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# Traveling waves for a microscopic model of traffic flow

• * Corresponding author: Wen Shen
• We consider the follow-the-leader model for traffic flow. The position of each car $z_i(t)$ satisfies an ordinary differential equation, whose speed depends only on the relative position $z_{i+1}(t)$ of the car ahead. Each car perceives a local density $ρ_i(t)$. We study a discrete traveling wave profile $W(x)$ along which the trajectory $(ρ_i(t), z_i(t))$ traces such that $W(z_i(t)) = ρ_i(t)$ for all $i$ and $t>0$; see definition 2.2. We derive a delay differential equation satisfied by such profiles. Existence and uniqueness of solutions are proved, for the two-point boundary value problem where the car densities at $x\to±∞$ are given. Furthermore, we show that such profiles are locally stable, attracting nearby monotone solutions of the follow-the-leader model.

Mathematics Subject Classification: Primary: 35L02, 35L65; Secondary: 34B99, 35Q99.

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• Figure 1.  Typical graphs of $G(\lambda)$ and location of the zeros

Figure 2.  Numerical simulations for the approximate sequence $W_n(x)$ for various values of $\hat x_n$. We use $\rho_- = 0.3$, $\rho_+ = 0.7$, $\ell = 0.5$, $V = 1$, and $\phi(\rho) = 1-\rho$. The solid curve is the graph of $\psi(x) = \rho_+-0.2 e^{-\lambda_+x}$, plotted on the interval $0\le x\le 2$. The dotted curves are plots for $W_n(x)$ with $\hat x_n = 0, 0.1, 0.25, 0.5, 1$

Figure 3.  Numerical simulations of stationary profiles $W(x)$ with various values of $(\rho_-, \rho_+)$

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