Jacobian Elliptic Research Articles

Investigating the higher dimensional Kadomtsev–Petviashvili–Sawada–Kotera–Ramani equation: exploring the modulation instability, Jacobi elliptic and soliton solutions

Investigating the higher dimensional Kadomtsev–Petviashvili–Sawada–Kotera–Ramani equation: exploring the modulation instability, Jacobi elliptic and soliton solutions

Bifurcation Analysis, Sensitivity Analysis, and Jacobi Elliptic Function Structures to a Generalized Nonlinear Schrödinger Equation

Bifurcation Analysis, Sensitivity Analysis, and Jacobi Elliptic Function Structures to a Generalized Nonlinear Schrödinger Equation

New analytical wave structures for generalized B-type Kadomtsev–Petviashvili equation by improved modified extended tanh function method

Abstract Recently, solving the complicated nonlinear partial differential equations has become very important demand in order to simulate their physical phenomena. This manuscript focuses on extracting the wave solutions of (3 + 1)-dimensional generalized B-type Kadomtsev-Petviashvili equation (GBKPE), which demonstrates the behavior of nonlinear waves in fluid mechanics. The improved modified extended Tanh function (IMETF) method is the suggested method to do this task as it gives different types of solutions. This method enables us to obtain many solutions, such as Jacobi elliptic, dark soliton, and singular soliton, exponential, and singular periodic wave solutions. Additionally, for more illustrations graphical visual representations of some solutions are provided.

Optical dromions for M-fractional Kuralay equation via complete discrimination system approach along with sensitivity analysis and quasi-periodic behavior

This paper investigates new analytical wave solutions for the fractional Kuralay-IIB equations (FK-IIBE), including a new definition of the fractional derivative. This model works in disciplines such as ferromagnetic materials, optical fibers, wave mixing, and nonlinear optics. The integrable motion of space curves can be determined by our governing model. This research holds great importance in the area of optical fibers, exact and analytical solitons solution. The polynomial method’s complete discrimination system (CDSPM) is used to identify a variety of exact and analytical soliton solutions such as periodic, hyperbolic, rational, Jacobian elliptic (JE), and trigonometric functions. Additionally, we also convert the JE function into a solitary wave (SW) solution. To provide a comprehensive understanding of the dynamic aspects of the system, we also address bifurcation points, quasi-periodic behavior, sensitivity analysis, Poincarè maps, time series profile, and critical solution conditions. The research offers straightforward, efficient algorithms that can consistently and effectively address FK-IIBE. Machine learning technologies are also utilized to satisfy the criteria defined by dynamic analysis. Python regression learner tools are used for reverse engineering to analyze the mathematical model’s equilibrium based on parameter intervals and thresholds determined by dynamical analysis.

Chaos analysis and traveling wave solutions for fractional (3+1)-dimensional Wazwaz Kaur Boussinesq equation with beta derivative

Wazwaz Kaur Boussinesq (WKB) equation can effectively simulate the behavior of water waves in shallow water, including the nonlinear effect and dispersion phenomenon of waves, which is of great significance for understanding the dynamic process of ocean, river and other water bodies. To enrich the wave equation theory, the (3+1)-dimensional integer order derivative of WKB equation is changed to the fractional one with beta derivative. The current work deals with the fractional (3+1)-dimensional WKB equation for discussing its chaotic behavior and establishing some new analytic solutions. The chaotic properties of the equation are verified by the trend of evolution along with time, Lyapunov exponents and initial sensitivity analysis. And then complete discrimination system for polynomial method is applied to derive some trigonometric, hyperbolic, Jacobi elliptic and other solutions. The graphical demonstrations are provided for part of these solutions. From these visualized graphs, the solitary, periodic and quasi-periodic wave are shown and the effect of fractional derivatives on the equation can be seen intuitively.

New Processing Technique of Jacobian Elliptic Equation and Its Application to the (3+1)-Dimensional Modified Korteweg de Vries–Zakharov–Kuznetsov Equation

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.

Comment on “Optical solitons for 2D-NLSE in multimode fiber with Kerr nonlinearity and its modulation instability”

Most recently, Baber et al. (Optical solitons for 2D-NLSE in multimode fiber with Kerr nonlinearity and its modulation instability, Mod. Phys. Lett. B (2024) 2450341, doi:10.1142/S021798492450341X2450341-1) claimed to have obtained exact solutions of a two-dimensional nonlinear Schrödinger equation with an instantaneous Kerr nonlinearity that governs beam movement within a multimode optical fiber featuring a parabolic index shape, using the Jacobi Elliptic Function Expansion. Additionally, they have analyzed the modulation instability for the underlying model. It is the aim of this comment to point out that all results found in that work and published in the journal Mod. Phys. Lett. B (doi:10.1142/S021798492450341X2450341-1) are all erroneous, from the exact solutions of the considered equation to the modulation instability of the underlying model.

