An infinite-dimensional Kolmogorov theorem and the construction of almost periodic breathers

TL;DR

This paper proves infinite-dimensional Kolmogorov theorems ensuring the persistence of full-dimensional KAM tori with prescribed frequencies and applies these results to demonstrate the existence of frequency-preserving almost periodic breathers in perturbed oscillator networks.

Contribution

It introduces new infinite-dimensional Kolmogorov theorems under weaker non-resonance conditions and applies them to establish frequency-preserving breathers, advancing the Aubry--MacKay conjecture.

Findings

Persistence of full-dimensional KAM tori with prescribed frequencies.

Existence of frequency-preserving almost periodic breathers in perturbed networks.

First frequency-preserving result related to the Aubry--MacKay conjecture.

Abstract

In this paper, we present two infinite-dimensional Kolmogorov theorems based on non-resonant frequencies of Bourgain's Diophantine type or even weaker conditions. To be more precise, under a Legendre-type nondegeneracy condition for an infinite-dimensional Hamiltonian system, we prove the persistence of a full-dimensional KAM torus with a universally prescribed frequency independent of any spectral asymptotics. As an application, we prove that for a class of perturbed networks with weakly coupled oscillators described by \[\frac{{{{\rm d}^2}{x_n}}}{{{\rm d}{t^2}}} + V'\left( {{x_n}} \right) = \varepsilon_n {W'\left( {{x_{n + 1}} - {x_n}} \right) - \varepsilon_{n-1}W'\left( {{x_n} - {x_{n - 1}}} \right)} ,\quad n \in \mathbb{Z},\] or even for more general perturbed networks, frequency-preserving almost periodic breathers do persist, provided that the local potential $ V $ and the coupling potential $ W $ satisfy certain assumptions. In particular, this yields the first frequency-preserving result for the Aubry--MacKay conjecture [MA94,Aub95].