Non-local time evolution equation with singular integral and its application to traffic flow model

We consider an integro-differential equation model for traffic flow which is an extension of the Burgers equation model. To discuss the model, we first examine general settings for integrable integro-differential equations and find that they are obtained through a simple residue formula from integrable eqations in a complex domain. As demonstration of the efficiency of this approach, we list several integrable equations including a difference equation with double singular integral and an equation with elliptic singular integral. Then, we discuss the traffic model with singular integral and show that the model exhibits interaction between free flow region and congested region depending on the parameter of non-locality.