Search Results (357)

This work investigates the optical soliton solutions and their dynamical behaviors of the fractional stochastic Biswas–Arshed equation (FSBAE). By employing the definition of truncated M-fractional derivative and a wave transformation, we reduce the FSBAE to an integrable ordinary differential equation. Applying the qualitative and bifurcation theory of planar dynamical systems, we analyze its bifurcations and construct five explicit types of exact traveling wave solutions, namely solitary wave solutions, kink-type exponential function solutions, Jacobi elliptic sn-type periodic solutions, Jacobi elliptic cn-type periodic solutions, and exponential-form solitary wave solutions. Furthermore, to explore the model’s sensitivity to perturbations, we introduce a time-periodic perturbation term, then the emergence of chaotic behavior is demonstrated by providing 2D and 3D phase portraits. Full article

This paper examines the (2 + 1)-dimensional complex modified Korteweg–de Vries equation, which describes the intricate motion of water particles from the surface to the bottom. By applying an appropriate wave transformation that reduces the governing nonlinear partial differential equations to an ordinary differential system, we conduct a comprehensive bifurcation analysis that identifies equilibrium points and characterizes their phase-space properties. Under periodic perturbations, the system exhibits bifurcations, quasi-periodicity, multistability, and chaotic behavior, supported by Lyapunov exponents, time-series evolution, phase portraits, and Poincaré sections. To obtain exact soliton solutions of various types, we employ both the dynamical system method and the generalized double auxiliary equation method. The physical features of the solutions are depicted using 2D and 3D representations of their real, imaginary, and absolute components. A comparison with related studies shows that the proposed approaches not only reproduce previously reported solutions but also yield broader and more general soliton families. Full article

This study investigates the forced transverse vibration of a thin rotating annular disc subjected to time-varying point forces that are fixed in space and act perpendicular to the disc surface. Earlier analytical treatments of this problem have almost always been restricted to a single support condition, most often the clamped–free disc of a hard-disk drive. The boundary conditions of the disc have not been treated as a design variable of the forced response. The contribution of this work is to remove that restriction: the same generalised Galerkin formulation is applied to clamped–clamped, clamped–free and free–clamped rotating annular discs, and the sensitivity of the forced response to the excitation parameters is compared across all three. The governing differential equation, which includes gyroscopic coupling and the membrane stresses induced by rotation, is nondimensionalised and solved by the Galerkin method with polynomial radial trial functions. The modal equations are then integrated in state-space form with light modal damping. The physical transverse response at a fixed observation point is characterised by power spectral density diagrams assembled into waterfall plots, and the corresponding steady-state harmonic responses are obtained in closed form from the frequency-domain resolvent of the same state-space model. The formulation is verified against an independent radial finite-element model for all three boundary conditions and against published rotating clamped–free natural frequencies. The central finding concerns how the excitation parameters act on the response. The excitation frequency, the radial position of a force, and the angular separation and phase of a pair of forces act as largely independent levers. The quantitative sensitivity to each lever, however, is set by the boundary conditions, as is the force placement that minimises a chosen travelling-wave family. The radial position that minimises the excitation of a chosen radial family is governed by the interior node of that mode, which lies at $r\approx 52$, 68 and 44 mm for the clamped–clamped, clamped–free and free–clamped discs, respectively, and does not in general coincide with a free edge. Angular separation, by contrast, suppresses a nodal-diameter family in a manner that is essentially boundary-condition independent. The parameter dependences are shown to be steady-state properties: the driven spectral line of the finite-duration records reproduces the resolvent solution to within $0.09$ dB. Full article

This article uses the enhanced direct algebraic method (EDAM) to study the exact traveling wave solutions of the paraxial wave equation with conformable fractional derivative. This method is selected due to its algorithmic simplicity, computational efficiency, and unique capability to yield diverse solution types within a unified algebraic framework. Firstly, the paraxial wave equation with conformable fractional derivative is transformed into an ordinary differential equation through traveling wave transformation. Then, the EDAM is systematically applied to construct accurate traveling wave solutions including twisted solitons, bell-shaped solitons, singular solitons, Weierstrass elliptic function solutions, and Jacobi elliptic function solutions, demonstrating the method’s superiority in handling complex nonlinear structures compared to standard expansion techniques. Finally, based on Matlab software to draw three-dimensional and two-dimensional graphs of partial solutions, these graphs intuitively demonstrate the regulatory effect of the fractional-order parameter $\alpha$ on waveform localization and amplitude. Specifically, as $\alpha$ increases, the soliton localization weakens and the wave packet broadens, providing insights into the dispersion management in optical fibers. The method used in this article is characterized by simple operation and rich solution types, providing an effective approach for studying fractional nonlinear partial-differential equations. Full article

