Jacobi Elliptic Functions Research Articles

Abundant optical solutions for the Sasa-Satsuma equation with M-truncated derivative

Here, we look at the Sasa-Satsuma equation with M-truncated derivative (SSE-MTD). The analytical solutions in the form of trigonometric, hyperbolic, elliptic, and rational functions are constructed using the Jacobi elliptic function and generalizing Riccati equation mapping methods. Because the Sasa–Satsuma equation is applied to explain the propagation of femtosecond pulses in optical fibers, the acquired solutions can be employed to explain a wide range of important physical phenomena. Moreover, we apply the MATLAB tool to generate a series of graphs to address the effect of the M-truncated derivative on the exact solution of the SSE-MTD.

A generalized nonlinear Schrödinger involving the weak nonlocality: Its Jacobi elliptic function solutions and modulational instability

The present paper investigates the features of specific waves in a weakly nonlocal nonlinear medium. More precisely, a generalized nonlinear Schrödinger equation (gNLSE) involving a high-order dispersion term, a high-order nonlinear term, and the effect of a term that accounts for the weak nonlocality is considered. By applying the Jacobi elliptic function (JEF) method, a wide range of Jacobi elliptic function solutions to the governing model is first derived. The characteristics of modulation instability (MI) and modulated wave (MW) topological and non-topological solitons are then analyzed by giving a series of two- and three-dimensional graphs. A major accomplishment of the study is the discovery of stable and unstable zones, as well as the control of the amplitude of topological and non-topological solitons.

Jacobi elliptic function approach to a conformable fractional nonlinear Schrödinger–Hirota equation

In this present paper we propound the following equation iDtαu+a2Dx2αu+ia3Dx3αu+a4Dx4α+ibDxα|u|2u+iσ|u|2Dxαu+iλDxα|u|2=c1|u|2u+c2|u|4,which is called the generalized nonlinear fractional Schrödinger–Hirota equation with conformable derivative. Actually the principal idea in this study is to use modified Jacobi elliptic function approach to find analytical solutions to the generalized nonlinear fractional Schrödinger–Hirota equation. Moreover, a graphical analysis of the fluctuations of the answers is considered using Mathematica software, which helps to better understand the solutions.

The Analytical Fractional Solutions for Coupled Fokas System in Fiber Optics Using Different Methods

The Fokas system with M-truncated derivative (FS-MTD) was considered in this study. To get analytical solutions of FS-MTD in the forms of elliptic, rational, hyperbolic, and trigonometric functions, we employed the extend F-expansion approach and the Jacobi elliptic function method. Since nonlinear pulse transmission in monomode optical fibers is explained by the Fokas system, the derived solutions may be utilized to analyze a broad range of important physical processes. In order to comprehend the impacts of MTD on the solutions, the dynamic behavior of the various generated solutions are shown using 2D and 3D figures.

Physical wave propagation and dynamics of the Ivancevic option pricing model

In this study, the Ivancevic option pricing model (IOPM), a nonlinear wave alternative to the traditional Black–Scholes option pricing model, is investigated. The Ivancevic option pricing model is formalized by adaptive nonlinear Schrodinger equations that characterize the option-pricing wave function in terms of stock price and time. This model exhibits the characteristic regulated Brownian behavior observed in financial markets. We use the unified auxiliary equation method, which is one of the most powerful methods to explore analytical solutions of the Ivancevic option pricing model. Via this method, we obtain several new exact solutions such as singular, periodic, bright, dark, hyperbolic, trigonometric, exponential and Jacobi elliptic functions solutions. In addition, we simulate 3D surface 2D graphs and counter plots graphics for some obtained solutions by giving specific values for the involved parameters. To further analyze the obtained solutions, we conduct simulations and generate visual representations in the form of 3D surface plots, 2D graphs, and counter plots. By assigning specific values to the parameters involved in the model, we are able to visualize and examine the dynamics of the solutions in a more tangible and intuitive manner. These graphical representations provide valuable insights into the behavior and patterns exhibited by the solutions, enhancing our understanding of the model’s implications and potential applications Furthermore, we present the option price wave functions of the dependent variable in 2D graphs under the effect of the time variable t.

Solitary and Periodic Wave Solutions of the Space-Time Fractional Extended Kawahara Equation

The extended Kawahara (Gardner Kawahara) equation is the improved form of the Korteweg–de Vries (KdV) equation, which is one of the most significant nonlinear evolution equations in mathematical physics. In that research, the analytical solutions of the conformable fractional extended Kawahara equation were acquired by utilizing the Jacobi elliptic function expansion method. The given expansion method was applied to different fractional forms of the extended Kawahara equation, such as the fraction that occurs in time, space, or both time and space by suitably changing the variables. In addition, various types of fractional problems are exhibited to expose the realistic application of the given method, and some of the obtained solutions were illustrated in two- or three-dimensional graphics as proof of the visualization.

