This research aims to examine the (1+1)-dimensional Schrödinger–Maxwell–Bloch (SMB) model, which is of great significance for the study of the dynamic properties of optical solitons in the media. We obtain the exact analytical soliton solutions for the given model using the new Kudryashov’s method, which helps to reveal various forms of the nonlinear waves. The proposed approach yields explicit localized wave structures, including bright soliton and dark-bright composite profiles associated with the field components of the SMB system. These solutions describe stable localized pulses and their accompanying field interactions in resonant nonlinear optical media. These solutions are shown using 3D plots, 2D plots, contour plots, as well as polar plots to gain a better understanding of their spatiotemporal properties. These plots also help to show the overlap of the temporal pulses. In addition to the solution development, the dynamical properties of the system are also rigorously investigated. Bifurcation plots are used to explore the stability properties of the system for varying important parameters. Chaotic properties are explored using numerical techniques such as simulations, including properties such as multistability, initial value sensitivity, recurrence plots, and power spectra. These topics illustrate in more detail the complex properties that are inherent in the SMB system. These topics not only discuss properties of optical solitons but could also provide research opportunities in the area of nonlinear optics, optical communication, or laser physics for which controllable soliton propagation is critical. This study provides a uniting methodology that embeds exact solutions and nonlinear dynamical systems analysis to present a well-rounded approach for the study of solitons and related phenomena in the SMB system.

A simple and efficient one-dimensional (1D) discrete Boltzmann method is developed for compressible flows with tunable specific heat ratios by incorporating extra degrees of freedom. To guarantee Galilean invariance in numerical simulations, a discrete velocity set is constructed with high spatial symmetry. Furthermore, an operator-splitting scheme is proposed to extend the 1D kinetic formulation to simulations of one-, two-, and three-dimensional flow systems within a unified framework. The proposed model and numerical method are verified and validated against several benchmark problems, including the Sod shock tube, Lax shock tube, two-dimensional Riemann problem, uniform translational flow, and acoustic wave propagation. The results demonstrate the accuracy, robustness, and flexibility of the present approach for compressible flow simulations.

In the field of nonlinear integrable systems, the investigation of the multi-component extensions of these systems is of great significance. In this paper, we present a novel method for constructing multi-component integrable hierarchies, with the Korteweg–de Vries (KdV) case used as an illustration. Based on a specific spectral problem, we propose a three-component generalized KdV hierarchy by applying the zero-curvature equation. Note that in the first system of this hierarchy, one of the equations is $u_t = {\frac{1}{32}}(p+q+2u)_{xxx}-{\frac{3}{16}}(p+q+2u)(p_x+q_x+2u_x)$ $+w_{21,x}$, where $w_{21} = w_{21}(p,q,u,p_x,q_x,u_x,\cdots)$ is an arbitrary function. Upon choosing a specific function, this system reduces to the KdV equation.

Neural networks, with their high expressiveness in function approximation, have become an important research area for finding exact solutions of partial differential equations (PDEs). This paper employs a neural network-based analytical solutions (NNAS) method, which uses explicit neural networks models as trial functions for PDEs. By substituting the trial functions into the PDEs and solving the resulting algebraic systems symbolically, exact solutions can be obtained. However, this method requires designing specific activation functions for each problem, which often limits the diversity of solutions. To overcome this limitation, this paper extends the Riccati sub-equation neural networks (RSENNs) framework by using different solution forms of the Riccati equation as activation functions in the first hidden layer, and constructing trial functions through feedforward computation. Compared with the NNAS method, the RSENNs method employs the Riccati solutions family as a unified activation function library, reducing the time required for the complex derivation of activation functions. To verify the effectiveness of this approach, the method is applied to the Burgers equation, Fokker–Planck equation, and KdV–mKdV equation, obtaining various solutions forms, including generalized interaction solutions, generalized rational function solutions, and generalized hyperbolic function solutions. These solutions correspond to wave interaction, singular behavior, and localized structures in nonlinear systems, demonstrating the advantages of the method in solutions diversity. Compared with the NNAS method, the RSENNs method also reduces the time required to find suitable activation functions and improves the efficiency of solving PDEs. This work represents an extension and further development of the RSENNs method, providing an effective approach for finding exact solutions to PDEs based on neural networks.