The generalized Burgers–Huxley (gBH) equation arises as a canonical model in nerve-pulse propagation (generalizing the Hodgkin–Huxley/FitzHugh–Nagumo excitable-media framework), in population dynamics with Allee-threshold reaction kinetics, and in nonlinear wave propagation in dispersive media; accurate benchmark solutions are essential for quantitative predictions in these domains. We couple the fourth-order exponential time differencing scheme ETDRK4 with a Chebyshev collocation spatial discretization and a linear boundary-lifting procedure to solve the gBH equation on a bounded interval with non-homogeneous Dirichlet data. On the canonical Ismail–Raslan–Rabboh travelling-wave benchmark the scheme attains $L^{\infty }$ errors at the level of floating-point round-off (∼10⁻¹⁹ absolute, ∼10⁻¹⁵ relative) with as few as $N=2$ collocation points and a single time step of size $\Delta t=1.0$—that is, three total nodes and one ETDRK4 advance. In strongly nonlinear regimes ($\gamma =0.1, 0.3, 0.5, 0.9$) the scheme exhibits approximately $\mathcal{O}(\Delta t^{2.45})$ temporal convergence across all four parameter values, consistent with the classical Hochbruck–Ostermann order reduction for exponential integrators on parabolic PDEs with non-homogeneous Dirichlet data. Used as a high-accuracy probe, the scheme provides a diagnostic of independent interest: the wave-speed formula of Wang, Zhu and Lu, still appearing as the exact-solution benchmark in numerical studies as recently as 2020, does not satisfy the partial differential equation. The corrected formula stated by Deng and verified symbolically by Appadu and Tijani is the unique value that makes the travelling-wave ansatz a genuine solution. We derive the residual associated with Wang’s formula in closed form, $R=\frac{\gamma A_1}{2}(A_2-A_2^{\mathrm{W}})(1-v^2)$, and show both analytically and numerically that reported errors for schemes benchmarked against Wang’s formula coincide with the analytical wave-profile gap $\frac{\gamma A_1}{2}|A_2-A_2^{\mathrm{W}}|$ rather than with true scheme accuracy. At the Ismail benchmark this gap equals $3.748\times {10}^{-7}$, which matches the N- and $\Delta t$-independent plateau observed when the scheme is measured against Wang’s profile. In the nerve-pulse and excitable-media interpretation, the two formulas correspond to action-potential propagation speeds of opposite sign at the Ismail benchmark, underscoring that the correction is not a mere algebraic curiosity but changes the qualitative physical prediction of the model. Full article

Kadomtsev–Petviashvili (KP)-type nonlinear dispersive wave equations play a fundamental role in the description of weakly nonlinear waves in plasmas, fluids, and nonlinear optical systems. In strongly turbulent or heterogeneous media, however, transport processes often become nonlocal and exhibit anomalous scaling behavior. Such phenomena are naturally described using fractional differential operators. Recent work has significantly advanced the mathematical understanding of fractional Kadomtsev–Petviashvili models by proving the existence of periodically modulated solitary waves and lump solutions, and by analyzing instability and other significant properties. The present paper complements this line of research by combining a plasma-oriented modeling motivation with a variational existence framework, structural properties of traveling profiles, and a Fourier spectral computational pipeline for profile construction and dynamical validation. The mathematical properties of the resulting equation are investigated. In particular, we prove the existence of traveling-wave solutions using a variational formulation and concentration-compactness arguments. To compute these coherent structures numerically, we develop a Fourier pseudospectral Petviashvili iteration scheme adapted to the fractional KP operator. The computed profiles are validated through direct time integration of the governing equation using an exponential time-differencing spectral method. The results demonstrate that fractional dispersive effects (e.g., the dependence on $\alpha$) significantly modify the structure of nonlinear plasma waves and provide a natural framework for describing wave dynamics in turbulent plasma environments. The numerical results include a detailed verification of the predicted algebraic decay law, a convergence study of the Petviashvili iteration, validation against the exact KP soliton, and dynamical stability tests over long time intervals. A quantitative comparison with existing results in the literature is also provided. Full article

