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KAM for the nonlinear beam equation

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Abstract

In this paper we prove a KAM theorem for small-amplitude solutions of the non linear beam equation on the d-dimensional torus

$$u_{tt}+\Delta^2 u+m u + \partial_u G(x,u)=0, \quad t \in {\mathbb{R}}, x \in {\mathbb{T}^d}, \quad (*)$$

where \({G(x,u)=u^4+ O(u^5)}\). 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

$$u_t=-v,\quad v_t=\Delta^2 u+mu,$$

persist as invariant tori of the nonlinear equation \({(*)}\), re-written similarly. The persisted tori are filled in with time-quasiperiodic solutions of \({(*)}\). If \({d\ge2}\), 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-multidimensional Hamiltonian PDEs behave in a chaotic way.

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Authors and Affiliations

  1. Université Paris Diderot, Sorbonne Paris Cité, Institut de, Mathémathiques de Jussieu-Paris Rive Gauche, UMR 7586, CNRS, Sorbonne Universités, UPMC Univ. Paris 06, 75013, Paris, France

    L. Hakan Eliasson

  2. Laboratoire de Mathématiques Jean Leray, Université de Nantes, UMR CNRS 6629, 2, Rue de la Houssinière, 44322, Nantes Cedex 03, France

    Benoît Grébert

  3. CNRS, Institut de Mathémathiques de Jussieu-Paris Rive Gauche, UMR 7586, Université Paris Diderot, Sorbonne Paris Cité, Sorbonne Universités, UPMC Univ. Paris 06, 75013, Paris, France

    Sergei B. Kuksin

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  1. L. Hakan Eliasson
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  2. Benoît Grébert
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  3. Sergei B. Kuksin
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Correspondence to Sergei B. Kuksin.

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Eliasson, L.H., Grébert, B. & Kuksin, S.B. KAM for the nonlinear beam equation. Geom. Funct. Anal. 26, 1588–1715 (2016). https://doi.org/10.1007/s00039-016-0390-7

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  • DOI: https://doi.org/10.1007/s00039-016-0390-7

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