Geom. Funct. Anal. Vol. 26 (2016) 1588–1715 DOI: 10.1007/s00039-016-0390-7 Published online November 4, 2016 c 2016 Springer International Publishing GAFA Geometric And Functional Analysis KAM FOR THE NONLINEAR BEAM EQUATION L. Hakan Eliasson, Benoˆ ıt Gr´ ebert and Sergei B. Kuksin 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 +Δ 2 u + mu + ∂ u G(x, u)=0, t ∈ R,x ∈ T d , (∗) 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, v t =Δ 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 ≥ 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-multidimensional Hamiltonian PDEs behave in a chaotic way. Contents 1 Introduction ....................................... 1590 1.1 The beam equation and KAM for PDE’s. ................... 1590 1.2 Beam equation in real and complex variables. ................. 1592 1.3 Invariant tori and admissible sets. ........................ 1594 1.4 The Birkhoff normal form. ............................ 1596 1.5 The KAM theorem. ............................... 1597 1.6 Small amplitude solutions for the beam equation. ............... 1598 Part I. Some Functional Analysis ............................. 1601 2 Matrix Algebras and Function Spaces ........................ 1601 2.1 The phase space. ................................. 1601 2.2 A matrix algebra. ................................. 1603 2.3 Functions. ..................................... 1606 2.4 Flows ....................................... 1610 Keywords and phrases: Beam equation, KAM theory, Hamiltonian systems
KAM FOR THE NONLINEAR BEAM EQUATION
Jun 19, 2023
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