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In this study, we obtained optical soliton solutions of the perturbed nonlinear Schrödinger–Hirota equation with generalized anti‐cubic law nonlinearity in the presence of spatio‐temporal dispersion. This equation models the propagation of optical pulses in fiber optic cables. Due to the anti‐cubic nonlinearity, perturbation, and spatio‐temporal dispersion present in the model, it provides more accurate results for high‐speed and long‐distance transmissions. Given the significant developments in the field of optics, studies on complex equations such as this model are of great importance. With the increase in real‐life applications, obtaining solutions to optical equations has become crucial. In this study, we used the improved F‐expansion method to derive the optical soliton solutions for the relevant model. This technique allows for obtaining various solutions through the Jacobi elliptic auxiliary functions it employs. The obtained solutions consist of trigonometric and hyperbolic functions. As a result of the application, 10 solutions were obtained, and 2D and 3D graphics of these solutions are included. These graphs illustrate the motion directions of optical solitons and the effect of the nonlinearity parameter p$$ p $$ and spatio‐temporal dispersion parameter β$$ \beta $$ on soliton behavior. No restrictions were encountered during the study. Finally, the originality of the study lies in the first application of this technique to the relevant model and in examining the effect of the parameters p$$ p $$ and β$$ \beta $$ on this model. [ABSTRACT FROM AUTHOR]

This work aims to look into the dynamic research of coupled NLS-type equations with three components. The optical solitons, including the periodic function, trigonometric function, exponential function, solitary wave, and elliptic function solutions are built using the Jacobi elliptic function (JEF) method. The investigations will aid in improving comprehension of the soliton dynamics system's overall illustration. Using Mathematica software, we visually represent some solutions found in 3D, contour, and 2D graphs for tangible demonstration and visual presentation. These results are helpful in optical fiber, signal processing and data transmission. [ABSTRACT FROM AUTHOR]

In this paper, with the multi-scale analysis and Wentzel–Kramers–Brillouin (WKB) theory, a variable-coefficient modified Zakharov–Kuznetsov (mZK)–Burgers equation is obtained via the quasi-geostrophic potential vorticity model. The equation can be described the propagation of the nonlinear long wave and solitary eddy. The exact solutions are given by the Jacobi elliptic function expansion method to analyze wave propagation characteristics, and the elliptical modulus k affects the amplitude and period of waves. [ABSTRACT FROM AUTHOR]

We investigate the two-component BKP hierarchy equation for its Lie point symmetries. To obtain a complete classification of the group-invariant solution, we derive the one-dimensional optimal system of subalgebras of A 3 , 3 (D ⨂ s T 2). By employing those subalgebras, we construct the invariant solutions which are represented through Weierstrass elliptic functions, Jacobi elliptic functions, and soliton solutions. Also, we analyse the existence of a bounded travelling-wave solution for the equation. Moreover, through the utilization of scale-invariant symmetry, we derive a self-similar solution. Additionally, by conducting singularity analysis, the solution can be expressed in the form of a right Painlevè series. • The two-component BKP hierarchy equation is studied under the lens of Lie symmetry. • We find the Lie point symmetries and perform a symmetry reduction of this equation. • Group invariant solutions are constructed. [ABSTRACT FROM AUTHOR]

In this paper, we study the spectrum σ (L) of the Lamé operator L = d 2 d x 2 − 12 ℘ (x + z 0 ; τ) in L 2 (R , C) , where ℘ (z ; τ) is the Weierstrass elliptic function with periods 1 and τ , and z 0 ∈ C is chosen such that L has no singularities on R. We prove that a point λ ∈ σ (L) is an intersection point of different spectral arcs but not a zero of the spectral polynomial if and only if λ is a zero of the following cubic polynomial: 4 15 λ 3 + 8 5 η 1 λ 2 − 3 g 2 λ + 9 g 3 − 6 η 1 g 2 = 0. We also study the deformation of the spectrum as τ = 1 2 + i b with b > 0 varying. We discover 10 different types of graphs for the spectrum as b varies around the double zeros of the spectral polynomial. [ABSTRACT FROM AUTHOR]

Let f \colon X \to B be a nonisotrivial complex elliptic surface and let \mathcal {D} \subset X be an integral divisor dominating B. We study finiteness related properties of generalized (S, \mathcal {D})-integral sections \sigma \colon B \to X of X where S \subset B is arbitrary. These integral sections \sigma are algebraic and satisfy the geometric condition f (\sigma (B) \cap \mathcal {D})\subset S. For S \subset B finite, the set of (S, \mathcal {D})-integral sections of X is finite by the well-known Siegel theorem. In this article, we establish a general quantitative finiteness result for large unions of (S, \mathcal {D})-integral sections in which both the subset S and the divisor \mathcal {D} are allowed to vary in families where notably S is not necessarily finite or countable. Some applications to generalized unit equations over function fields are also obtained. [ABSTRACT FROM AUTHOR]

