Traveling Waves for Nonlocal Models of Traffic Flow

TL;DR

This paper investigates non-local traffic flow models, analyzing both microscopic and macroscopic formulations, and establishes the existence, uniqueness, and asymptotic behavior of traveling wave solutions.

Contribution

It introduces a unified analysis of traveling waves in non-local traffic models, deriving delay equations and proving their existence and stability.

Findings

Existence and uniqueness of stationary traveling wave profiles

Traveling wave profiles are time asymptotic limits for the models

Derivation of delay differential and integro-differential equations for profiles

Abstract

We consider several non-local models for traffic flow, including both microscopic ODE models and macroscopic PDE models. The ODE models describe the movement of individual cars, where each driver adjusts the speed according to the road condition over an interval in the front of the car. These models are known as the FtLs (Follow-the-Leaders) models. The corresponding PDE models, describing the evolution for the density of cars, are conservation laws with non-local flux functions. For both types of models, we study stationary traveling wave profiles and stationary discrete traveling wave profiles. We derive delay differential equations satisfied by the profiles for the FtLs models, and delay integro-differential equations for the traveling waves of the nonlocal PDE models. The existence and uniqueness (up to horizontal shifts) of the stationary traveling wave profiles are established. Furthermore, we show that the traveling wave profiles are time asymptotic limits for the corresponding Cauchy problems, under mild assumptions on the smooth initial condition.