Wen-Long Li, Xiaojun Cui
We study a generalized Frenkel-Kontorova model. Using minimal and Birkhoff solutions as building blocks, we construct a lot of homoclinic solutions and heteroclinic solutions for this generalized Frenkel-Kontorova model under gap conditions. These new solutions are not minimal and Birkhoff any more. We use constrained minimization method to prove our results.
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In this article, the Frenkel-Kontorova model for dislocation dynamics is considered, where the on-site potential consists of quadratic wells joined by small arcs, which can be spinodal (concave) as commonly assumed in physics. The existence of heteroclinic waves ---making a transition from one well of the on-site potential to another--- is proved by means of a Schauder fixed point argument. The setting developed here is general enough to treat such a Frenkel-Kontorova chain with smooth ($C^2$) on-site potential. It is shown that the method can also establish the existence of two-transition wav…
Phonon gap analysis for equilibria of perturbed almost-periodic Frenkel-Kontorova models
本文研究了扰动近周期弗伦克尔-康托罗瓦模型的平衡态，发现KAM方法得到的平衡态无声子间隙，而反可积极限方法得到的平衡态有声子间隙。
Anti-integrable limits for generalized Frenkel-Kontorova models on almost-periodic media
该研究针对几乎周期介质中的广义Frenkel-Kontorova模型，通过建立反可积极限理论，证明了在足够大势能下存在具有任意指定旋转数的局部唯一双曲平衡态，适用于短程和长程模型及其多维推广。
On the existence of solutions for Frenkel-Kontorova models on quasi-crystals
该研究针对准晶体势场下的Frenkel-Kontorova模型，提出斐波那契准晶体驱动的一维显式势场模型，证明给定正数$θ$下存在多个旋转数为$θ$的平衡态，并通过数值实验验证了结论。
We study heteroclinic solutions of a generalized Frenkel-Kontorova model. Using the methods of Rabinowitz and Stredulinsky, we prove that if the rotation vector of the configuration is rational and if there is an adjacent pair of periodic configurations, then there is a solution that is heteroclinic in one fixed direction and periodic in other directions. Furthermore, if the above heteroclinic solutions have an adjacent pair, then there is a solution that is heteroclinic in two directions and periodic in other directions. The procedure can be repeated to produce more complex solutions. Thus we…
We consider a generalized one-dimensional chain in a periodic potential (the Frenkel-Kontorova model), with dissipative, pulsating (or ratchet) dynamics as a model of transport when the average force on the system is zero. We find lower bounds on the transport speed under mild assumptions on the asymmetry and steepness of the site potential. Physically relevant applications include explicit estimates of the pulse frequencies and mean spacings for which the transport is non-zero, and more specifically the pulse frequencies which maximize work. The bounds explicitly depend on the pulse period an…
We study a generalized Frenkel-Kontorova model and obtain periodic and heteroclinic mountain pass solutions. Heteroclinic mountain pass solution in the second laminations is new to the generalized Frenkel-Kontorova model. Our proof follows that of Bolotin and Rabinowitz for an Allen-Cahn equation, which is different with heat flow method for finding critical point of Frenkel-Kontorova model in the literature. The proofs depend on suitable choices of functionals and working spaces. We also study the multiplicity of these mountain pass solutions.
The Frenkel-Kontorova model describes how an infinite chain of atoms minimizes the total energy of the system when the energy takes into account the interaction of nearest neighbors as well as the interaction with an exterior environment. An almost-periodic environment leads to consider a family of interaction energies which is stationary with respect to a minimal topological dynamical system. We introduce, in this context, the notion of calibrated configuration (stronger than the standard minimizing condition) and, for continuous superlinear interaction energies, we show the existence of thes…
We consider generalized Frenkel-Kontorova models on higher dimensional lattices. We show that the invariant tori which are parameterized by continuous hull functions can be destroyed by small perturbations in the $C^r$ topology with $r<1$.
