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

Zhicheng Tong, Yong Li

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].

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

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 or even for more general perturbed networks, frequency-preserving almost periodic breathers do persist, provided that the local potential and the coupling potential satisfy certain assumptions. In particular, this yields the first frequency-preserving result for the Aubry--MacKay conjecture [MA94,Aub95].

Paper Structure

This paper contains 23 sections, 141 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Persistence of full-dimensional frequency-preserving tori
  3. Infinite-dimensional non-resonance condition, spatial structure and norms
  4. KAM with infinite-dimensional Diophantine frequency-preserving
  5. KAM with infinite-dimensional weak Diophantine frequency-preserving
  6. Further discussion on the spatial structure
  7. Application to the Aubry--MacKay conjecture
  8. Discrete breathers
  9. On the Aubry--MacKay conjecture: Persistence of frequency-preserving almost periodic breathers
  10. Broader applications: More general networks of coupled oscillators
  11. Special cases and asymptotic analysis
  12. The case of $V(x)=x^{2p}$ with $p \in \mathbb{N}^+$
  13. The case of $V\left( x \right) = 2{x^2} + {x^4}$
  14. Some notes on Assumptions \ref{['con:a2']}--\ref{['con:a4']}
  15. Comparison with previous literature
  16. ...and 8 more sections

Figures (3)

  • Figure 1: Weak coupling in system \ref{['MACXT']}: only neighboring oscillators are coupled.
  • Figure 2: General coupling in system \ref{['MACXIT2']}: coupling occurs exclusively among each consecutive triplet of oscillators.
  • Figure 3: General coupling in system \ref{['MACXIT2']}: coupling occurs exclusively among each consecutive quartet of oscillators.

Theorems & Definitions (7)

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