Weierstrass Elliptic Function Research Articles

Weierstrass elliptic function solutions and degenerate solutions of a variable coefficient higher-order Schrödinger equation

In this paper, the auxiliary equation method is used to study the Weierstrass elliptic function solutions and degenerate solutions of the variable coefficient higher order Schrödinger equation, including Jacobian elliptic function solutions, trigonometric function solutions and hyperbolic function solutions. The types of solutions of the variable coefficient higher-order Schrödinger equation are enriched, and the method of seeking precise and accurate solutions is extended. It is concluded that the types of degenerate solutions are related to the coefficients of the equation itself when the degenerate solutions are obtained from the solutions of the Weierstrass elliptic functions. In addition, the solutions form of the equation is extended from the power series expansion form to the Laurent series expansion form, and the corresponding solutions are obtained. After the conversion formula between the Weierstrass elliptic function solutions and the Jacobian elliptic function solutions is constructed, the Jacobian elliptic function solutions of the higher order Schrödinger equation with variable coefficients are also obtained. These have not been previously studied.

Dynamics of newly created soliton solutions via Atangana–Baleanu Fractional (ABF) for system of (ISALWs) equations

In this work, newly created soliton solutions for ion sound and Langmuir waves with an Atangana–Baleanu fractional (ABF) are given. Symbolic software is used to perform the unified solver method (USM) and Weierstrass elliptic function method (WEFM) in order to solve this model. In terms of hyperbolic functions, extended trigonometric functions, and so forth. Single-wave solutions that were entirely novel and universal were attained. The way the soliton solutions behaved in connection with the two-dimensional (2D) and three-dimensional (3D) graphics was also investigated. The dynamic investigation of the newly created soliton solutions reveals that they have several soliton forms like single-soliton, bell-shaped and mixed-form soliton profiles. This strategy is viewed as promising for handling a range of ABF evolution systems.

Analytical solutions and dynamic behaviors to synchronous oscillation of same bubbles at vertices of cuboid and rectangle

The present work focuses on the nonlinear dynamics of the synchronous oscillating multiple bubbles in two typical spatial locations, namely, cuboid and rectangle arrangements. The governing equation for such synchronous oscillating multiple bubbles is derived from a modified Rayleigh–Plesset equation. Theoretical results including the collapse time and analytical solution (in three forms) for multiple vapor bubbles, as well as the maximum/minimum radii, oscillation period, and analytical solution in the form of Weierstrass elliptic function for multiple gas-filled ones, are provided. On the basis of these results, we not only study the dynamic characteristics of multi-bubbles straightforwardly but also carefully observe a series of evolution behaviors of bubbles when the number of bubbles decreases gradually on the order of 8→4→2→1. It should be pointed out that we also compare the multi-bubble behaviors between the general cuboid/rectangle arrangements and the corresponding cube/square arrangements under two reasonable restrictions, respectively. Furthermore, the limiting behaviors of the synchronous oscillating multiple gas-filled bubbles are discussed as the initial pressure of the gas in bubble approaches to zero.

Kerr geodesics in terms of Weierstrass elliptic functions

Kerr geodesics in terms of Weierstrass elliptic functions

Solution Structures of an Electrical Transmission Line Model with Bifurcation and Chaos in Hamiltonian Dynamics

Employing the Riccati–Bernoulli sub-ODE method (RBSM) and improved Weierstrass elliptic function method, we handle the proposed [Formula: see text]-dimensional nonlinear fractional electrical transmission line equation (NFETLE) system in this paper. An infinite sequence of solutions and Weierstrass elliptic function solutions may be obtained through solving the NFETLE. These new exact and solitary wave solutions are derived in the forms of trigonometric function, Weierstrass elliptic function and hyperbolic function. The graphs of soliton solutions of the NFETLE describe the dynamics of the solutions in the figures. We also discuss elaborately the effects of fraction and arbitrary parameters on a part of obtained soliton solutions which are presented graphically. Moreover, we also draw meaningful conclusions via a comparison of our partially explored areas with other different fractional derivatives. From our perspectives, by rewriting the equation as Hamiltonian system, we study the phase portrait and bifurcation of the system about NFETLE and we also for the first time discuss sensitivity of the system and chaotic behaviors. To our best knowledge, we discover a variety of new solutions that have not been reported in existing articles [Formula: see text], [Formula: see text]. The most important thing is that there are iterative ideas in finding the solution process, which have not been seen before from relevant articles such as [Tala-Tebue et al., 2014; Fendzi-Donfack et al., 2018; Ashraf et al., 2022; Ndzana et al., 2022; Halidou et al., 2022] in seeking for exact solutions about NFETLE.

