Abstract In this paper, the parameter-constrained long wave limit method is proposed to investigate the (2+1)-dimensional Korteweg–de Vries system. This method overcomes the limitation of the conventional approach caused by the special cross-term structure and enables the analytical construction of the lump and one-lump- M -stripe solutions. For M = 1 and M = 2 , explicit formulas are presented and the corresponding dynamic behaviors are analyzed. The lump wave moves from the position determined by one polynomial function to another after collision, with only a phase shift. The two stripe waves exhibit either crossing or overtaking behavior. This work extends the applicability of the long wave limit method and deepens the understanding of elastic interaction in high-dimensional integrable systems.

As modern transportation systems face increasing complexity, with challenges such as increased vehicle volumes, limited road resources, and rising safety concerns, there is an urgent need for innovative solutions. Cooperative driving, which enables vehicles to share information and collaborate through communication technologies, presents a promising solution to enhance safety, reduce congestion, and improve mobility. However, the validation of cooperative driving systems is hindered by a critical scarcity of real-world data. To address this challenge, we introduce CoDEA (Cooperative Driving Extraction and Augmentation), a comprehensive three-stage pipeline designed to generate robust and realistic cooperative driving datasets. First, a systematic method is developed to extract cooperative lane-changing behaviors from large-scale Naturalistic Driving Data (NDD), ensuring that the extracted data captures the key kinematic and cooperative features of real-world scenarios. Next, to effectively generate realistic cooperative lane-changing scenarios, we enhance the DiffTraj framework by introducing our Interaction-Aware Context Encoding (IA-CE) module. This module allows the diffusion model to condition its generation process on the nuanced interactions between vehicles, leading to the creation of more realistic and diverse cooperative trajectories. Finally, the effectiveness of the generated trajectories is evaluated using computational metrics such as RMSE and MAE, and by comparing key feature distributions between real and generated trajectories. The results show a strong similarity between the generated data and real-world cooperative lane-changing patterns, while also introducing greater diversity in certain features. Ultimately, the proposed CoDEA approach lays a solid foundation for advancing cooperative lane change control algorithms by providing a robust dataset for both training and evaluation, effectively bridging the gap between real-world complexity and algorithm testing environments.

This paper investigates an extended (2+1)-dimensional modified Korteweg-de Vries-Calogero-Bogoyavlenskii-Schiff (mKdV-CBS) equation. The integrability properties of the equation are given, including the bilinear form, the Bäcklund transformation, and the Lax pair. The multi-soliton solutions are constructed via the Hirota bilinear method. By virtue of the long-wave limit technique, the second- and third-order positon solutions are derived, and the interaction solutions of n second-order positons are further obtained accordingly. The expression for the interaction solution between n-soliton and second-order positon is presented for the first time. More general and asymmetric asymptotic trajectories are proposed, with distinct forms applicable to multi-positon solutions and positon-soliton interaction solutions, respectively. Furthermore, the detailed analysis of elastic interaction phenomena is provided, such as soliton-positon, breather-positon, and soliton-breather. This study significantly enriches the known solution structure of this integrable system and provides deep insights into complex nonlinear wave dynamics in higher-dimensional space.

