkhoff–Lewis type for the nonlinear wave equation
, 2007
"... We prove the existence of infinitely many periodic solutions accumulating to zero for the one–dimensional nonlinear wave equation (vibrating string equation). The periods accumulate to zero and are both rational and irrational multiples of the string length. ..."
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Cited by 2 (1 self)
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We prove the existence of infinitely many periodic solutions accumulating to zero for the one–dimensional nonlinear wave equation (vibrating string equation). The periods accumulate to zero and are both rational and irrational multiples of the string length.
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.