• The effectiveness of a three-layer annular coaxial shroud (TACS) was experimentally validated for laser cladding in open environments. • Oxygen concentration and molten pool protection were ensured when shielding gas flowrate exceeded 30 L/min. • The cross-section profiles of single-track clads were fitted by parabola, sine, and elliptic functions, with the elliptic model showing best accuracy under shielding gas. • The flat-top overlapping model (FOM) was employed to optimize the overlapping coefficient of multi-track clads. • The optimal overlapping coefficient was determined as 0.542, enabling improved surface flatness and reduced pile-up. Cladding active metal for surface repair and complex components fabrication in open environment remains a ground challenge and the advanced laser cladding technique with coaxial shroud protection has exhibited some new promises compared to other traditional deposition methods. In this study, comprehensive and thorough experiments have been carried out, focusing on the protectiveness of a three-layer annular coaxial shroud with qualitative measurements and quantitative data analysis. The experimental results have shown that the Ti6Al4V cladding can be effectively protected when the shielding gas flowrate is greater than 30 L/min. The optimal cross-section profiles of three frequently used models (Parabola, Sine and Elliptic functions) are measured with post-processing. It shows that the parabola model produces the most accurate results without extra shielding gas in terms of R-square values compared with the original cladding profile, whilst the elliptic model is more accurate in describing the cladding profile with extra shielding gas over the same test range. Applying both elliptic function model and flat-top overlapping model, a generalized formulae is then derived to evaluate the horizontal coordinate of the cladding centre point, which is used to determine an optimal overlapping coefficient of 0.542 under an extra shielding gas flowrate of 30 L/min. Further examinations consider four different scenarios, corresponding to four different overlapping ratio ranges. Results are in good agreement with experimental observations. It is worth noting that few literatures have documented to predict the crucial overlapping ratio using the EF model by determining the horizontal coordinate of point C value. Moreover, the elliptic function model proves more precisely representation of experimental data profile while compared to those fitted by Sine Function and Parabola Function models. The observations and findings of this study have shed some lights on current understanding of advanced laser cladding techniques.

Applications of the Jacobi elliptic function expansion method to the dimensionless time-dependent paraxial equation and the Biswas–Milovic equation

This work investigates dispersive optical solitons governed by a perturbed cubic-quartic nonlinear Schrödinger equation with parabolic self-phase modulation, a model of direct relevance to high-capacity fiber-optic systems where simultaneous higher-order dispersion and nonlinear perturbations shape pulse dynamics. The model is physically motivated by fibers with intensity-dependent refractive index profiles, where the interplay between fourth-order chromatic dispersion and parabolic (cubic-quintic) nonlinearity generates wave structures that the standard Kerr approximation cannot capture. To extract exact traveling-wave solutions, we employ the improved modified extended tanh-function method (IMETFM), which is selected for its ability to handle multi-parameter auxiliary equations and yield a wider diversity of solution families than classical expansion methods such as the tanh-function or [Formula: see text]-expansion approaches, without requiring the integrability of the underlying system. Our analysis produces five families of exact solutions: bright solitons, dark solitons, exponential-type solutions, singular periodic waves, and solutions expressed in terms of Weierstrass elliptic functions. For each family, explicit existence conditions and free-parameter restrictions are stated. The parametric constraints governing solution validity are derived and physically interpreted in terms of the dispersion, nonlinearity, and perturbation coefficients. Graphical representations of the spatial and temporal profiles illustrate the distinct propagation features of each solution type. A linear stability analysis, conducted via perturbation theory, yields an explicit eigenvalue dispersion relation and identifies a critical wavenumber threshold at which modulational instability sets in. The stability criteria provide actionable guidelines for maintaining soliton integrity under weak disturbances in practical optical environments. The results have direct implications for optical fiber communications, ultrafast signal processing, and dispersion-engineered photonic waveguides. The novelty lies in the simultaneous treatment of the parabolic law nonlinearity, fourth-order dispersion, and perturbative effects within a unified algebraic framework, yielding solution families including Weierstrass elliptic solutions that have not previously been reported for this model. Future work will address numerical validation, extension to stochastic and variable-coefficient models, and higher-dimensional soliton dynamics.

