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Depinning transition of travelling waves for particle chains
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Baesens, C., MacKay, R. S., Qin, W.-X. and Zhou, T. (2023) Depinning transition of travelling waves for particle chains. Nonlinearity, 36 (2). pp. 878-901. doi:10.1088/1361-6544/aca94b ISSN 0951-7715.
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WRAP-Depinning-transition-of-travelling-waves-for-particle-chains-MacKay-2022.pdf - Accepted Version - Requires a PDF viewer. Available under License Creative Commons Attribution Non-commercial No Derivatives 4.0. Download (329Kb) | Preview |
Official URL: http://dx.doi.org/10.1088/1361-6544/aca94b
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 $F\in (A,B)$. It is already known that there exist transition thresholds $F_c^-\leqslant F_c^+$ of the driving force such that the particle chain has stationary fronts but no travelling fronts for $F_c^-\leqslant F\leqslant F_c^+$ and travelling fronts but no stationary fronts if $A\lt F\lt F_c^-$ or $F_c^+\lt F\lt B$. 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 $F\in (F_c^-,F_c^+)$, 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 $F_c^\pm$ are continuous with respect to potential functions in C1 topology.
Item Type: | Journal Article | ||||||||||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||||||
Journal or Publication Title: | Nonlinearity | ||||||||||||
Publisher: | Institute of Physics Publishing Ltd. | ||||||||||||
ISSN: | 0951-7715 | ||||||||||||
Official Date: | February 2023 | ||||||||||||
Dates: |
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Volume: | 36 | ||||||||||||
Number: | 2 | ||||||||||||
Page Range: | pp. 878-901 | ||||||||||||
DOI: | 10.1088/1361-6544/aca94b | ||||||||||||
Status: | Peer Reviewed | ||||||||||||
Publication Status: | Published | ||||||||||||
Re-use Statement: | This is an author-created, un-copyedited version of an article accepted for publication/published in Nonlinearity. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at https://doi.org/10.1088/1361-6544/aca94b | ||||||||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||||||||
Copyright Holders: | IOP Publishing Ltd & London Mathematical Society | ||||||||||||
Date of first compliant deposit: | 20 December 2022 | ||||||||||||
Date of first compliant Open Access: | 20 December 2022 | ||||||||||||
RIOXX Funder/Project Grant: |
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