The usage of information and communication technology in the intelligent transportation system (ITS) has provided drivers with a vast amount of valuable traffic information. To characterize driver behavior and vehicle interactions in the ITS environment by extensively harnessing vehicle-to-vehicle (V2V) data from arbitrary number of adjacent vehicles, we propose an enhanced car-following model incorporating the steady-state control effect. Our proposed model aims to expound on how V2V data influences the car-following dynamics of a connected and autonomous vehicle (CAV) platoon, thus shedding light on its intrinsic mechanisms for enhancing travel smoothness and operational flexibility. This proposition is discussed through a rigorous analysis involving both analytical and numerical methodologies. Initial findings from linear stability analysis indicate a marked enhancement in anti-interference capabilities compared to conventional autonomous vehicle platoons. Specific phenomena such as traffic bottlenecks and spontaneous instability are subsequently investigated by employing the reductive perturbation method and Hopf bifurcation theory. The results demonstrate that bottlenecks are effectively mitigated and subdued in the enhanced model, as evidenced by soliton solutions with reduced amplitude and increased wave velocity. The enhanced CAV platoon demonstrates a broader adjustable range for expected headway, thereby mitigating the instability risks stemming from Hopf bifurcation. Numerical simulations are then conducted under varied V2V usage scenarios in a circular road setting, exhibiting the detailed steady-state control performance across diverse real-world traffic conditions. Our findings of the enhanced model reveal the inherent mechanism in which V2V communication information prevents the potential CAV platoon instabilities and alleviates the traffic congestion on roadways, and can be served as a fundamental principle in the future era of intelligent driving.

Compared to freeway mainline sections, freeway tunnels present a higher risk of collisions and pose greater challenges for rescue operations, owing to the distinctive features of tunnel lighting and structures. Moreover, the safety demands for tunnel groups, which consist of multiple tunnels, are more pronounced. Recent studies have demonstrated the promising application of reinforcement learning in implementing variable speed limit (VSL) control on freeways to enhance safety and efficiency. However, existing approaches cannot adequately capture the interaction of traffic flows across different tunnels in group scenarios, nor can they efficiently train policies under highly complex environments. This study proposes a VSL strategy based on a two-level collaborative multi-agent reinforcement learning (TCMARL) framework to improve the safety and efficiency of freeway tunnel groups. The framework realizes coordinated control at two levels: (1) capturing potential spatial dependencies among tunnel agents through a spatial graph model, thereby achieving synergy at the information input level; and (2) constructing a mixing network that fuses features of different tunnel agents and introducing an importance weight vector to optimize the global Q-value output, achieving synergy at the action output level. To improve training efficiency and policy generalization, a model-based reinforcement learning mechanism is further incorporated to generate short-horizon virtual rollouts. Simulation experiments are conducted in four tunnel-group scenarios, including two-, three-, four-, and five-tunnel configurations. Comparative results show that the proposed framework achieves faster convergence and better safety and efficiency performance than benchmark methods. Additional ablation and sensitivity analyses confirm the complementary roles of the two-level collaboration design and the model-based planning mechanism, while out-of-training testing under unseen demand profiles demonstrates robust transferability. Beyond algorithmic performance, the findings provide policy relevant evidence for corridor level speed management in smart freeway tunnel systems, informing practical guidelines on coordinated VSL deployment and operational governance.

In this paper, we propose a novel framework of physics-informed deep learning for the forward and inverse problems of generalized Manakov systems including reverse-spacetime nonlocal Manakov equations and the Dirac-Manakov equation, which can describe the self-phase modulation and the cross-phase modulation effects of nonlinear light propagation in a random birefringent optical fiber and the potential interaction of two incoherent light beams in a wavelength division-multiplexed system. Self-adaptive loss coefficients are embedded into the parallelized neural networks to constrain the prediction error of data and physical residuals, so as to achieve efficient training and multi-object collaborative optimization. Based on different initial and boundary conditions, we have derived diverse types of numerical solutions, e.g., analytical line solitons, singular solitons and exponentially decaying solitons. Compared with classic physics-informed neural networks, our model achieves advanced baselines with lower solution error especially in high-order coupling cases. Setting coefficients of nonlinear terms in equations as dependent parameters to be trained, we can deduce the state of systems via inverse problem studies with noise data, which further verifies the robustness of the proposed algorithm. Abundant numerical tests are shared here to promote the innovation of physics-informed deep learning in nonlinear coupled systems of optical communications.

