Stationary wave profiles for nonlocal particle models of traffic flow on rough roads

  • Jereme Chien
  • Wen ShenEmail author


We study a nonlocal particle model describing traffic flow on rough roads. In the model, each driver adjusts the speed of the car according to the condition over an interval in the front, leading to a system of nonlocal ODEs which we refer to as the Follow-the-Leaders model. Assuming that the road condition is discontinuous, we seek stationary wave profiles (see Definition 1.1) for the system of ODEs across this discontinuity. We derive a nonlocal delay differential equation with discontinuous coefficient, satisfied by the profiles, together with conditions on the asymptotic values as \(x\rightarrow \pm \infty \). Results on existence, uniqueness, and local stability are established for all cases. We show that, depending on the case, there might exist a unique profile, infinitely many profiles, or no profiles at all. The stability result also depends on cases. Various numerical simulations are presented. Finally, we establish convergence of these profiles to those of a local particle model, as well as those of a nonlocal PDE model.


Traffic flow Follow-the-Leaders Particle model Traveling waves Stationary wave profiles Existence and uniqueness Stability Convergence 

Mathematics Subject Classification

Primary 56M20 35L02 35L65 Secondary 34B99 35Q99 



The authors are grateful to the anonymous reviewer for useful remarks that led to an improvement of the manuscript.


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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Mathematics DepartmentPennsylvania State UniversityUniversity ParkUSA

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