Exact solutions, conservation laws, and shock wave propagation of two-lanes traffic flow model via Lie symmetry

In this study, we consider a hyperbolic system of quasi-linear partial differential equations, governed by the traffic flow model on two lanes. We employ symmetry analysis and establish one-dimensional optimal subalgebras. Subsequently, we reduce the model into a system of ordinary differential equations for each optimal subalgebra and construct some new exact solutions; some of them are presented graphically. Further, by imposing the traveling wave transformation, we derive solutions including peakon-type solitons and upward parabola solitons. Furthermore, we demonstrate the existence of the nonlinear self-adjointness property of the model and formulate conservation laws. Finally, we discussed the evolutionary behavior of C1-waves, characteristic shock, and their interactions through one of the obtained exact solutions.