Displaying similar documents to “Invariant manifold of hyperbolic-elliptic type for nonlinear wave equation.”

KAM theory for the hamiltonian derivative wave equation

Massimiliano Berti, Luca Biasco, Michela Procesi (2013)

Annales scientifiques de l'École Normale Supérieure

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We prove an infinite dimensional KAM theorem which implies the existence of Cantor families of small-amplitude, reducible, elliptic, analytic, invariant tori of Hamiltonian derivative wave equations.

Periodic orbits close to elliptic tori and applications to the three-body problem

Massimiliano Berti, Luca Biasco, Enrico Valdinoci (2004)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We prove, under suitable non-resonance and non-degeneracy “twist” conditions, a Birkhoff-Lewis type result showing the existence of infinitely many periodic solutions, with larger and larger minimal period, accumulating onto elliptic invariant tori (of hamiltonian systems). We prove the applicability of this result to the spatial planetary three-body problem in the small eccentricity-inclination regime. Furthermore, we find other periodic orbits under some restrictions on the period...

Diffusion times and stability exponents for nearly integrable analytic systems

Pierre Lochak, Jean-Pierre Marco (2005)

Open Mathematics

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For a positive integer n and R>0, we set B R n = x ∈ ℝ n | x ∞ < R . Given R>1 and n≥4 we construct a sequence of analytic perturbations (H j) of the completely integrable Hamiltonian h r = 1 2 r 1 2 + . . . 1 2 r n - 1 2 + r n on 𝕋 n × B R n , with unstable orbits for which we can estimate the time of drift in the action space. These functions H j are analytic on a fixed complex neighborhood V of 𝕋 n × B R n , and setting ε j : = h - H j C 0 ( V ) the time of drift of these orbits is smaller than (C(1/ɛ j)1/2(n-3)) for a fixed constant c>0. Our unstable orbits stay close to a doubly...

Galerkin averaging method and Poincaré normal form for some quasilinear PDEs

Dario Bambusi (2005)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We use the Galerkin averaging method to construct a coordinate transformation putting a nonlinear PDE in Poincaré normal form up to finite order. We also give a rigorous estimate of the remainder showing that it is small as a differential operator of very high order. The abstract theorem is then applied to a quasilinear wave equation, to the water wave problem and to a nonlinear heat equation. The normal form is then used to construct approximate solutions whose difference from true...

Quasi-periodic solutions of PDEs

Massimiliano Berti (2011-2012)

Séminaire Laurent Schwartz — EDP et applications

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The aim of this talk is to present some recent existence results about quasi-periodic solutions for PDEs like nonlinear wave and Schrödinger equations in 𝕋 d , d ≥ 2 , and the 1 - d derivative wave equation. The proofs are based on both Nash-Moser implicit function theorems and KAM theory.