In this paper, the (4 + 1)-dimensional Fokas model is investigated analytically with extensive applications in nonlinear wave theory including the evolution of a three-dimensional wave packet in water with a finite depth, and surface waves and internal waves in straits or channels of varied depth and width. Applying the Hirota bilinear approach, the lump wave solitons and interaction of lump with periodic waves, the interactions of lump wave along single, double-kink solitons as well as the interactions of lump, periodic along double-kink solitons of the governing model are developed. Additionally, by using the extended tanh function technique, certain new traveling wave solitons are established. The physical nature of several solitons is illustrated by drawing the 3D, contours and 2D plots. Consequently, a set of periodic, bright, dark, rational as well as elliptic function solitons are constructed. The employed techniques seem to be more effective and well organized to build some interesting analytical wave solitons to several appealing nonlinear models. [ABSTRACT FROM AUTHOR]

We consider the shadow Gierer-Meinhardt model { ∂ t U = ε 2 U x x − U + U p ξ q for 0 < x < 1 , 0 < t < T , τ ∂ t ξ = ∫ 0 1 (− ξ + U r ξ s ) d x for 0 < t < T , U x (0 , t) = U x (1 , t) = 0 , where ε > 0 , τ > 0 , p > 1 , q > 0 , r > 0 , s ≥ 0 and q r / (p − 1) (s + 1) > 1. Wei and Winter showed that if (p , r) = (3 , 2) or (2 , 2) , then as τ increases, a stable stationary monotone solution is destabilized by Hopf bifurcation, and hence periodic solutions appear. In this paper we consider two cases (p , r) = (3 , 1) and (3 , 3). We show that a Hopf bifurcation occurs for n -mode stationary solutions, n ≥ 1 , in a rigorous way, studying eigenvalues in detail. Exact periods and exact critical values of τ can be written by using complete elliptic integrals. A relationship between a period and a shape of a stationary solution is also studied. In particular, a maximum point of the period is studied. [ABSTRACT FROM AUTHOR]

With the rapid development of modern information technology and network technology, the construction of tax informatization plays an important role in improving the level of tax management and work efficiency. However, in the centralized data processing mode, the establishment of large data center and the efficient, stable, and secure connection of server bandwidth have become key challenges. Moreover, the openness and interconnectivity of the tax system network pose various network security threats, seriously threatening the integrity and security of data. For the above issues, this article proposes a tax system network security protection scheme based on elliptic curve cryptography. This scheme combines hash function and elliptic curve cryptography to achieve user anonymity and secure generation of session keys, effectively ensuring the secure transmission of confidential information. The security and effectiveness of the scheme are verified through informal and formal security analysis based on security proof, BAN logic and AVISPA tool. And a comprehensive evaluation of the scheme is conducted, and the results show that the scheme significantly reduced the costs while providing security features. • This article proposes a tax system network authentication scheme. • Based on BAN logic and AVISPA,the security of the new scheme is further verified. • This article comprehensively analyzes the performance of the designed scheme. [ABSTRACT FROM AUTHOR]

In this research, the integrable Zhanbota-IIA equation which plays an important role in nonlinear optical dynamics is analytically investigated. Based on the extended sinh-Gordon equation expansion method, its bright, dark, combined bright-dark, singular soliton, combined singular soliton and singular periodic solutions are constructed under some constraint conditions. In addition, by using the modified Jacobi elliptic function expansion method, the Jacobi elliptic function solutions and the other soliton solutions are derived. Further, the modulational instability is studied based on the standard linear stability analysis. The graphical interpretations of the obtained results are demonstrated graphically. To the best of our knowledge, this study represents a unique and significant contribution to investigating the integrable Zhanbota-IIA equation. In addition, the extracted soliton solutions open doors to understanding the nonlinear systems more deeply and connecting their potential across a range of scientific and technological domains. [ABSTRACT FROM AUTHOR]

This paper deals with the soliton solutions for beam movement within a multimode optical fiber featuring a parabolic index shape. It is considered that a Two-Dimensional Nonlinear Schrödinger Equation (2D-NLSE) with an instantaneous Kerr nonlinearity of the kind can represent the beam dynamics. Nonlinear Multimode Optical Fibers (MMFs) of this kind are gaining popularity because they provide novel approaches to control the spectral, temporal, and spatial characteristics of ultrashort light pulses. We gain the optical soliton solutions for the nonlinear evolution beam dynamics using the Jacobi Elliptic Function Expansion (JEFE) method. The exact analytical solution of Nonlinear Partial Differential Equations (NLPDEs) can be achieved with wide application using the effective JEFE approach. These solutions are obtained in the form of dark, bright, combined dark–bright, complex combo, periodic, and plane wave solutions. Additional solutions for Jacobi elliptic functions, encompassing both single and dual function solutions, have been acquired. This approach is based on Jacobi elliptic functions, which will provide us the exact soliton solutions to nonlinear problems. Additionally, we will analyze the Modulation Instability (MI) for the underlying model. Moreover, we show the physical behavior of the beam propagation in a multimode optical fiber the three-dimensional, two-dimensional, and their corresponding contour plots are dispatched using the different values of parameters. [ABSTRACT FROM AUTHOR]

