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Article Dans Une Revue Geometric And Functional Analysis Année : 2016

KAM for the nonlinear beam equation

Résumé

In this paper we prove a KAM result for the non linear beam equation on the d-dimensional torus utt+∆2u+mu+g(x,u)=0, t∈R,x∈Td, (∗) where g(x,u) = 4u3 +O(u4). Namely, we show that, for generic m, many of the small amplitude invariant finite dimensional tori of the linear equation (∗)g=0, written as the system ut =−v, vt =∆2u+mu, persist as invariant tori of the nonlinear equation (∗), re-written similarly. If d ≥ 2, then not all the persisted tori are linearly stable, and we construct explicit examples of partially hyperbolic invariant tori. The unstable invariant tori, situated in the vicinity of the origin, create around them some local instabilities, in agreement with the popular belief in the nonlinear physics that small-amplitude solutions of space-multidimensonal hamiltonian PDEs behave in a chaotic way.

Dates et versions

hal-01480367 , version 1 (01-03-2017)

Identifiants

Citer

L. Hakan Eliasson, Benoît Grébert, Sergei B. Kuksin. KAM for the nonlinear beam equation. Geometric And Functional Analysis, 2016, 26 (6), pp.1588 - 1715. ⟨10.1007/s00039-016-0390-7⟩. ⟨hal-01480367⟩
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