The theoretical analysis of the lattice hydrodynamic models for traffic flow theory

The lattice hydrodynamic model is not only a simplified version of the macroscopic hydrodynamic model, but also connected with the microscopic car following model closely. The modified Korteweg-de Vries (mKdV) equation related to the density wave in a congested traffic region has been derived near the critical point since Nagatani first proposed it. But the Korteweg-de Vries (KdV) equation near the neutral stability line has not been studied, which has been investigated in detail for the car following model. We devote ourselves to obtaining the KdV equation from the original lattice hydrodynamic models and the KdV soliton solution to describe the traffic jam. Especially, we obtain the general soliton solution of the KdV equation and the mKdV equation. We review several lattice hydrodynamic models, which were proposed recently. We compare the modified models and carry out some analysis. Numerical simulations are conducted to demonstrate the nonlinear analysis results.     

The “backward looking” effect in the lattice hydrodynamic model

The novel lattice hydrodynamic model is presented by incorporating the “backward looking” effect. The stability condition for the the model is obtained using the linear stability theory. The result shows that considering one following site in vehicle motion leads to the stabilization of the system compared with the original lattice hydrodynamic model and the cooperative driving lattice hydrodynamic model. The Korteweg-de Vries (KdV, for short) equation near the neutral stability line is derived by using the reductive perturbation method to show the traffic jam which is proved to be described by KdV soliton solution obtained from the KdV equation. The simulation result is consistent with the nonlinear analysis.     

Two velocity difference model for a car following theory

In the light of the optimal velocity model, a two velocity difference model for a car-following theory is put forward considering navigation in modern traffic. To our knowledge, the model is an improvement over the previous ones theoretically, because it considers more aspects in the car-following process than others. Then we investigate the property of the model using linear and nonlinear analyses. The Korteweg-de Vries equation (for short, the KdV equation) near the neutral stability line and the modified Korteweg-de Vries equation (for short, the mKdV equation) around the critical point are derived by applying the reductive perturbation method. The traffic jam could be thus described by the KdV soliton and the kink-anti-kink soliton for the KdV equation and mKdV equation, respectively. Numerical simulations are made to verify the model, and good results are obtained with the new model.     

Density waves in a lattice hydrodynamic traffic flow model with the anticipation effect

By introducing the traffic anticipation effect in the real world into the original lattice hydrodynamic model, we present a new anticipation effect lattice hydrodynamic (AELH) model, and obtain the linear stability condition of the model by applying the linear stability theory. Through nonlinear analysis, we derive the Burgers equation and Korteweg-de Vries (KdV) equation, to describe the propagating behaviour of traffic density waves in the stable and the metastable regions, respectively. The good agreement between simulation results and analytical results shows that the stability of traffic flow can be enhanced when the anticipation effect is considered.     

Density waves in traffic flow model with relative velocity

The car-following model of traffic flow is extended to take into account the relative velocity. The stability condition of this model is obtained by using linear stability theory. It is shown that the stability of uniform traffic flow is improved by considering the relative velocity. From nonlinear analysis, it is shown that three different density waves, that is, the triangular shock wave, soliton wave and kink-antikink wave, appear in the stable, metastable and unstable regions of traffic flow respectively. The three different density waves are described by the nonlinear wave equations: the Burgers equation, Korteweg-de Vries (KdV) equation and modified Korteweg-de Vries (mKdV) equation, respectively.     

Analysis of stability and density waves of traffic flow model in an ITS environment

By introducing relative velocities of arbitrary number of cars ahead into the full velocity difference models (FVDM), we present a forward looking relative velocity model (FLRVM) of cooperative driving control system. To our knowledge, the model is an improvement over the similar extension in the forward looking optimal velocity models (FLOVM), because it is more reasonable and realistic in implement of incorporating intelligent transportation system in traffic. Then the stability criterion is investigated by the linear stability analysis with finding that new consideration theoretically lead to the improvement of the stability of traffic flow, and the validity of our theoretical analysis is confirmed by direct simulations. In addition, nonlinear analysis of the model shows that the three waves: triangular shock wave, soliton wave and kink-antikink wave appear respectively in stable, metastable and unstable regions. These correspond to the solutions of the Burgers equation, Korteweg-de Vries (KdV) equation and modified Korteweg-de Vries (mKdV) equation.     

