广义Frenkel-Kontorova模型中的有序结构和无序结构

As one of the most important examples of monotone recurrence relations, the Frenkel-Kontorova (F-K) model is by now used to describe a wide range of physical systems. The generalized F-K model is the generalization of the classical F-K model, which has more dimensions and longer interactions. The ordered structures in the generalized F-K model are foliations and laminations, and the disordered structures are the invariant subsets without being ordered structures. The ordered structures are closely related to the stability of the system, and the disordered structures are used to describe chaotic behaviors of the system. In the project, we investigate the ordered and disordered structures in the generalized Frenkel-Kontorova model..The main topic of the project is the following. First, we apply the properties of monotone system, together with the variational method to give a criterion of the existence of foliation. We shall prove that the foliation with prescribed rotation vector exists if and only if the corresponding depinning force vanishes. The depinning force is the critical value of the driven force in the tilted model, below which the system is pinned, and above which the system is sliding. Second, we shall construct uncountable many laminations when the foliation disintegrate. Finally, we construct some orbits in every unstable area with theω-limit sets contained in one ordered structure and the α-limit sets contained in another one.

Frenkel-Kontorova 模型 (F-K模型) 是单调回复关系的一个重要的实例，它在物理学中有着广泛的应用。广义F-K模型是对经典F-K模型的一般化，它还包含了高维和长程耦合的情形。广义F-K模型中的有序结构是指叶状结构和层状结构。无序结构是指系统的除有序结构外的不变集。有序结构与系统的稳定性有着密切关系，无序结构是对系统的混沌行为的刻画。在此项目中，我们来研究广义F-K模型中的有序结构和无序结构。.本项目主要完成如下工作：第一，我们将利用系统的单调性和变分方法给出叶状结构存在的判定准则。我们拟证明，系统的某个叶状结构存在当且仅当相应的脱钉力为零。脱钉力是斜置模型中外力F的临界值，当F大于这个值时，系统存在平衡态；当F小于这个值时，系统会滑动起来。第二，我们将在叶状结构消失时，构造不可数无穷多个层状结构。最后，我们将在广义模型的不稳定区域中构造有序结构的连接轨道。

Frenkel-Kontorova(F-K)模型是Hamilton系统中一个非常重要实例，在固体物理学中占有非常重要的地位。本项目利用将单调梯度流来研究F-K模型中的有序结构和无序结构。这些结构对系统的稳定性起到重要作用。项目负责人及其合作者研究了多种F-K模型中的有序结构和无序结构，给出了叶状结构存在的判定准则和层状结构存在的条件，并研究了层状结构的形态。另外，还构造了系统的异宿轨道，并建立了连续情形下的变分原理。这些结果将会对进一步了解F-K模型起到重要作用。在本项目的支持下，项目负责人及其合作者在国际著名杂志《Journal of Differential Equations》、《Advances in Mathematics》和《Ergodic Theory of Dynamical Systems》上发表学术论文4篇。