Development of kinks in car-following models

[1612.08500] Development of kinks in car-following models Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Nonlinear Sciences > Pattern Formation and Solitons arXiv:1612.08500 (nlin) [Submitted on 27 Dec 2016] Title: Development of kinks in car-following models Authors: Douglas A. Kurtze View a PDF of the paper titled Development of kinks in car-following models, by Douglas A. Kurtze View PDF HTML (experimental) Abstract: Many car-following models of traffic flow admit the possibility of absolute stability, a situation in which uniform traffic flow at any spacing is linearly stable. Near the threshold of absolute stability, these models can often be reduced to a modified Korteweg-deVries (mKdV) equation plus small corrections. The hyperbolic-tangent "kink" solutions of the mKdV equation are usually of particular interest, as they represent transition zones between regions of different traffic spacings. Solvability analysis then shows that only a single member of the one-parameter family of kink solutions is preserved by the correction terms, and this is interpreted as a kind of selection. We point out that one cannot extend the solvability analysis to a multiple-time-scales calculation, so that the solvability analysis does not point the way to any dynamical mechanism by which the "selected" kink might actually be selected. On the other hand, we display a two-parameter family of traveling wave solutions of the mKdV equation which describe regions of one traffic spacing embedded in traffic of a different spacing; this family includes the kink solutions as a limiting case. We carry out the multiple-time-scales calculation and find conditions under which the inclusions decay, conditions that lead to a selected inclusion, and conditions for which the inclusion evolves into a pair of kinks. Finally, we show that the usual "solvability" calculation for kink solutions does not in fact identify a selected kink, but -- for all kink solutions -- merely gives a first-order correction to the relation between the traffic spacings far behind and far ahead of the kink. Comments: 4 figures. Submitted to Phys. Rev. E Subjects: Pattern Formation and Solitons (nlin.PS) Cite as: arXiv:1612.08500 [nlin.PS] (or arXiv:1612.08500v1 [nlin.PS] for this version) https://doi.org/10.48550/arXiv.1612.08500 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Phys. Rev. E 95, 032221 (2017) Related DOI : https://doi.org/10.1103/PhysRevE.95.032221 Focus to learn more DOI(s) linking to related resources Submission history From: Douglas Kurtze [ view email ] [v1] Tue, 27 Dec 2016 05:21:33 UTC (24 KB) Full-text links: Access Paper: View a PDF of the paper titled Development of kinks in car-following models, by Douglas A. 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