On the propagation of cnoidal wave and overtaking collision of slow shear Alfvén solitons in low β magnetized plasmas

The overtaking collisional phenomenon of slow shear Alfvén solitons are studied in a low beta (β = kinetic pressure/magnetic pressure) collisionless, magnetized plasma consisting of electron and ion fluids. By employing a reductive perturbation technique, the Korteweg–de Vries (KdV) equation is deduced for investigating the nonlinear slow shear Alfvén wave. Before embarking on the study of the overtaking collisions, the stability analysis of the KdV equation is studied using the bifurcation theory. Also, a nonlinear periodic solution of the KdV equation is derived for the first time in the Weierstrass elliptic function formula. Moreover, the condition for converting the Weierstrass elliptic function expression to soliton is discussed. Furthermore, it is found that only density dip (rarefactive) solitons are formed in the super-Alfvénic regime. The next step includes the use of the Hirota bilinear method, which results in two and three shear Alfvén soliton solutions and their subsequent phase shifts. The influence of the plasma parameters on the amplitude as well as width of the slow shear Alfvén wave solitons are examined analytically and numerically. We also find out the profiles of overtaking interaction of slow shear Alfvén dip solitons having different amplitudes and speeds numerically. This study is important for understanding the phenomena of nonlinear slow shear Alfvén wave structures both in space and in laboratory plasmas.

Cutting the Coon amplitude

The Coon amplitude is a q-deformed generalization of the Veneziano amplitude exhibiting a semi-infinite sequence of poles that converge on an accumulation point, from which a branch cut emerges. A number of recent papers have provided compelling evidence that the residues of this amplitude satisfy the positivity requirements imposed by unitarity. This paper investigates whether positivity is also satisfied along the branch cut. It is demonstrated for a wide range of q-values that positivity violations occur in a region of the branch cut exponentially close to the accumulation point according to a scale set by q. The closing section of the paper discusses possible interpretations of this fact and strategies for excising negativity from the partial wave coefficients.An appendix presents derivations of instrumental identities relating the q-gamma and q-polygamma functions to the Weierstrass elliptic and quasiperiodic functions.

Operations for D-Algebraic Functions

A function is differentially algebraic (or simply D-algebraic) if there is a polynomial relationship between some of its derivatives and the indeterminate variable. Many functions in the sciences, such as Mathieu functions, the Weierstrass elliptic functions, and holonomic or D-finite functions are D-algebraic. These functions form a field, and are closed under composition, taking functional inverse, and derivation. We present implementation for each underlying operation. We also give a systematic way for computing an algebraic differential equation from a linear differential equation with D-finite function coefficients. Each command is a feature of our Maple package NLDE available at https://mathrepo.mis.mpg.de/DAlgebraicFunctions/NLDEpackage.

A Malmquist–Steinmetz Theorem for Difference Equations

It is shown that if the equation f(z+1)n=R(z,f),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} f(z+1)^n=R(z,f), \\end{aligned}$$\\end{document}where R(z, f) is rational in both arguments and degf(R(z,f))≠n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\deg _f(R(z,f))\ ot =n$$\\end{document}, has a transcendental meromorphic solution, then the equation above reduces into one out of several types of difference equations where the rational term R(z, f) takes particular forms. Solutions of these equations are presented in terms of Weierstrass or Jacobian elliptic functions, exponential type functions or functions which are solutions to a certain autonomous first-order difference equation having meromorphic solutions with preassigned asymptotic behavior. These results complement our previous work on the case degf(R(z,f))=n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\deg _f(R(z,f))=n$$\\end{document} of the equation above and thus provide a complete difference analogue of Steinmetz’ generalization of Malmquist’s theorem.

