The density wave in a car-following model

According to the optimal-velocity model, the condition for stable traffic flow is deduced. Nonlinear analysis shows that the density fluctuation in traffic flow itself induces two types of local density waves. A weak fluctuation occurring near the stability state in a wide range of headways forms a soliton determined by the Korteweg-de Vries (KdV) equation. The appearance of such a soliton shows that drivers tend to reach the safety distance when they are away from it. This density wave degenerates to a travelling wave at the critical point. A strong fluctuation occurring around the critical point forms a kink or a soliton determined by the modified Korteweg-de Vries (MKdV) equation.

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