Analysis of the stability and density waves for traffic flow

In this paper,the optimal velocity model of traffic is extended to take into account the relative velocity.The stability and density waves for traffic flow are investigated analytically with the perturbation method.The stability criterion is dervied by the linear stability analysis.It is shown that the triangular shock wave,soliton wave and kink wave appear respectivley in our model for density waves in the three regions;stable,metastable and unstable regions.Theses Correspond to the solutions of the Burgers equation,Korteweg-de Vries equation and modified Korteweg-de Vires equation.The analytical resulsts are confirmed to be in good agreement with those of numerical simulation.All the results indicate that the interaction of a car with relative velocity can affect the stability of the traffic flow and raise critical density.