Time-dependent Ginzburg-Landau equation in a car-following model considering the driver’s physical delay

In this paper, an extended car-following model considering the delay of the driver’s response in sensing headway is proposed to describe the traffic jam. It is shown that the stability region decreases when the driver’s physical delay in sensing headway increases. The phase transition among the freely moving phase, the coexisting phase, and the uniformly congested phase occurs below the critical point. By applying the reductive perturbation method, we get the time-dependent Ginzburg-Landau (TDGL) equation from the car-following model to describe the transition and critical phenomenon in traffic flow. We show the connection between the TDGL equation and the mKdV equation describing the traffic jam.     

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.     

The Korteweg-de Vries soliton in the lattice hydrodynamic model

The lattice hydrodynamic model is not only a simplified version of the macroscopic hydrodynamic model, but is also closely connected with the microscopic car following model. The modified Korteweg-de Vries (mKdV) equation about the density wave in congested traffic 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 in the car following model. So we devote ourselves to obtaining the KdV equation from the lattice hydrodynamic model and obtaining the KdV soliton solution describing the traffic jam. Numerical simulation is conducted, to demonstrate the nonlinear analysis result.     

Lattice models of traffic flow considering drivers' delay in response

This paper proposes two lattice traffic models by taking into account the drivers' delay in response. The lattice versions of the hydrodynamic model are described by the differential-difference equation and difference-difference equation, respectively. The stability conditions for the two models are obtained by using the linear stability theory. The modified KdV equation near the critical point is derived to describe the traffic jam by using the reductive perturbation method, and the kink--antikink soliton solutions related to the traffic density waves are obtained. The results show that the drivers' delay in sensing headway plays an important role in jamming transition.     

A new car-following model with two delays

A new car-following model is proposed by taking into account two different time delays in sensing headway and velocity. The effect of time delays on the stability analysis is studied. The theoretical and numerical results show that traffic jams are suppressed efficiently when the difference between two time delays decreases and those can be described by the solution of the modified Korteweg–de Vries (mKdV) equation. Traffic flow is more stable with two delays in headway and velocity than in the case with only one delay in headway. The impact of local small disturbance to the system is also studied.     

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.     

A simple and direct method for generating travelling wave solutions for nonlinear equations

We propose a simple and direct method for generating travelling wave solutions for nonlinear integrable equations. We illustrate how nontrivial solutions for the KdV, the mKdV and the Boussinesq equations can be obtained from simple solutions of linear equations. We describe how using this method, a soliton solution of the KdV equation can yield soliton solutions for the mKdV as well as the Boussinesq equations. Similarly, starting with cnoidal solutions of the KdV equation, we can obtain the corresponding solutions for the mKdV as well as the Boussinesq equations. Simple solutions of linear equations can also lead to cnoidal solutions of nonlinear systems. Finally, we propose and solve some new families of KdV equations and show how soliton solutions are also obtained for the higher order equations of the KdV hierarchy using this method.     

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.     

Analyses of driver’s anticipation effect in sensing relative flux in a new lattice model for two-lane traffic system

In this paper, a new lattice hydrodynamic traffic flow model is proposed by considering the driver’s anticipation effect in sensing relative flux (DAESRF) for two-lane system. The effect of anticipation parameter on the stability of traffic flow is examined through linear stability analysis and shown that the anticipation term can significantly enlarge the stability region on the phase diagram. To describe the phase transition of traffic flow, mKdV equation near the critical point is derived through nonlinear analysis. The theoretical findings have been verified using numerical simulation which confirms that traffic jam can be suppressed efficiently by considering the anticipation effect in the new lattice model for two-lane traffic.     

多前车速度差的车辆跟驰模型的稳定性与孤波

通过线性稳定性分析,得到了多前车速度差模型的稳定性条件, 并发现通过调节多前车信息,使交通流的稳定区域明显扩大. 通过约化摄动方法 研究了该模型的非线性动力学特性:在稳定流区域,得到了描述密度波的Burgers方程;在交 通流的不稳定区域内,在临界点附近获得了描述车头间距的修正的Korteweg-de Vries (modified Korteweg-de Vries, mKdV)方程; 在亚稳态区域内,在中性稳定曲线附近获得了描述车头间距 的KdV方程. Burgers的孤波解、mKdV方程的扭结-反扭结波解及KdV方程的 孤波解描述了交通流堵塞现象.     

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.     

Complex solitary waves and soliton trains in KdV and mKdV equations

We demonstrate the existence of complex solitary wave and periodic solutions of theKorteweg-de Vries (KdV) and modified Korteweg-de Vries (mKdV) equations. The solutions ofthe KdV (mKdV) equation appear in complex-conjugate pairs and are even (odd) under thesimultaneous actions of parity (??) and time-reversal (??) operations. The corresponding localized solitons arehydrodynamic analogs of Bloch soliton in magnetic system, with asymptotically vanishingintensity. The ????-odd complex soliton solution is shown to beiso-spectrally connected to the fundamental sech² solution through supersymmetry. Physically, thesecomplex solutions are analogous to the experimentally observed grey solitons of non-liner Schödinger equation, governing the dynamics of shallow waterwaves and hence may also find physical verification.     