This paper proposes a discrete deep learning method, termed the traveling-wave constrained physics-informed neural networks (TC-PINNs), for addressing both the forward and inverse problems of discrete nonlinear lattice equations (NLEs). The discrete mKdV equation is used as a representative example to validate the effectiveness of the proposed approach. We respectively carry out numerical simulations and parameter inversion for one- and two-soliton solutions of the discrete mKdV equation by employing the traveling-wave transformation and propagation properties through soft constraint and hard constraint schemes, and further extend this approach to rational soliton solutions. Numerical experiments show that the TC-PINNs consistently outperform traditional PINNs and symmetric difference data enhancement physics-informed neural networks (SDE-PINNs) in both the interior and extrapolation predictions of soliton solutions as well as parameter inversion. Compared with soft constraints, TC-PINNs under hard constraints achieve a higher prediction accuracy. This method can also be applied to solve other discrete NLEs and perform parameter inversion.

This study investigates the dynamics of a monoatomic lattice featuring cubic and nonlinear on-site interactions, relevant to optical solitons. From the lattice Hamiltonian, we derive a generalized (2+1)-dimensional nonlinear evolution equation in which the dispersion and nonlinear coefficients are explicitly derived from the underlying microscopic lattice parameters. This establishes a fully theoretical and microscopic foundation for the model, as opposed to conventional Davey–Stewartson (D–S) equations that employ phenomenological coefficients. Analytical one and two-soliton solutions are obtained via the Hirota bilinear method, revealing that the cubic coupling $J_3$ and the on-site potential $J_6$ enable controlled transitions between elastic and inelastic collisions, producing $X$ and $Y$-type interaction patterns accompanied by amplitude redistribution and trajectory shifts. Furthermore, modulational instability (MI) analysis reveals the gain spectrum and unstable bandwidth, modifying peak growth rates, and introduces anisotropic mode amplification in higher-dimensional lattices. This microscopic formulation reveals collision transitions ($X \leftrightarrow Y$ types) and anisotropic MI structures that cannot be captured by continuum D–S equations. The results establish lattice nonlinearities $J_3$ and $J_6$ as controllable physical parameters for soliton steering and energy localization.

The unbound calcium cations in the cytoplasmic matrix are imperative for numerous visceral and cellular functions. Changes in calcium concentration result in neuronal death, initiating the primary symptoms of the neuronal disease. Considering the above-mentioned points, the main objective of this research is to create a two-dimensional model in order to examine the spatial distribution patterns of calcium in neuronal cells. This model includes key parameters including buffer concentration, diffusion coefficient, and endoplasmic reticulum (ER). The proposed model is classified as an imprecise boundary value problem, with boundary conditions represented by triangular fuzzy functions. Additionally, research suggests that establishing a single solution for a specific issue effectively eliminates inherent ambiguity within the problem by employing a strategy based on linear transformation principles. We derived an estimated calcium profile using the fuzzy undetermined coefficient approach and compared the solutions with linear transformation principles. The significant impact of the buffer and ER has been obtained using a two-dimensional fuzzy boundary value problem. The solution was executed using MATLAB, and numerical results were obtained using simulation. The proposed model provides novel insights into the effect of dysregulation of calcium profile in the fuzzy environment, leading to the neurological disorder Alzheimer's disease.

We investigate exact soliton solutions, bifurcation structures, and chaotic behavior in the Akbota-Myrzakulov-Tolkynay-Zhaidary (AMTZ) equation, which governs nonlinear wave propagation in multi-dimensional dispersive systems. Using the improved modified Sardar sub-equation method, we develop hyperbolic, trigonometric, and rational function solutions, along with full visualizations illustrating wave stability characteristics. Bifurcation analysis yields critical equilibrium points such as saddle, center, and cuspidal configurations, and phase portraits reveal complicated nonlinear transitions. Adding external periodic perturbations reveals chaotic attractors confirmed using Poincaré sections and positive Lyapunov exponents, establishing sensitivity to initial conditions. The AMTZ equation is used extensively in the field of optical fiber communications, plasma physics, and fluid dynamics. This is the first comprehensive chaos analysis of the AMTZ equation, connecting exact soliton construction with nonlinear dynamical characterization, providing a theoretical framework for modeling and controlling nonlinear wave processes across disciplines.

