  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.