Accurate and fast fault detection is essential to ensure the stability and reliability of High Voltage AC (HVAC) transmission systems. Conventional protection methods, including impedance-based and traveling wave techniques, may exhibit reduced performance under noisy operating conditions, system uncertainties, and complex fault scenarios while often requiring separate approaches for fault classification, location detection, and stability assessment. This paper proposes a unified Artificial Neural Network (ANN) based framework for simultaneous fault classification, location detection, and stability assessment using Critical Clearing Time (CCT) within a single HVAC transmission line model. A detailed MATLAB Simulink model is developed to generate a structured dataset comprising twelve fault scenarios, including single-line, double-line, three-phase, and ground faults at different locations along the transmission line. Three-phase voltages and currents, along with zero-sequence components, are used as input features. The ANN model is trained using the Levenberg–Marquardt (LM) optimization algorithm, which was comparatively evaluated against Bayesian Regularization (BR) and Scaled Conjugate Gradient (SCG) and demonstrated faster convergence, lower prediction error, and higher regression accuracy. To further evaluate the robustness of the proposed framework under high-impedance fault conditions, supplementary simulations were performed using fault resistance values of 10 Ω and 50 Ω in addition to the baseline 0.01 Ω case. The resulting datasets were combined to form an expanded training and evaluation dataset, enabling comprehensive validation of the proposed LM-trained ANN under varying fault resistance conditions. Using the baseline dataset, the proposed framework achieved a high regression coefficient (R = 0.9882) and low mean squared error (MSE = 0.1386), demonstrating accurate fault classification and precise per-kilometer fault location estimation. Furthermore, the integration of fault inception time and duration enables direct computation of CCT, allowing the model to distinguish between stability-critical and non-critical fault conditions. The results confirm that the proposed framework provides a comprehensive and efficient solution for real-time fault analysis by combining classification, localization, temporal analysis, and stability-aware decision support within a single model. Full article

We consider a generalization of the modified Korteweg–de Vries–Burgers equation with a nonlinear source. Using the Painlevé test for partial differential equation with the Kruskal variable, we show that the corresponding Cauchy problem cannot be solved by the inverse scattering transform. However, the equation admits a two-wave solution, which is obtained by means of the Cole–Hopf transformation. Taking into account the traveling wave reduction, we derive the resulting nonlinear ordinary differential equation and determine the parameter conditions under which it passes the Painlevé test. This finding suggests the possible existence of the general solution for the ordinary differential equation, which can be reduced to the linear third-order equation. The general solution of the resulting linear equation is expressed in terms of the hypergeometric function. Full article

We study a special case of the extended Korteweg–de Vries (eKdV) equation, arising in the description of weakly nonlinear long waves with higher-order dispersive effects. The model incorporates both third- and fifth-order dispersion and quadratic nonlinearity and describes steeper and shorter waves than the classical KdV equation. First, we determine the Lie point symmetry algebra of the equation and show that it reduces to space–time translations, which in turn motivates a traveling-wave reduction. The reduced fifth-order ODE is then analyzed by means of a calibrated $(G^{\prime }/G)$-expansion ansatz. Although homogeneous balance suggests a degree $M=4$ for exact solutions, a degree-$M=2$ truncation already yields three coherent families of traveling waves—hyperbolic (solitary), trigonometric (periodic), and rational—distinguished by the discriminant of the auxiliary linear equation. Using the direct multiplier method, we construct four conservation laws, corresponding to mass, momentum, energy, and a higher-order dispersion invariant, with all ${\alpha }^2$ contributions retained. Direct substitution and numerical diagnostics demonstrate that, once the algebraic wave speed is imposed, the $M=2$ profiles satisfy the PDE with residuals of order ${10}^{-3}$ and preserve the conserved quantities to machine precision (below ${10}^{-13}\%$ relative variation) over extended integration domains. These results extend the known solution structure of the special eKdV equation and illustrate the effectiveness of the $(G^{\prime }/G)$ framework for higher-order dispersive models. Full article

We investigate the dynamical stability of traveling-wave solutions in a scalar, age-structured model that incorporates nonlocal dispersal. By combining a comparison argument with a weighted-energy framework tailored to the convolution-type operator, we prove that traveling waves of non-critical speed are exponentially stable under appropriately weighted perturbations after integration over age. Numerical experiments illustrate the qualitative content of the decay result and the structure of the resulting rate bound. Full article

In this study, a detailed analytical-numerical study of the complex modified Korteweg–De Vries (mKdV) model with truncated M-fractional derivative is carried out to investigate the effects of the fractional order on nonlinear wave propagation. The fractional partial differential equation is solved by an appropriate fractional traveling wave transformation, which transforms it into a nonlinear ordinary differential equation. Two very powerful analytical methods are then used: the modified sub-equation method and the Kumar–Malik method, which give the exact closed-form solutions. The obtained semi-analytical numerical approximations are then obtained from the Differential Transformation Method (DTM). Bright and dark solitons, kink-type waves, periodic and rational solutions, exponential solutions, and Jacobi elliptic functions are found for a variety of parametric regimes. Explicit compatibility conditions and parametric constraints, which control the amplitude, width, and propagation, are derived. The DTM approximations are found to converge to the exact solutions with good accuracy, and the absolute errors are almost negligible, which validates the accuracy of the approximations and reliability of the solution. The three-dimensional visualizations of surface plots, two-dimensional profiles, and contour visualization further illustrate the dispersive dynamics and stability properties. Significance: This study shows that the truncated M-fractional derivative is a good operator to model memory-dependent nonlinear wave propagation. A new precise solution and reliable validation methods have been obtained for high-dimensional fractional nonlinear evolution equations in the hybrid analytical-numerical framework, which can be useful in plasma physics, nonlinear optics, and complex media. The present study contains restrictions for constant coefficients, a specific parametric regime, one fractional derivative definition, and experimental validation is not included. Future directions are limitations on constant coefficients, specific parametric regimes, one fractional derivative definition, and experimental validation is not included. The approach is to be extended in the future to variable coefficients, other fractional operators (Caputo, Riemann–Liouville), and to higher-order nonlinearities, and then to be experimentally tested in optical or plasma systems. Full article

