Modified KdV equation for jamming transition in the continuum models of traffic

Continuum models of traffic are proposed to describe the jamming transition in traffic flow on a highway. They are the simplified versions of the hydrodynamic model of traffic. Two continuum models are presented: one is described by the partial differential equations and the other is the discrete lattice version. The linear stability theory and the nonlinear analysis are applied to the continuum models. The modified Korteweg–de Vries equation (KdV) near the critical point is derived using the reduction perturbation method. It is shown that the jamming transition and the density wave in the congested traffic flow are described by the modified KdV equation. The solutions of the KdV equations obtained from the two models are compared with that of the optimal velocity model (car following model).