The general solutions for nonlinear waves in a bubble-liquid mixture are obtained from the Kudryashov–Sinelshchikov (KS) equation under different parametric regimes. The Chiellini integrability condition has been used for constructing exact general solutions. In the non-dissipative case, we find that the solutions may be expressed in terms of Jacobi elliptic and Weierstrass functions. On the other hand, in the dissipative case, only implicit solutions are obtained for the KS equation. For an alternative perspective, the notion of the Jacobi Last Multiplier has been utilized to obtain the corresponding Lagrangian and Hamiltonian description of the reduced equation for the bubble-liquid mixture system. [ABSTRACT FROM AUTHOR]

In this contribution, heterogeneous micropolar elastic fiber‐reinforced composites (FRCs) with a periodic structure are analyzed using the two‐scale asymptotic homogenization method (AHM). We focus on predicting the antiplane effective properties of micropolar two‐phase FRCs with parallelogram‐like unit cells. The periodic structure is defined by unidirectional, infinitely long, and concentric cylindrical fibers embedded in a homogeneous matrix. Constituent materials are assumed centro‐symmetric isotropic materials, and perfect interface conditions are considered. The AHM allows us to address the local problems on the periodic cell and determine the corresponding effective properties. This is achieved by employing two‐scale asymptotic expansions for the displacement and microrotation fields, which depend on both macro‐ and micro‐scales. The complex variable theory, combined with the complex‐potential method and doubly periodic Weierstrass elliptic functions, is applied to determine the solution of the antiplane local problems. Simple closed‐form formulas are provided for the antiplane stiffness and torque effective properties of two‐phase micropolar elastic FRCs, which depend on the physical properties and volume fractions of constituents. Finally, numerical examples are reported and discussed. Comparisons with other theoretical models are also presented, and good agreements are obtained. [ABSTRACT FROM AUTHOR]

In this article, we determine various analytical solutions for the (4 + 1) -dimensional Fokas equation, a significant model in mathematical physics with numerous applications in nonlinear dynamics. Utilizing multiple integration techniques such as the improved F-expansion technique and the Jacobi elliptic function method, we retrieve an array of solution types, including traveling waves, periodic solutions, bell-shaped waves, rational functions, and both kink and anti-kink structures. We further explore the complex nature of these solutions through their graphical representations. By applying Maple, we visualize our results in three-dimensional (3D), and two-dimensional (2D) formats to illustrate the dynamic behavior of these solutions across various parameters and initial conditions. Our findings provide deeper insights into the properties of the Fokas equation and offer a valuable reference for further studies in nonlinear wave phenomena. [ABSTRACT FROM AUTHOR]

This article presents an analytical investigation performed on a generalized geophysical Korteweg–de Vries model with nonlinear power law in ocean science. To start with, achieving diverse solitary wave solutions to the generalized power‐law model involves using wave transformation, which reduces the model to a nonlinear ordinary differential equation. A direct integration approach is adopted to construct solutions in the beginning. This brings the emergence of interesting solutions like non‐topological solitons, trigonometric functions, exponential functions, elliptic functions, and Weierstrass functions in general structures. Besides, in a bid to secure more general exact solutions to the model, one adopts the extended Jacobi elliptic function expansion technique (for some specific cases of n$$ n $$). Thus, various cnoidal, snoidal, and dnoidal wave solutions to the understudied model are attained. The copolar trio explicated in a tabular form reveals that these solutions can be relapsed to various hyperbolic and trigonometric functions under certain criteria. Additionally, diverse graphical exhibitions of the dynamical attributes of the gained results are presented in a bid to have a sound understanding of the physical phenomena of the underlying model. Later, one gives the conserved vectors of the aforementioned equation by employing the standard multiplier approach. [ABSTRACT FROM AUTHOR]

Costa first constructed a family of complete minimal surfaces which have genus 1 and 4 planar ends by use of Weierstrass-℘ functions. They are Willmore tori of Willmore energy 16π. In this paper, the authors consider the geometry of conjugate surfaces of these surfaces. It turns out that these conjugate surfaces are doubly periodic minimal surfaces with flat ends in ℝ3. Moreover, the authors can also perform a Lorentzian deformation on these Costa's minimal tori, which produce a family of complete space-like stationary surfaces (i.e., of zero mean curvature) with genus 1 and 4 planar ends in 4-dimensional Lorentz-Minkowski space ℝ14. [ABSTRACT FROM AUTHOR]

The present work concerns with the nonlinear dynamics for the synchronous pulsating bubble clusters uniformly distributed on a spherical surface. First, the governing equation for such clusters with 4/6/8/12/20 coupled bubbles are established. Second, the maximum and minimum radii for the gas-filled bubble clusters are analyzed according to the first integral. Third, by introducing suitable nonlocal transformations, two novel equivalent parametric analytical solutions in the form of Weierstrass elliptic function are constructed for the gas-filled bubble clusters for a specific polytropic exponent κ = 3 / 2 without considering the surface tension, and based on which we immediately derive the parametric analytical solution for the corresponding vapor bubble clusters. Further, to consider the case of arbitrary polytropic exponent and surface tension, we develop a direct approach to construct the parametric analytical solution using Jacobi elliptic function for gas-filled bubble clusters. It is shown that, the behaviors and results for the bubble clusters will degenerate to the corresponding ones for single bubbles as the radius of the bubble cluster approaches infinity. In addition, on the basis of the analytical results, dynamic properties and motion laws of the bubble clusters are also discussed. [ABSTRACT FROM AUTHOR]

