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Anti-integrable limits for generalized Frenkel-Kontorova models on almost-periodic media

Jianxing Du, Xifeng Su

TL;DR

The paper develops an anti-integrable-limit approach for generalized Frenkel-Kontorova models in almost-periodic media, contrasting with KAM methods. By formulating a unified framework of invariant operators and the Aubry criterion, it proves the existence of locally unique, hyperbolic equilibria for large potentials $\lambda$ with prescribed rotation data, valid for short-range, long-range, and multidimensional variants. The core contribution is a constructive fixed-point argument that couples a Lipschitz invariant operator $\Delta$ with a pointwise Aubry operator $\Psi$ to produce equilibria satisfying $(\Delta+\lambda\Psi)u=0$; hyperbolicity follows via a twist-map interpretation. The results are complemented by concrete classes of Aubry-satisfying functions—periodic, pattern-equivariant, and almost-periodic—showing broad applicability, including the almost-periodic case where $\nabla f$ is Aubry (Theorem APisAubry). Overall, the work extends anti-integrable-limit theory to highly nonperiodic media and provides a rigorous path to hyperbolic equilibria in generalized FK models.

Abstract

We study the equilibrium configurations for generalized Frenkel-Kontorova models subjected to almost-periodic media. By contrast with the spirit of the KAM theory, our approach consists in establishing the other perturbation theory for fully chaotic systems far away from the integrable, which is called "anti-integrable" limits. More precisely, we show that for large enough potentials, there exists a locally unique equilibrium with any prescribed rotation number/vector/plane, which is hyperbolic. The assumptions are general enough to satisfy both short-range and long-range Frenkel-Kontorova models and their multidimensional analogues.

Anti-integrable limits for generalized Frenkel-Kontorova models on almost-periodic media

TL;DR

The paper develops an anti-integrable-limit approach for generalized Frenkel-Kontorova models in almost-periodic media, contrasting with KAM methods. By formulating a unified framework of invariant operators and the Aubry criterion, it proves the existence of locally unique, hyperbolic equilibria for large potentials with prescribed rotation data, valid for short-range, long-range, and multidimensional variants. The core contribution is a constructive fixed-point argument that couples a Lipschitz invariant operator with a pointwise Aubry operator to produce equilibria satisfying ; hyperbolicity follows via a twist-map interpretation. The results are complemented by concrete classes of Aubry-satisfying functions—periodic, pattern-equivariant, and almost-periodic—showing broad applicability, including the almost-periodic case where is Aubry (Theorem APisAubry). Overall, the work extends anti-integrable-limit theory to highly nonperiodic media and provides a rigorous path to hyperbolic equilibria in generalized FK models.

Abstract

We study the equilibrium configurations for generalized Frenkel-Kontorova models subjected to almost-periodic media. By contrast with the spirit of the KAM theory, our approach consists in establishing the other perturbation theory for fully chaotic systems far away from the integrable, which is called "anti-integrable" limits. More precisely, we show that for large enough potentials, there exists a locally unique equilibrium with any prescribed rotation number/vector/plane, which is hyperbolic. The assumptions are general enough to satisfy both short-range and long-range Frenkel-Kontorova models and their multidimensional analogues.
Paper Structure (9 sections, 8 theorems, 45 equations, 1 figure)

This paper contains 9 sections, 8 theorems, 45 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Invariant operators
  4. Aubry criterion
  5. Main theorem and its proof
  6. Examples of functions satisfying the Aubry criterion
  7. Periodic functions
  8. Pattern-equivariant functions
  9. Almost-periodic functions

Key Result

Theorem 1.1

Let $V$ be an almost-periodic function of class $C^2$ with at least one non-degenerate critical point. Then, there exists a constant $\lambda_0>0$, determined by $V$, such that for any $\lambda>\lambda_0$ and $\rho\in\mathbb{R}$, there exists a sequence $\{u_n\}_{n\in\mathbb{Z}}$ that satisfies the Moreover, the sequence $\{(u_n,u_n-u_{n-1})\}_{n\in\mathbb{Z}}$ constitutes a hyperbolic orbit of t

Figures (1)

  • Figure 1: The partial graph of a one-dimensional function satisfying the Aubry criterion is illustrated. The portion within $O+[-r,r]$ is represented by a solid line, while the portion outside $O+[-r,r]$ is depicted with a dotted line.

Theorems & Definitions (26)

  • Theorem 1.1
  • Remark 2.1
  • Definition 2.2: Invariant operator
  • Lemma 2.3
  • proof
  • Definition 2.4: Lipschitz continuous operator
  • Example : Long-range Frenkel-Kontorova models
  • Definition 2.5: Pointwise operator
  • Definition 2.6: Aubry criterion
  • Remark 2.7
  • ...and 16 more