In this work, the wave solutions of the generalized (3+1)-dimensional P-type equation, a significant model for describing the evolution of waves in plasma physics, are investigated. The modified extended mapping method (MEMM) is applied as an effective analytical tool to get these solutions. Through the use of this method, a large variety of exact solutions is successfully derived, including Jacobi elliptic function solutions, bright and dark solitons, singular solitons, exponential forms, and singular periodic waveforms solutions. These solutions provide additional insight into the complex dynamics of the used equation. Furthermore, a linear stability analysis is performed to examine the stability of the steady-state solutions. The dispersion relation shows that the perturbation growth rate is purely imaginary for generic parameters, indicating neutral stability and the absence of modulation instability. Moreover, graphical representations of some of the solutions are given in order to disclose their physical behavior and better understand the corresponding wave phenomena.

Steady hydroelastic cnoidal wave on a thin flexural plate floating on a shallow water surface

The present study aims to investigate the magnetohydrodynamic flow and heat transfer characteristics of a Jeffrey fluid between two permeable flat disks under the combined effects of Hall current and a modified Darcy law. The primary objective is to obtain exact analytical solutions for both accelerating and decelerating radial flow configurations and to analyze the influence of key physical parameters on velocity and temperature fields. Thermal effects arising from radiation and internal heat generation or absorption are also incorporated to enhance the physical realism of the model. Closed-form solutions of the nonlinear governing equations are derived using Jacobi elliptic functions, enabling an accurate description of the flow behavior. The results reveal that the Hall parameter significantly enhances both accelerating and decelerating velocities, whereas magnetic and porous medium resistances suppress the flow. The temperature distribution is found to increase with heat generation and decrease with stronger thermal radiation. Streamline patterns and sensitivity analysis further validate the robustness of the obtained solutions. The findings of this study provide valuable insights into non-Newtonian transport phenomena and are relevant to applications in energy systems, cooling technologies, rotating machinery, and porous media engineering.

Stability, sensitivity, chaotic behavior, and soliton solution of the perturbed Kaup-Newell equation

Abstract Constructed herein are the dark breathers for the sinh-Gordon equation which characterize the interaction between a dark soliton and an oscillatory travelling wave. It is worth noting that this family of solutions cannot degenerate into any two-soliton solution. Utilizing the eigenvalues and corresponding eigenfunctions associated with the Lax operator of the sinh-Gordon equation, we apply the Darboux transformation to the oscillatory wave background to analytically express these solutions in terms of the Jacobi elliptic and theta functions. Parameterizing the zero-pole structure of these Jacobi functions within the spectral gap, we simplify the analysis of the breathers. It is revealed that dark breathers achieve the maximum localization for a particular value of the parameter and never surpass the rightward-propagating background waves. Since the amplitude of the dark breathers diminishes as the continuous spectral gap narrows, these solutions are precisely approximated by the dark soliton in the defocusing nonlinear Schrödinger equation.

ABSTRACT This paper takes the integrable Kuralay equations as the research object, aiming to derive various types of soliton solutions and explore the integrable motion of space curves induced by the equations, so as to support the research on nonlinear spin dynamics in the field of magnetic materials. In this paper, the unified ‐expansion method is used to systematically derive soliton solutions expressed by Jacobian elliptic functions. Through parameter degeneration (degenerating into hyperbolic function solutions when the modulus and trigonometric function solutions when ), the evolutionary relationship among different solutions is revealed. Eight types of soliton solutions are obtained in this paper, including periodic trigonometric function solutions, parabolic function solutions, singular solutions, and M‐shaped/W‐shaped solitons (corresponding to Figures 1–8). The parameter configurations and 2D/3D graphical characteristics of each solution are clarified (e.g., kink waves show unidirectional step‐like transitions, and M‐shaped bright waves possess symmetric double peaks). All solitons have clear boundaries without diffusion. For numerical verification, the fourth‐order Runge‐Kutta method combined with Richardson extrapolation is adopted, reducing the calculation error from to . In addition, phase portrait, bifurcation, and initial condition sensitivity analyses are supplemented, and the stability of equilibrium points is classified by the eigenvalues of the Jacobian matrix. In terms of physical implications, the soliton solutions are deeply associated with magnetic spin systems. For instance, kink waves correspond to the migration of spin domain walls, supporting the reading and writing operations of magnetic storage; M‐shaped/W‐shaped solitons contribute to the realization of multistate and high‐density storage. The quantitative influences of parameters on the low‐power consumption and high‐capacity performance of devices are clarified, providing theoretical support and practical guidance for the research on nonlinear spin dynamics and the design of magnetic storage and magneto‐optical modulation devices.

