## Abstract

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 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), z_{i}(t)) traces such that W(z_{i}(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) |
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Pages (from-to) | 2571-2589 |

Number of pages | 19 |

Journal | Discrete and Continuous Dynamical Systems- Series A |

Volume | 38 |

Issue number | 5 |

DOIs | |

State | Published - May 2018 |

## All Science Journal Classification (ASJC) codes

- Analysis
- Discrete Mathematics and Combinatorics
- Applied Mathematics