Asymptotic formulas of the eigenvalues for the linearization of a one-dimensional sinh-Poisson equation

We are concerned with a Neumann problem of a one-dimensional sinh-Poisson equation u′′+λsinhu=0for0<x<1,u′(0)=u′(1)=0,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} u''+\\lambda \\sinh u=0 &{} \ ext {for}\\ 0<x<1,\\\\ u'(0)=u'(1)=0, \\end{array}\\right. } \\end{aligned}$$\\end{document}where lambda >0 is a parameter. A complete bifurcation diagram of this problem is obtained. We also consider the linearized eigenvalue problem at every nontrivial solution u. We derive exact expressions of all the eigenvalues and eigenfunctions, using Jacobi elliptic functions and complete elliptic integrals. Then, we also derive asymptotic formulas of eigenvalues as lambda rightarrow 0. Exact eigenvalues and eigenfunctions for a Dirichlet problem are presented without proof. The main technical tool is an ODE technique.

A continuum description of the buckling of a line of spheres in a transverse harmonic confining potential

A line of contacting hard spheres, placed in a transverse confining potential, buckles under compression or when tilted away from the horizontal, once a critical tilt angle is exceeded. This interesting nonlinear problem is enriched by the combined application of both compression and tilt. In a continuous formulation, the profile of transverse sphere displacement is well described by numerical solutions of a second-order differential equation (provided that buckling is not of large amplitude). Here we provide a detailed discussion of these solutions, which are approximated by analytic expressions in terms of Jacobi, Whittaker and Airy functions. The analysis in terms of Whittaker functions yields an exact result for the critical tilt for buckling without compression.

Retrieval of optical solitons in fiber Bragg gratings for high-order coupled system with arbitrary refractive index

With the rapid development of communication technology, fiber Bragg gratings are used in all optical communication systems. Bragg gratings are a technological spectacle that sustained balance between dispersion and nonlinear effects that leads to a stable transmission of solitons over huge distances. In this paper, the extended F-expansion technique is used for determining the solitons and other invariant solutions to the higher-order coupled system of nonlinear partial differential equations (NLPDEs) with arbitrary refractive index in fiber Bragg gratings. Various solutions, such as bright solitons, dark solitons, hyperbolic solutions, periodic solutions, Jacobi elliptic functions solutions, and rational solutions are extracted. The extracted solutions confirmed the efficacy and strength of the current technique. Moreover, the graphs for some solutions are presented to demonstrate their physical nature.

Optical spatiotemporal solitary wave solutions of the generalized (3+1)-dimensional Gross–Pitaevskii equation with distributed coefficients

In this work, the extended Jacobi elliptic function expansion approach is used to analyze a generalized [Formula: see text]-dimensional Gross–Pitaevskii equation with distributed time-dependent coefficients because of its use in the Bose–Einstein condensation. The Gross–Pitaevskii equation plays a significant role in Bose–Einstein condensation, where it characterizes the dynamics of the condensate wave function. By using this approach with a homogeneous balance principle, the spatiotemporal soliton solutions and exact extended traveling-wave solutions of governing equation have been successfully obtained. A few double periodic, trigonometric and hyperbolic function solutions from the Jacobi elliptic function solutions have been found under specific constraints on a parameter. It is obvious that the proposed approach is the most straightforward, efficient and useful way to handle numerous nonlinear models that arise in applied physics and mathematics in order to generate various exact solutions. A case with variable gain, constant diffraction and parabolic potential strength has been considered in this study to derive exact solutions. Numerous novel varieties of traveling-wave solutions have been revealed in this work, including the double periodic singular, the periodic singular, the dark singular, the dark kink singular, the periodic solitary singular and the singular soliton solutions and these newly obtained results differ from those previously investigated for this governing equation. In addition to addressing a scientific explanation of the analytical work, the results have been graphically presented by 3D plots and contour plots for some suitable parameter values to understand the physical meaning of the derived solutions. Due to their applicability to a variety of quantum systems, the acquired solutions are of considerable importance.

Analytical Calculation of Instantaneous Liquefaction of a Seabed around Buried Pipelines Induced by Cnoidal Waves

Cnoidal wave theory perfectly describes nearshore wave characteristics. However, cnoidal wave theory is not widely applied in practical engineering because the formula for the wave profile involves a complex Jacobian elliptic function. In this paper, the approximate cnoidal wave theory is presented. Based on the Biot consolidation theory and the approximate cnoidal wave theory, an analytical solution for the pore water pressure around buried pipelines caused by waves is derived. In addition, based on the principle of effective stress, a theory of soil liquefaction around pipelines is proposed. The theoretical results were virtually identical to the results obtained in a practical flume test. Thus, the analytical method proposed in this paper is feasible. Further, the theory is applied to analyze the instantaneous liquefaction of the seabed around buried pipelines and the stability of the pipeline in the Chengdao oilfield.

