Some exact solutions to the Lighthill-Whitham-Richards-Payne traffic flow equations. II. Moderate congestion (Q2927696)

The paper addresses the problem of solving of a system of the Lighthill-Whitham-Richards-Payne equations, which constitute a known model of the traffic flow. The system is analyzed in the regime of moderate congestion, in which the local density of cars is much smaller than the critical value that leads to the stoppage of the traffic. The accordingly simplified system, which has some similarity to the Korteweg-de Vries (KdV) equation, but is different from it, can be solved (transformed into a linear form) by means of using Lagrangian coordinates (instead of original Eulerian ones). The transformation is composed of two stages. The respective solutions are obtained either explicitly, or in a parametric form. They demonstrate that a traffic lineup splits into two ``offshoots'', which resemble the two-soliton solution of the KdV equation.