This paper develops a stochastic variant of the non-Boussinesq wavepacket model, an nonlinear Schrödinger (NLS) type envelope description for internal wave packets in strongly stratified media. Multiplicative Stratonovich noise is introduced in the carrier phase to represent environmental fluctuations. A gauge transformation with a traveling wave reduction yields a deterministic Hamiltonian envelope system that admits periodic wavetrains, localized pulse profiles, and kink type fronts across relevant parameter regimes. A modulational instability analysis identifies the sideband growth condition and shows how higher order dispersion shifts the unstable band, while phase noise does not change the leading growth rate. To probe complexity under weak periodic forcing, a Melnikov analysis predicts transverse manifold intersections and the onset of chaotic dynamics, supported by numerical sections, bifurcation diagrams, and Lyapunov exponents. The framework links stochastic modeling and nonlinear wave dynamics and provides practical diagnostics for stability, coherence, and complexity in non Boussinesq media and in related NLS type systems.

In this paper, the modulation of nonlinear waves governed by a $(3+1)$-dimensional variable-coefficient Hirota bilinear system in fluids is studied through the analysis of characteristic lines and phase differences. The transition condition for breather-to-soliton is derived by employing characteristic line analysis. From the perspective of the phase difference, we demonstrate that the waveforms of transformed waves can be effectively modulated by the appropriate choice of coefficients in the system under study, yielding both time-varying and time-invariant transformed waves, where the latter are absent in constant-coefficients systems. Moreover, long- and short-lived collisions between two nonlinear waves are obtained. Utilizing the velocity resonance method, we construct various molecular waves including some new types which contain time-invariant transformed waves as atoms, such as a breather molecule, a transformed wave-breather molecule and a transformed wave molecule. The propagation direction of these molecular waves is shown to be controllable through coefficient selection. These results indicate that this systematic framework can precisely regulate transformed waves in variable-coefficient systems.

In this paper, we construct integrable couplings for the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy on a space scale. Using a unified reduction approach, we derive the integrable couplings of the nonlinear Schrödinger (NLS) and modified Korteweg-de Vries (mKdV) equations on a space scale, thereby establishing a bridge between continuous and semi-discrete integrable systems. We further develop a Darboux transformation framework for integrable couplings on a space scale. By applying this framework to the AKNS integrable coupling, we obtain soliton solutions for the integrable couplings of the NLS and mKdV equations.

A piezoelectric ceramic transforms external forces into electric signals, generating a varying output voltage when subjected to certain deformations. In this study, a two-neuron network is constructed by coupling two FitzHugh-Nagumo neurons through a memristor-based synapse. Neural systems and neuron models are developed by simulating various brain process firing patterns using variations in external sound waves. Animals have the ability to synchronize the activity of two auditory neurons by using both ears to effectively receive and encode external sound information simultaneously. The process of energy supply and emission is an essential component of both the neuronal network and the individual neuron. The effective energy of piezoelectric sensing neurons influenced by external forces is investigated.

This study presents an innovative investigation of two nonlinear partial differential equations: the dimensionless time-dependent paraxial equation (DTDPE) and the Biswas–Milovic equation (BME). The DTDPE has important applications in nonlinear optics and quantum mechanics, while the BME has particular significance in the field of fiber optics. The Jacobi elliptic function expansion method is employed to construct exact analytical solutions for both equations. As a result, several types of exact solutions are obtained, including dark soliton solutions, singular soliton solutions, periodic Jacobi elliptic function solutions, and combined hyperbolic function solutions. All obtained solutions satisfy the equations under consideration. Furthermore, the characteristics and behaviors of these solutions are thoroughly visualized using numerical schemes, which enhances our understanding of their potential applications in real-world physical systems. To the best of our knowledge, this work provides the first unified analytical treatment of these equations using the Jacobi elliptic function expansion method to generate multiple distinct classes of exact solutions.

The Sasa–Satsuma (S–S) equation, an integrable extension of the nonlinear Schrödinger equation, could provide a more accurate description of ultrashort optical pulse propagation by incorporating third-order dispersion, self-steepening, and delayed nonlinear response. In this work, we establish a quantitative correspondence between the spectral degeneracies of two types of related linear non-Hermitian matrices and the localized wave solutions of the nonlinear Hermitian systems. Compared with the standard nonlinear Schrödinger equation system, the S–S equation displays richer spectral and dynamical structures due to its multi-band Hamiltonian. In particular, we show that exceptional points correspond to rogue waves in the modulational instability regime and to W-shaped solitons in the modulational stability regime. Moreover, higher-order degeneracies, such as third-order exceptional points, reside at the boundary between modulational stability and modulational instability. This configuration enables a coexistence mechanism of W-shaped solitons and rogue waves, where two W-shaped solitons can interact to generate a rogue wave. These results provide a unified non-Hermitian perspective on localized wave phenomena in higher-order integrable systems and reveal the fundamental link between spectral degeneracies and nonlinear wave dynamics.