Abstract We present a novel approach for constructing quasi-isospectral higher-order Hamiltonians from time-independent Lax pairs by reversing the conventional interpretation of the Lax pair operators. Instead of treating the typically second-order $L$-operator as the Hamiltonian, we take the higher-order $M$-operator as the starting point and construct a sequence of quasi-isospectral operators via intertwining techniques. This procedure yields a variety of new higher-order Hamiltonians that are isospectral to each other, except for at least one state. We illustrate the approach with explicit examples derived from the KdV equation and its extensions, discussing the properties of the resulting operators based on rational, hyperbolic, and elliptic function solutions. In some cases, we present infinite sequences of quasi-isospectral Hamiltonians, which we generalise to shape-invariant differential operators capable of generating such sequences. Our framework provides a systematic mechanism for generating new candidate integrable structures/integrable operator families associated with known Lax pairs.

The Weierstrass elliptic function is presented in the light of the astrodynamical equation. We synchronise the Weierstrass elliptic function with the elliptic curve which relates the set of Sato weights with the genus $(n,s)$, where $n<s$ and $n$, $s$ are co-prime, making all equations homogeneous. The duplication formula of the Weierstrass formula, as previously used in Uwamusi [14], is introduced as an indicator of how the function behaves near and far away from the origin of a complex number. The differential equation satisfied by the Weierstrass function is explained, and the invariant discriminant function of the Weierstrass elliptic function is taken as an important tool. A method for speeding up the computation process in the radial anomaly in Kepler’s equation, which provides the time of root passage between a pseudo-time and a stable variable time, is introduced via the Chebyshev–Halley iteration formula of third order in the light of the Weierstrass elliptic function. This thus provides in-depth information regarding the radius of peri-center passage and the real root closest to the exact solution. Also in the paper is a computation of the Schwarzian derivative for the Weierstrass elliptic function. Numerical examples are demonstrated with these methods, and the results obtained are quite impressive.

Optimization of asymmetric gyrostatic satellite kinematics in a resistive medium: A novel elliptic function solution.

In this research, a [Formula: see text]-model approximation method is employed to investigate the localized wave solutions for the Lakshmanan-Porsezian-Daniel equation with beta-derivative. This equation integrates the fundamental phenomena such as Space-Time Dispersion (STD), Group Velocity Dispersion (GVD) and parabolic-law-governed by nonlinear behavior. A variety of optical soliton waves are obtained by applying the [Formula: see text]-model expansion approach. These waves are expressed as Jacobi elliptic functions [Formula: see text] that based on the particular values of the parameter A, that can be converted into solutions of trigonometric or hyperbolic functions. This technique provides variety of solutions, including dark soliton solutions, hyperbolic solutions, periodic waves solutions, bright solitons, singular soliton solution and singular periodic waves solutions. To further explore the system's behavior, bifurcation analysis is done. For this analysis planar dynamical system is obtained by using Galilean transformation. This analysis offers deep understanding of the phase portraits, time series, chaotic behavior and sensitivity analysis of the equation to external perturbations. The sensitivity and dynamics of optical solitons are thoroughly investigated that offers significant insights into their behavior within fractional models.

Abstract We show that in an elliptic function based formulation for fields with large amplitudes, the rolling down of solitons towards the degenerate vacua can be described as transitions through local meta-stable potentials; the kink-like solitons constructed locally around the maxima of the potential will roll down into the vacua without changing their profiles and their energies, but amplifying their amplitudes and their topological charges. Furthermore, the cusp-like solitons trapped at vacua will admit a nonlinear decomposition in terms of fundamental solitons with different profiles, energies, amplitudes, and topological charges; a scale-probe is proposed in order to reveal the different scales for those vacuum fundamental solitons. The possibility for a vacuum-vacuum tunneling for solitons is addressed.

Spectral stability of elliptic function solutions for the short pulse equation

The Calogero–Bogoyavlenskii–Schiff (CBS) equation serves as a cornerstone for nonlinear integrable partial differential systems that govern complex wave phenomena. This study investigates the [Formula: see text]-dimensional CBS-negative-order CBS (CBS–nCBS) model, unveiling novel soliton solutions. Using the bilinear neural network method, a bilinear representation of the model is derived, and multi-soliton solutions are identified through various neural network architectures. The bilinear neural framework employs single-layer architectures (4–3–1, 4–4–1) combined with diverse test functions to capture kink–breather waves, M-lumps, and lump–kink interaction solitons. A novel Jacobi elliptic activation function is applied to explore rogue wave soliton solutions. Spatiotemporal visualizations, such as 3D with contour, density, and 2D plots, generated using MATLAB for selected parameters, offer valuable insights into the structural evolution of these solutions. Moreover, the stability analysis of the governing equation is investigated using linear stability analysis under small perturbations, and the results are illustrated with dispersion analysis graphs. These visualizations offer a quantitative representation of the intricate dynamics at play. The bilinear neural methodology, utilizing established test functions, provides a systematic and robust approach for analyzing complex nonlinear phenomena. The findings enhance the understanding of soliton structures and expand the applicability of the CBS–nCBS equation in nonlinear wave theory.

