KAM for the nonlinear beam equation

• Published in: Geometric and functional analysis Vol. 26; no. 6; pp. 1588 - 1715
• Main Authors: Eliasson, L. Hakan, Grébert, Benoît, Kuksin, Sergei B.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.12.2016; Springer Nature B.V; Springer Verlag
• Subjects: Analysis; Analysis of PDEs; Invariants; Mathematics; Mathematics and Statistics; Nonlinear equations; Toruses; Hamiltonian systems; Beam equation; KAM theory; hamiltonian systems
• ISSN: 1016-443X; 1420-8970
• DOI: 10.1007/s00039-016-0390-7

In this paper we prove a KAM theorem for small-amplitude solutions of the non linear beam equation on the d-dimensional torus u t t + Δ 2 u + m u + ∂ 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 + m u , 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.