Narrow Results

Firstly twenty-four types of doubly-periodic solutions of the reduction mKdV equation are given. Secondly based on the reduction mKdV equation and its solutions, a systemic transformation method (called the reduction mKdV method) is developed to construct new doubly-periodic solutions of nonlinear equations. Thirdly with the aid of symbolic computation, we choose the KdV equation, the coupled variant Boussinesq equation and the cubic nonlinear Schrödinger equation to illustrate our method. As a result many types of solutions are obtained. These show that this method is simple and powerful to obtain more exact solutions including doubly-periodic solutions, soliton solutions and singly-periodic solutions to a wide class of nonlinear wave equations. Finally we further extended the method to a general form.

Two extended cooperative driving lattice hydrodynamic models are proposed by incorporating the intelligent transportation system and the backward-looking effect in traffic flow under certain conditions. They are the lattice versions of the hydrodynamic model of traffic: one (model A) is described by the differential-difference equation where time is a continuous variable and space is a discrete variable, and the other (model B) is the difference-difference equation in which both time and space variables are discrete. In light of the real traffic situations, the appropriate forward and backward optimal velocity functions are selected, respectively. Then the stability conditions for the two models are investigated with the linear stability theory and it is found that the new consideration leads to the improvement of the stability of traffic flow. The modified Korteweg-de Vries equations (the mKdV equation, for short) near the critical point are derived by using the nonlinear perturbation method to show that the traffic jam could be described by the kink-antikink soliton solutions for the mKdV equations. Moreover, the anisotropy of traffic flow is further discussed through examining the negative propagation velocity as the effect of following vehicle is involved.

In view that drivers would pay attention to the variation of headway on roads, an extended optimal velocity model is proposed by considering anticipation driving behavior. A stability criterion is given through linear stability analysis of traffic flows. The mKdV equation is derived with the reductive perturbation method for headway evolution which could be used to describe the stop-and-go traffic phenomenon. The results show a good effect of anticipation driving behavior on the stabilization of car flows and the anticipation driving behavior can improve the numerical stability of the model as well. In addition, the fluctuation of kinetic energy and the consumption of average energy in congested traffic flows are systematically analyzed. The results show that the reasonable level of anticipation driving behavior can save energy consumption in deceleration process effectively and lead to an associated relation like a "bow-tie" between the energy-saving and the value of anticipation factor.

In this paper, a new lattice model for bidirectional pedestrian flow on single path which involves the effect of friction parameter is presented. Linear stability analysis is used to obtain the stability condition. The modified Korteweg–de Vries (mKdV) equation and time-dependent Ginzburg–Landan (TDGL) equation are deduced by means of the reductive perturbation method respectively. Further, the influence of the friction parameters upon pedestrian flow has been discussed. Our results also indicate that pedestrians moving along both directions uniformly are most stable.

A new lattice model is proposed by taking the average density difference effect into account for two-lane traffic system according to Transportation Cyber-physical Systems. The influence of average density difference effect on the stability of traffic flow is investigated through linear stability theory and nonlinear reductive perturbation method. The linear analysis results reveal that the unstable region would be reduced by considering the average density difference effect. The nonlinear kink–antikink soliton solution derived from the mKdV equation is analyzed to describe the properties of traffic jamming transition near the critical point. Numerical simulations confirm the analytical results showing that traffic jam can be suppressed efficiently by considering the average density difference effect for two-lane traffic system.

In this paper, an extended optimal velocity model is proposed to simulate single-file dense pedestrian flow by considering asymmetric interaction (i.e. attractive force and repulsive force), which depends on the different distances between pedestrians. The stability condition of this model is obtained by using the linear stability theory. The phase diagram comparison and analysis show that asymmetric effect plays an important role in strengthening the stabilization of system. The modified Korteweg–de Vries (mKdV) equation near the critical point is derived by applying the reductive perturbation method. The pedestrian jam could be described by the kink–antikink soliton solution for the mKdV equation. From the simulation of space-time evolution of the pedestrians distance, it can be found that the asymmetric interaction is more efficient compared to the symmetric interaction in suppressing the pedestrian jam. Furthermore, the simulation results are consistent with the theoretical analysis as well as reproduce experimental phenomena better.