For the ℤ -lift X ℤ of a monoid scheme X of finite type, Deitmar et al. calculated its absolute zeta function by interpolating # X ℤ ( q) for all prime powers q using the Fourier expansion. This absolute zeta function coincides with the absolute zeta function of a certain polynomial. In this paper, we characterize the polynomial as a ceiling polynomial of the sequence (# X ℤ ( q)) q , which we introduce independently. Extending this idea, we introduce a certain pair of absolute zeta functions of a separated scheme X of finite type over ℚ by means of a pair of Puiseux polynomials which estimate " # X ( p m ) " for sufficiently large p. We call them the ceiling and floor Puiseux polynomials of X. In particular, if X is an elliptic curve, then our absolute zeta functions of X do not depend on its isogeny class. [ABSTRACT FROM AUTHOR]

In this work, we explore the Riemann problem of the Fokas–Lenells (FL) equation given initial data in the form of a step discontinuity by employing the Whitham modulation theory. The periodic wave solutions of the FL equation are characterized by elliptic functions along with the Whitham modulation equations. Moreover, we find that the ±$\pm$ signs for the velocities of the periodic wave solutions remain unchanged during propagation. Thus, when analyzing the propagation behavior of solutions, it is necessary to separately consider the clockwise (negative velocity) and counterclockwise (positive velocity) cases. In this regard, we present the classification of the solutions to the Riemann problem of the FL equation in both clockwise and counterclockwise cases for the first time. [ABSTRACT FROM AUTHOR]

The uniformity and efficiency of ultrasonic needle peen-forming (UNPF) cannot be achieved by a handheld single-needle peening device. Thus, a novel robotized UNPF system is designed in this paper with a multi-needle peening device. By taking the aluminum alloy 2024-T3 specimens as an example, the peening device's travel velocity and standoff distance are first determined to balance processing efficiency and safety. Thereafter, the resulting arc height for Almen-size specimens with respect to other manipulated variables (MVs), such as the vibration amplitude, the air pressure, and the number of peening passes, are fitted by the data-driven models. As the radius of curvature at the saturation state is identical for small and large specimens with similar shapes, its relationship with the arc height and at the saturation state is represented by a function of elliptic paraboloid for understanding the limit of UNPF capability. It is shown that specimens of different sizes but the same aspect ratio have a consistent radius of curvatures after being treated with the same MVs. Notably, by increasing the vibration amplitude from 21 to 35 μ m , the ratios of peening time required for achieving full coverage to its counterpart for achieving saturation state reduced from 76 to 8.3%, showing a significant improvement in forming productivity and capability. Furthermore, increasing the number of peening passes, from 1 to 32 significantly reduces surface roughness by up to 86.4%, approaching the as-received specimen's smoothness. Moreover, the maximum compressive residual stress after UNPF is - 500 MPa at a depth of 1.2 mm, and the hardness is drastically increased by 45.4% for specimens of thickness 1.5 mm, improving their fatigue life. [ABSTRACT FROM AUTHOR]

The coupled higher-order nonlinear Schrödinger equations (CHNLSEs) are investigated in this paper. This model mimic the actuation of the solitons through multimode fibers. The technique of improved modified extended tanh function is used to conduct the study. Many exact solutions are achieved like bright solitons, dark solitons and singular solitons. Moreover, singular periodic solutions, Weierstrass elliptic function solutions, hyperbolic and Jacobi elliptic solutions are raised. Using 3D and 2D graphs, the physical behavior of the produced solutions is illustrated. [ABSTRACT FROM AUTHOR]

In this paper, exact wave propagation patterns and chaotic dynamical behaviors of the extended (3+1)-dimensional nonlinear Schrödinger equation are studied. The topological structure of the dynamic system of the equation is studied by the complete discrimination system for the cubic polynomial method, in which the existence conditions of the soliton solutions and periodic solutions are obtained. Then, by the trial equation method, thirteen exact solutions are obtained, including solitary wave solutions, triangular function solutions, rational solutions and the elliptic function double periodic solutions, especially the elliptic function double periodic solutions. Finally, it is found that the system has chaotic behaviors when given the appropriate perturbations. [ABSTRACT FROM AUTHOR]

Costa first constructed a family of complete minimal surfaces which have genus 1 and 4 planar ends by use of Weierstrass-℘ functions. They are Willmore tori of Willmore energy 16π. In this paper, the authors consider the geometry of conjugate surfaces of these surfaces. It turns out that these conjugate surfaces are doubly periodic minimal surfaces with flat ends in ℝ3. Moreover, the authors can also perform a Lorentzian deformation on these Costa's minimal tori, which produce a family of complete space-like stationary surfaces (i.e., of zero mean curvature) with genus 1 and 4 planar ends in 4-dimensional Lorentz-Minkowski space ℝ14. [ABSTRACT FROM AUTHOR]

