We consider the follow-the-leader model for traffic flow. The position of each car zi(t) satisfies an ordinary differential equation, whose speed depends only on the relative position zi+1(t) of the car ahead. Each car perceives a local density ?i(t). We study a discrete traveling wave profile W(x) along which the trajectory (?i(t), zi(t)) traces such that W(zi(t)) = ?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 ? ±? are given. Furthermore, we show that such profiles are locally stable, attracting nearby monotone solutions of the follow-the-leader model.
|Original language||English (US)|
|Number of pages||19|
|Journal||Discrete and Continuous Dynamical Systems- Series A|
|State||Published - May 2018|
All Science Journal Classification (ASJC) codes
- Discrete Mathematics and Combinatorics
- Applied Mathematics