Considering the effect of density difference, an extended lattice hydrodynamic model for bidirectional pedestrian flow is proposed in this paper. The stability condition is obtained by the use of linear stability analysis. It is shown that the stability of pedestrian flow varies with the reaction coefficient of density difference. Based on nonlinear analysis method, 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, metastable and unstable regions, respectively. The results show that jams may be alleviated by considering the effect of density difference. The findings also indicate that in the process of building and subway station design, a series of auxiliary facilities should be considered in order to alleviate the possible pedestrian jams.

Understanding the pedestrian behavior contributes to traffic simulation and facility design/redesign. In practice, the interactions between individual pedestrians can lead to virtual honk effect, such as urging surrounding pedestrians to walk faster in a crowded environment. To better reflect the reality, this paper proposes a new lattice hydrodynamic model for bidirectional pedestrian flow with consideration of pedestrians’ honk effect. To this end, the concept of critical density is introduced to define the occurrence of pedestrians’ honk event. In the linear stability analysis, the stability condition of the new bidirectional pedestrian flow model is given based on the perturbation method, and the neutral stability curve is also obtained. Based on this, it is found that the honk effect has a significant impact on the stability of pedestrian flow. In the nonlinear stability analysis, the modified Korteweg–de Vries (mKdV) equation of the model is obtained based on the reductive perturbation method. By solving the mKdV equation, the kink-antikink soliton wave is obtained to describe the propagation mechanism and rules of pedestrian congestion near the neutral stability curve. The simulation example shows that the pedestrians’ honk effect can mitigate the pedestrians crowding efficiently and improve the stability of the bidirectional pedestrian flow.

A Lie algebra which consists of linear combinations of one basis of the Lie algebra A₁ is presented for which an isospectral Lax pair is exhibited. By using the zero curvature equation, the generalized mKdV equation, Liouville equation and sine-Gordon equation, sinh-Gordon equation are generated via polynomial expansions. Finally, we investigate a kind of formal Bäcklund transformation for the generalized sine-Gordon equation. The explicit Bäcklund transformation of the standard sine-Gordon equation is presented. The other equations given in the paper are obtained similarly.

Kerner and Konhäuser study moving jam dynamics first discovered in 1993 in Ref. 1. In light of their previous work, a new lattice hydrodynamic model is presented with consideration of the effect of multiple optimal current difference. To investigate the influences of new consideration on traffic jams, the linear stability analysis of the new model is conducted by employing the linear stability theory. Theoretical analysis result shows that the new consideration can stabilize traffic flow. By means of nonlinear analysis method, a modified Korteweg–deVries (mKdV) equation near the critical point is constructed and solved. The propagation behavior of traffic jam can thus be described by the kink–antikink soliton solution for the mKdV equation. Numerical simulation result shows that the effect of the multiple optimal current differences can suppress the emergence of traffic jams and the result is in good agreement with the analytical results.

In this paper, a new car-following model is proposed by considering driver’s desired velocity according to Transportation Cyber Physical Systems. The effect of driver’s desired velocity on traffic flow has been investigated through linear stability theory and nonlinear reductive perturbation method. The linear stability condition shows that driver’s desired velocity effect can enlarge the stable region of traffic flow. From nonlinear analysis, the Burgers equation and mKdV equation are derived to describe the evolution properties of traffic density waves in the stable and unstable regions respectively. Numerical simulation is carried out to verify the analytical results, which reveals that traffic congestion can be suppressed efficiently by taking driver’s desired velocity effect into account.

To reveal the impact of the current vehicle’s interruption information on traffic flow, a new car-following model with consideration of the current vehicle’s interruption is proposed and the influence of the current vehicle’s interruption on traffic stability is investigated through theoretical analysis and numerical simulation. By linear analysis, the linear stability condition of the new model is obtained and the negative influence of the current vehicle’s interruption on traffic stability is shown in the headway-sensitivity space. Through nonlinear analysis, the modified Korteweg–de Vries (mKdV) equation of the new model near the critical point is derived and it can be used to describe the propagating behavior of the traffic density wave. Finally, numerical simulation confirms the analytical results, which shows that the current vehicle’s interruption information can destabilize traffic flow and should be considered in real traffic.

