Search Results (509)

This study introduces a high-order numerical scheme for solving 1D second-order hyperbolic telegraph equations (HTEs) with variable coefficients. We employ a generalized temporal discretization (TD) of order p via the Rothe approach, combined with a spatial spectral collocation (SCM) method using generalized shifted Jacobi polynomials (GSJPs). By utilizing a Galerkin-type basis that structurally satisfies homogeneous boundary conditions (HBCs)—including Dirichlet or Neumann types—we achieve a global error bound of $\mathcal{O}({\left(\Delta \tau \right)}^p+N^{-s})$, where $\Delta \tau$ denotes the temporal step size and s represents the spatial regularity of the exact solution (ExaS). The proposed algorithm, Rothe-GSJP, allows for an optimal balance between the temporal and spatial parameters, minimizing computational effort for high-precision engineering applications such as Phase-Locked Loop (PLL) modeling. Numerical experiments performed on an i9-10850 workstation show that the scheme always reaches the machine precision floor of ${10}^{-16}$. While the framework supports temporal orders up to $p=6$, the results indicate that $p\in \{2,3,4\}$ provides an optimal balance between high-order precision and absolute stability. The Rothe-GSJP method proves to be a robust, efficient, and highly accurate alternative to traditional solvers for hyperbolic systems. Full article

This paper addresses the position control problem for a Lagrangian pendulum. Using a strict Lyapunov function, a rigorous analysis is presented to prove that the closed-loop system equilibrium point composed of the pendulum dynamics and a classical linear PID control is globally asymptotically stable. Motivated by these results, the theoretical proposal is extended to analyze a novel hyperbolic PID-type control scheme; reformulating the Lyapunov function, global asymptotic stability of the equilibrium point for the corresponding closed-loop equation is demonstrated. The proposed hyperbolic scheme is a rational function with bounded control action composed of a suitable combination of hyperbolic sine and cosine functions. The hyperbolic structure is used in the proportional, integral, and derivative terms of the control algorithm to drive the position error and joint velocity to zero. Experimental results of both a linear PID and a novel hyperbolic PID-type controller on a direct-drive pendulum are presented to illustrate the effectiveness and performance of the proposed control algorithm. Full article

The stabilization of rock mass vibrations in underground excavations presents a critical engineering challenge due to the interplay of viscoelastic dynamics, impulsive shocks from blasting or rock bursts, and time-varying delays induced by wave propagation and sensor–actuator networks. In this paper, an integral sliding mode control scheme is developed for a Kelvin–Voigt type hyperbolic system subject to such impulsive effects and time-varying delays. To preserve sliding surface continuity under impulsive disturbances, the impulse information is explicitly incorporated into the design of the integral sliding function. The resulting sliding mode dynamics, which include discrete state jumps, are analyzed using a piecewise Lyapunov functional combined with inequality techniques; sufficient conditions are derived to guarantee asymptotic stability. Moreover, a sliding mode control law is synthesized to ensure that the system trajectories reach and remain on the sliding manifold from the initial time onward, despite parameter uncertainties and external disturbances. Numerical simulations with parameters reflecting realistic mining scenarios verify the effectiveness of the proposed control strategy, demonstrating its potential for practical rock mass vibration stabilization in geotechnical engineering. Full article

The purpose of this study is to construct diverse forms of exact soliton solutions and conduct a comprehensive qualitative analysis. For this aim, we use the Gross–Pitaevskii system, which belongs to the family of nonlinear Schrödinger equations. This model is considered to be iconic and significant because it has potential applications in applied sciences, such as in physics, where it is used to exemplify quantum systems like Bose–Einstein condensates and illustrate the propagation of waves in optical fibers. Employing analytical techniques, the modified sine–cosine/sinh–cosh and extended rational sinh–Gordon expansion methods, we extract several waves from solutions in the shape of trigonometric, hyperbolic, and rational forms. To further deepen our insights related to the system’s behavior, we execute a detailed dynamical analysis, including sensitivity, bifurcation, and chaos, using the corresponding Hamiltonian structure. We also derive the instability modulation using linear stability theory. Using Mathematica, we systematically simulate and verify all constructed results and present some solutions for appropriate parameter values using 2D, 3D, and contour plots. The outcomes provide fruitful insights relevant to multiple scientific domains, including optical fiber technology, plasma, and condensed matter physics. This work contributes to the ongoing study of nonlinear models by applying novel solution techniques and offering a broader perspective on the complex behavior of such systems. The novelty of this study lies in the fact that the proposed model has not been previously explored using the aforementioned advanced methods and comprehensive dynamical analyses. Full article

