KdV and kink–antikink solitons in car-following models

The jams in the congested traffic are related with various density waves, which might be governed by the nonlinear wave equations, such as the Korteweg–de-Vries (KdV) equation, the Burgers equation and the modified Korteweg–de-Vries (mKdV) equation. Three different versions of optimal velocity models are examined. The stability conditions of the models are obtained by using the linear stability theory. The KdV equation near the neutral stability line and the mKdV equation around the critical point are derived by applying the reductive perturbation method, respectively. The traffic jams could be thus described by the KdV and kink–antikink soliton solutions for the two kinds of equations. The general solutions are given for, which can lead to specific solutions in previous work. Moreover, they are applied to solve a new model—the full velocity difference model and the corresponding KdV and kink–antikink soliton solutions could be quickly obtained, which demonstrates the general solutions presented herein are useful.