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The purpose of this study is to form a linear structure of components of the modified Korteweg–De Vries (mKdV) hierarchy. The new model includes 3rd order standard mKdV equation, 5th order and 7th order mKdV equations.

The authors investigate Painlevé integrability of the constructed linear structure.

The Painlevé analysis demonstrates that established sum of integrable models retains the integrability of each component.

The research also presents a set of rational schemes of trigonometric and hyperbolic functions to derive breather solutions.

The authors also furnish a variety of solitonic solutions and complex solutions as well.

The work formally furnishes algorithms for extending integrable equations that consist of components of a hierarchy.

The paper presents an original work for developing Painlevé integrable model via using components of a hierarchy.

This study aims to consider the influence of density difference integral and relative flow difference on traffic flow, a novel two-lane lattice hydrodynamic model is proposed. The stability criterion for the new model is obtained through the linear analysis method.

The modified Korteweg de Vries (KdV) (mKdV) equation is derived to describe the characteristic of traffic jams near the critical point. Numerical simulations are carried out to explore how density difference integral and relative flow difference influence traffic stability. Numerical and analytical results demonstrate that traffic congestions can be effectively relieved considering density difference integral and relative flow difference.

The traffic congestions can be effectively relieved considering density difference integral and relative flow difference.

Novel two-lane lattice hydrodynamic model is presented considering density difference integral and relative flow difference. Applying the linear stability theory, the new model’s linear stability is obtained. Through nonlinear analysis, the mKdV equation is derived. Numerical results demonstrate that the traffic flow stability can be efficiently improved by the effect of density difference integral and relative flow difference.

Initial value problems for the one‐dimensional third‐order dispersion equations are investigated using the reliable Adomian decomposition method (ADM).

The solutions are obtained in the form of rapidly convergent power series with elegantly computable terms.

It was found that the technique is reliable, powerful and promising. It is easier to implement than the separation of variables method. Modifications of the ADM and the noise terms phenomenon are successfully applied for speeding up the convergence of non‐homogeneous equations.

The method is restricted to initial value problems in which the space variable fills the whole real axis. Modifications are required to deal with initial boundary value problems. Further, the input initial condition is required to be an infinitely differentiable function and obviously, the convergence radius of the decomposition series depends on the input data.

The method was mainly illustrated for linear partial differential equations occuring in water resources research, but the natural extension of the ADM to solving nonlinear problems is extremely useful in nonlinear studies and soliton theory.

The study undertaken in this paper provides a reliable approach for solving both linear and nonlinear dispersion equations and new explicit or recursively‐based exact solutions are found.

The purpose of this paper is to explore how curved road and lane-changing rates affect the stability of traffic flow.

An extended two-lane lattice hydrodynamic model on a curved road accounting for the empirical lane-changing rate is presented. The linear analysis of the new model is discussed, the stability condition and the neutral stability condition are obtained. Also, the mKdV equation and its solution are proposed through nonlinear analysis, which discusses the stability of the extended model in the unstable region. Furthermore, the results of theoretical analysis are verified by numerical simulation.

The empirical lane-changing rate on a curved road is an important factor, which can alleviate traffic congestion.

This paper does not take into account the factors such as slope, the drivers’ characters and so on in the actual traffic, which will have more or less influence on the stability of traffic flow, so there is still a certain gap with the real traffic environment.

The curved road and empirical lane-changing rate are researched simultaneously in a two-lane lattice hydrodynamic models in this paper. The improved model can better reflect the actual traffic, which can also provide a theoretical reference for the actual traffic governance.

The purpose of this paper is to develop a scheme to study numerical solution of time fractional nonlinear evolution equations under initial conditions by reduced differential transform method.

The paper considers two models of special interest in physics with fractional‐time derivative of order, namely, the time fractional mKdV equation and time fractional convection diffusion equation with nonlinear source term.

The numerical results demonstrate the significant features, efficiency and reliability of the proposed method and the effects of different values are shown graphically.

The paper shows that the results obtained from the fractional analysis appear to be general.

The purpose of this paper is concerned with developing two-mode higher-order modified Korteweg-de Vries (KdV) equations. The study shows that multiple soliton solutions exist for essential conditions related to the nonlinearity and dispersion parameters.