Autoencoders for a manifold learning problem with a jacobian rank constraint

We formulate the manifold learning problem as the problem of finding an operator that maps any point to a close neighbor that lies on a “hidden” k-dimensional manifold. We call this operator the correcting function. Under this formulation, autoencoders can be viewed as a tool to approximate the correcting function. Given an autoencoder whose Jacobian has rank k, we deduce from the classical Constant Rank Theorem that its range has a structure of a k-dimensional manifold. A k-dimensionality of the range can be forced by the architecture of an autoencoder (by fixing the dimension of the code space), or alternatively, by an additional constraint that the rank of the autoencoder mapping is not greater than k. This constraint is included in the objective function as a new term, namely a squared Ky-Fan k-antinorm of the Jacobian function. We claim that this constraint is a factor that effectively reduces the dimension of the range of an autoencoder, additionally to the reduction defined by the architecture. We also add a new curvature term into the objective. To conclude, we experimentally compare our approach with the CAE+H method on synthetic and real-world datasets.

Dynamical Study of Coupled Riemann Wave Equation Involving Conformable, Beta, and M-Truncated Derivatives via Two Efficient Analytical Methods

In this study, the Jacobi elliptic function method (JEFM) and modified auxiliary equation method (MAEM) are used to investigate the solitary wave solutions of the nonlinear coupled Riemann wave (RW) equation. Nonlinear coupled partial differential equations (NLPDEs) can be transformed into a collection of algebraic equations by utilising a travelling wave transformation. This study’s objective is to learn more about the non-linear coupled RW equation, which accounts for tidal waves, tsunamis, and static uniform media. The variance in the governing model’s travelling wave behavior is investigated using the conformable, beta, and M-truncated derivatives (M-TD). The aforementioned methods can be used to derive solitary wave solutions for trigonometric, hyperbolic, and jacobi functions. We may produce periodic solutions, bell-form soliton, anti-bell-shape soliton, M-shaped, and W-shaped solitons by altering specific parameter values. The mathematical form of each pair of travelling wave solutions is symmetric. Lastly, in order to emphasise the impact of conformable, beta, and M-TD on the behaviour and symmetric solutions for the presented problem, the 2D and 3D representations of the analytical soliton solutions can be produced using Mathematica 10.

Complete discrimination System method for finding exact solutions, dynamical properties of combined Zakharsov-Kuznetsov-modified Zakarsov-Kuznetsov equation

The (2+1)-dimensional combined Zakharsov-Kuznetsov-modified Zakarsov-Kuznetsov (ZK-mZK) equation with constant co-efficients are solved by employing the complete discrimination system for polynomial Method (CDSPM). Use of it reduces the key Nonlinear Differential Equation (NLPDE) to an Ordinary Differential Equation (ODE) in the integral form under the traveling wave transformation, and the solutions are classified. Exact traveling wave solutions are derived using CDSPM of order four. Based on CDSPM, Liu’s approach provides a set of new solutions including Jacobi elliptic functions for the Combined-(ZK-mZK) equation. These analytic solutions exhibit different nonlinear wave structures viz., solitary wave, shock wave, and periodic wave. Among them, the finite amplitude periodic solutions are very important wave features for Combined-(ZK-mZK) equation and their existence is confirmed from the phase portrait of the said equation. Especially the finite amplitude periodic solutions are acquired from the system with the inclination of Jacobean elliptic functions. Numerical results confirm the reliability and effectiveness of the method. Finally, the phase portrait of the dynamical system ensures the entity and exactness of different topological structures of the solutions. Based on this analysis, we conclude that this method is appropriate one to classify the solutions, and can be expanded throughout the vast areas of application.

New optical soliton solutions to the coupled fractional Lakshmanan–Porsezian–Daniel equations with Kerr law of nonlinearity

This article studies the optical soliton solutions of the coupled fractional Lakshmanan–Porsezian–Daniel equations with kerr’s law nonlinearity based on the complete discriminant system of cubic polynomials. By means of traveling wave transformation, the fractional Lakshmanan–Porsezian–Daniel equations are simplified into two ordinary differential equations. Then, according to the classification of the roots of cubic polynomials, the optical soliton solutions of the fractional model with Kerr’s law nonlinearity are traversed. Then new traveling wave solutions are obtained, including the rational solutions, triangle function solutions, implicit function solutions and Jacobian elliptic function solutions, and the corresponding solutions are shown in two-dimensional and three-dimensional figures. Compared with previous related research, the exact optical soliton solutions of the LPD equation system for nonlinear Kerr’s law presented in this study are more completely and clearly classified, and researchers are easier to obtain these solutions.

