arXiv subjects

Jianxing Du

Publications and source records attributed to Jianxing Du.

3 recordsLinked to original sources

arXiv2025

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

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.

math.DS↗

arXiv2024

On the existence of solutions for Frenkel-Kontorova models on quasi-crystals

This article focuses on recent investigations on equilibria of the Frenkel-Kontorova models subjected to potentials generated by quasi-crystals. We present a specific one-dimensional model with an explicit potential driven by the Fibonacci quasi-crystal. For a given positive number $θ$, we show that there are multiple equilibria with rotation number $θ$, e.g., a minimal configuration and a non-minimal equilibrium configuration. Some numerical experiments verifying the existence of such equilibria are provided.

math.DS↗

arXiv2024

Singular dynamics for discrete weak K.A.M. solutions of exact twist maps

For any exact twist map $f$ and any cohomology class $c\in\mathbb{R}$, let $u_c$ be any associated discrete weak K.A.M. solution, and we introduce an inherent Lipschitz dynamics $Σ_+$ given by the discrete forward Lax-Oleinik semigroup. We investigate several properties of $Σ_+$ and show that the non-differentiable points of $u_c$ are globally propagated and forward invariant by $Σ_+$. In particular, such propagating dynamics possesses the same rotation number $α'(c)$ as the associated Aubry-Mather set at cohomology class $c$. As applications, we provide via $Σ_+$ {a discrete analogue of Bernard's regularization theorem \cite{Ber07} and} a detailed exposition of Arnaud's observation \cite{Arnaud_2011}. Furthermore, we construct and analyze the corresponding dynamics on the full pseudo-graphs of discrete weak K.A.M. solutions.

math.DS↗