Understanding how the multi-layer structure of biological membranes influences neural electrical activity remains an important issue in computational neuroscience and neural circuit modeling. Some dual-membrane neuron models characterize the electrophysiological properties of the cell membrane, but the complete hierarchical structure of biological membranes is not addressed well. Based on the ‘sandwich-like' membrane structure as the Davson–Danielli model, this paper constructs a new neuron model consisting of three-layers of membranes, which are described by three capacitors coupled via memristors. An additive capacitor is used to shunt current from the inductive channel and energy level is regulated. The tri-capacitors are connected by two different memristors, and the material property of inner media between capacitive membranes is considered. The proposed neural circuit can maintain signal processing capabilities even when one capacitor breakdowns. The three-layers of membranes exhibit significant inter-layer non-uniform electrical responses under external stimulation, which reveals the possible source mechanism for the negative resting potential of biological cell membranes. Further, energy function is defined to describe the intrinsic relationship between energy distribution and different firing modes, and explores the occurrence conditions for stochastic resonance under noisy excitation. Numerical results show that the proposed model can reproduce multiple firing patterns and reveal the relationship between membrane energy redistribution and firing mode transition. An adaptive control scheme under energy regulation can actively achieve firing mode conversion by adjusting the bifurcation parameter. The proposed three-membrane neuron model not only can reproduce captures possible hierarchical electrical responses in membrane-related circuits of multi-layer membranes and their ion channel encoding effects, but also provides a new theoretical framework for revealing the relationship between membrane structure, energy mechanism, and neural signal regulation, thereby highlighting its potential significance for understanding membrane electrophysiology and for designing artificial neural circuits with multi-modal response capabilities.

In this study, we develop the inverse scattering transform of the focusing Lakshmanan–Porsezian–Daniel equation with a one-sided nonzero boundary condition, where the asymptotic amplitude tends to zero on one side at spatial infinity while remaining nonzero on the other. For the direct scattering problem, single-valued eigenfunctions are rigorously defined, and their analyticity, symmetries, and asymptotic behavior are rigorously analyzed. Furthermore, the Marchenko integral equations and the Riemann–Hilbert problems (both right and left) on a single sheet of the scattering variable are constructed and utilized to formulate the inverse problem. Additionally, a proper uniformization variable is defined to formulate both the direct and inverse scattering problems, which maps the two-sheeted Riemann surface of the spectral parameter onto a single complex plane. Ultimately, the time evolutions of the eigenfunctions are derived, revealing the nontrivial time dependence of the scattering data.

Constructing nonlinear wave solutions for integrable systems has always been a significant work in applied mathematics, physics, and engineering. As one of the frontier works in integrable systems, doubly localized rogue waves, which are localized in time and all space variables, appear only in local and low-dimensional complex nonlocal systems, and remain open in high-dimensional real nonlocal systems. In this paper, the semi-rational solutions of nonlocal Kadomtsev–Petviashvili (KP) equation formed by the partial resonant collision of solitons and lumps, including single soliton and single lump, multiple lumps and single soliton, multiple lumps and multiple solitons, have been proposed. The partial resonant collisions of these solutions are semi-localized in time (i.e., lumps appear only when t → −∞ or t → +∞), and the dynamics and asymptotic behaviors of these solutions are discussed. Additionally, the semi-rational solutions of nonlocal KP equation formed by the full resonant collision of solitons and lumps are also presented, where the lump waves are localized in space variable (x, y) and time t. These lump waves are called doubly localized rogue waves, and their generation mechanism and various characteristics are analyzed in detail. The results of this paper provide reference for predicting rogue waves and future research on other high-dimensional real nonlocal integrable systems.