This paper investigates the bifurcation, phase portrait, and traveling wave solutions of the Aizhan–Gudekli–Nurshuak–Zhanbota equation. By performing a simple linear transformation on the solution of the equation and applying the traveling wave transformation, the original equations are reduced to a system of ordinary differential equations. Through a qualitative analysis of the resulting two-dimensional dynamical system, the types and stability of equilibrium points are classified under various parameter conditions. Using both the dynamical system method and the complete discriminant system, multiple families of exact traveling wave solutions including solitary, kink, and periodic wave solutions are derived for different parameter ranges. Numerical simulations using Maple software are performed to visualize selected solutions. The obtained solutions are novel and provide a theoretical foundation for understanding the physical applications of the equation. Full article

In this study, we utilize the ${\varphi }^6$-model expansion method to derive a diverse set of Jacobi elliptic function solutions for the conformable resonant Nonlinear Schrödinger Equation (NLSE) with parabolic law nonlinearity. As the modulus of the Jacobi elliptic functions approaches 1 and 0, the solutions transform into hyperbolic and trigonometric functions, respectively. This methodology yields various exact traveling wave solutions, including kink solitons, singular solitons, periodic solutions, and singular periodic solutions. Notably, this work represents the first investigation into identifying Jacobi elliptic function solutions for the conformable resonant NLSE. These results enhance the understanding of the nonlinear dynamical properties intrinsic to the NLSE. We use graphical illustrations to highlight the dynamical features of the solutions. Moreover, our approach showcases versatility in addressing other nonlinear partial differential equations, offering insights applicable to nonlinear optics, fluid dynamics, and quantum physics. Full article

In industrial applications such as in situ leaching and uranium mining, permanent magnet synchronous motors (PMSMs) for submersible pumps are frequently connected to frequency converters via long cables. During this long-distance transmission, traveling wave reflections induced by high-frequency pulse width modulation (PWM) generate severe transient overvoltages that threaten motor insulation. Because installation space at deep-well motor terminals is severely restricted, overvoltage suppression must be implemented at the inverter output. Here, the parameter design and optimization of a passive LC filter specifically developed for 250 Hz high-frequency PMSMs are presented. The optimal inductance and capacitance parameters were determined by balancing multiple operational constraints, including fundamental voltage drop, high-frequency harmonic attenuation, and the avoidance of low-order harmonic resonance. Furthermore, the anti-saturation performance of the magnetic core material, evaluated thermal characteristics through electromagnetic-thermal co-simulation, and analyzed the risk of self-excited oscillation between the filter capacitors and the motor was analyzed. Finally, hardware experiments conducted on a 20 m cable test bench validate that the designed LC filter effectively mitigates terminal overvoltage. The peak terminal voltage was reduced from 900 V to 505 V, and total harmonic distortion (THD) was limited to below 5%. This design provides a highly reliable, space-efficient solution for overvoltage suppression in high-speed, long-cable motor drive systems. Full article

This study investigates the ($3+1$)-dimensional extended Kadomtsev–Petviashvili equation via traveling-wave phase-space geometry. The equation is reduced to a planar Hamiltonian system with cubic nonlinearity, whose conserved energy partitions the phase space into periodic orbits, separatrices, and unbounded trajectories. Closed-form profiles for the gradient variable $\phi =U_{\xi }$ are obtained through separation of variables; the corresponding field U is recovered by quadrature and must satisfy a zero-mean condition for periodic reconstruction. In particular, for $h_1>0$, the reconstructed field exhibits kink/antikink-type rather than localized-pulse behavior. Under weak periodic forcing, an explicit Melnikov amplitude factor is derived. Its exponential decay with the forcing frequency implies that the leading-order separatrix splitting distance $\mu A\left(\omega \right)$ becomes exponentially small at high frequency, while the simple-zero condition still predicts transverse intersections of stable and unstable manifolds and the onset of horseshoe chaos. Applying the complete discriminant method yields eight distinct solution families—hyperbolic, trigonometric, rational, and Jacobi elliptic—each associated with a unique orbital topology. These results enrich both the dynamical theory and the exact solution framework of higher-dimensional nonlinear evolution equations. Full article