KAM for the nonlinear beam equation

TL;DR

This paper proves a KAM theorem for small-amplitude solutions of a nonlinear beam equation on a torus, showing the persistence of many invariant tori and the existence of quasiperiodic solutions, with some tori being unstable.

Contribution

It establishes the persistence of invariant tori for a nonlinear beam PDE on a torus, including explicit examples of partially hyperbolic tori, extending KAM theory to this setting.

Findings

Many small amplitude invariant tori persist under nonlinearity.

Existence of quasiperiodic solutions on these invariant tori.

Presence of partially hyperbolic, unstable invariant tori in higher dimensions.

Abstract

In this paper we prove a KAM theorem for small-amplitude solutions of the non linear beam equation on the d-dimensional torus $$u_{tt}+\Delta^2 u+m u + \partial_u G(x,u)=0\ ,\quad t\in { \mathbb{R}} , \; x\in \ { \mathbb{T}}^d, \qquad \qquad (*) $$ 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.