A systematic scheme is designed to derive reduced equations for dissipative nonlinear evolution equations, based on a renormalization group (RG) approach to inertial manifolds, low-dimensional structures that govern long-term dynamics. Unlike traditional model reduction techniques that often presume specific analytic forms of these structures, the RG method constructs them directly from the equation of motion through a controlled, step-by-step expansion. The method is successfully demonstrated on typical 1D and 2D evolution equations on a finite spatial domain, which accurately captures different wave patterns—steady states, periodic oscillations, and chaos with a precise detection of bifurcation points upon parameter variation. The current approach preserves the underlying physics and symmetries, thereby sparing them from recalibration by data and carving out a rigorous, equation-based path to the construction of low-dimensional models that are both accurate and computationally efficient.

This paper introduces a novel analytical technique called the “Variable Coefficient Second Degree Generalized Abel equation Method” (VCSDGAE), designed to tackle the complex Ginzburg–Landau equation. Unlike conventional methods that depend on constant coefficient ordinary differential equations (ODEs) and auxiliary ODEs, our approach employs variable coefficient ODEs within a sub-equation framework. We demonstrate the versatility of this method by successfully applying it to the complex Ginzburg–Landau equation. Through the presentation of analytical solutions, we showcase the method's effectiveness and efficiency, positioning it as a valuable resource for addressing complex nonlinear partial differential equations in fields such as fluid dynamics and wave propagation. The stability of the obtained soliton solutions is established through a linear stability analysis, confirming their robustness against small perturbations. This research not only broadens the spectrum of available analytical techniques but also contributes significantly to the advancement of solutions for various mathematical physics models.

The potential modified Korteweg–de Vries (pmKdV) equation possesses an infinite number of symmetries; however, the physical implications of these symmetries remain unexplored. Recent improvements have successfully elucidated that the physical significance of the infinite symmetries are related to wave number translation and wave center translation invariance. Following the ideas of these results, this paper aims to investigate the physical meanings associated with the infinite symmetries linked to the n-soliton solutions of the pmKdV equation. The findings indicate that the K-symmetries are related to the translational invariance of the wave center, while the τ-symmetry is associated with combined translational symmetries of both the wave center and wave number. Furthermore, these infinitely many symmetries are not entirely independent, only some of the special symmetries are autonomous. These results offer a novel method for deriving n-wave solutions of the pmKdV equation. Using this method, some special exact solutions of the pmKdV equation, such as the soliton, complexiton, breather and double-pole solutions, are presented.

In this paper, a generalized (3+1)-dimension variable-coefficient nonlinear evolution equation is investigated, which serves as a model for describing nonlinear wave behaviors in shallow water, ion-acoustic wave fluid mechanics and plasma physics. The Painlevé integrability is tested by the Weiss, Tabor and Carnevale (WTC) method with the simplified form of Krustal. The bilinear form of the equation is derived through the application of the Hirota bilinear method. Building on the bilinear equation, a broad range of analytical solutions are then obtained, including X-shaped and Y-shaped soliton solutions, lump solution, breather solution, and interaction solutions. In addition, another type of soliton solution, periodic solution, and ratio of trigonometric functions are derived.

Explosive synchronization (ES) describes an abrupt and hysteretic transition from incoherence to collective order and has been widely studied in static networks. Here, we show that temporal variability of network connectivity can fundamentally reshape this transition. We investigate inertial Kuramoto oscillators evolving on stochastically rewired random networks, where links are continuously replaced at controlled rates, allowing us to tune the interplay between inertia and topological dynamics. Our results reveal that temporal rewiring can both induce and suppress ES depending on the network density and the switching timescale. Sparse networks display ES only under very slow or very rapid switching, whereas denser networks exhibit robust explosive transitions across a broad parameter range. Increasing the rewiring probability generally promotes abrupt synchronization, but excessively frequent rewiring weakens hysteresis and reduces bistability. A systematic exploration across different degrees confirms that ES is most prominent when moderate-to-high rewiring probability is combined with rapid switching, whereas small rewiring probability favors continuous transitions. These findings demonstrate that temporal randomness is not merely a perturbation but a key control mechanism for abrupt collective behavior, representing how time-varying connectivity governs the onset, robustness, and disappearance of ES in dynamical networks.