Lie Symmetry and Nonlinear Instability in Computation of KdV Solitons

The soliton calculation method put forward by Zabusky and Kruskal has played an important role in the development of soliton theory, however numerous numerical results show that even though the parameters satisfy the linear stability condition, nonlinear instability will also occur. We notice an exception in the numerical calculation of soliton, gain the linear stability condition of the second order Leap-frog scheme constructed by Zabusky and Kruskal, and then draw the perturbed equation with the finite difference method. Also, we solve the symmetry group of the KdV equation with the knowledge of the invariance of Lie symmetry group and then discuss whether the perturbed equation and the conservation law keep the corresponding symmetry. The conservation law of KdV equation satisfies the scaling transformation, while the perturbed equation does not satisfy the Galilean invariance condition and the scaling invariance condition. It is demonstrated that the numerical simulation destroy some physical characteristics of the original KdV equation. The nonlinear instability in the calculation of solitons is related to the breaking of symmetry.     

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.     

Nonliner Analysis of a Synthesized Optimal Velocity Model for Traffic Flow

We analyze a new car-following model described by a differential-difference equation with a synthesized optimal velocity function (SOVF), which depends on the front interactions between every two adjacent vehicles instead of the weighted average headway. The model is analyzed with the use of the linear stability theory and nonlinear analysis method. The stability and neutral stability condition are obtained. We also derive the modified KdV (Korteweg-de Vries) equation and the kink-antikink soliton solution near the critical point. A simulation is conducted with integrating the differential-difference equation by the Euler scheme. The results of the numerical simulation verify the validity of the new model.     

Solitary Density Waves for Improved Traffic Flow Model with Variable Brake Distances

Traffic flow model is improved by introducing variable brake distances with varying slopes.Stability of the traffic flow on a gradient is analyzed and the neutral stability condition is obtained.The KdV(Korteweg-de Vries)equation is derived the use of nonlinear analysis and soliton solution is obtained in the meta-stable region.Solitary density waves are reproduced in the numerical simulations.It is found that as uniform headway is less than the safety distance solitary wave exhibits upward form,otherwise it exhibits downward form.In general the numerical results are in good agreement with the analytical results.     

Engineering solitons and breathers in a deformed ferromagnet: Effect of localised inhomogeneities

We investigate the soliton dynamics of the electromagnetic wave propagating in an inhomogeneous or deformed ferromagnet. The dynamics of magnetization and the propagation of electromagnetic waves are governed by the Landau–Lifshitz–Maxwell (LLM) equation, a certain coupling between the Landau–Lifshitz and Maxwell's equations. In the framework of multiscale analysis, we obtain the perturbed integral modified KdV (PIMKdV) equation. Since the dynamic is governed by the nonlinear integro-differential equation, we rely on numerical simulations to study the interaction of its mKdV solitons with various types of inhomogeneities. Apart from simple one soliton experiments with periodic or localised inhomogeneities, the numerical simulations revealed an interesting dynamical scenario where the collision of two solitons on a localised inhomogeneity create a bound state which then produces either two separated solitons or a mKdV breather.     

The effect of the negative particle velocity in a soliton gas within Korteweg–de Vries-type equations

The effect of changing the direction of motion of a defect (a soliton of small amplitude) in soliton lattices described by the Korteweg–de Vries and modified Korteweg–de Vries integrable equations (KdV and mKdV) was studied. Manifestation of this effect is possible as a result of the negative phase shift of a small soliton at the moment of nonlinear interaction with large solitons, as noted in [1], within the KdV equation. In the recent paper [2], an expression for the mean soliton velocity in a “cold” KdV-soliton gas has been found using kinetic theory, from which this effect also follows, but this fact has not been mentioned. In the present paper, we will show that the criterion of negative velocity is the same for both the KdV and mKdV equations and it can be obtained using simple kinematic considerations without applying kinetic theory. The averaged dynamics of the “smallest” soliton (defect) in a soliton gas consisting of solitons with random amplitudes has been investigated and the average criterion of changing the sign of the velocity has been derived and confirmed by numerical solutions of the KdV and mKdV equations.     

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.     

Simple Soliton Solution Method for the Combined KdV and MKdV Equation

Malfliet first proposed a simple solution method for the multisoliton solutionofthe KdV equation. Abdel-Rahman used Malfliet's method in a slightly modifiedform, and gave the multisoliton solution of the mKdV equation, RLW equation,Boussinesq equation, and modified Boussinesq equation. In this paper, we solvethe soliton solution of the cKdV=nmKdV equation by using this method.