This work investigates the propagation of optical solitons in coupled Higgs equation which shows a system of neutral scalar mesons working with preserved scalar nucleons, using the extended hyperbolic function approach, the unified solver approach and the Jacobi elliptic function solutions in rational form. The suggested methods are described as being easier to understand, clearer, and more concise. They enable the rally of a wide variety of optical solitons, including 1-singular, multisingular, 2-singular, bell-shaped, periodic, dark, bright, and kink ones, as well as the broadening of the class of solutions. This demonstrates the significance of the outcomes. As these techniques have never been applied to obtain optical solitons for this model, and the Jacobi elliptic function solutions in rational form are the first time in literature to the best of our knowledge, the authors highlight their novelty. The visual representations of some of the solutions in the figures help to understand the physical characteristics of the numerous new soliton solutions and their corresponding behaviors.

In this study, the non-linear vibration behavior of multilayer cylindrical shells composed of non-homogeneous orthotropic (NHO) materials supported by a nonlinear Pasternak elastic foundation (NL-PF) is investigated using an exact analytical approach. Unlike the isotropic assumptions, linear basis models, and purely numerical methods used in the vast majority of existing studies, the proposed formulation simultaneously considers material orthotropy, functional grading of layers in the thickness direction, transverse shear deformations, and geometric nonlinearities. The governing equations are derived using von Kármán-type nonlinear strain-displacement relations within the Donnell shell theory and a generalized Hooke’s law for orthotropic layers. To find the exact solution in terms of the Jacobi elliptic function, the nonlinear equation of motion is transformed into the Jacobi elliptic function form, and the nonlinear frequency-amplitude relationship is obtained analytically. Parametric investigations are conducted for six- and eight-layered shell configurations. The results show that orthotropic grading, layer arrangement, transverse shear effects and NL-PF effects significantly alter the dynamic stiffness and frequency response of the system, providing original contributions to the literature on the nonlinear dynamics of advanced laminated orthotropic shell structures.

This study emphasizes the construction of various types of solitary wave solutions of the M-truncated fractional (2 + 1)-dimensional Kadomtsev–Petviashvili-modified equal-width (KPmEW) equation, which models various long-wavelength phenomena, such as water waves, ferromagnetic waves, and matter-wave pulses. A modified form of the trial equation method is employed to investigate the wave structures of the proposed fractional nonlinear model. Through this approach, a wide spectrum of rational, exponential, hyperbolic, and Jacobi elliptic function solutions is derived. The corresponding wave profiles are visualized in two and three dimensions to elucidate their physical characteristics. The analytical findings are verified using the symbolic software Mathematica. The obtained results extend previous studies and provide new insights into the dynamic behaviour of fractional nonlinear evolution equations.

A bstract In this paper, we explore the chamber dissection of the loop-geometry of Correlahedron, which encodes the loop integrand of four-point stress-energy correlators in planar $$ \mathcal{N} $$ N = 4 super Yang-Mills. We demonstrate that at four loops, continuing the pattern of lower loops, the integrand of the four-point correlation function can be written as a sum over products of chamber-forms and local loop integrands. The chambers and their associated forms are identical to those at three-loops, indicating that the dissection may be complete to all loop orders. Furthermore, this suggests that the leading singularities to all loops are simply linear combinations of these chamber forms. This is especially intriguing at four loops since it contains elliptic functions. Interestingly, each elliptic function appears in a subset of chambers. Our geometric approach motivates us to “diagonalize” the representation, where the local integrals only possess a single leading singularity or elliptic cut. In such a representation, all integrands must evaluate to pure functions, including a single pure elliptic integrand. Inspired by this picture, we also present a simplified form of the three-loop correlator in terms of two independent pure functions (weight-6 single-valued multiple polylogarithms), which are directly computed from local integrands with unit leading singularities, multiplied by the leading singularities from chamber forms.