# Stationary Wave Profiles for Nonlocal Particle Models of Traffic Flow on Rough Roads

## TL;DR

This paper investigates stationary wave profiles in a nonlocal particle traffic model on rough roads with discontinuous conditions, establishing existence, uniqueness, stability, and convergence results through analytical and numerical methods.

## Contribution

It introduces a nonlocal delay differential equation framework for traffic flow on rough roads with discontinuities, analyzing wave profiles and their stability.

## Key findings

- Existence and uniqueness of stationary wave profiles in various cases.
- Profiles may be unique, multiple, or nonexistent depending on conditions.
- Profiles converge to local particle and PDE models as parameters vary.

## Abstract

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 FtLs (follow-the-leaders) model. Assuming that the road condition is discontinuous at the origin, we seek stationary wave profiles (see Definition 1.1) for the system of ODEs across this discontinuity. We derive a non-local delay differential equation with discontinuous coefficient, satisfied by the profiles, together with conditions on the asymptotic values as $x\to\pm\infty$. Results on existence, uniqueness, and local stability are proved, for all cases. We show that, depending on the case, there might exist a unique profile, infinitely many profiles, or no profiles. 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.