The proposed technique for constructing a two-wave model, as presented in this work, has been shown to be very efficient. The employed approach formally derives the essential conditions for soliton solutions to exist.

The examined two-wave model features interesting results in propagation of waves and fluid flow.

The paper presents a new and efficient algorithm for constructing and studying two-wave-mode higher-order modified KdV equations.

A two-wave model was constructed for higher-order modified KdV equations. The essential conditions for multiple soliton solutions to exist were derived.

The work shows the distinct features of the standard equation and the newly developed equation.

The work is original and this is the first time for two-wave-mode higher-order modified KdV equations to be constructed and studied.

This paper aims to put forward an extended lattice hydrodynamic model, explore its effects on alleviating traffic congestion and provide theoretical basis for traffic management departments and traffic engineering implementation departments.

The control method is applied to study the stability of the new model. Through nonlinear analysis, the mKdV equation representing kink-antikink soliton is acquired.

The predictive effect and the control signal can enhance the traffic flow stability and reduce the energy consumption.

The predictive effect and feedback control are first considered in lattice hydrodynamic model simultaneously. Numerical simulations demonstrate that these two factors can enhance the traffic flow stability.

The purpose of this paper is to explore the impact of the mixed traffic flow, self-stabilization effect and the lane changing behavior on traffic flow stability.

An extended two-lane lattice hydrodynamic model considering mixed traffic flow and self-stabilization effect is proposed in this paper. Through linear analysis, the stability conditions of the extended model are derived. Then, the nonlinear analysis of the model is carried out by using the perturbation theory, the modified Kortweg–de Vries equation of the density of the blocking area is derived and the kink–antikink solution about the density is obtained. Furthermore, the results of theoretical analysis are verified by numerical simulation.

The results of numerical simulation show that the increase of the proportion of vehicles with larger maximum speed or larger safe headway in the mix flow are not conducive to the stability of traffic flow, while the self-stabilization effect and lane changing behavior is positive to the alleviation of traffic congestion.

This paper does not take into account the factors such as curve and slope in the actual road environment, which will have more or less influence on the stability of traffic flow, so there is still a certain gap with the real traffic environment.

The existing two-lane lattice hydrodynamic models are rarely discussed in the case of mixed traffic flow. The improved model proposed in this paper can better reflect the actual traffic, which can also provide a theoretical reference for the actual traffic governance.

In an intelligent transportation system (for short, ITS) environment, a vehicle’s motion is affected by the information in a large scale. The purpose of this paper is to study the integration effect of multiple vehicles’ delayed velocities on traffic flow.

This paper constructed a new car-following model to study the integration effect of multiple vehicles’ delayed velocities on traffic flow. The new model is analyzed by linear and nonlinear perturbation method theoretically and also verified by simulation.

It is found out that the integration of preceding vehicles’ delayed velocities affect the stability of traffic flow importantly, and three preceding vehicles’ delayed velocities information should be considered in real traffic.

The new car-following model by considering the integration effect of multiple vehicles’ delayed velocities is firstly proposed in this paper. The research result shows that three preceding vehicles’ delayed velocities information is the best choice to stabilizing traffic flow.

The purpose of this study is to explore the influence of the electronic throttle (ET) dynamics and the average speed of multiple preceding vehicles on the stability of traffic flow.

An extended car-following model integrating the ET dynamics and the average speed of multiple preceding vehicles is presented in this paper. The novel model’s stability conditions are obtained by using the thought of control theory, and the modified Korteweg–de Vries equation is inferred in terms of the nonlinear analysis method. In addition, some simulation experiments are implemented to explore the properties of traffic flow, and the results of these experiments confirm the correctness of theoretical analysis.

In view of the results of theoretical analysis and numerical simulation, traffic flow will become more stable when the average speed and ET dynamics of multiple preceding vehicles are considered, and the stability of traffic flow will also be enhanced by increasing the number of preceding vehicles considered.

This study leaves the factors such as the mixed traffic flow, the multilane and so on out of account in real road environment, which more or less influences the traffic flow’s stability, so the real traffic environment is not fully reflected.

There is little research integrating ET dynamics and the average velocity of multiple preceding vehicles to study the properties of traffic flow. The enhanced model constructed in this study can better reflect the real traffic, which can also give some theoretical reference for the development of connected and autonomous vehicles.