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.

For the superoperators that are linear in bosonic operators, we prove Wick’s theorem rigorously by applying stable partition theory. Higher-order time-ordered superoperator terms can be decomposed into sums of products of second-order time-ordered terms. By applying the theorem, explicit expressions for all even-order terms of the reduced density matrix are derived, with vanishing odd-order terms. We obtain the set of equations for the reduced dynamics of a system interacting with a bosonic bath initially in thermal equilibrium.

We investigate the impact of including a dynamical charm quark on the properties of light hadrons. Our study compares the calculations performed on 2+1+1 flavor (highly improved staggered quark) ensembles at four lattice spacings to those on 2+1 flavor (clover fermion) ensembles at six lattice spacings, with both sets of ensembles employing the identical Symanzik gauge action. For the light, strange and charm flavor observables, we employ the same tadpole-improved clover fermion action. From numerical results for light and strange quark masses, pion and kaon decay constants, and $\Omega$ and $\Omega_{ccc}$ baryon masses, we find that the values obtained after continuum, chiral, and infinite-volume extrapolations are consistent within uncertainties. Even though the mixed action setup can introduce additional discretization effects, our calculation shows evidence that these effects can cancel with the discretization error in the unitary setup, resulting in better convergence in the continuum extrapolation.

We investigate the fine-tuning of radiative alpha-particle capture on carbon, ${} ^{12}\textrm{C}(\alpha,\gamma){}^{16}\textrm{O}$, at astrophysical energies. Utilizing results from cluster effective field theory for this reaction, we find that the low-energy data of the astrophysical S-factor allow for only very small variations in the electromagnetic fine-structure constant $\alpha$, namely $|\delta \alpha/\alpha| \unicode{x2A7D} 0.2\,$‰, in both $E1$ and $E2$ radiative capture.

The mechanism and the observables (including cross section, angular distribution, energy spectrum and double-differential cross section (DDCS)) for the p+$ ^{28}$Si reaction below incident energy 200 MeV are systematically studied. A set of optimal optical potential parameters for the proton is obtained by simultaneously fitting the reaction cross sections and elastic scattering angular distributions. Utilizing the derived optical potential parameters, the direct inelastic scattering cross sections and their angular distributions based on the distorted wave Born approximation theory are calculated. Then, the energy spectra and DDCSs of the light charged particles are self-consistently derived on the basis of the exciton model, evaporation model and Hauser–Feshbach theory with fluctuation correction. The calculated results are in good agreement with the existing experimental data except several structural peaks of outgoing deuteron. Additionally, a comparative analysis has been performed between our calculated results and those derived from TALYS-2.0.

We investigate the extremality relations by examining perturbative corrections to both the entropy of Schwarzschild-de Sitter black holes and their extremality bounds under the Nariai limit within the frame of the generalized uncertainty principle (GUP) and the extended uncertainty principle (EUP) respectively. The GUP-corrected or EUP-corrected horizons bring about the additional terms finally in the Goon–Penco relation to violate the universal ones and these extra terms cannot be canceled by adding the adequate conditions. We argue that the corrected uncertainty principles including GUP and EUP violate the validity of extremality relations because no matching condition can be imposed to support the Goon–Penco relation unless the influences from GUP and EUP disappear.

The characterization of few-body interactions is fundamental for understanding the properties and collisional stability of ultracold quantum gases in reduced dimensions. In this work, we theoretically investigate the atom-dimer scattering and three-body recombination processes in a two-dimensional quantum gas. By generalizing the Skorniakov–Ter-Martirosian equations to two dimensions, we obtain the exact atom-dimer scattering amplitude and systematically evaluate the validity of common theoretical approximations, revealing a distinct feature of two-dimensional (2D) collisions: while mean-field theory fails rapidly with increasing interaction strength, the single-pole approximation remains remarkably accurate across a broad interaction regime. Furthermore, we extend our exact framework to non-zero incident momenta to calculate the three-body recombination rate. Our results demonstrate that the lifetime of the 2D quantum gas exhibits a power-law dependence on the 2D gas parameter $na_\textrm{2D}^2$ with an exponent of approximately $-2.45$, showing a significant reduction in stability as interactions are enhanced. This study bridges the gap between exact few-body theory and practical many-body approximations, offering a robust benchmark for predicting the stability of cold atomic mixtures and guiding future experimental explorations of universal few-body physics in highly controllable 2D environments.