Nonlinear option pricing models are of significant research value as they better reflect the realities of financial markets, yet their numerical solution remains highly challenging. On the one hand, such models typically involve strong nonlinearity, multi-scale features of small parameters, and high sensitivity to initial data, which often make it difficult for traditional numerical methods to maintain stability and accuracy. On the other hand, many deep learning methods rely on boundary data, while in real financial markets boundary conditions are often unavailable, thereby limiting their applicability. Thus, deep learning methods that rely solely on initial data still face significant challenges in efficiently solving nonlinear option pricing models and achieving effective numerical predictions. To address these difficulties, this work employs the respecting causality physics-informed neural network (RCPINN), which depends solely on initial data and respects the inherent spatiotemporal causal structure of system evolution, enabling it to effectively handle the complex characteristics of nonlinear financial models. In the Ivancevic nonlinear option pricing model, the RCPINN successfully predicts the formation and evolution of both low-order and high-order financial rogue waves, revealing the underlying dynamical mechanisms of extreme localized waves. In the nonlinear Black–Scholes transaction cost model, the RCPINN effectively captures the dynamical evolution of European call and put option prices. This work not only validates the effectiveness and applicability of RCPINN in predicting financial rogue waves and European option prices within the framework of nonlinear option pricing models, but also demonstrates its advantages in handling initial data sensitivity, multi-scale features, and strong nonlinearity in financial problems. More importantly, this approach provides a novel technical pathway for both early warning of extreme financial risks and the simulation of European option price evolution, while the findings offer valuable insights for applications in derivative pricing, exchange rate forecasting and financial market risk monitoring and early warning.

The Navier–Stokes (NS) equation with Coriolis force and density-dependent viscosity is an important physical model, which has been widely used to understand and analyze a wide array of phenomena, including behaviors of the Gulf stream, dynamics of hurricanes, operation of chemical reactors and functionality of rotating machines. In this paper, based on the matrix and curve integration techniques, we build a sufficient condition for the existence of Cartesian vector solutions u = b(t) + A(t)x for the N-dimensional NS equation, in which A satisfies appropriate matrix equations. Then, we discuss two special cases of A and thereby explicit analytical solutions are obtained. To shed light on these solutions, we give some illustrative examples. Among them, some examples form the generalization previously obtained by other authors and some examples are quite new. Finally, we analyze the properties of Cartesian vector solutions in a special case.

This paper explores the application of Whitham modulation theory to the third-order focusing Kaup–Newell model, offering a complete classification of solutions for step-like initial value problems. Using the finite-gap integration method, we derive periodic solutions and the corresponding Whitham modulation equations. By analyzing the distribution of Riemann invariants, we identify the fundamental wave structures emerging from the step-like initial value problem. Furthermore, we provide a complete classification of solutions for this problem. Our results demonstrate that Whitham modulation theory serves as an effective analytical framework for studying initial value discontinuities in the third-order focusing KN model, offering new insights into its nonlinear dynamical behavior. Moreover, the direct numerical simulations show remarkable agreement with the results from Whitham modulation theory.

In this work, we propose a new deep learning approach based on physics-informed neural networks (PINNs), termed parallel parity-time-symmetric PINNs (PPTS-PINNs), to address the inverse problem of determining the PT-symmetric potential function in the nonlinear Schrödinger equation (NLSE). By incorporating PT-symmetry constraints and a gradient enhancement strategy, the method effectively improves both the accuracy and stability of solving inverse problems, while preserving the consistency of the physical structure. We conduct systematic numerical experiments on two representative NLSEs under both noise-free and noisy conditions (with noise levels of 1% and 5%) to evaluate the reconstruction performance of the model given fixed observational data. The results demonstrate that PPTS-PINNs can robustly reconstruct the complex-valued potential function across different noise levels, achieving an overall error on the order of 10^-3. Notably, under high-noise conditions, the combination of PT-symmetry constraints and the gradient enhancement strategy significantly enhances the model's robustness by mitigating error propagation. Overall, the proposed method exhibits strong adaptability and generalization capabilities in PT-symmetric modeling and noise-resilient learning, offering a novel perspective for solving more complex physical inverse problems.

In this article, the large-time asymptotic wave dynamics of rogue curves are analytically investigated and numerically confirmed in the Davey-Stewartson (DS) I equation. We show that, when time in bilinear expressions of the rogue curves is large, a certain number of localized lump-shaped waves would arise on the uniform background, exhibiting various wave patterns. We further show that, as time increases, the individual lump-shaped wave asymptotically evolves into a line soliton on the constant background that persist at large time. By performing large-time asymptotic analysis, we reveal that such wave patterns as well as the numbers of lump-shaped waves can be analytically determined by the structure of nonzero roots of the Wronskian-Hermite polynomials. Our asymptotic predictions are compared to true solutions quantitatively and excellent agreement is obtained.

