Density waves in a traffic flow model with reaction-time delay

Density waves are investigated analytically and numerically in the optimal velocity model with reaction-time delay of drivers. The stability condition of this model is obtained by using the linear stability theory. The results show that the decrease of reaction-time delay of drivers leads to the stabilization of traffic flow. The Burgers, Korteweg–de Vries (KdV) and modified Korteweg–de Vries (mKdV) equations are derived to describe the density waves in the stable, metastable and unstable regions respectively. The triangular shock waves, soliton waves and kink–antikink waves appearing respectively in the three distinct regions are derived to describe the traffic jams. The numerical simulations are given.