数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (2): 564-582.doi: 10.1007/s10473-023-0207-3

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MINIMAL FOLIATIONS FOR THE HIGH-DIMENSIONAL FRENKEL-KONTOROVA MODEL*

Xueqing Miao1,†, Jianhua Ge1, Wenxin Qin2, Yanan Wang3   

  1. 1. School of Sciences, Nantong University, Nantong 226019, China;
    2. Department of Mathematics, Soochow University, Suzhou 215006, China;
    3. School of Mathematical Sciences, Nanjing Normal University, Nanjing 210097, China
  • 收稿日期:2021-09-07 修回日期:2021-12-21 出版日期:2023-03-25 发布日期:2023-04-12
  • 通讯作者: †Xueqing Miao, E-mail: ntitmxq@163.com.
  • 作者简介:Jianhua Ge, E-mail: jsgjh2009@163.com; Wenxin Qin, E-mail: qinwx@suda.edu.cn; Yanan Wang, E-mail: kuangbiao14@126.com
  • 基金资助:
    The first author was supported by the National Natural Science Foundation of China (11701298).

MINIMAL FOLIATIONS FOR THE HIGH-DIMENSIONAL FRENKEL-KONTOROVA MODEL*

Xueqing Miao1,†, Jianhua Ge1, Wenxin Qin2, Yanan Wang3   

  1. 1. School of Sciences, Nantong University, Nantong 226019, China;
    2. Department of Mathematics, Soochow University, Suzhou 215006, China;
    3. School of Mathematical Sciences, Nanjing Normal University, Nanjing 210097, China
  • Received:2021-09-07 Revised:2021-12-21 Online:2023-03-25 Published:2023-04-12
  • Contact: †Xueqing Miao, E-mail: ntitmxq@163.com.
  • About author:Jianhua Ge, E-mail: jsgjh2009@163.com; Wenxin Qin, E-mail: qinwx@suda.edu.cn; Yanan Wang, E-mail: kuangbiao14@126.com
  • Supported by:
    The first author was supported by the National Natural Science Foundation of China (11701298).

摘要: For the high-dimensional Frenkel-Kontorova (FK) model on lattices, we study the existence of minimal foliations by depinning force. We introduce the tilted gradient flow and define the depinning force as the critical value of the external force under which the average velocity of the system is zero. Then, the depinning force can be used as the criterion for the existence of minimal foliations for the FK model on a $\mathbb{Z}^d$ lattice for $d>1$.

关键词: Aubry-Mather theory, Frenkel-Kontorova model, minimal foliation, depinning force, gradient flow

Abstract: For the high-dimensional Frenkel-Kontorova (FK) model on lattices, we study the existence of minimal foliations by depinning force. We introduce the tilted gradient flow and define the depinning force as the critical value of the external force under which the average velocity of the system is zero. Then, the depinning force can be used as the criterion for the existence of minimal foliations for the FK model on a $\mathbb{Z}^d$ lattice for $d>1$.

Key words: Aubry-Mather theory, Frenkel-Kontorova model, minimal foliation, depinning force, gradient flow