A New Lattice Model of Two-Lane Traffic Flow with the Consideration of the Honk Effect

In this paper, a new lattice model of two-lane traffic flow with the honk effect term is proposed to study the influence of the honk effect on wide moving jams under lane changing. The linear stability condition on two-lane highway is obtained by applying the linear stability theory. The modified Korteweg-de Vries (KdV) equation near the critical point is derived and the coexisting curves resulted from the modified KdV equation can be described, which shows that the critical point, the coexisting curve and the neutral stability line decrease with increasing the honk effect coefficient. A wide moving jam can be conceivably described approximately in the unstable region. Numerical simulation is performed to verify the analytic results. The results show that the honk effect could suppress effectively the congested traffic patterns about wide moving jam propagation in lattice model of two-lane traffic flow.     

A new lattice model of traffic flow with the consideration of the traffic interruption probability

In this paper, we present a new lattice model which involves the effects of traffic interruption probability to describe the traffic flow on single lane freeways. The stability condition of the new model is obtained by the linear stability analysis and the modified Korteweg-de Vries (KdV) equation is derived through nonlinear analysis. Thus, the space will be divided into three regions: stable, metastable and unstable. The simulation results also show that the traffic interruption probability could stabilize traffic flow.     

A new lattice hydrodynamic traffic flow model with a consideration of multi-anticipation effect

We present a new multi-anticipation lattice hydrodynamic model based on the traffic anticipation effect in the real world.Applying the linear stability theory,we obtain the linear stability condition of the model.Through nonlinear analysis,we derive the modified Korteweg-de Vries equation to describe the propagating behaviour of a traffic density wave near the critical point.The good agreement between the simulation results and the analytical results shows that the stability of traffic flow can be enhanced when the multi-anticipation effect is considered.     

Nonplanar Ion-Acoustic Solitons in Electron-Positron-Ion Quantum Plasmas

The propagation of nonplanar quantum ion-acoustic solitary waves in a dense, unmagnetized electron-positronion （e-p-i） plasma are studied by using the Korteweg-de Vries （KdV） model. The quantum hydrodynamic （QHD） equations are used taking into account the quantum diffraction and quantum statistics corrections. The analytical and numerical solutions of KdV equation reveal that the nonplanar ion-acoustic solitons arc modified significantly with quantum corrections and positron concentration, and behave differently in different geometries.     

A Coupled Hybrid Lattice: Its Related Continuous Equation and Symmetries

The hybrid lattice, known as a discrete Korteweg-de Vries (KdV) equation, is found to be a discrete modified Korteweg-de Vries (mKdV) equation in this paper. The coupled hybrid lattice, which is pointed to be a discrete coupled KdV system, is also found to be discrete form of a coupled mKdV systems. Delayed differential reduction system and pure difference systems are derived from the coupled hybrid system by means of the symmetry reduction approach. Cnoidal wave, positon and negaton solutions for the coupled hybrid system are proposed.     

Soliton and kink jams in traffic flow with open boundaries

Soliton density wave is investigated numerically and analytically in the optimal velocity model (a car-following model) of a one-dimensional traffic flow with open boundaries. Soliton density wave is distinguished from the kink density wave. It is shown that the soliton density wave appears only at the threshold of occurrence of traffic jams. The Korteweg-de Vries (KdV) equation is derived from the optimal velocity model by the use of the nonlinear analysis. It is found that the traffic soliton appears only near the neutral stability line. The soliton solution is analytically obtained from the perturbed KdV equation. It is shown that the soliton solution obtained from the nonlinear analysis is consistent with that of the numerical simulation.     

A New Lattice Model of Two-Lane Trafc Flow with the Consideration of the Honk Efect

In this paper,a new lattice model of two-lane trafc flow with the honk efect term is proposed to study the influence of the honk efect on wide moving jams under lane changing.The linear stability condition on two-lane highway is obtained by applying the linear stability theory.The modified Korteweg-de Vries(KdV)equation near the critical point is derived and the coexisting curves resulted from the modified KdV equation can be described,which shows that the critical point,the coexisting curve and the neutral stability line decrease with increasing the honk efect coefcient.A wide moving jam can be conceivably described approximately in the unstable region.Numerical simulation is performed to verify the analytic results.The results show that the honk efect could suppress efectively the congested trafc patterns about wide moving jam propagation in lattice model of two-lane trafc flow.     

