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On stop-and-go waves in dense traffic. (English) Zbl 1165.90374
Summary: From a Vlasov-type kinetic equation with nonlocal braking and acceleration forces, taken as a traffic model for higher densities, we derive macroscopic equations generalizing the second order model of conservation laws suggested by A. Aw and M. Rascle [SIAM J. Appl. Math. 60, No. 3, 916–938 (2000; Zbl 0957.35086)] and H. M. Zhang [Chin. Ann. Math., Ser. B 26, No. 2, 275–290 (2005; Zbl 1067.17002)]. The nonlocality remains present in these equations, but more conventional, local equations are derived by using suitable Taylor expansion. A second order model of this type is discussed in some detail and is shown to possess traveling wave solutions that resemble stop-and-go waves in dense traffic. A phase space analysis suggests that inside the class of such traveling waves there are steady solutions that are stable.

MSC:
90B20Traffic problems
34D10Stability perturbations of ODE
35J15Second order elliptic equations, general