Nonlinear analysis of traffic flow on a gradient highway

We investigate the slope effects upon traffic flow on a single lane gradient (uphill/downhill) highway analytically and numerically. The stability condition, neutral stability condition and instability condition are obtained by the use of linear stability theory. It is found that stability of traffic flow on the gradient varies with the slopes. The Burgers, Korteweg–de Vries (KdV) and modified Korteweg–de Vries (mKdV) equations are derived to describe the triangular shock waves, soliton waves and kink–antikink waves in the stable, meta-stable and unstable region respectively. A series of simulations are carried out to reproduce the triangular shock waves, kink–antikink waves and soliton waves. Results show that amplitudes of the triangular shock waves and kink–antikink waves vary with the slopes, the soliton wave appears in an upward form when the average headway is less than the safety distance and a downward form when the average headway is more than the safety distance. Moreover both the kink–antikink waves and the solitary waves propagate backwards. The numerical simulation shows a good agreement with the analytical result.