Dynamics of the Frenkel–Kontorova model with irrational mean spacing

Abstract

In this paper we focus on the dynamics of the overdamped limit case of the Frenkel-Kontorova (F-K) model with irrational mean spacing. We extend the results of Baesens and MacKay on rational mean spacing case to irrational case: there exists a depinning force F d 0 such that the F-K model possesses an equilibrium with a given mean spacing if the constant driving force F ∈ [0, F d ] or it admits a uniform sliding solution provided F > F d . Moreover, the average velocity of the particles exists and is unique, and it is non-decreasing and continuous with respect to F ∈ [0, +∞). In particular, we prove that the average velocity depends continuously upon the mean spacing.

Existence and Modulation of Uniform Sliding States in Driven and Overdamped Particle Chains

“…We should mention here that similar conclusions in Theorem C for (1.5) with harmonic interaction potential have been reported in a previous work [31].…”

“…We should mention here that we have already discussed in [31] the irrational mean spacing case for the overdamped limit case of (1.1):…”

“…If the interaction potential in (1.1) is harmonic, then by reducing to a fixed point theorem, we applied the Schauder fixed point theorem to discuss in [31] the uniform sliding states for the overdamped limit case (1.5). However, for interaction potentials of general form, the approach in [31] seems unworkable.…”

“…We reduced the problem of existence of modulation functions to a fixed point problem in a suitable Sobolev space which was solved by the Schauder fixed point theorem. However, we emphasize that the approach in [31] depends heavily on the linearity of coupling term so that we can define a linear operator. It seems unworkable for anharmonic interaction potentials.…”

Dynamics of incommensurate structures of the\\ Frenkel-Kontorova model

“…文 献 [16] 则利用单调系统理论研究了在周期边界条件下常外力驱动情形的动力学. 文献 [17] 讨论了无 理平均间距的情形. 周期外力驱动情形在文献 [4] (周期边界条件) 和 [9] (无理平均间距) 得到讨论.…”