Superposed and Superposed-type Double Periodic Jacobi Elliptic Function Solutions of Variable Coefficients KdV Equation

Superposed and Superposed-type Double Periodic Jacobi Elliptic Function Solutions of Variable Coefficients KdV Equation

Generation of optical dromions to generalized stochastic nonlinear Schrödinger equation with Kerr effect and higher order nonlinearity

Stochastic nonlinear Schrödinger equation is a mathematical model, used in the study of complex systems especially nonlinear optics and quantum mechanics. It explains the evolution of a quantum field with stochastic and nonlinear terms. A stochastic or random process is a mathematical representation that outlines the progression of a system in a way that incorporates probability. This is in contrast to a deterministic process, where the future states are strictly defined by initial conditions. In mathematical model, stochastic term is usually represent by variable or a function that involves random variable or a noise. In this paper, we study the generalized nonlinear Schrödinger equation by adding stochastic term for the first time to retrieve different types of optical solitons. To the best of our knowledge, no one work on this model with stochastic term. Our aim is to find optical solitons like bright and dark solitons, also solitary wave solutions for this model like bell type, hyperbolic functions, Jacobi Elliptic Function, Weierstrass Elliptic Function, periodic and rational solutions. These solutions will be obtained using the sub-ode method, subject to certain constraint conditions. Finally, we will use Jupyter as a machine learning tool and also mathematica to plot some obtained solutions in various dimensions like 3D, 2D and contour plots by using some libraries of PYTHON like numpy, scipy.integrate and matplotlib.pyplot. For the first time, we use this approach for this model and our results are new and novel [1].

Exploring exact solitary wave solutions of Kuralay-II equation based on the truncated M-fractional derivative using the Jacobi Elliptic function expansion method

Exploring exact solitary wave solutions of Kuralay-II equation based on the truncated M-fractional derivative using the Jacobi Elliptic function expansion method

Optical solitons for 2D-NLSE in multimode fiber with Kerr nonlinearity and its modulation instability

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.

Bounds for Weierstrass Elliptic Function and Jacobi Elliptic Integrals of First and Second Kinds

The Weierstrass elliptic function is presented in connection with the Jacobi elliptic integrals of first and second kinds leading to comparing coefficients appearing in the Laurent series expansion with those of Eisenstein series for the cubic polynomial in the meromorphic Weierstrass function. It is unified in the formulation the Weierstrass elliptic function with Jacobi elliptic integral by considering motion of a unit mass particle in a cubic potential in terms of bounded and unbounded velocities and the time of flight with imaginary part in the complex function playing a major role. Numerical tools box used are the Konrad-Gauss quadrature and Runge-Kutta fourth order method.

A detailed analysis of the improved modified Korteweg-de Vries equation via the Jacobi elliptic function expansion method and the application of truncated M-fractional derivatives

Using the Jacobi elliptic function expansion method, which is improved by the novel use of truncated M-fractional derivatives, we thoroughly analyze the improved modified Korteweg-de Vries problem in this paper. Not only does this method provide new insight into the improved modified Korteweg-de Vries Equation, but it also represents a giant leap forward for nonlinear differential equations. Periodic, solitary, and trigonometric waves are among the answers found by us; all of them have important consequences for mathematical physics and engineering.The MATLAB-generated visual representations of our results are crucial because they provide aconcrete shape to the once intangible mathematical ideas. In addition to showing the solutions, these images highlight the waves' dynamic behaviors and stability under various conditions, providing a better understanding of the physical processes that the improved modified Korteweg-de Vries equation attempts to explain. The graphic representations in this study demonstrate how the equation can be used to simulate complicated systems in many scientific disciplines. The relevance of our findings goes beyond only theoretical studies. Our study lays the groundwork for more accurate predictive models in fields including quantum physics, optical fibers, and fluid dynamics by giving exact and concrete solutions to the improved modified Korteweg-de Vries equation. The combination of the Jacobi Elliptic Function Expansion Method with truncated M-fractional derivatives provides a robust framework for solving comparable complicated differential equations, which in turn opens up new research and development opportunities. