The time-dependent Ginzburg–Landau equation for the two-velocity difference model

A thermodynamic theory is formulated to describe the phase transition and critical phenomenon in traffic flow. Based on the two-velocity difference model, the time-dependent Ginzburg—Landau (TDGL) equation under certain condition is derived to describe the traffic flow near the critical point through the nonlinear analytical method. The corresponding two solutions, the uniform and the kink solutions, are given. The coexisting curve, spinodal line and critical point are obtained by the first and second derivatives of the thermodynamic potential. The modified Korteweg de Vries (mKdV) equation around the critical point is derived by using the reductive perturbation method and its kink—antikink solution is also obtained. The relation between the TDGL equation and the mKdV equation is shown. The simulation result is consistent with the nonlinear analytical result.