The KdV–Burgers equation in speed gradient viscous continuum model

Based on the microscopic two velocity difference model, a macroscopic model called speed viscous continuum model is developed to describe traffic flow. The relative velocities are added to the motion equation, which leads to viscous effects in continuum model. The viscous continuum model overcomes the backward travel problem, which exists in many higher-order continuum models. Nonlinear analysis shows that the density fluctuation in traffic flow leads to density waves. Near the onset of instability, a small disturbance could lead to solitons described by the Korteweg–de Vries–Burgers (KdV–Burgers) equation, which is seldom found in other traffic flow models, and the soliton solution is derived.