As deep learning models are increasingly embedded as critical components within complex socio-technical systems, understanding and evaluating their systemic robustness against adversarial perturbations has become a fundamental concern for system safety and reliability. Deep neural networks (DNNs) are highly effective in visual recognition tasks but remain vulnerable to adversarial perturbations, which can compromise their reliability in safety-critical applications. Existing attack methods often distribute perturbations uniformly across the input, ignoring the spatial heterogeneity of model sensitivity. In this work, we propose the Spatially Distributed Perturbation Strategy with Smoothed Gradient Sign Method (SD-SGSM), a adversarial attack framework that exploits decision-dependent regions to maximize attack effectiveness while minimizing perceptual distortion. SD-SGSM integrates three key components: (i) decision-dependent domain identification to localize critical features using a deterministic zero-out operator; (ii) spatially adaptive perturbation allocation to concentrate attack energy on sensitive regions while constraining background disturbance; and (iii) gradient smoothing via a hyperbolic tangent transformation to enable fine-grained and continuous perturbation updates. Extensive experiments on CIFAR-10 demonstrate that SD-SGSM achieves near-perfect attack success rates (ASR 99.9%) while substantially reducing ${\ell }_2$ distortion and preserving high structural similarity (SSIM 0.947), outperforming both single-step and momentum-based iterative attacks. Ablation studies further confirm that spatial distribution and gradient smoothing act as complementary mechanisms, jointly enhancing attack potency and visual fidelity. These findings underscore the importance of spatially aware, decision-dependent adversarial strategies for system-level robustness assessment and the secure design of AI-enabled systems. Full article

With the rapid development of laser processing and infrared imaging, the demand for flat-top beams with high uniformity has become increasingly urgent. Conventional beam-shaping techniques based on bonded aspheric lenses are inherently bulky and inflexible, which limits their compatibility with modern optical systems. In this work, we propose a dielectric metasurface for laser beam shaping operating at 1064 nm, where the target phase distribution is derived by the given initial phase and is represented by a hyperbolic phase. An inverse optimization algorithm is proposed to optimize the unit cell consisting of silicon carbide (SiC) nanopillars and the silicon dioxide (SiO₂) substrate. Numerical results show that, after transmission through the designed metasurface, the beam forms a circular flat-top spot with a radius of 2 μm at the target plane, exhibiting an intensity uniformity of 0.1021 and an energy efficiency of 76.3%. This study offers a compact and highly efficient solution for the flat-top beam shaping, demonstrating significant potential for applications in precision-laser processing, optical trapping, and bioimaging. Full article

Accurate prediction of post-construction settlement in high-speed railway (HSR) soft foundations is critical for operational safety yet challenging due to the non-equidistant and non-stationary nature of observation data. This study systematically evaluated the robustness and accuracy of settlement prediction models using a Monte Carlo simulation approach. A numerical model incorporating the permeability characteristics of soft foundations was established to simulate stochastic system responses. Furthermore, an innovative multi-metric evaluation framework was constructed using the entropy weight method, integrating goodness-of-fit, prediction accuracy (systematic error), and stability (random error). Four classical empirical models—Hyperbolic, Exponential Curve, Asaoka, and Hoshino—were assessed. The results indicate that: (1) The Hyperbolic Method significantly outperformed other models ($p<0.01$) in goodness-of-fit (mean correlation coefficient: 0.983 ± 0.006) and accuracy (systematic error: 3.2% ± 1.1%); (2) The Hoshino Method exhibited optimal stability, characterized by the lowest random error (3.8 ± 2.0 mm); and (3) Model performance showed a significant positive correlation with the permeability coefficient ($R^2>0.92$). Validated by five distinct engineering cases, the comprehensive performance ranking was determined as: Hyperbolic > Hoshino > Exponential Curve > Asaoka. These findings provide a scientific strategy for model selection under non-stationary conditions and offer theoretical support for refining railway deformation monitoring standards. Full article

Negative refraction of nanolight (e.g., polaritons, hybrid light, and matter excitation) provides a promising building block for nanophotonics, as it paves the way for developing cutting-edge nanoscale applications, such as super-resolution and subwavelength imaging. In the visible regime, negative refraction of surface plasmon polaritons has been extensively studied in conventional plasmonic and metamaterial systems; however, the inherent metallic losses remain a challenge that hinders their practical applications. Herein, we demonstrate negative refraction of low-loss and highly confined hyperbolic plasmon polaritons (HPPs) in a lateral heterojunction of a natural hyperbolic van der Waals material, molybdenum dioxide chloride (MoOCl₂). Owing to the exotic and ray-like propagating properties of HPPs, the negative refraction-inspired superlens can easily reach into the deep subwavelength scale, with spatial confinement of 800 nm near-infrared light wavelengths to below 150 nm focal spots. By elaborately adjusting the orientation directions of two-sided MoOCl₂, the mirror-symmetric superlensing effect can be tilted, and therefore, the focal spots are tuned and steered to deviate from the vertical interfacial lines. Our results applying the concepts of in-plane negative refraction with vdW materials achieve deep subwavelength light confinement and manipulation, offering new possibilities for constructing efficient and compact nanophotonic and opto-electronic devices. Full article