Community detection, which aims to reveal the underlying structure of communities within a network, is a fundamental task in network analysis. In this paper, we employ Free Energy Machine (FEM) framework to maximize modularity, thereby identifying optimal community partitions. FEM is grounded in the principles of statistical physics, and the concept of free-energy minimization, and integrates ideas from mean-field theory and simulated annealing, while leveraging modern computational techniques such as automatic differentiation and gradient-based optimization. Experimental results on a wide range of real-world and synthetic networks show that FEM achieves an improvement in modularity compared with other prominent algorithms. These findings demonstrate that FEM is an effective tool for solving the modularity optimization problem.

The network dismantling problem (NDP), which seeks to fragment a network into subcritical components by removing a minimal set of vertices, is a fundamental challenge in complex network science with broad applications in infrastructure protection and epidemic control. Physically, the NDP maps to the problem of finding the ground state of a disordered spin system, a task known to be NP-hard. Consequently, exact solutions have remained computationally intractable for all but the smallest graphs, leaving a critical gap in evaluating the true optimality of widely used heuristic algorithms. In this work, we bridge this gap by presenting an exact dismantling algorithm based on a branch-and-bound framework. By efficiently pruning the solution space based on connected component formation, our method determines the exact ground states for regular random graphs up to 70 nodes and Erdős–Rényi graphs up to 80 nodes. These rigorous results serve as a definitive benchmark, revealing the precise optimality gap of popular approximation methods such as Collective Influence and CoreHD. Furthermore, we leverage our exact approach to optimize heuristic strategies, demonstrating a 1%–4% improvement in dismantling efficiency on larger networks.

The translational dynamics usually decouple earlier than the rotational in supercooled liquids as the temperature decreases, whereas the inverted scenario remains scarcely reported. In this work, starting from the coarse-grained ortho-terphenyl model, we build a rigid tetrahedral structure model. It exhibits earlier decoupling in rotational dynamics than in translational. The Stokes–Einstein–Debye relation breaks down while the Stokes–Einstein relation remains intact. The decoupling happens at approximately 2T_g similar to that observed in supercooled water. The rotation shows more heterogeneous dynamics than the translation at all temperatures. Our results suggest that the steric hindrance plays an important role in determining the decoupling between translational and rotational dynamics.

Recent advancements in deep neural networks have improved phase transition analysis. However, their performance depends heavily on the labeling method, and traditional approaches based on critical values for labeling lack physical interpretability and overlook the behaviors of different configurations under fluctuations. To address these issues, we propose a labeling strategy based on the average value of the observable at the critical point. In the percolation model, the threshold is determined by the average relative size of the largest connected cluster, while in the Ising model, it is based on the average absolute magnetization and energy. We validate the strategy on classical models, including site and bond percolation on 2D square and triangular lattices, bond percolation on Erdõs–Rényi (ER) random networks, and the Ising model on 2D square and triangular lattices. Convolutional neural networks are used for lattice data, and graph convolutional networks for ER network data. The results demonstrate that this observable-based labeling strategy can accurately identify critical points and exponents, offering universality and stability across various phase transition types.

Exploring anisotropic dynamics in exotic superconductors or superfluids with spin–orbit coupling (SOC) is essential for understanding the role of SOC in these systems. We theoretically calculate the density dynamical structure factor of a two-dimensional Fermi superfluid with Raman-type SOC and investigate its primary anisotropic dynamical characteristics under varying SOC strengths during the Lifshitz phase transition. Owing to the recoil momentum of the SOC laser beam, both the collective phonon mode and the single-particle excitations exhibit distinct anisotropic behavior. In contrast to superfluids without SOC, the phonon mode under SOC not only has a shorter lifetime due to stronger competition with single-particle excitations but also exhibits an anisotropic sound velocity. The sound velocity initially decreases and then increases as the SOC strength increases. Moreover, three types of threshold energies for single-particle excitations to break Cooper pairs are identified, which elucidate the complex edge curves of the dynamical structure factor.

