# American Institute of Mathematical Sciences

July  2019, 39(7): 4001-4040. doi: 10.3934/dcds.2019161

## Traveling waves for nonlocal models of traffic flow

 Mathematics Department, Pennsylvania State University, University Park, PA 16802, USA

* Corresponding author: Wen Shen

Received  August 2018 Revised  December 2018 Published  April 2019

We consider several nonlocal 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 nonlocal flux functions. For both types of models, we study stationary traveling wave profiles and stationary discrete traveling wave profiles. (See definitions 1.1 and 1.2, respectively.) 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.

Citation: Johanna Ridder, Wen Shen. Traveling waves for nonlocal models of traffic flow. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 4001-4040. doi: 10.3934/dcds.2019161
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Typical profiles $P(\cdot)$ with $v(\rho) = 1-\rho, h = 0.2, \ell = 0.01$ and various $[\rho^-, \rho^+]$ values given in the legends. In the left we use the weight function $w(x) = \frac{2}{h}-\frac{2x}{h^2}$ on $(0, h)$ where $w'<0$, while in right plot we use $w(x) = \frac{2x}{h^2}$ on $(0, h)$ where $w'>0$
Typical solutions of the FtLs model $(z_i(t), \rho_i(t))$ at various time $t$, with oscillatory initial condition. Above: $w(x) = \frac{2}{h}-\frac{2x}{h^2}$ and the solution approaches some profile as $t$ grows. Below: $w(x) = \frac{2x}{h^2}$ and the solution oscillates more as $t$ grows
Left: Graphs of $Q(x)$ and $\widehat Q(x)$ on $[x_1, x_1+h]$. Right: Graphs of shifted functions $Q(s+x_1)$ and $\widehat Q(s+x_2)$
Typical profiles $Q(\cdot)$ with $v(\rho) = 1-\rho, h = 0.2$, and various asymptotic values $[\rho^-, \rho^+]$ given in the legends. For the left plot we use $w(x) = \frac{2}{h}-\frac{2x}{h^2}$ for $x\in(0, h)$, and for right plot we use $w(x) = \frac{2x}{h^2}$ for $x\in(0, h)$
Solution for the nonlocal conservation law (1.1) with oscillatory initial condition, at $t = 0, 0.4, 0.8$. Top: With $w(x) = \frac{2}{h}-\frac{2x}{h^2}$, i.e., $w'<0$, the solution quickly approaches the traveling wave profile $Q(x)$ as $t$ grows. Bottom: With $w(x) = \frac{2x}{h^2}$, i.e., $w'>0$, the solution becomes more oscillatory as $t$ grows
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