The Nonlinear Equations For Traffic Flow Density Waves

Traffic problem has become a major hot issue of global concern. It has been conductedresearch by variety of disciplines scholars including physics mechanics and mathematics. In thisdissertation, by using the relationships between macroscopic and microscopic traffic models, thenew model is given. Based on the existing models for traffic flow, the corresponding theoreticalanalysis and numerical simulation are performed, focusing on the investigation of the variousnonlinear density waves in traffic flows. The main contents are as follows.Ⅰ.Modified Korteweg-de-Vries(mKdV) for traffic flow is analytically examined from themacroscopic hydrodynamic model.Based on a car-following model with the consideration of the driver’s forecast effect (DFE)proposed by Tang et.al., the characteristics of the linear and nonlinear density wave are studied.Using an asymptotic approximation between headway and density given by Berg et.al., we get aviscous continuum model. The linear stability theory is applied to derive the neutral stabilitycondition. The Korteweg-de Vries (KdV) equation to describe the traffic jam near the neutralstability line is given by nonlinear analysis and the corresponding solution for the traffic densitywave is derived.Ⅱ. The time-dependent Ginzburg-Landau (TDGL) equation and modified Korteweg-de-Vries(mKdV) equation for two velocity difference model.In the light of the thermodynamic theory can be applied to describe the phase transition andcritical phenomenon of the traffic flow, the thermodynamic property through constructing a certainrelation between the weight p and λ in TVDM is investigated. The coexisting curve, thespinodal line and the critical point are given by the derivatives of the thermodynamic potentialwhich is related to the delay time. The TDGL equation and the mKdV equation near the criticalpoint are obtained from TVDM. The connection between the TDGL equation and the mKdVequation has been shown. The description of nonlinear characteristics of the model is supplement.Ⅲ.The TDGL equation for lattice hydrodynamic model describing pedestrian flowBased on the extended lattice hydrodynamic pedestrian model taking the interaction of thenext-nearest-neighbor persons into account which is proposed by Wen et.al., the TDGL equation isderived to describe the pedestrian flow near the critical point through the nonlinear analysis method.And the corresponding two solutions, the uniform and the kink solutions, are given. The coexistingcurve, spinodal line and critical point are obtained by derivatives of the thermodynamic potential.The numerical simulation is carried out.