### 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 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

*Discrete and Continuous Dynamical Systems- Series A*,

*38*(5), 2571-2589. https://doi.org/10.3934/dcds.2018108

}

*Discrete and Continuous Dynamical Systems- Series A*, vol. 38, no. 5, pp. 2571-2589. https://doi.org/10.3934/dcds.2018108

**Traveling waves for a microscopic model of traffic flow.** / Shen, Wen; Shikh-Khalil, Karim.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Traveling waves for a microscopic model of traffic flow

AU - Shen, Wen

AU - Shikh-Khalil, Karim

PY - 2018/5

Y1 - 2018/5

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85043584633&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85043584633&partnerID=8YFLogxK

U2 - 10.3934/dcds.2018108

DO - 10.3934/dcds.2018108

M3 - Article

AN - SCOPUS:85043584633

VL - 38

SP - 2571

EP - 2589

JO - Discrete and Continuous Dynamical Systems

JF - Discrete and Continuous Dynamical Systems

SN - 1078-0947

IS - 5

ER -