Stickiness of KAM tori for higher dimensional beam equation

TL;DR

This paper proves that KAM tori for higher dimensional beam equations exhibit stickiness, meaning solutions starting nearby remain close for polynomially long times, by constructing a suitable partial normal form.

Contribution

It introduces a higher-order partial normal form with p-tame property to establish the stickiness of KAM tori in higher dimensional beam equations.

Findings

Solutions near KAM tori stay close for polynomial long times.

Constructs a p-tame partial normal form around KAM tori.

Demonstrates the stickiness property for higher dimensional beam equations.

Abstract

This paper is concerned with the stickiness of invariant tori obtained by KAM technics (so-called KAM tori) for higher dimensional beam equation. We prove that the KAM tori are sticky, i.e. the solutions starting in the $\delta$-neighborhood of KAM torus still stay close to the KAM torus for a polynomial long time such as $|t|\leq \delta^{-\mathcal{M}}$ with any $\mathcal{M}\geq 0$, by constructing a partial normal form of higher order, which satisfies $p$-tame property, around the KAM torus.