Models attempting an analytical description of free-electron laser (FEL) devices have been proposed in the past. They provided interesting results, leading either to a deeper understanding of the FEL dynamics and to semi-analytical formulae, useful for the preliminary design of self amplified spontaneous emission and oscillator FELs. Most of these models work well until the level of mild saturation. In this paper, we comment on the so-called logistic model and a more recent analysis describing the FEL evolution in terms of Jacobi elliptic functions. Both models are shown to be suited to describe the evolution from the low signal to the onset of saturation. We attempt therefore an extension of these theoretical formulations using a delayed logistic model, capable of including characteristic features like the post saturation power oscillations. [ABSTRACT FROM AUTHOR]

We discuss the problem of uniqueness of a meromorphic function f (z), which shares a1(z), a2(z), and a3(z) CM with its shift f (z + c), where a1(z), a2(z), and a3(z) are three c-periodic distinct small functions of f (z) and c ∈ C \{0}. The obtained result improves the recent result of Heittokangas, et al. [Complex Var. Elliptic Equat., 56, No. 1-4, 81 (2011)] by omitting the assumption about the order of f (z). In addition, we introduce a way of characterization of elliptic functions in terms of meromorphic functions sharing values with two of their shifts. Moreover, by a number of illustrative examples, we show that our results are, in certain senses, best possible. [ABSTRACT FROM AUTHOR]

We obtain two closed-form nonrecurrent formulas for the coefficients of the Taylor series of the Weierstrass sigma function related to elliptic function theory. These results are derived from the differential-recurrence and two-dimensional recurrence relations that arise when searching for the analytic solutions of the system of partial differential equations constructed by Weierstrass for the sigma function. [ABSTRACT FROM AUTHOR]

This work delves into the behavior of cold bosonic atoms within a zigzag optical lattice which offers a rich platform for studying quantum many‐body physics and has potential applications in various fields including quantum simulation, quantum computing, and precision measurement. The study aims to derive exact solutions and explore qualitative properties through dynamical analysis. Exact solutions of the nonlinear differential equation model can provide insights into the behavior of bosonic atoms in complex optical lattice configurations, allowing researchers to simulate and study phenomena such as quantum phase transitions, many‐body localization, and exotic states of matter. The invariant analysis has been performed for the first time on the specified equation, yielding a reduced system of equations with forms that are easier to handle. Novel techniques are applied to extract solutions from the reduced system of equations obtained via invariant analysis. The results reveal a rich set of solutions, including various traveling wave solutions and doubly periodic functions in the form of Jacobian elliptic functions. Bifurcation analysis, conducted through phase portraits, provides insights into the system's long‐term behavior under different parameter values. This work contributes to a deeper understanding of the dynamics of cold bosonic atoms in optical lattices via 2D and 3D plots of obtained solutions depicting the change in the behavior of soliton solutions with fractional derivative parameter. [ABSTRACT FROM AUTHOR]

Through the modified deformation algorithm related to conservation laws, the (1+1)-dimensional AKNS(-1) equations are extended to a (4+1)-dimensional AKNS(-1) system. When one, two, or three of the independent variables are removed, the (4+1)-dimensional AKNS(-1) system degenerates to some novel (3+1)-dimensional, (2+1)-dimensional, and (1+1)-dimensional AKNS(-1) systems, respectively. Under a simple dependent transformation, the (1+1)-dimensional AKNS(-1) equations turn into the classical sine-Gordon equation. Then using the same deformation procedure, the (1+1)-dimensional sine-Gordon equation is generalized to a (3+1)-dimensional version. By introducing the deformation operators to the Lax pairs of the original (1+1)-dimensional models, the Lax integrability of both the (4+1)-dimensional AKNS(-1) system and the (3+1)-dimensional sine-Gordon equation is proven. Finally, the traveling wave solutions of the (4+1)-dimensional AKNS(-1) system and the (3+1)-dimensional sine-Gordon equation are implicitly given and expressed by tanh function and incomplete elliptic integral, respectively. These results may enhance our understanding of the complex physical phenomena described by the nonlinear system discussed in this paper. [ABSTRACT FROM AUTHOR]

This article introduces two kinds of processing techniques to solve Jacobian elliptic equations and obtain rich periodic wave solutions. Then, the equation was used as an auxiliary equation to solve the (3+1)-dimensional modified Korteweg de Vries–Zakharov–Kuznetsov (mKDV-ZK) equation. Combined with the mapping method, a large number of new types of exact periodic wave solutions were obtained, many of which were rarely found in previous research. Numerical simulations have demonstrated the evolution of various periodic waves in (3+1)-dimensional mKDV-ZK. The solutions and wave phenomena obtained in this article will help expand our understanding of the equation. [ABSTRACT FROM AUTHOR]

This paper presents an approach to summing a few families of infinite series involving hyperbolic functions, some of which were first studied by Ramanujan. The key idea is based on their contour integral representations and residue computations with the help of some well-known results of Eisenstein series given by Ramanujan, Berndt, et al. As our main results, several series involving hyperbolic functions are evaluated and expressed in terms of z = F 1 2 (1 / 2 , 1 / 2 ; 1 ; x) and z ′ = d z / d x . When a certain parameter in these series is equal to π , the series are expressed in closed forms in terms of some special values of the Gamma function. Moreover, many new illustrative examples are presented. [ABSTRACT FROM AUTHOR]

