Density waves in traffic flow

Density waves are investigated in the car-following model analytically and numerically. This work is a continuation of our previous investigation of t raffic flow in the metastable and unstable regions [Phys. Rev. E 58, 5471 ( 1998); 60, 180 (1999)]. The Burgers equation is derived for the density wav e in the stable region of traffic flow by the use of nonlinear analysis. It is shown, numerically, that the triangular shock wave appears as the densi ty wave at the late stage in the stable region. The decay rate of the shock wave is calculated and compared with the analytical result. It is shown th at the density waves out of the coexisting curve, near the spinodal line, a nd within the spinodal line appear, respectively, as the triangular shock w ave, the soliton, and the kink-antikink wave. The density waves are describ ed, respectively, by the Burgers, Korteweg-de Vries, and modified Korteweg- de Vries equations.