Multi-mode solitons in a long-short range traffic lattice model with time delay

Abstract

We consider a new form of solutions of a special lattice model for traffic system. By analyzing nearest sites’ interactions, time delay, and bumpy effects, we deduce the bifurcation lines and surfaces for stable and unstable regions and show how they vary as parameters vary. It shows that keeping other conditions unchanged, as the incoming flow increases, the traffic flow becomes unstable, opposite to when outgoing flow increases, it becomes stable. Besides, considering delayed optimal flow, multiple sites effect or artificial parameters can also help stabilize the traffic road condition. Moreover, by putting it into the framework of mKdV equations, we obtain the kink–antikink solitons involving all parameters, which show the feature of the traffic congestion. The result is original, and our model in differential or difference form can be reduced into the previous ones by choosing appropriate parameters. Since the optimal velocity function we considered involves finitely or infinitely many sites, the density waves can be in multi-mode and high dimension forms and can also be quasi-periodic, we show a new feature of the traffic lattice system.