Kink soliton characterizing traffic congestion.

We study traffic congestion by analyzing a one-dimensional traffic flow model. Developing an asymptotic method to investigate the long time behavior near a critical point, we derive the modified Korteweg--de Vries (MKdV) equation as the lowest-order model. There is an infinite number of kink solitons to the MKdV equation, while it has been found by numerical simulations that the kink pattern arising in traffic congestion is uniquely determined irrespective of initial conditions. In order to resolve this selection problem, we consider higher-order corrections to the MKdV equation and find that there is a kink soliton that can deform continuously, with the perturbation represented by the addition of these corrections. With numerical confirmation, we show that this continuously deformable kink soliton characterizes traffic congestion. We also discuss the relationship between traffic congestion and the slugging phenomenon in granular flow.