scholarly journals An example for nonexistence of minimal foliations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Xue-Qing Miao
Keyword(s):  
Local Potential ◽  
High Dimensional ◽  
Minimal Foliations ◽  
Minimal Laminations ◽  
Kontorova Model

AbstractFor the high-dimensional Frenkel–Kontorova model on lattices, we have concluded that there are heteroclinic connections between neighboring Birkhoff minimizers which are more periodic. This conclusion is based on the existence of neighboring elements, i.e., the existence of gaps. By adding a large enough oscillation to the local potential, I prove that all minimal foliations can be destroyed into minimal laminations, and hence there always exist gaps.

Minimizers with Bounded Action for the High-Dimensional Frenkel–Kontorova Model

2015 ◽  
Vol 25 (08) ◽  
pp. 1550098
Author(s):  
Xue-Qing Miao ◽  
Ya-Nan Wang ◽  
Wen-Xin Qin
Keyword(s):  
Nearest Neighbor ◽  
Global Minimizer ◽  
High Dimensional ◽  
Minimal Energy ◽  
One Dimensional ◽  
Mather Theory ◽  
Twist Maps ◽  
The One ◽  
Kontorova Model

In Aubry–Mather theory for monotone twist maps or for one-dimensional Frenkel–Kontorova (FK) model with nearest neighbor interactions, each global minimizer (minimal energy configuration) is naturally Birkhoff. However, this is not true for the one-dimensional FK model with non-nearest neighbor interactions or for the high-dimensional FK model. In this paper, we study the Birkhoff property of minimizers with bounded action for the high-dimensional FK model.

The observation of solutions in Ni3Nb

1988 ◽  
Vol 46 ◽  
pp. 540-541
Author(s):  
Z.M. Wang ◽  
J.P. Zhang
Keyword(s):  
Electron Microscopy ◽  
High Resolution ◽  
Phase Modulation ◽  
High Resolution Electron Microscopy ◽  
Antiphase Domain ◽  
Boundary Area ◽  
Matrix Type ◽  
Resolution Electron ◽  
Domain Boundaries ◽  
Kontorova Model

High resolution electron microscopy reveals that antiphase domain boundaries in β-Ni3Nb have a hexagonal unit cell with lattice parameters ah=aβ and ch=bβ, where aβ and bβ are of the orthogonal β matrix. (See Figure 1.) Some of these boundaries can creep “upstairs” leaving an incoherent area, as shown in region P. When the stepped boundaries meet each other, they do not lose their own character. Our consideration in this work is to estimate the influnce of the natural misfit δ{(ab-aβ)/aβ≠0}. Defining the displacement field at the boundary as a phase modulation Φ(x), following the Frenkel-Kontorova model [2], we consider the boundary area to be made up of a two unit chain, the upper portion of which can move and the lower portion of the β matrix type, assumed to be fixed. (See the schematic pattern in Figure 2(a)).

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