Kink Antikink Soliton Solution Research Articles

Traffic model for the dynamical behavioral study of a traffic system imposing push and pull effects

In this study, a novel traffic model, namely the push-pull model (PPM), has been developed considering not only forward-looking effects but also how focal vehicles are influenced by their following vehicle. The developed model represents a traffic flow field whose flow is significantly different from the conventional full velocity difference model in the congestion region but is analogous to the actual traffic flow system. We focused on linear analysis, nonlinear analysis, and numerical simulation to validate our model. Linear stability analysis determines the stability condition, delineating stable and unstable regions of traffic flow. The phase diagram of stability analysis demonstrates that the PPM has a crucial positive impact on traffic flow stabilization. In the case of nonlinearity, a kink-antikink soliton solution is generated from the modified Korteweg-de Vries equation, which is derived from the push-pull effect model to interpret traffic congestion. Furthermore, to check the validity of the theoretical results of this model, a numerical simulation has been conducted, demonstrating significant suppression of traffic congestion by considering push and pull effects.

Painlevé integrability, auto-Bäcklund transformations, new abundant analytic solutions including multi-soliton solutions for time-dependent extended KdV8 equation in nonlinear physics

In this paper, a new eighth-order (1+1)-dimensional time-dependent extended KdV equation has been developed. This considered equation is being found completely integrable by using the Painlevé analysis method. Moreover, three auto-Bäcklund transformations have been generated by truncating the Painlevé series at a constant level. These auto-Bäcklund transformations have been used to derive various analytic solution families for the newly developed equation. These solutions include the kink-antikink soliton, kink-soliton, antikink-soliton, periodic-soliton, complex kink-soliton and complex periodic-soliton solutions. Multi-soliton solutions including N-soliton solution, have been obtained by using the simplified Hirota’s method for the considered equation. All the results are being expressed graphically to signify the physical importance of the considered equation.

A modified lattice hydrodynamic model considering the driver’s predictive and honk effect

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.

Exact solutions of semi-discrete sine-Gordon equation

In this paper, we study a semi-discrete sine-Gordon (sd-SG) equation and compute various types of solutions analytically. We apply Darboux transformation to the associated spectral problem and construct N-soliton solutions of sd-SG equation in terms of ratio of ordinary determinants. In addition, we also construct explicit expressions of discrete one-kink, two-kink, kink-antikink, breather and degenerate soliton solutions of sd-SG equation in zero background.

Mean-field flow difference model with consideration of on-ramp and off-ramp

In this paper, an extended lattice hydrodynamic model is proposed by incorporating the effects of mean-field flow difference, on-ramp and off-ramp. The linear stability condition is obtained through linear stability theory and the mKdV equation is derived by using nonlinear analysis method. Therefore, the propagation behavior of traffic jam can be described by the kink–antikink soliton solution of the mKdV equation. Furthermore, the theoretical findings are verified with the use of numerical simulation which confirms that the stability of traffic flow can be efficiently improved with the consideration of the effects of mean-field flow difference, on-ramp and off-ramp.

A new lattice model of traffic flow considering driver’s anticipation effect of the traffic interruption probability

In this paper, a new lattice model is proposed by considering the driver’s anticipation effect of traffic interruption probability. The stability condition of the extended model is obtained by the linear stability analysis, which shows that the driver’s anticipation effect of traffic interruption probability can enlarge the stable area of traffic flow. The kink–antikink soliton solution of the modified Korteweg–de Vries (KdV) equation is derived from nonlinear analysis, which can describe the coexisting phase. The simulation results also show that driver’s anticipation effect of the traffic interruption probability could stabilize traffic flow.

Effects of speed deviation and density difference in traffic lattice hydrodynamic model with interruption

An extended lattice hydrodynamic model is proposed by incorporating the effects of speed deviation, traffic interruption probability and the density difference between the leading and the following lattice. The stability condition of the extended model is obtained by using the linear stability theory. Based on nonlinear analysis method, the mKdV equation is derived. Therefore, the propagation behavior of traffic jam can be described by the kink-antikink soliton solution of the mKdV equation. Numerical simulation demonstrated that the traffic flow will be more stable and it is more consistent with real traffic with the consideration of effects of speed deviation and traffic interruption probability and the density difference.

Phase transition of a new lattice hydrodynamic model with consideration of on-ramp and off-ramp

A new traffic lattice hydrodynamic model with consideration of on-ramp and off-ramp is proposed in this paper. The influence of on-ramp and off-ramp on the stability of the main road is uncovered by theoretical analysis and computer simulation. Through linear stability theory, the neutral stability condition of the new model is obtained and the results show that the unstable region in the phase diagram is enlarged by considering the on-ramp effect but shrunk with consideration of the off-ramp effect. The mKdV equation near the critical point is derived via nonlinear reductive perturbation method and the occurrence of traffic jamming transition can be described by the kink–antikink soliton solution of the mKdV equation. From the simulation results of space-time evolution of traffic density waves, it is shown that the on-ramp can worsen the traffic stability of the main road but off-ramp is positive in stabilizing the traffic flow of the main road.

Prevision of vehicle headway effect on urban traffic with a new car-following model

In this study, a new car-following model is established aiming to predict the variation of vehicle headways on urban road. The linear stability condition is derived corresponding to the prevision of headway in moving. The modified Korteweg–de Vries (mKdV) equation is deduced through the nonlinear analysis. The kink–antikink soliton solution of the mKdV equation can interpret the urban traffic jams near the critical point under the prevision of vehicle headway. Moreover, it is clear that the prevision of headway effect did improve the stability of urban traffic flow since the traffic jams are alleviated efficiently by taking into account the prevision of headway term in numerical simulations, which are consistent with the theoretical analysis.