We construct an algebra that is an elliptic generalization of |$A_1$| spherical DAHA acting on its finite-dimensional module at |$t=-q^{-K/2}$| with |$K=2$|⁠. We prove that |$PSL(2,\mathbb{Z})$| acts by automorphisms of the algebra we constructed, and provide an explicit representation of automorphisms and algebra operators alike by |$3 \times 3$| matrices with matrix elements given by products of elliptic functions. A relation of this construction to the |$K$| -theory character of affine Laumon space is conjectured. We point out two potential applications, respectively to |$SL(3,\mathbb{Z})$| symmetry of Felder–Varchenko functions and to new elliptic invariants of torus knots and Seifert manifolds. [ABSTRACT FROM AUTHOR]

We discuss the p -adic Weierstrass zeta functions associated with elliptic curves defined over the field of algebraic numbers and linear relations for their values in the p -adic domain. These results are extensions of the p -adic analogues of results given by Wüstholz in the complex domain [see A. Baker and G. Wüstholz, Logarithmic Forms and Diophantine Geometry , New Mathematical Monographs, 9 (Cambridge University Press, Cambridge, 2007), Theorem 6.3] and also generalise a result of Bertrand to higher dimensions ['Sous-groupes à un paramètre p -adique de variétés de groupe', Invent. Math. 40 (2) (1977), 171–193]. [ABSTRACT FROM AUTHOR]

This study delves into the exploration of the dynamics of (3 + 1) -dimensional Painlevé integrable generalized model from different perspectives, which delineates the evolution of nonlinear phenomena in three spatial dimensions and one temporal dimension, displaying the remarkable Painlevé integrability property. The ϕ 6 -expansion technique is applied to extract traveling wave solutions in the form of Jacobi elliptic functions. To give physical insights of obtained solutions, we present these solutions through graphs such as 2D, 3D and contour plots. Further, to understand the planar dynamical system, we employ the concepts of bifurcation, chaos theory and sensitivity analysis. Bifurcation analysis reveals the dependence on the solution of a planar dynamical system at critical points. Additionally, the detection of chaotic movements in the perturbed dynamical system is achieved by detecting tools. Also the sensitivity analysis of the model is investigate by three distinct initial conditions. The findings are innovative, valuable and captivating for the readers in exploring this model. [ABSTRACT FROM AUTHOR]

Regularity for elliptic equations with oscillatory coefficients was concerned. Problem domains were periodic and consisted of a connected region with normal permeability and a disconnected matrix block subset with high permeability. Coefficients of the elliptic equations depending on the permeability of the domains were highly oscillatory. Let be the periodic size of domain, the size ratio of a matrix block to the whole domain, and the permeability ratio of the disconnected matrix block subset to the connected sub-region of the domains. This work presented Lipschitz estimate uniformly in for the Green's functions and the solutions of the elliptic equations. [ABSTRACT FROM AUTHOR]

This study is focusing on the integrable (3+1)-dimensional equation that combines the potential Kadomtsev-Petviashvili (pKP) equation with B-type Kadomtsev-Petviashvili (BKP) equation, also known as the pKP-BKP equation. The idea of combining integrable equations has the potential to produce a variety of unexpected outcomes such as resonance of solitons. This article provides a wide range of alternative exact solutions for the pKP-BKP equation in three dimensional form, including dark solitons, singular solitons, singular periodic solutions, Jacobi elliptic function (JEF) solutions, rational solutions and exponential solution. The improved modified extended (IME) tanh function method is employed to investigate these solutions. All of the obtained solutions for the investigated model are presented using the Wolfram Mathematica program. To further help in understanding the solutions' physical characteristics and dynamic structure, the article provides visual representations of some derived solutions using 2D representation in addition to the 3D graphs via symbolic computation. This article aims to use a potent strategy using a powerful scheme to derive different solutions with various structures. Additionally, the results greatly improve and enhance the literature's solutions to a combined pKP-BKP equation and allow deep understanding of the nonlinear dynamic system through different exact solutions. [ABSTRACT FROM AUTHOR]

This paper thoroughly investigates the integrable Akbota equation, a Heisenberg ferromagnet-type equation that plays a crucial role in exploring curve and surface geometry. The ϕ6-model expansion method, known for its proficiency and reliability, is applied to generate generalized solitonic wave profiles, spanning a diverse range of soliton families. Prior to this study, there is no existing study in which this technique is utilized that ensured the existence of solution via development of conditions. On the other hand, this method is provided with the Jacobi elliptic function-based solutions with the limiting cases. The analytical strategy presented a notable advantage by specifying a constraint for each solution, ensuring its existence. Consequently, the solitonic wave structures exhibit various attributes, including the Jacobi elliptic function, periodicity, brightness, dark-brightness, singularity, exponential, trigonometry, and rational solitonic structures. These characteristics emerge under previously unexplored existence conditions. The results are visually depicted through 2D, 3D, and contour plots, providing a clear illustration of the behavioral responses to pulse propagation and allowing for the inference of fitting values for system parameters. This visualization offers valuable insights into the characteristics and dynamics of soliton solutions derived from the integrable Akbota equation. [ABSTRACT FROM AUTHOR]