Sliding mode control (SMC) is a robust nonlinear control framework that enforces system trajectories onto predefined manifolds, providing strong robustness guarantees against uncertainties. However, SMC inherently introduces unwanted transients or chattering in system state trajectories, which may cause issues especially for sensitive applications such as regulation of drug administration. This paper proposes a multi-input smooth sliding mode control (MISSMC) strategy that combines radiotherapy and chemotherapy for a nonlinear tumor–immune dynamical system described by ordinary differential equations. The closed-loop system is first analyzed to establish key qualitative properties: all state variables remain positive and bounded, the sliding surfaces exhibit asymptotic convergence, and explicit analytical upper bounds on the cumulative therapy doses are derived under clinically motivated constraints. On this basis, a smooth hyperbolic-tangent sliding manifold and associated control law are designed to regulate the radiation and drug infusion rates. While the use of a hyperbolic-tangent smoothing function effectively suppresses chattering, it introduces a small steady-state error due to the presence of a boundary layer. To address this limitation, integral action is incorporated into the sliding surfaces, ensuring asymptotic convergence of tumor state and reducing residual steady-state error, while enhancing robustness against model uncertainties and parameter variations. Numerical simulations, based on a brain-tumor case study, show that the proposed smooth SMC markedly suppresses transient overshoots in both states and control inputs, while preserving effective tumor reduction. Compared with a conventional (non-smooth) SMC scheme, the MISSMC controller reduces baseline radiation and chemotherapy intensities on average by roughly 70%. Similarly, MISSMC lowers the overall cumulative doses on average by about 40%, without degrading the therapeutic outcome. The resulting integral smooth SMC framework therefore offers a rigorous nonlinear-systems approach to designing combined radio-chemotherapy protocols with guaranteed positivity, boundedness, and asymptotic stabilization of the closed-loop system, together with explicit bounds on the control inputs. Full article

We investigate the Lagrange formulation for the one-dimensional Saint Venant–Exner system. The system describes shallow-water equations with a bed evolution, for which the bedload sediment flux depends on the velocity, $Q\left(t,x\right)=A_g{u}^m,m\ge 1$. In terms of the Lagrange variables, the nonlinear hyperbolic system is reduced to one master third-order nonlinear partial differential equation. We employ Lie’s theory and find the Lie symmetry algebra of this equation. It was found that for an arbitrary parameter m, the master equation possesses four Lie symmetries. However, for $m=3$, there exists an additional symmetry vector. We calculate a one-dimensional optimal system for the Lie algebra of the equation. We apply the latter for the derivation of invariant functions. The invariants are used to reduce the number of the independent variables and write the master equation into an ordinary differential equation. The latter provides similarity solutions. Finally, we show that the traveling-wave reductions lead to nonlinear maximally symmetric equations which can be linearized. The analytic solution in this case is expressed in closed-form algebraic form. Full article

Controlled-release systems must translate material design choices into predictable pharmacokinetic (PK) profiles, yet purely mechanistic or purely data-driven models often underperform when tuning complex polymer networks. Here, we develop tunable poly(vinyl formal) membranes (PVFMs) for diclofenac delivery and integrate classical kinetic analysis with a Generalized Regression Neural Network (GRNN) to connect formulation variables to release behavior and PK-relevant targets. PVFMs were synthesized across a gradient of crosslink densities by varying HCl content; diclofenac release was quantified under standardized conditions with geometry and dosing rigorously controlled (thickness, effective area, surface-area-to-volume ratio, and areal drug loading are reported to ensure reproducibility). Release profiles were fitted to Korsmeyer–Peppas, zero-order, first-order, Higuchi, and hyperbolic tangent models, while a GRNN was trained on material descriptors and time to predict cumulative release and flux, including out-of-sample conditions. Increasing crosslink density monotonically reduced swelling, areal release rate, and overall release efficiency (strong linear trends; r ≈ 0.99) and shifted transport from anomalous to Super Case II at the highest crosslinking. Classical models captured regime transitions but did not sustain high accuracy across the full design space; in contrast, the GRNN delivered superior predictive performance and generalized to conditions absent from training, enabling accurate interpolation/extrapolation of release trajectories. Beyond prior work, we provide a material-to-PK design map in which crosslinking, porosity/tortuosity, and hydrophobicity act as explicit “knobs” to shape burst, flux, and near-zero-order behavior, and we introduce a hybrid framework where mechanistic models guide interpretation while GRNN supplies robust, data-driven prediction for formulation selection. This integrated PVFM–GRNN approach supports rational design and quality control of controlled-release devices for diclofenac and is extendable to other therapeutics given appropriate descriptors and training data. Full article