We investigate the topological and localization properties of a quasiperiodic one-dimensional Su–Schrieffer–Heeger (SSH) model modulated by an interpolating Aubry–André–Fibonacci function. In the Aubry–André (AA) limit, the system undergoes a topological phase transition from a topological insulator to a topological Anderson insulator by increasing the quasiperiodic potential strength. Moreover, we show the existence of three distinct pure phases including the extended, localized and critical phases. By calculating the inverse participation ratio, normalized participation ratio, and fractal dimension, we demonstrate that the multiple localization transitions occur when tuning the quasiperiodic potential strength at a moderate value of interpolating parameter $\beta$. These reentrant phenomena which are absent in the standard AA–modulated SSH chain, develop gradually during the interpolation and eventually vanish for large $\beta$. Furthermore, two different types of mobility edges emerge in the intermediate phase, which separate the extended from the localized states, and the localized from the critical states.

Quantum fluctuations can give rise to a singular quantum critical point (QCP) in the ground state, whose influence extends to finite temperatures, forming a quantum critical regime (QCR). Recently, it has been shown that in the quantum Ising model, the symmetry-breaking, longitudinal field can induce a quantum supercritical regime (QSR) emanating from the QCP, which hosts a universally enhanced quantum supercritical magnetocaloric effect (Lv et al 2025 Nat. Commun. 16 10646). In this paper, we show that the QSR also emerges in the spin-1/2 XXZ model, in both the forms of Ising and Berezinskii–Kosterlitz–Thouless (BKT) supercriticality. Using ground-state and finite-temperature tensor-network methods, we investigate quantum supercritical phenomena near a BKT QCP. We reveal a quantum supercritical crossover scaling $T \propto h^{2/3}$ and a Grüneisen ratio scaling $\Gamma_h \propto T^{-3/2}$ for the BKT QCP, which differ from the corresponding Ising supercritical scalings. Nevertheless, we find that the scaling function $\phi_{\Gamma}(x)$ of the singular Grüneisen ratio for both BKT and Ising cases can be approximately described by the same expression $\phi_{\Gamma}(x) \approx x/(1+x^2)$. Our work extends the study of quantum supercritical phenomena from the Ising to the XXZ Heisenberg model, thereby revealing the presence of BKT quantum supercriticality and broadening the scope of quantum supercritical physics.

Continuous memristor nonlinear systems and discrete memristor nonlinear systems are two branches for studying complex dynamical behaviors in nonlinear fields. The nonlinear oscillator modeling based on memristor circuits and the circuit design of memristor nonlinear oscillators have achieved mature research results. However, the modeling of discrete systems based on memristor circuits and the memristor circuit design of map systems are open issues. Therefore, this review systematically elaborates on the precise modeling methods from memristor circuits to discrete dynamical models, and vice versa, to memristor circuits based on the discrete dynamical systems. This review first describes the modeling method from the memristor circuits to the memristor map models. We will construct the equivalent nonlinear map models based on the continuous dynamical oscillators of the memristor oscillation circuits. This process reveals that the connection topology of circuit elements (nonlinear resistors, capacitors, inductors, memristors) directly determines the specific form of the discrete dynamical system. Moreover, this review presents an inverse method from discrete dynamical models to memristor circuits. We demonstrate how to decompose and map the iterative equations of a discrete system into specific memristor circuits. This process reveals that the specific discrete dynamical system can be equivalent to different connection topologies of memristor circuits consisting of nonlinear resistors, capacitors, inductors, and memristors. The two-way framework established in this review profoundly reveals the intrinsic connection between continuous and discrete dynamics in memristor circuits and provides a systematic design methodology for specialized nonlinear hardware systems applicable to fields such as secure communication, true random number generation, and neuromorphic computing.