The Klein–Gordon equation plays an important role in mathematical physics, such as plasma and, condensed matter physics. Exploring its exact solution helps us understand its complex nonlinear wave phenomena. In this paper, we first propose a new extended Jacobian elliptic function expansion method for constructing rich exact periodic wave solutions of the (2+1)-dimensional Klein–Gordon equation. Then, we introduce a novel wave transformation for constructing nonlinear complex waves. To demonstrate the effectiveness of this method, we numerically simulated several sets of complex wave structures, which indicate new types of complex wave phenomena. The results show that this method is simple and effective for constructing rich exact solutions and complex nonlinear wave phenomena to nonlinear equations. [ABSTRACT FROM AUTHOR]

The current paper undertakes an in-depth exploration of the dynamics of nonlinear waves governed by a 3D-modified nonlinear wave equation, a significant model in the study of complex wave phenomena. To this end, the study employs both classical and nonclassical Lie symmetries for rigorously deriving invariant solutions of the governing equation. These symmetries enable the formal construction of exact solutions, which are crucial for understanding the complex behavior of the model. Furthermore, the research extends into the realm of bifurcation analysis through the application of planar dynamical system theory. Such an analysis reveals the conditions under which the 3D-modified nonlinear wave equation admits Jacobi elliptic function solutions. The study also delves into the impact of the nonlinear parameter on the physical characteristics of bright and kink solitary waves as well as continuous periodic waves using Maple. Overall, the comprehensive analysis presented not only enhances the understanding of complex nonlinear wave dynamics but also sets the stage for future advancements in vast areas of fluid dynamics and plasma physics. [ABSTRACT FROM AUTHOR]

In this study, we apply the extended direct method to seek new similarity solutions and reduction equations of the (2+1)-dimensional Kadomtsev–Petviashvili (KP) equation. Through this method, we can reduce the KP equation to a system of ordinary differential equations, and we divide the discussion into four cases via calculation. In some cases, new similarity reductions and exact solutions are obtained, such as elliptic function solutions, polynomial solutions. In other cases, similarity solutions are consistent with the present solutions. Futhermore, a similarity solution we get has more clear and concise form than that obtained by the generalized Clarkson–Kruskal direct method. Therefore, it demonstrates that our study is correct and more efficient than before which can also be applied to the other nonlinear physical models. [ABSTRACT FROM AUTHOR]

Some identities that involve the elliptic version of the Cauchy matrices are presented and proved. They include the determinant formula, the formula for the inverse matrix, the matrix product identity and the factorization formula. [ABSTRACT FROM AUTHOR]

Recently, we established and used the generalized Littlewood theorem concerning contour integrals of the logarithm of analytical function to obtain new criteria equivalent to the Riemann hypothesis. The same theorem was subsequently applied to calculate certain infinite sums and study the properties of zeroes of a few analytical functions. In this article, we discuss what are, in a sense, inverse applications of this theorem. We first prove a Lemma that if two meromorphic on the whole complex plane functions f(z) and g(z) have the same zeroes and poles, taking into account their orders, and have appropriate asymptotic for large |z|, then for some integer n, d n ln (f (z)) d z n = d n ln (g (z)) d z n . The use of this Lemma enables proofs of many identities between elliptic functions, their transformation and n-tuple product rules. In particular, we show how exactly for any complex number a, ℘(z)-a, where ℘(z) is the Weierstrass ℘ function, can be presented as a product and ratio of three elliptic θ 1 functions of certain arguments. We also establish n-tuple rules for some elliptic theta functions. [ABSTRACT FROM AUTHOR]

We investigated the extended sixth-order Korteweg–de Vries (KdV6) equation which describes many nonlinear phenomena. The improved modified (IM) extended tanh-function method is used to conduct the study. Many exact solutions, like dark, combo singular-dark and singular soliton are achieved. Moreover, hyperbolic solutions, singular periodic solutions, rational solutions, exponential solutions and Jacobi elliptic function (JEF) solutions are derived. Graphical illustrations are presented to visually depict the dynamics of selected solutions. [ABSTRACT FROM AUTHOR]

Propellantless propulsion systems, such as the well-known photonic solar sails that provide thrust by exploiting the solar radiation pressure, theoretically allow for extremely complex space missions that require a high value of velocity variation to be carried out. Such challenging space missions typically need the application of continuous thrust for a very long period of time, compared to the classic operational life of a space vehicle equipped with a more conventional propulsion system as, for example, an electric thruster. In this context, an interesting application of this propellantless thruster consists of using the solar sail-induced acceleration to artificially precess the apse line of a planetocentric elliptic orbit. This specific mission application was thoroughly investigated about twenty years ago in the context of the GeoSail Technology Reference Study, which analyzed the potential use of a spacecraft equipped with a small solar sail to perform an in situ study of the Earth's upper magnetosphere. Taking inspiration from the GeoSail concept, this study analyzes the performance of a solar sail-based spacecraft in (artificially) precessing the apse line of a high elliptic orbit around Venus with the aim of exploring the planet's induced magnetotail. In particular, during flight, the solar sail orientation is assumed to be Sun-facing, and the required thruster's performance is evaluated as a function of the elliptic orbit's characteristics by using both a simplified mathematical model of the spacecraft's planetocentric dynamics and an approximate analytical approach. Numerical results show that a medium–low-performance sail is able to artificially precess the apse line of a Venus-centered orbit in order to ensure the long-term sensing of the planet's induced magnetotail. [ABSTRACT FROM AUTHOR]