Toll plaza diverging area is a typical non-lane-based high-risk area characterized by frequent weaving and complex vehicle interactions. While observation-based approaches are effective for analyzing current safety conditions, they lack the flexibility in evaluating the safety impacts of infrastructure designs and traffic control strategies under future scenarios. To address this limitation, this study proposes a microsimulation-based approach to analyze the safety performance of toll plaza diverging areas by simulating the realistic conflict distributions under various traffic conditions. Based on the perception-decision-action (PDA) framework, the proposed approach improves the conflict simulation accuracy by more accurately modeling the weak-constraint driving behavior, including non-lane-based perception, dynamic toll lane selection, and car-following under weak-constraint conditions. Validated on real-world trajectory data from two distinct toll plaza diverging areas, the simulated conflict distributions by the PDA approach closely align with the observed data, while SUMO significantly underestimates the safety risks in diverging areas. Furthermore, a simulation platform is developed based on the PDA approach to quantitatively analyze the safety performance of toll plaza diverging areas under different diverging lengths and traffic volumes. Results indicate that insufficient diverging lengths increase severe conflicts, whereas excessively long diverging areas lead to inefficiencies without substantial safety benefits. This study provides novel insights into safety performance analysis in non-lane-based areas, offering a reliable simulation tool for optimizing management strategies in complex weaving scenarios.

In this paper, we investigate the long-time diffusion behavior of fractional reaction-diffusion models with variable coefficients that emerge in biological contexts. Specifically, we focus on the spatial fractional Gray-Scott (GS) model and the FitzHugh-Nagumo (FHN) model. By employing the Fourier spectral method, we validate its effectiveness in solving these complex three-dimensional variable-coefficient fractional reaction-diffusion models. Furthermore, we analyze the influence of various forms of diffusion coefficients such as trigonometric, polynomial and combined functions, on pattern formation within these models in both two-dimensional and three-dimensional spaces. In the GS model, the variability of the diffusion coefficient leads to non-uniform pattern formation speeds across different regions. The influence of the variable diffusion coefficient causes the initially symmetrical pattern to become asymmetrical. When the variable coefficients are functions associated with time t, the final pattern still exhibits a certain degree of symmetry at the Takens-Bogdanov point, which has not been previously reported. In the FHN model, when the variable coefficient is a function of the spatial variables, the line wave exhibits a tilt. In contrast, when the variable coefficient is a function of time t, the distance between two line waves increases. These variations do not influence the formation of spiral waves. The results in this paper hold promise for opening novel horizons in the investigation of pattern dynamics.

Variable-coefficient equations are crucial in the field of fluid dynamics as they accurately capture the spatial and temporal properties of fluid. In many cases, there exist some constraints among the coefficients and embedding these constraints into neural networks poses a challenge. In this paper, we design a variable coefficient-informed neural network (VCINN) to address the inverse problem of variable-coefficient partial differential equation in fluid dynamics. The VCINN framework integrates the physics-informed neural network (PINN) with the constraints among multiple coefficients, encoding both constraints and physics information into the neural networks. Compared to classical PINN, VCINN enjoys such advantages as parallelization capacity, embedding constraint information and efficient hyperparameter adjustment. Through a series of examples, the capability of the approach to recover coefficients from observations has been validated. Numerical results indicate that the present method achieves higher accuracy and lower training error compared to classical PINN.

In this paper, we focus on the Hirota equation appearing in communications and finance. In the field of communications, the Hirota equation is used to describe the ultrashort pulse transmission in optical fibers, while model the generalized option pricing problem in finance. The data-driven solutions are derived and the parameters are calibrated through physics-informed neural networks (PINNs), where various complex initial conditions on a continuous wave background are considered and compared. PINNs define the loss function based on the strong form via partial differential equations (PDEs), while it is subject to the diminished accuracy when the PDEs enjoy high-order derivatives or the solutions contain complex functions. We hereby propose a PINN with weak form (PINN-wf), where the weak form residual of PDEs is embedded into the loss function accounting for data errors effectively. The proposed algorithm involves domain decomposition to derive the weak form function, assigning distinct test functions to each sub-domain based on the selected sample points. Two schemes of computational experiments are carried out to provide valuable insights into the dynamic characteristics of solutions to the Hirota equation. These experiments serve as a robust reference for understanding and analyzing the behavior of solutions in practical scenarios.

