Traveling Waves for a Microscopic Model of Traffic Flow

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 $\rho_i(t)$. We study a discrete traveling wave profile $W(x)$ along which the trajectory $(\rho_i(t),z_i(t))$ traces such that $W(z_i(t))=\rho_i(t)$ 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 $x\to\pm\infty$ are given. Furthermore, we show that such profiles are locally stable, attracting nearby monotone solutions of the follow-the-leader model.