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

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→ ± ∞. 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.