Gevrey KAM equilibria for quasi-periodic long-range Frenkel-Kontorova models
本文针对准周期长程Frenkel-Kontorovich模型，将KAM定理从解析势推广至Gevrey正则势，通过拟牛顿法与精细估计证明近似解附近存在保持准周期性的真解。
Rigidity of the Suris' potential in the Frenkel-Kontorova Model
本文研究了Frenkel-Kontorova模型中Suris势的局部刚性及谱刚性，通过发展作用角坐标并构造里斯基底，证明了该势的可积性在局部的刚性特性。
We rigorously show that dissipatively driven Frenkel-Kontorova models with either uniform or time-periodic driving asymptotically synchronize for a wide range of initial conditions. The main tool is a new Lyapunov function, as well as a 2D representation of the attractor. We then characterize dynamical phase transitions and outline new algorithms for determining them.
In this paper, we consider the Frenkel-Kontorova model of a one dimensional chain of atoms submitted to a potential. This potential splits into an interaction potential and a potential induced by an underlying substrate which is a quasicrystal. Under standard hypotheses, we show that every minimal configuration has a rotation number, that the rotation number varies continuously with the minimal configuration, and that every non negative real number is the rotation number of a minimal configuration. This generalizes well known results obtained by S. Aubry and P.Y. le Daeron in the case of a cry…
We construct new traveling wave solutions of moving kink type for a modified, driven, dynamic Frenkel-Kontorova model, representing dislocation motion under stress. Formal solutions known so far are inadmissible for velocities below a thresh- old value. The new solutions fill the gap left by this loss of admissibility. Analytical and numerical evidence is presented for their existence; however, dynamic simulations suggest that they are probably unstable.
Equilibrium configurations for generalized Frenkel-Kontorova models on quasicrystals
本文研究准晶背景下广义Frenkel-Kontorova模型，利用模式等变几乎周期函数刻画势，在温和的C²条件与非退化假设下证明任意指定旋转数/向量/平面的平衡构型存在性，方法基于反可积极限。
The Frenkel-Kontorova model for dislocation dynamics from 1938 is given by a chain of atoms, where neighbouring atoms interact through a linear spring and are exposed to a smooth periodic on-site potential. A dislocation moving with constant speed corresponds to a heteroclinic travelling wave, making a transition from one well of the on-site potential to another. The ensuing system is nonlocal, nonlinear and nonconvex. We present an existence result for a class of smooth nonconvex on-site potentials. Previous results in mathematics and mechanics have been limited to on-site potentials with har…
We study numerically the "analyticity breakdown" transition in 1-dimensional quasi-periodic media. This transition corresponds physically to the transition between pinned down and sliding ground states. Mathematically, it corresponds to the solutions of a functional equation losing their analyticity properties. We implemented some recent numerical algorithms that are efficient and backed up by rigorous results so that we can compute with confidence even close to the breakdown. We have uncovered several phenomena that we believe deserve a theoretical explanation: A) The transition happens in a …
Exploiting the monotone property of the gradient dynamics of the Frenkel-Kontorova model, we locate in the space of (p,q)-configurations the ordered and unordered stationary states, as well as forbidden regions for such states. Moreover we show that some generalized Frenkel--Kontorova models (associated to multiharmonic standard maps) can have ordered (p,q)--configurations that are neither action minimizing nor minimaximizing, and give their location with respect to the set of (p,q)--minimizers and minimaximizers.
Depinning of discommensurations for tilted Frenkel-Kontorova chains
本文研究了倾斜 Frenkel-Kontorova 链中 discommensurations 的去钉扎现象，证明了在阈值 $F_d(p/q)$ 以下存在平衡和周期滑动 discommensurations，并揭示了其与扭转映射旋转不变圆的等价关系。
We consider Frenkel-Kontorova models corresponding to 1 dimensional quasicrystals. We present a KAM theory for quasi-periodic equilibria. The theorem presented has an \emph{a-posteriori} format. We show that, given an approximate solution of the equilibrium equation, which satisfies some appropriate non-degeneracy conditions, then, there is a true solution nearby. This solution is locally unique. Such a-posteriori theorems can be used to validate numerical computations and also lead immediately to several consequences a) Existence to all orders of perturbative expansion and their convergence b…
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