| Abstract: |
| We investigate nonlinear wave structures in the Optimal Velocity (OV) model, a fundamental microscopic traffic flow model describing car-following dynamics on a circuit. By introducing a traveling-wave representation of vehicle headways, the original ordinary differential system is reduced to a difference-differential equation. We focus on steep optimal velocity functions close to a step function, which generate sharp transition layers in the headway profile.
In the singular limit, explicit heteroclinic transition layer solutions connecting two uniform traffic states can be constructed. Motivated by related exactly solvable queue models in the literature, we rigorously prove the existence of heteroclinic traveling waves for sufficiently steep optimal velocity functions. We further establish the existence of homoclinic solutions formed by the interaction of increasing and decreasing transition layers and derive a necessary condition on the amplitude parameter for their existence.
To construct periodic stop-and-go waves on a circuit, we impose a global constraint reflecting conservation of total road length. Within this constrained framework, we prove the existence of large-period periodic solutions consisting of alternating transition layers and quasi-uniform states. These results provide a rigorous foundation for nonlinear congestion wave patterns beyond local bifurcation analysis. |
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