Depinning transition of travelling waves for particle chains

Abstract

Abstract In this paper we apply Aubry–Mather theory for equilibria of 1D Hamiltonian lattice systems and the theory of invariant ordered circles to investigate the depinning transition of travelling waves for particle chains. Assume A < B are two critical values such that the particle chain has three homogeneous equilibria if the driving force F ∈ ( A , B ) . It is already known that there exist transition thresholds F c − ⩽ F c + of the driving force such that the particle chain has stationary fronts but no travelling fronts for F c − ⩽ F ⩽ F c + and travelling fronts but no stationary fronts if A < F < F c − or F c + < F < B . The novelty of our approach is that we prove the transition threshold F c + ( F c − ) coincides with the upper (lower) limit of the upper (lower) depinning force as the rotation number tends to zero from the right. Based on this conclusion, we demonstrate that when the driving force F ∈ ( F c − , F c + ) , besides stationary fronts there are various kinds of equilibria with rotation numbers close to zero such that the spatial shift map has positive topological entropy on the set of equilibria. Furthermore, we give a necessary and sufficient condition for the absence of propagation failure, i.e. F c − = F c + , in terms of a minimal foliation. Finally we show that F c ± are continuous with respect to potential functions in C 1 topology.