The effect of diffusion in a new viscous continuum traffic model

In this Letter, we propose a new continuum traffic model with a viscous term. The linear stability condition for viscous shock waves is derived. We derive the Korteweg-de Vries (KdV) equation near the neutral stability line. Then we investigate the effect of the viscous term by numerical simulations. The results show that viscosity may induce oscillations and the amplitude of the oscillation increases as the viscosity coefficient increases. This agrees with the linear stability condition. The local clusters are compressed by increasing the viscosity coefficient in the cluster study.     

Dynamics of localized waves with large amplitude in a weakly dispersive medium with a quadratic and positive cubic nonlinearity

The dynamics of localized waves is analyzed in the framework of a model described by the Korteweg-de Vries (KdV) equation with account made for the cubic positive nonlinearity (the Gardner equation). In particular, the interaction process of two solitons is considered, and the dynamics of a “breathing” wave packet (a breather) is discussed. It is shown that solitons of the same polarity interact as in the case of the Korteweg-de Vries equation or modified Korteweg-de Vries equation, whereas the interaction of solitons of different polarity is qualitatively different from the classical case. An example of “unpredictable” behavior of the breather of the Gardner equation is discussed.     

A Discrete Lax-Integrable Coupled System Related to Coupled KdV and Coupled mKdV Equations

A modified Korteweg-de Vries （mKdV） lattice is found to be also a discrete Korteweg-de Vries （KdV） equation. A discrete coupled system is derived from the single lattice equation and its Lax pair is proposed. The coupled system is shown to be related to the coupled KdV and coupled mKdV systems which are widely used in physics.     

Analysis of car-following model considering driver’s physical delay in sensing headway

An extended car-following model is proposed by taking into account the delay of the driver’s response in sensing headway. The stability condition of this model is obtained by using the linear stability theory. The results show that the stability region decreases when the driver’s physical delay in sensing headway increases. The KdV equation and mKdV equation near the neutral stability line and the critical point are respectively derived by applying the reductive perturbation method. The traffic jams could be thus described by soliton solution and kink-antikink soliton solution for the KdV equation and mKdV equation respectively. The numerical results in the form of the space-time evolution of headway show that the stabilization effect is weakened when the driver’s physical delay increases. It confirms the fact that the delay of driver’s response in sensing headway plays an important role in jamming transition, and the numerical results are in good agreement with the theoretical analysis.     

Density waves in a traffic flow model with reaction-time delay

Density waves are investigated analytically and numerically in the optimal velocity model with reaction-time delay of drivers. The stability condition of this model is obtained by using the linear stability theory. The results show that the decrease of reaction-time delay of drivers leads to the stabilization of traffic flow. The Burgers, Korteweg-de Vries (KdV) and modified Korteweg-de Vries (mKdV) equations are derived to describe the density waves in the stable, metastable and unstable regions respectively. The triangular shock waves, soliton waves and kink-antikink waves appearing respectively in the three distinct regions are derived to describe the traffic jams. The numerical simulations are given.     

On the theory of acoustic solitons in a liquid with distributed gas bubbles

A close relation is established between numerical solutions to two systems of equations, viz., the two-level nonlinear wave dynamic model of a liquid with gas bubbles and the Korteweg-de Vries (KdV) equation. This model is used for deriving the KdV equation in the long-wave approximation for any dependent variable of the gas-liquid mixture. The KdV equations derived earlier using radically different approximations are particular cases of our equations.     

考虑驾驶员预估效应的交通流格子模型与数值仿真

考虑驾驶员的预估效应对车流的影响,提出了一个改进的一维交通流格子模型.基于线性稳定性理论得到了该模型的线性稳定性判据;运用非线性分析方法导出了描述交通阻塞相变时的mKdV方程.应用数值仿真验证了mKdV方程的解,研究表明适当考虑车流中预估效应的作用能够增强交通流稳定性,从而能有效抑制交通阻塞的形成. 关键词： 预估效应 交通流 格子模型 数值仿真