Modulus of continuity for depinning force at rational rotation symbols and application

Abstract

The depinning force for the Frenkel-Kontorova chain is a critical value $F_d\left(\omega \right)$ of the driving force F up to which there continue to be Birkhoff equilibria of rotation symbol ω and above which there are none. In this paper we investigate the modulus of continuity for the depinning force at rational rotation symbols $p/q+$ and $p/q-$ and obtain the estimate$\left|F_d\right(p/q+)-F_d(\omega \left)\right|\le C|q{\omega }^{⁎}-p|,\text{ for }\omega >p/q+,$ where C is a constant and ${\omega }^{⁎}$ denotes the underlying number associated to the rotation symbol ω. A similar conclusion for $p/q-$ also holds true.

As an application, we give an open and dense result for $F_d(0/1+)>0$, a threshold of driving force such that there exist stationary fronts for $F\le F_d(0/1+)$ and traveling fronts for $F>F_d(0/1+)$.