On the Applications of the Generalized Littlewood Theorem Concerning Integrals of the Logarithm of Analytical Functions to Elliptic Functions

Recently, we have 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. Later, the same theorem was applied to calculate certain infinite sums and study the properties of zeroes of a few analytical functions. In this study, we apply this approach to elliptic functions of Jacobi and Weierstrass. Numerous sums over inverse powers of zeroes and poles are calculated, including some results for the Jacobi elliptic functions sn(z, k) and others understood as functions of the index k. The consideration of the case of the derivative of the Weierstrass rho-function, ℘z(z,τ), leads to quite easy and transparent proof of numerous equalities between the sums over inverse powers of the lattice points m+nτ and “demi-lattice” points m+1/2+nτ, m+(n+1/2)τ, m+1/2+(n+1/2)τ. We also prove theorems showing that, in most cases, fundamental parallelograms contain exactly one simple zero for the first derivative θ1′(z|τ) of the elliptic theta-function and the Weierstrass ζ-function, and that far from the origin of coordinates such zeroes of the ζ-function tend to the positions of the simple poles of this function.

Symmetry analysis and exact Jacobi elliptic solutions for the nonlinear couple Drinfeld Sokolov Wilson dynamical system arising in shallow water waves

We adopted the Lie symmetry technique to obtain some new exact waveform solutions for the coupled Drinfeld Sokolov Wilson (DSW) system. This system emerges from phenomena such as water waves, theoretical physics, fluid dynamics, biology, and chemical sciences. We use the Jacobi elliptic function (JEF) method to deal with some of the reduced systems during the symmetry reductions, which reveals rational, exponential, and hyperbolic functions based waveform solutions. The properties of these solutions indicate that these are periodic waves, solitary waves, double periodic waves, and shock wave solutions. Several of the answers that have been found are entirely new and might be important for researchers to recognize. By assigning specific values to parameters, some chosen solutions are graphically validated. This method can be applied more broadly to handle several different types of nonlinear evolution systems.

Dynamics and optical solitons in polarization-preserving fibers for the cubic–quartic complex Ginzburg–Landau equation with quadratic–cubic law nonlinearity

In this paper, the cubic–quartic complex Ginzburg–Landau (CGL) equation is investigated by using the trial function method. The traveling wave hypothesis is applied to convert the CGL equation to an ordinary differential equation (ODE), which is equivalent to a dynamic system. The qualitative behavior, bifurcation of the phase portraits for this system is studied. Furthermore, the chaotic motions in system involving external periodic perturbation are considered. Some new optical solitons, such as Jacobian elliptic function solutions, for CGL equation with quadratic–cubic law nonlinearity are constructed using the complete discrimination system for polynomial method. Moreover, two-dimensional graphs, three-dimensional, corresponding contour and density plots for some acquired solutions are depicted to carry out the physical properties of optical pulse propagation.

A Tether System at the L1, L2 Collinear Libration Points of the Mars–Phobos System: Analytical Solutions

This paper is dedicated to identifying stable equilibrium positions of the tether systems attached to the L1 or L2 libration points of the Mars–Phobos system. The orbiting spacecraft deploying the tether is at the L1 or L2 libration point and is held at one of these unstable points by the low thrust of its engines. In this paper, the analysis is performed assuming that the tether length is constant. The equation of motion for the system in the polar reference frame is obtained. The stable equilibrium positions are found and the dependence of the tether angular oscillation period on the tether length is determined. An analytical solution in the vicinity of the stable equilibrium positions for small angles of deflection of the tether from the local vertical is obtained in Jacobi elliptic functions. The comparison of the numerical and analytical solutions for small angles of deflection is performed. The results show that the dependencies of the oscillation period on the length of the tether are fundamentally different for L1 and L2 points. Analytical expressions for the tether tension are derived, and the influence of system parameters on this force is investigated for static and dynamic cases.

Optical solitons with DNA dynamics arising in oscillator-chain of Peyrard–Bishop model

In this paper, we introduce the Noether approach for studying the Peyrard–Bishop (PB) DNA dynamic model equation. The Jacobi elliptic function expansion (JEFE) method is used to analyze this model. The obtained soliton solutions are hyperbolic, trigonometric, and rational functions. The dynamics of the obtained solutions conform to the properties of shock waves, periodic waves, solitary waves, double periodic waves, and rational solutions. A comparison between the results obtained by the JEFE method and the results existing in the literature shows that our results are more general, and novel, and have never been reported earlier. The wave profiles are investigated by Wolfram Mathematica simulations for the physical wave behavior.