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 p(i)(t). We study a discrete traveling wave profile W(x) along which the trajectory (p(i)(t), z(i)(t)) traces such that W(z(z)(t)) = p(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 -> +/-infinity are given. Furthermore, we show that such profiles are locally stable, attracting nearby monotone solutions of the follow-the-leader model.