In this study, we propose that many different thermodynamic modeling approaches, including the general equation for the non-equilibrium reversible-irreversible coupling (GENERIC), Onsager's variational principle, the energetic variational approach, and classical irreversible thermodynamics, can all be cast into the gradient-conservative structure (GCS). GCS enjoys many nice mathematical properties, has close connection with the large deviations principle and gradient flows in Wasserstein space, and fulfills laws of thermodynamics. Our results demonstrate that the GCS may serve as a unified theoretical framework to model various non-equilibrium thermodynamic processes.

Under study in this paper is a nonlinear Schrödinger equation with local and nonlocal nonlinearities, which originates from the parity-symmetric reduction of the Manakov system and has applications in some physical systems with the parity symmetry constraint between two fields/components. Via the Riemann-Hilbert method, the theory of inverse scattering transform with the presence of double poles is extended for this equation under nonzero boundary conditions (NZBCs). Also, the double-pole soliton solutions with NZBCs are derived in the reflectionless case. It is shown that the quasi-periodic beating solitons can be obtained when the double pole lies off the circle $\Gamma$ centered at the origin with radius $\sqrt{2}{q}_{0}$ (where q₀ is the modulus of NZBCs) on the spectrum plane. Moreover, using the improved asymptotic analysis method, the asymptotic solitons are found to be located in some logarithmic curves of the xt plane.

We establish a quantum dynamics framework for curved submanifolds embedded in higher-dimensional spaces. Through rigorous dimensional reduction, we derive the first complete Schrödinger and Klein-Gordon equations incorporating non-perturbative geometric interactions-resolving ambiguities in constrained quantization. Crucially, extrinsic curvature of the ambient manifold governs emergent low-dimensional quantum phenomena. Remarkably, this mechanism generates scalar field masses matching Kaluza-Klein spectra while eliminating periodic compactification Requirements. We hypothesize that geometric induction can produce Higgs-mechanism-type potentials. Under this working hypothesis, particle masses arise solely from submanifold embedding geometry and matter-field couplings are encoded in curvature invariants. If this hypothesis holds, it would enable experimental access to higher-dimensional physics at all energy scales through geometric induction. We also discuss the Higgs vacuum near small-mass black holes.

This paper presents a geometric perspective that connects reciprocal transformations with multidimensional integrable deformations. By interpreting conservation laws as closed 1-forms, we formalize reciprocal transformations as induced local diffeomorphisms on the jet bundle. This allows us to characterize higher-dimensional deformations as systematic fiber bundle extensions, where fiber coordinates are generated by potential functions of the conservation laws. This perspective provides an interpretation for the covariant lifting of Lax pairs to higher dimensions and reveals that auto-Bäcklund transformations are composite diffeomorphisms. These results are applied to several classical integrable models.

This study investigates the dimensionless quasi-geostrophic potential vorticity (QG-PV) equation with external sources. Employing the Gardner–Morikawa transformation and weakly nonlinear perturbation expansion, we derive the nonlinear Boussinesq equation with external sources. We demonstrate the existence of explicit zero-order and first-order Wronskian solutions for the model equation when α₄ = 0. Furthermore, using a modified Jacobi elliptic function method, we obtain soliton-like solutions for both α₄ = 0 and α₄ ≠ 0. Analysis of these solutions reveals that the generalized β-plane approximation and shear flow are significant factors in inducing nonlinear Rossby waves, and that external sources play a crucial role in influencing Rossby wave behavior.

The paper considers applications of Rota–Baxter algebras to renormalization in quantum field theory and quantum integrability to obtain new solutions for the Yang–Baxter equations which can be studied by the method of Renormalization Group.

In this paper, we investigate data-driven bright soliton solutions of the nonlocal reverse-time nonlinear Schrödinger (NLS) equation and the parameter identification using the physically informed neural networks (PINNs) algorithm. Accurate simulations and comparative analyses of relative and absolute errors are performed for two-soliton and four-soliton solutions including linear solitary waves and periodic waves. In the training process, the standard PINNs scheme is employed for linear solitary wave solutions, while the prior information is added at local sharp regions for periodic wave solutions due to the complicated collision behaviors. For the parameter identification, we accurately recognize the nonlinear coefficients of the nonlocal NLS equation from known solutions with different noises. These results reinforce the application of deep learning with the PINNs framework to successfully study nonlocal integrable systems.

