Minimizers And Foliations For The High-Dimensional Frenkel-Kontorova Model

We consider the generalized Frenkel-Kontorova (F-K) model on high-dimensional lattice Zd, which describes wide range and non-linear coupling between particles of a d-dimensional crystal lying in the periodic background potential. In this paper, we use the main tools of the maximum-minimum principle, the anti-integrable limit method and the gradient flow to discuss the classification and properties of Birkhoff minimizers for high-dimensional lattice systems, depinning forces, and minimal foliations.In Chapter 2, we study the classification by introducing secondary invariants of Birkhoff minimizers with rotation vector ω which is rationally dependent. First for the admissible secondary invariants (α1,…αt), we prove that the set of Birkhoff minimiz-ers.M(α1,…αt) is totally ordered. The set M(α1) UM(α1,α2)∪…∪(α1,…αt) is also totally ordered. Then similar to the d= 1 case, if there are neighboring ele-ments, there exist heteroclinic orbits. That is:if there are neighboring elements x-and x+ with the same secondary invariants (α1,…αt-1), then there exists a Birkhoff minimizer x with secondary invariants (α1,…αt) connecting x- and x+.Further, in Chapter 3 we discuss Birkhoff properties of the minimizers with bound-ed action for the high-dimensional lattice systems. We get the following conclusions: if x is a minimizer with bounded action, there is a sequence of its translation tending to a Birkhoff minimizer. If x= (xk) is a, minimizer with{xk-k·ω} being bounded and ω rationally independent, then x is Birkhoff.Finally, in Chapter 4 we use gradient flow method to study the tilted F-K model (i. e. the F-K model with external driving force). By introducing the concept of depinning force, we get a criterion for the existence of minimal foliations. For the F-K model of high-dimensional lattice Zd, we conclude that if depinning force is zero, M(a1) is a minimal foliation.