Nonlinear analysis of a new car-following model accounting for the global average optimal velocity difference

In this paper, a new car-following model is proposed by considering the global average optimal velocity difference effect on the basis of the full velocity difference (FVD) model. We investigate the influence of the global average optimal velocity difference on the stability of traffic flow by making use of linear stability analysis. It indicates that the stable region will be enlarged by taking the global average optimal velocity difference effect into account. Subsequently, the mKdV equation near the critical point and its kink–antikink soliton solution, which can describe the traffic jam transition, is derived from nonlinear analysis. Furthermore, numerical simulations confirm that the effect of the global average optimal velocity difference can efficiently improve the stability of traffic flow, which show that our new consideration should be taken into account to suppress the traffic congestion for car-following theory.

Effect of optimal estimation of flux difference information on the lattice traffic flow model

Abstract In this paper, a new lattice model is proposed by considering the optimal estimation of flux difference information. The effect of this new consideration upon the stability of traffic flow is examined through linear stability analysis. Furthermore, a modified Korteweg–de Vries (mKdV) equation near the critical point is constructed and solved by means of nonlinear analysis method, and thus the propagation behavior of traffic jam can be described by the kink–antikink soliton solution of the mKdV equation. Numerical simulation is carried out under periodical condition with results in good agreement with theoretical analysis, therefore, it is verified that the new consideration can enhance the stability of traffic systems and suppress the emergence of traffic jams effectively.

Analysis of a new two-lane lattice hydrodynamic model with consideration of the global average flux

A new two-lane traffic lattice hydrodynamic model is proposed with the consideration of the global average-and-optimal flux difference effect based on the local relative flux two-lane lattice model. First, the influence of the global average-and-optimal flux difference on the stability of traffic flow is investigated through linear stability theory. The results reveal that the unstable region will be shrunk by taking the global average-and-optimal flux difference effect into account. Additionally, by using the reductive perturbation method, the mKdV equation near the critical point is derived and traffic jam transition can be described by its kink–antikink soliton solution. The good agreement between the numerical simulations and the analytical results shows that traffic congestion can be suppressed efficiently by considering the global average-and-optimal flux difference and the local relative flux effects in two-lane traffic system and the local relative flux is more important than the global average-and-optimal flux difference in stabilizing traffic flow.

Analysis of average density difference effect in a new two-lane lattice model

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.

Asymmetric effect on single-file dense pedestrian flow

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.

Analysis of two-lane lattice hydrodynamic model with consideration of drivers’ characteristics

A new lattice hydrodynamic model for two-lane traffic system is proposed with consideration of drivers’ timid or aggressive characteristics. The effect of drivers’ timid or aggressive characteristics on the stability of traffic flow is studied via linear analysis theory and nonlinear reductive perturbation method. The linear analysis results show that the stable region for aggressive drivers is much larger than that for timid drivers, which means that aggressive drivers are much more prompt than timid drivers in stabilizing traffic flow. Moreover, we derive the mKdV equation near the critical point and obtain its kink–antikink soliton solution to describe the feature of traffic jamming transition. The good agreement between numerical simulations and analytical results reveals that drivers’ characteristics have an important role in the stability of traffic flow and traffic jamming transition in two-lane traffic system.

Jamming transitions and the effect of interruption probability in a lattice traffic flow model with passing

A new lattice hydrodynamic model is proposed by considering the interruption probability effect on traffic flow with passing and analyzed both theoretically and numerically. From linear and non-linear stability analysis, the effect of interruption probability on the phase diagram is investigated and the condition of existence for kink–antikink soliton solution of mKdV equation is derived. The stable region is enhanced with interruption probability and the jamming transition occurs from uniform flow to kink flow through chaotic flow for higher and intermediate values of non-interruption effect of passing. It is also observed that there exists conventional jamming transition between uniform flow and kink flow for lower values of non-interruption effect of passing. Numerical simulations are carried out and found in accordance with the theoretical findings which confirm that the effect of interruption probability plays an important role in stabilizing traffic flow when passing is allowed.

Multiple optimal current difference effect in the lattice traffic flow model

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.

Stability Analysis of a Car-Following Model on Two Lanes

Considering lateral influence from adjacent lane, an improved car-following model is developed in this paper. Then linear and nonlinear stability analyses are carried out. The modified Korteweg-de Vries (MKdV) equation is derived with the kink-antikink soliton solution. Numerical simulations are implemented and the result shows good consistency with theoretical study.

A Lattice Hydrodynamic Model Considering the Following Lattice and its Stability Analysis

Incorporating the ITS in traffic flow, two lattice hydrodynamic models considering the following lattice are proposed to study the influence of the following lattice on traffic flow stability. The results from the linear stability theory show that considering the following lattice could lead to the improvement of the traffic flow stability. 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.

A Lattice Hydrodynamic Model Considering the Following Lattice and its Stability Analysis

Incorporating the ITS in traffic flow, two lattice hydrodynamic models considering the following lattice are proposed to study the influence of the following lattice on traffic flow stability. The results from the linear stability theory show that considering the following lattice could lead to the improvement of the traffic flow stability. 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.