Based on the two velocity difference model (TVDM), an extended car-following model is developed to investigate the effect of driver’s memory and jerk on traffic flow in this paper. By using linear stability analysis, the stability conditions are derived. And through nonlinear analysis, the time-dependent Ginzburg–Landau (TDGL) equation and the modified Korteweg–de Vries (mKdV) equation are obtained, respectively. The mKdV equation is constructed to describe the traffic behavior near the critical point. The evolution of traffic congestion and the corresponding energy consumption are discussed. Numerical simulations show that the improved model is found not only to enhance the stability of traffic flow, but also to depress the energy consumption, which are consistent with the theoretical analysis.

In this paper, a new lattice two-lane hydrodynamic model is proposed by considering the lane changing and the optimal current change with memory effect. The linear stability condition of the model is obtained through the linear stability analysis, which depends on both the lane-changing rate and the memory step. A modified Korteweg–de Vries (mKdV) equation is derived through nonlinear analysis to describe the propagating behavior of traffic density wave near the critical point. To verify the analytical findings, numerical simulation was carried out, which confirms that the optimal current change with memory of drivers and the memory step contribute to the stabilization of traffic flow, and that traffic congestion can be suppressed efficiently by taking the lane-changing behavior into account in the lattice model.

In this paper, a new two-lane lattice hydrodynamic model is presented by accounting for the “backward looking” effect and the relative flow information. Linear analysis is applied to deduce the linear stability condition. With this method, we can demonstrate that “backward looking” and relative flow information have great positive significance in improving traffic flow stability. Nonlinear analysis is performed to derive the mKdV equation, which can represent transmission characteristic of density waves. The results achieved by the numerical simulation are consistent with theoretical analytical results. Numerical results indicate that both “backward looking” effect and relative flow information are helpful to heighten the traffic flow stability efficiently in two-lane traffic model.

The difference between the optimal current difference and the actual current difference will be used as the correction item. The dynamic multiple current information about the front lattice will be considered. A modified lattice traffic hydrodynamics model is established by considering the downstream traffic conditions in the two-lane system. Through the stability analysis, it is found that the downstream traffic condition can be added as a correction term to increase the stability of the system. The area of the stable region on the phase diagram is enlarged by the derived stability. The mKdV equation, which can describe density wave, is derived by nonlinear analysis. Finally, the phase diagram of stability condition in linear analysis and the kink wave diagram of mKdV equation in nonlinear analysis are obtained by numerical simulation, which verifies the theoretical derivation of this paper. The results show that in the two-lane traffic flow expansion model, considering the downstream traffic conditions can effectively suppress traffic jams and make the traffic flow stable.

Vehicular honk behavior is not uncommon in real traffic. Meanwhile, with the advent of sensors and communication technology, the real-time traffic state can be readily available to drivers. With such information, drivers can estimate the future traffic status and take countermeasures in advance. In this study, we propose a new lattice hydrodynamic model by jointly considering the vehicular honk and driver’s predictive effect. In the linear stability analysis section, neutral stability curve of the model is given, and the modified Korteweg de Vries (mKdV) equation is obtained in nonlinear stability analysis. The kink–antikink soliton solution is obtained by solving the above mKdV equation, which can be used to describe the phase transition of the traffic flow near the critical point. Results show that the honk and the driver’s predictive effect contribute to the traffic flow stability.

Using the Inverse Scattering Method with a nonvanishing boundary condition, we obtain an explicit breather solution with nonzero vacuum parameter b of the focusing modified Korteweg–de Vries (mKdV) equation. Moreover, taking the limiting case of zero frequency, we obtain a generalization of the double pole solution introduced by M. Wadati et al.

The actual road traffic is a heterogeneous traffic flow composed of various types of vehicles. Ensuring safe operation of different types of vehicles while improving traffic stabilization has become a very important issue. Inspired by this, taking into account the discrepancy between the safety headway and the maximal velocity of different types of vehicles, an improved dual-lane heterogeneous lattice hydrodynamic model (DLHLHM) considering curved road is put forward in this research. The improved model’s stabilization condition is inferred by reductive perturbation method. The mKdV equation is also inferred to describe the evolutionary processes of traffic density wave. Numerical examples probed into the impact of lane-changing rate, angle of curved road and different maximum speeds and safety distances on traffic stabilization. The hysteresis loop for new model also is explored to study traffic flow stabilization. Numerical simulation results accord with theoretical analysis, which indicates the effectiveness and feasibility of the DLHLHM. This study provides a new perspective on the dynamic evolution of heterogeneous traffic flow.