KAM for the nonlinear beam equation

International audienceIn this paper we prove a KAM result for the non linear beam equation on the d-dimensional torusutt+∆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 systemut =−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