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 traffic flow in the metastable and unstable regions [Phys. Rev. E 58, 5471 (1998); 60, 180 (1999)]. The Burgers equation is derived for the density wave 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 density 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 that the density waves out of the coexisting curve, near the spinodal line, and within the spinodal line appear, respectively, as the triangular shock wave, the soliton, and the kink-antikink wave. The density waves are described, respectively, by the Burgers, Korteweg-de Vries, and modified Korteweg-de Vries equations.