This paper is concerned with a generalized (3 + 1)-dimensional variable-coefficient Fokas-typed equation, which is used to describe the interaction of nonlinear waves in ocean dynamics, shallow water waves, etc. By virtue of the Hirota bilinear method, the one-, two-, and threesoliton solutions are derived. Three kinds of bilinear B & auml;cklund transformation (BT) and the Bell-polynomial-typed BT are constructed based on the bilinear form. In addition, we obtain the lump solution and breather solution via the test function method. Three sets of variable coefficients are selected and the dynamic behavior of the corresponding lump solution is observed and analyzed. The main physics characteristics and spatial structure of these exact solutions are graphically discussed. Our results can be helpful in explaining certain related nonlinear phenomena.

Accurate pricing of barrier options is essential for facilitating informed investment decisions, optimizing resource allocation, and promoting market stability. The dynamics of interest rate and volatility significantly influence the barrier option pricing. Traditional equations associated with constant parameters may fail to capture these complexities. In this paper, the underlying asset is assumed to follow an extended geometric Brownian motion incorporating varying interest rate and volatility, and then a coupled pricing system for the up-and-out call option is derived based on the Kolmogorov forward equation and backward equation, enabling the analysis of volatility fluctuations. Higher volatility indicates a greater level of risk, which may correspond to higher potential returns. The fusion framework, the physics-informed neural network (PINN), is introduced to solve this coupled system, consisting of two subnetworks: one dedicated to estimating the expected values of barrier option prices, and another for capturing the volatility surface associated with the option prices. Experimental results based on the closing price data of CSI 300ETF options show that PINN offers an effective and efficient framework for evaluating the prices of financial derivatives, achieving high precision and interpretability, even in cases where closed-form analytical solutions are unavailable.

Barrier options, a type of path-dependent financial derivative, play a crucial role in modern markets for customizable risk management. Accurate pricing and efficient computation are essential for making informed decisions and developing effective investment strategies. In this paper, the underlying asset is assumed to follow a generalized geometric Brownian motion with adaptive varying interest rate and volatility, and a modified Black-Scholes-Merton equation is derived to price barrier options based on the martingale theory, which can fit different market volatility structures. The pricing formulas for eight barrier options are outlined including up/down-and-in/out calls/puts. We propose an extended physics-informed neural network (ePINN) framework integrating a prior model that encapsulates the risk-neutral pricing structure to ensure fair valuation of barrier options. The dynamics of the volatility surface are modeled through incorporating historical and implied volatility. Experiments conducted using CSI 300ETF options data from 2019 to 2025 show that ePINN outperforms PINN in predictive accuracy with a 69% improvement as measured by the mean absolute error metric. Enhanced performance indicates that the ePINN exhibits superior fitting capabilities, greater stability and improved interpretability, even in the presence of volatility fluctuations.

Multimodal synchronization has become the research highlight of the ITS, where complex driving scenarios, various types of vehicles and diverse data sources are crucial constituents. As real-time microscopic traffic characteristics can be vividly represented by graph data, we strive to achieve accurate trajectory predictions via graph-structured series for the stability and the efficiency of transportation. Although the existing data-driven algorithms have achieved fabulous accuracy in various simulation tasks, there are limitations in the distribution and the number of vehicles under investigation. With car-following patterns applied as physical information, we derive the adjacency matrix and design graph filters to explore the spatial dependence between vehicles via the graph-represented multi-lane traffic. The multi-head attention layer is attached to the spatiotemporal convolutional network as an extension. The rationality and the superiority of our model are validated on two calibrated datasets. Through error comparisons, we discuss the role of changeable hyper-parameters to deduce the optimal model for one-step and multi-step predictions. Novel ideas are shared in this paper to simplify the complexity of trajectory prediction in the synchronized transportation system.