We have introduced various novel soliton waves and other analytic wave solutions for nonlinear Schrödinger equation with cubic, quintic, septic, and nonic nonlinearities. The modified extended direct algebraic method governs the transmission of various solitons with different effects. The combination of this system enables the obtaining of analytical soliton solutions with some unique behaviors, including bright, dark, and mixed dark-bright soliton solutions; singular soliton solutions; singular periodic, exponential, rational wave solutions; and Jacobi elliptic function solutions. These results realize the stability of the nonlinear waves' propagation in a high-nonlinear-dispersion medium that is illustrated using 2D and 3D visuals and contour graphical diagrams of the output solutions. This research focused on determining exact soliton solutions under certain parameter conditions and evaluating the stability and reliability of the soliton solutions based on the used modified extended direct algebraic method. This will be useful for many various domains in technology and physics, such as biology, optics, and plasma physical science. At the end, we use modulation instability analysis to assess the stability of the wave solutions obtained. [ABSTRACT FROM AUTHOR]

In this study, we obtain solitary wave solutions and other exact wave solutions for Davey-Stewartson equation (DSE), which explains how waves move through water with a finite depth while being affected by gravity and surface tension. The study is conducted with the aid of the modified extended mapping method (MEMM). A variety of distinct traveling wave solutions are furnished. The obtained solutions comprise dark, bright, and singular solitary wave solutions. Additionally, Jacobi elliptic function solutions, exponential wave solutions, singular periodic wave solutions, rational wave solutions, and periodic wave solutions are also offered. To help readers physically grasp the acquired solutions, graphical representations of some of the extracted solutions are provided. [ABSTRACT FROM AUTHOR]

The cubic–quartic perturbed Biswas–Milovic equation, which contains Kudryashov's nonlinear form and two generalized nonlocal laws, has been explored qualitatively and quantitatively, as demonstrated in the present work. The research methods used include the complete discrimination system for polynomial method and the trial equation method. The results show that the Hamiltonian has the conservation property, and the global phase diagrams obtained via the bifurcation method reveal the existence of periodic and soliton solutions. Furthermore, we fully classify all the single traveling wave solutions to substantiate our findings, covering singular solutions, solitons, and Jacobian elliptic function solutions. We analyze their topological stabilities and present two‐dimensional graphs of solutions. We also delve deeper into the dynamic system by incorporating the perturbation item to explore the chaotic phenomena associated with the equation. These outcomes are valuable for studying the propagation of high‐order dispersive optical solitons and have potential applications in optimizing optical communication systems to improve efficiency. [ABSTRACT FROM AUTHOR]

Secure routing and communication with confidentiality based on encryption of texts in multiple natural languages are challenging issues in wireless sensor networks which are widely used in recent applications. The existing works on Elliptic Curve Cryptography based secured routing algorithms are focused only on the encryption and decryption of single language text encrypted over a Prime finite field. In this article, a new algorithm called Multi‐Language ECC encrypted Secure Routing algorithm with trust management is proposed, in order to ensure confidentiality and integrity which focuses on the encryption of plain text using Riemann's zeta function and Elliptic Curve Cryptography for improving the key strength which is applied for encryption over a range of multi languages namely Tamil, English, Hindi French and German which are supported by Unicode and routing the text security. From the experiments conducted using the proposed multi‐lingual encryption algorithm with network routing, we prove that the suggested method provides greater security than the current secure routing algorithms due to the use of Zeta function and Gamma function with ECC key and trust management. but also boasts reduced complexity compared to other existing multi‐lingual encryption algorithms. [ABSTRACT FROM AUTHOR]

We consider the problem of determining the steady-state temperature on the upper boundary of a layer under known conditions at the lower boundary. The existence and uniqueness of a solution of this problem are proved. [ABSTRACT FROM AUTHOR]

In this research, we are examining the stochastic modified Korteweg-de Vries (SMKdV) equation forced in the Itô sense by multiplicative noise. We use an appropriate transformation to convert the SMKdV equation to another MKdV equation with random variable coefficients (MKdVRVCs). We use the generalizing Riccati equation mapping and Jacobi elliptic functions methods in order to acquire new trigonometric, hyperbolic, and rational solutions for MKdV-RVCs. After that, we can get the solutions to the SMKdV equation. To our knowledge, this is the first time we have assumed that the solution of the wave equation for the SMKdV equation is stochastic, since all earlier research assumed that it was deterministic. Furthermore, we provide different graphic representations to show the influence of multiplicative noise on the exact solutions of the SMKdV equation. [ABSTRACT FROM AUTHOR]