Two velocity difference model for a car following theory

In the light of the optimal velocity model, a two velocity difference model for a car-following theory is put forward considering navigation in modern traffic. To our knowledge, the model is an improvement over the previous ones theoretically, because it considers more aspects in the car-following process than others. Then we investigate the property of the model using linear and nonlinear analyses. The Korteweg–de Vries equation (for short, the KdV equation) near the neutral stability line and the modified Korteweg–de Vries equation (for short, the mKdV equation) around the critical point are derived by applying the reductive perturbation method. The traffic jam could be thus described by the KdV soliton and the kink–anti-kink soliton for the KdV equation and mKdV equation, respectively. Numerical simulations are made to verify the model, and good results are obtained with the new model.