TDGL and MKdV equations for jamming transition in the lattice models of traffic

The lattice models of traffic are proposed to describe the jamming transition in traffic flow on a highway in terms of thermodynamic terminology of phase transitions and critical phenomena. They are the lattice versions of the hydrodynamic model of traffic. Two lattice models are presented: one is described by the differential-difference equation where time is a continuous variable and space is a discrete variable, and the other is the difference equation in which both time and space variables are discrete. We apply the linear stability theory and the nonlinear analysis to the lattice models. It is shown that the time-dependent Ginzburg–Landau (TDGL) equation is derived to describe the traffic flow near the critical point. A thermodynamic theory is formulated for describing the phase transitions and critical phenomena. It is also shown that the perturbed modified Korteweg-de Vries (MKdV) equation is derived to describe the traffic jam.