Multi-peak and rational soliton propagations for (3 + 1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup Kupershmidt model in fluid mechanics, ocean dynamics and plasma physics

This paper retrieves the investigation of rational solitons via symbolic computation with logarithmic transformation and ansatz functions approach for the [Formula: see text]-dimensional generalized Konopelchenko–Dubrovsky–Kaup-Kupershmidt (GKDKK) equation in fluid mechanics, ocean dynamics and plasma physics. We find two categories of M-shaped rational solitons and their dynamics will be revealed through graphs by choosing the suitable values of involved parameters. In addition, two categories of M-shaped rational solitons and their interactions with kink waves will be analyzed. Furthermore, homoclinic breathers, multi-wave and kink cross rational solitons will be investigated. The periodic, rational, dark, bright, Weierstrass elliptic function and positive soliton solutions will also be retrieved with the aid of Sub-ODE approach. Moreover, stability characteristics of solutions will be evaluated.

Parameters and dynamical graphs analysis about the new dynamics traveling wave solutions of longitudinal bud equation in a magneto-electro-elastic round rod

In this paper, we investigate the nonlinear longitudinal wave equation (LWE) which involves mathematical physics in a magneto-electro-elastic (MEE) circular rod. By three different methods, we found that various forms for exact traveling wave solutions of longitudinal bud equation among an MEE round rod. These new exact and solitary wave solutions are derived in the form of rational, single periodic function, double periodic Jacobian elliptic functions (JEF), especially about Weierstrass elliptic function (WEF). Additionally, certain interesting (3D), (2D) figures represent the LWE provides the corporal information to explain the physically phenomena. By choosing different suitable values of the free parameters, we made some charts and we analyzed these charts to get a lot of valuable information to understand the MEE circular rod. Our work describes the dynamics of the MEE circular rod. Finally, we conclude with some perspectives for our future research work.

Spherical photon orbits around a rotating black hole with quintessence and cloud of strings

In this paper, we calculate the analytical solutions for the radii of planar and polar spherical photon orbits around a rotating black hole that is associated with quintessential field and cloud of strings. This includes a full analytical treatment of a quintic that describes orbits on the equatorial plane. Furthermore, The radial profile of the impact parameters is studied and the radii corresponding to the extreme cases are derived. For the more general cases, we also discuss the photon regions that form around this black hole. To simulate the orbits that appear in different inclinations, we analytically solve the latitudinal and azimuth equations of motion in terms of the Weierstrassian elliptic functions, by considering the radii of spherical orbits, in their general form, as the initial conditions. The period and the stability conditions of the orbits are also obtained analytically.

Lie symmetry analysis, particular solutions and conservation laws for the dissipative (2 + 1)- dimensional AKNS equation

<abstract><p>The dissipative (2 + 1)-dimensional AKNS equation is considered in this paper. First, the Lie symmetry analysis method is applied to the dissipative (2 + 1)-dimensional AKNS and six point symmetries are obtained. Symmetry reductions are performed by utilizing these obtained point symmetries and four differential equations are derived, including a fourth-order ordinary differential equation and three partial differential equations. Thereafter, the direct integration approach and the $ (G'/G^{2})- $expansion method are employed to solve the ordinary differential respectively. As a result, a periodic solution in terms of the Weierstrass elliptic function is obtained via the the direct integration approach, while six kinds of including the hyperbolic function types and the hyperbolic function types are derived via the $ (G'/G^{2})- $expansion method. The corresponding graphical representation of the obtained solutions are presented by choosing suitable parametric values. Finally, the multiplier technique and the classical Noether's theorem are employed to derive conserved vectors for the dissipative (2 + 1)-dimensional AKNS respectively. Consequently, eight local conservation laws for the dissipative (2 + 1)-dimensional AKNS equation are presented by utilizing the multiplier technique and five local conservation laws are derived by invoking Noether's theorem.</p></abstract>

Dispersive optical solitons in magneto-optic waveguides for perturbed stochastic NLSE with generalized anti-cubic law nonlinearity and spatio-temporal dispersion having multiplicative white noise

Object:The current article studies optical solitons solutions for the dimensionless form of the perturbed stochastic NLSE in magneto-optic waveguides with generalized anti-cubic (GAC) law nonlinearity and spatio-temporal dispersion (STD) having multiplicative noise in the itô sense. Methods:The generalized Jacobi-elliptic function expansion method and the addendum Kudryashov’s method are investigated. Results:The generalized Jacobi elliptic function expansion method secures the Jacobi’s elliptic functions solutions, the dark and singular soliton solutions as long as the Weierstrass elliptic functions solutions and periodic wave solutions. The addendum Kudryashov’s method deduces the combo bright-singular soliton, bright soliton and singular soliton solutions.

