Constructing set-valued fundamental diagrams from jamiton solutions in   second order traffic models
  Fundamental diagrams of vehicular traffic flow are generally multi-valued in the congested flow regime. We show that such set-valued fundamental diagrams can be constructed systematically from simple second order macroscopic traffic models, such as the classical Payne-Whitham model or the inhomogeneous Aw-Rascle-Zhang model. These second order models possess nonlinear traveling wave solutions, called jamitons, and the multi-valued parts in the fundamental diagram correspond precisely to jamiton-dominated solutions. This study shows that transitions from function-valued to set-valued parts in a fundamental diagram arise naturally in well-known second order models. As a particular consequence, these models intrinsically reproduce traffic phases. 
Benjamin Seibold, Morris R. Flynn, Aslan R. Kasimov, Rodolfo Ruben   Rosales
math.AP nlin.PS physics.soc-ph