高维Frenkel-Kontorova模型的第二类有序结构与异宿轨

The Frenkel-Kontorova model (FK model) with one-dimensional nearest neighbor coupling is a key example of classical Aubry-Mather theory. It has been widely used in physics. High-dimensional FK model is a generalization of one-dimensional FK model to high dimensional lattice systems. Since this generalization corresponds to further coupled-range of a more-dimensional crystal lying in a periodic background potential. The generalization can be used to describe a wide range of physical systems, so it has important significance. The ordered structures of the high-dimensional FK model are foliations and laminations, corresponding to invariant curves and Denjoy minimal sets respectively of the one-dimensional nearest neighbor coupling model. The ordered structures are closely related to the stability of the system. An orbit is heteroclinic if it connects two neighboring elements of same period.. The main aims of this project are as follows：First we will discuss secondary ordered structures of the high-dimensional FK model by introducing secondary invariants of Birkhoff minimizers. We plan to give a criterion of the existence of foliations by using monotone properties of the system, depinning force, and other tools. Then we shall study laminations when foliations disintegrate. Finally，we will investigate the ordering of the set of periodic configurations, the existence of neighboring elements, and heteroclinic orbits connecting neighboring elements after introducing the secondary invariants.

一维近邻耦合的Frenkel-Kontorova模型(FK模型)是经典Aubry-Mather理论的关键例子，在物理学中有着广泛的应用。高维FK模型是一维FK模型向高维格点系统的推广，这样的推广对应于周期势能场中更高维晶体的更大耦合范围。从而，它可以描述更广泛的物理系统，有重要的意义。高维FK模型的有序结构是指叶状结构和层状结构，分别对应于一维近邻耦合模型的不变曲线和Denjoy极小集。有序结构与系统的稳定性有密切的联系。异宿轨则是具有相同周期的相邻元之间的连接轨。. 本项目拟完成如下主要工作：首先，我们对高维FK模型的Birkhoff最小能量构型引入第二不变量，讨论系统的第二类有序结构。希望利用系统的单调性和脱钉力等工具给出叶状结构的判定准则。然后在叶状结构消失时，讨论层状结构的情况。最后，我们还将研究引入第二不变量后，周期构型集合的有序性，相邻元以及连接它们的异宿轨的存在性。

本项目主要讨论高维FK模型，完成如下主要工作：首先，我们对高维FK模型的Birkhoff最小能量构型引入第二不变量，讨论系统的第二类有序结构。利用系统的单调性和脱钉力等工具给出叶状结构的判定准则。然后在叶状结构消失时，讨论层状结构的情况。最后，我们还研究了周期构型集合的有序性，相邻元以及连接它们的异宿轨的存在性。

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