Jamming transitions and the modified Korteweg–de Vries equation in a two-lane traffic flow

The two lattice models are presented to simulate the traffic flow on a two-lane highway. They are the lattice versions of the hydrodynamic model of traffic: the one (model A) is described by the differential-difference equation where time is a continuous variable and space is a discrete variable, and the other (model B) is the difference equation in which both time and space variables are discrete. The jamming transitions among the freely moving phase, the coexisting phase, and the uniform congested phase are studied by using the nonlinear analysis and the computer simulation. The modified Korteweg–de Vries (MKdV) equations are derived from the lattice models near the critical point. The traffic jam is described by a kink–antikink solution obtained from the MKdV equation. It is found that the critical point, the coexisting curve, and the neutral stability line decrease with increasing the rate of lane changing. Also, the computer simulation is performed for the model B. It is shown that the coexisting curves obtained from the MKdV equation are consistent with the simulation result.