New soliton molecules to couple of nonlinear models: ion sound and Langmuir waves systems.

In this paper, we will study two various nonlinear models: the Atangana–Baleanu fractional system of equations for the ion sound and Langmuir waves (ISALWs) and Hirota Ramani equation to obtain variety of solitary wave solutions. We will obtain bright, dark, periodic wave and solitory wave for ISALWs equation. We will also retreived bell type, kink type, singular, Jacbion elliptic function, Weierstrass-elliptic function, hyperbolic functions, periodic functions and other solitary wave solutions for Hirota Ramani equation using Sub ODE technique under some constraint conditions. At the end we will present our solutions with the help of graphs in distinct dimensions.

Revisiting timelike and null geodesics in the Schwarzschild spacetime: general expressions in terms of Weierstrass elliptic functions

The theory of Schwarzschild geodesics is revisited. Basing on a result by Weierstrass and Biermann, we derive a formula describing all non radial, timelike and null trajectories in terms of Weierstrass elliptic functions. Quite remarkably, a single formula works for an entire geodesic trajectory, even if it passes through turning points. Using this formula, we derive expressions for the proper and coordinate time along the geodesic.

Novel multi-soliton solutions of (2 + 1)-dimensional breaking equation based on Weierstrass elliptic function

Novel multi-soliton solutions of (2 + 1)-dimensional breaking equation based on Weierstrass elliptic function

Analytic solutions of the long-wave-short-wave resonance system in fluid mechanics

Under investigation in this paper is the long-wave-short-wave resonance system, which can describe a variety of nonlinear wave phenomena such as the two-dimensional packets of capillary-gravity waves in hydrodynamics and the optical-terahertz waves. The intended aim is carried out via considering a traveling wave reduction, adopting a modified version of the Jacobi elliptic expansion method and employing the Weierstrass elliptic function method to derive such analytic solutions as the bright and dark soliton solutions, periodic solutions, trigonometric-function and elliptic-function solutions in fluid mechanics.

Exact solutions of a coupled space-time fractional nonlinear Schrödinger type equation in quantum mechanics

In this paper, we consider a coupled space-time fractional nonlinear Schrödinger type equation which can be used for describing nonrelativistic quantum mechanical behavior. Under the help of the fractional complex transform and the conformable fractional derivative sense, we apply the extended auxiliary equation method, the modified He’s semi-inverse method and the improved Weierstrass elliptic function method to obtain many exact solutions. These exact solutions include the bright, dark, bright-dark, bright-like and kinky-bright optical soliton solutions as well as trigonometric function solutions, the Jacobi elliptic function solutions and the general solutions in form of Weierstrass elliptic function. The behaviors of some obtained solutions are displayed through the 3D graphs for different fractional orders by using Matlab. The obtained results show that these proposed methods are straightforward, efficient and can provide more forms of exact solutions, which are helpful to understand the complex physical meaning of space-time fractional coupled nonlinear Schrödinger equation.

Calculation of the Cosmological Constant for the Planetary System in Schwarzschild’s Cosmological Model

In this work, the static cosmological model of the Schwarzschild solution for the solar system is proposed taking into account the cosmological constant in the equation of the general theory of relativity (GTR) proposed by A. Einstein. We found the nonlinear differential equation that describes the behavior of the planets around the Sun; this is solved exactly by the Jacobi and Weierstrass elliptic functions. The obtained solution allows for us to estimate the value of the cosmological constant knowing the perihelion of the different planets and from different mathematical approaches; that is, the inverse problem is solved. From the obtained results, the Schwarzschild static cosmological model for the solar system is proposed, establishing the Schwarzschild cosmological radius and the curvature limit of the solar system. From the curvature limit, different regions are proposed for the planets, exoplanets, and a region is predicted where the existence of new planets and exoplanets belonging to the solar system is possible. The proposed theory of the static Schwarzschild cosmological model may be of great interest to astronomers, cosmologists, and all those interested in the study of the universe.