We investigate the entanglement entropy in quantum states featuring repeated sequential excitations of unit patterns in momentum space. In the scaling limit, each unit pattern contributes independently and universally to the entanglement entropy, leading to a characteristic volume-law scaling. Crucially, this universal contribution remains identical for both free and interacting models, enabling decomposition of the total entanglement into pattern-specific components. Numerical verification in fermionic and bosonic chains confirms this volume-law fragmentation phenomenon. For fermionic systems, we derive analytical expressions where many-body entanglement becomes expressible through few-body entanglement components. Notably, this analytical framework extends to spin-1/2 XXZ chains through appropriate identifications.

This paper proposes an innovative form of group reduction or similarity transformation involving off-diagonal block matrices. The proposed method is applied to the Ablowitz–Kaup–Newell–Segur (AKNS) matrix spectral problem, leading to the generation of reduced matrix AKNS integrable hierarchies. As a result, a variety of reduced multiple-component integrable nonlinear Schrödinger and modified Korteweg–de Vries models are derived from the analysis of the reduced AKNS matrix spectral problem.

In this paper, we investigate the integrable fractional coupled Gerdjikov–Ivanov equation and derive its explicit form by employing the completeness relation of squared eigenfunctions. Based on the Riemann–Hilbert method, we construct the fractional N-soliton solutions. We find that as the power ε of the Riesz fractional derivative increases, the amplitudes of the fractional soliton solutions remain invariant, while their widths decrease and the absolute values of the wave velocity, group velocity, and phase velocity increase. Additionally, we examine the long-time asymptotic behavior of the fractional N-soliton solution. The results show that as t →±∞, the solution can be approximated by the sum of N fractional one-soliton solutions, with each soliton's amplitude and velocity remaining constant, whereas both position and phase shifts are observed.

Urban rail transit is an efficient and environmentally friendly mode of transport, which is an important means of transportation for passengers. From a holistic point of view, this paper constructs an urban rail transit interchange topology (URTIT) network based on the interchange relationships among lines. We investigate a unique influence propagation mechanism to explore the impact of applying new technologies on the passenger travel behavior of urban rail transit. We analyze the influence from three aspects: the influence range, the influence propagation path, and the influence intensity. Based on the Dijkstra algorithm, the influence propagation paths are found according to the shortest transfer time. The improved path−based gravity model is applied to measure the influence intensity. The case study on urban rail transit in Beijing, China is carried out. The influence propagation mechanism of a single line in the Beijing URTIT network is analyzed, considering that Beijing Subway Line S1 is equipped with magnetic levitation technology. We not only quantify the impact of technologies on passenger travel behavior of urban rail transit, but also perform the sensitivity analysis. To avoid randomness, the influence propagation mechanisms of all lines are explored in this paper. The research results correspond to the situation in reality, which can provide certain references for urban rail transit operation and planning.

In order to investigate physically meaning localized nonlinear waves on the periodic background defined by Weierstrass elliptic ℘-function for the (n + 1)-dimensional generalized Kadomtsev–Petviashvili equation by Darboux transformation, the associated linear spectral problem with the Weierstrass function as the external potential is studied by utilizing the Lamé function. The degenerate solutions of the nonlinear waves have also been obtained by approaching the limits of the half-periods ω₁ and ω₂ of ℘(x). At the same time, the evolution and nonlinear dynamics of various nonlinear waves under different parameter regimes are systematically discussed. The findings may open avenues for related experimental investigations and potential applications in various nonlinear science domains, such as nonlinear optics and oceanography.

This paper investigates the physical significance of the infinitely many K- and τ-symmetries associated with the soliton and complex solutions of the sine-Gordon (sG) equation. It is shown that the K-symmetries are linear combinations of wave center translation symmetries, while the τ-symmetries combine both wave center translation and wave number translation symmetries. Only a subset of the K- and τ-symmetries are independent, indicating that these symmetries are not incomplete. A special one-soliton solution of the sG equation is derived by using the generalized symmetries.