Existing research on decision-making of autonomous vehicles (AVs) has mainly focused on normal road sections, with limited exploration of decision-making in complex traffic environments without lane markings. Taking toll plaza diverging area as an example, this study proposes a lateral motion strategy for AVs based on deep reinforcement learning (DRL) algorithms. First, a microscopic simulation platform is developed to simulate the realistic diverging trajectories of human-driven vehicles (HVs), providing AVs with a high-fidelity training environment. Next, a DRL-based self-efficient lateral motion strategy for AVs is proposed, with state and reward functions tailored to the environmental features of the diverging area. Simulation results indicate that the strategy can significantly reduce the diverging time of single vehicles. In addition, considering the long-term coexistence of AVs and HVs, the study further explores how the varying penetration of AVs with self-efficient strategy impacts traffic flow in the diverging area. Findings reveal that a moderate increase in AV penetration can improve overall traffic efficiency and safety. But an excessive penetration of AVs with self-efficient strategy leads to intense competition for limited road resources, further deteriorating operational conditions in the diverging area.

This paper is concerned with an extended (3+1) -dimensional shallow water wave equation with variable coefficients, which is used to describe the interaction of nonlinear waves in ocean dynamics, shallow water waves, etc. Hereby, it is of further value to investigate the integrability characteristics of this model. Firstly, we conduct the Painlevé analysis and find it can pass the Painlevé test. Then, the one- and two-soliton solution are obtained by virtue of the Hirota bilinear method. Bäcklund transformation, Lax pair and infinitely many conservation laws are derived through the Hirota bilinear method and Bell polynomial approach. Particularly, we generate two type of interaction solutions in terms of a combination of quadratic function, exponential function and trigonometric function, namely, the lump-kink solution and the periodic lump solution. Finally, dynamics characteristics and evolution behaviors are exhibited for the obtained solution waves through particular plots with proper choices of different values for the parameters.

A new test function is proposed to construct the elastic one-lump-multi-stripe solutions to the (2+1)-dimensional nonlinear evolution equations via Hirota bilinear forms. The necessary and sufficient conditions for the elastic one-lump-one-stripe solutions, onelump-two-stripe solutions and one-lump-three-stripe solutions are given to reduce the number of algebraic equations to be solved. The application is made for the (2+1)dimensional Boiti-Leon-Manna-Pempinelli system in incompressible fluid. Different from the interaction solutions derived by previous test functions, all the collisions between the lump wave and stripe waves are elastic if we ignore the phase shift of the lump wave. The lump wave can pass through the stripe waves. After the collision, the shapes and velocities of the two types of waves remain unchanged. The new test function can be applied to construct elastic one-lump-multi-stripe solutions to other nonlinear evolution equations which cannot be solved by the long wave limit method. The diverse elastic interaction phenomena between one lump wave and stripe waves will be of great significance to discuss the dynamic properties of nonlinear waves.(c) 2023 Elsevier B.V. All rights reserved.

We focus on the M-coupled nonlinear Schrödinger system with variable coefficients, describing simultaneous pulse propagation of the M-field components in an inhomogeneous optical fiber. The Riemann–Hilbert problem of this system is investigated based on the (M+1)× (M+1) matrix spectral problem. The N-soliton solutions are obtained when the jump matrix is an identity matrix. A variable-coefficient nonlocal nonlinear Schrödinger equation of reverse-time type is proposed with a special reduction of the M-coupled nonlinear Schrödinger system with variable coefficients. The symmetry relations of eigenvectors for one- and two-soliton solutions are given. But it is challenging and difficult to derive such relations for N-soliton solutions when N≥ 3 . The nonlocal one- and two-solutions exhibit special dynamical properties, such as periodicity and amplitude reduction. The results in this paper might be helpful to study the related physical problem in the field of optical fiber communication.