Time-dependent Ginzburg–Landau equation in a car-following model considering the driver’s physical delay

In this paper, an extended car-following model considering the delay of the driver’s response in sensing headway is proposed to describe the traffic jam. It is shown that the stability region decreases when the driver’s physical delay in sensing headway increases. The phase transition among the freely moving phase, the coexisting phase, and the uniformly congested phase occurs below the critical point. By applying the reductive perturbation method, we get the time-dependent Ginzburg–Landau (TDGL) equation from the car-following model to describe the transition and critical phenomenon in traffic flow. We show the connection between the TDGL equation and the mKdV equation describing the traffic jam.