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

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 in...