This paper presents a novel method for the precise localization of remote radio-signal sources using a formation of unmanned aerial vehicles (UAVs). The approach is based on time-difference-of-arrival (TDoA) measurements and the geometric analysis of hyperbolas formed by pairs of UAVs. By studying the asymptotic intersections of these hyperbolas, the method ensures unique determination of the source position, even in the presence of multiple intersection points. Theoretical analysis confirms that the correct intersection point is located at a significantly larger distance from the UAV formation center compared to spurious intersections, providing a rigorous criterion for resolving localization ambiguity. The proposed framework also addresses secure inter-UAV communication via optical-fiber links and supports expansion of UAV groups with directional antennas and low-power signal relays. Additionally, the study discusses practical UAV configurations, including hybrid propulsion and jet-assisted kamikaze platforms, demonstrating the applicability of the method in contested environments. The results indicate that this approach provides a robust mathematical basis for unambiguous emitter localization and enables scalable, secure, and resilient multi-UAV systems, with potential applications in electronic-warfare scenarios, surveillance, and tactical operations. Full article

In this research, we establish a precise correspondence between the theory of logarithmic connections on principal G-bundles over compact Riemann surfaces and the geometric formulation of control systems on curved manifolds, providing a novel differential–geometric framework for analyzing optimal control problems with non-holonomic constraints. By characterizing control systems through the geometric structure of flat connections with logarithmic singularities at marked points, we demonstrate that optimal trajectories correspond precisely to horizontal lifts with respect to the connection. These horizontal lifts project onto geodesics on the punctured surface, which is equipped with a Riemannian metric uniquely determined by the monodromy representation around the singularities. The main geometric result proves that the isomonodromic deformation condition translates into a compatibility condition for the control system. This condition preserves the conjugacy classes of monodromy transformations under variations of the marked points, and ensures the existence and uniqueness of optimal trajectories satisfying prescribed boundary conditions. Furthermore, we analyze systems with non-holonomic constraints by relating the constraint distribution to the kernel of the connection form, showing how the degree of non-holonomy can be measured through the failure of integrability of the associated horizontal distribution on the principal bundle. As an application, we provide computational implementations for $\mathrm{SL}(2,\mathbb{C})$ connections over hyperbolic Riemann surfaces with genus $g\ge 2$, explicitly constructing the monodromy-induced metric via the Poincaré uniformization theorem and deriving closed-form expressions for optimal control strategies that exhibit robust performance characteristics under perturbations of initial conditions and system parameters. Full article

This paper presents the construction of exact wave solutions for the generalized nonlinear Schrödinger equation (NLSE) with second-order spatiotemporal dispersion using the modified exponential rational function method (mERFM). The NLSE plays a vital role in various fields such as quantum mechanics, oceanography, transmission lines, and optical fiber communications, particularly in modeling pulse dynamics extending beyond the traditional slowly varying envelope estimation. By incorporating higher-order dispersion and nonlinear effects, including cubic–quintic nonlinearities, this generalized model provides a more accurate representation of ultrashort pulse propagation in optical fibers and oceanic environments. A wide range of soliton solutions is obtained, including bright and dark solitons, as well as trigonometric, hyperbolic, rational, exponential, and singular forms. These solutions offer valuable insights into nonlinear wave dynamics and multi-soliton interactions relevant to shallow- and deep-water wave propagation. Conservation laws associated with the model are also derived, reinforcing the physical consistency of the system. The stability of the obtained solutions is investigated through the analysis of modulation instability (MI), confirming their robustness and physical relevance. Graphical representations based on specific parameter selections further illustrate the complex dynamics governed by the model. Overall, the study demonstrates the effectiveness of mERFM in solving higher-order nonlinear evolution equations and highlights its applicability across various domains of physics and engineering. Full article

The exact analytical solutions of a new combined Kairat-II-X differential equation are presented. The related model is investigated by combining the enhanced modified extended tanh function method and the modified $tan(\varphi /2)$-expansion method. Then, a wide range of solitary wave solutions with unknown coefficients are extracted in a variety of shapes, including dark, bright, bell-shaped, kink-type, combine, and complex solitons, exponential, hyperbolic, and trigonometric function solutions. To offer physical insight, some of the identified solutions are presented in figures. Also, 3D, 2D, and 2D density profiles of the obtained outcomes are illustrated in order to examine their dynamics with the choices of parameters involved. Based on the obtained findings, we can assert that the suggested computational approaches are efficient, dynamic, well-structured, and valuable for tackling complex nonlinear problems in several fields, including symbolic computations. The bifurcation analysis and sensitivity analysis are employed to comprehend the dynamical system. We assume that our findings will be very beneficial in improving our understanding of the waves that manifest in solids. Full article