Traveling Waves for a Microscopic Model of Traffic Flow

2017/09/26 by Wen Shen, Shen, Wen, Karim Shikh-Khalil +1
Engineering · Social Sciences · #34B99 #35L02 #35L65 #35Q99 #Analysis of PDEs (math.AP) #Evacuation and Crowd Dynamics #FOS: Mathematics #Traffic control and management #Transportation Planning and Optimization

paper · pdf · doi:10.48550/arxiv.1709.08892

openalex publication_date 2017/09/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the follow-the-leader model for traffic flow. The position of each car z_i(t) satisfies an ordinary differential equation, whose speed depends only on the relative position z_i+1(t) of the car ahead. Each car perceives a local density ρ_i(t). We study a discrete traveling wave profile W(x) along which the trajectory (ρ_i(t),z_i(t)) traces such that W(z_i(t))=ρ_i(t) for all i and t>0; see definition 2.2. We derive a delay differential equation satisfied by such profiles. Existence and uniqueness of solutions are proved, for the two-point boundary value problem where the car densities at x→±∞ are given. Furthermore, we show that such profiles are locally stable, attracting nearby monotone solutions of the follow-the-leader model.

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