Nonlinear wave interactions of the Riemann problem for a two-lane traffic flow model

In this work, we analyze all possible nonlinear wave interactions within a macroscopic traffic flow model for a two-lane system with a van der Waals gas pressure, two distinct velocities, and equal vehicle density in each lane. The Riemann solution is derived constructively by exploiting the characteristic analysis approach. Properties of possible elementary waves, namely, shock, rarefaction, and contact discontinuity waves, are thoroughly investigated. With respect to arbitrary initial data, we also prove that the exact solution to the Riemann problem uniquely exists. Furthermore, we explore in detail about the elementary wave interactions that occur between rarefaction and shock waves with and without involving contact discontinuity. At the end, following each wave interaction the global structure of the Riemann solution is explicitly established with an evident graphic illustration using MATHEMATICA software. Here, nonlinear wave interaction finds that the local solution of the initial value problem produces a unique result, where a shock wave can travel in the same direction or the opposite direction while a rarefaction wave also can be generated in either directions. Additionally, the contact discontinuity wave may or may not be developed following the interaction.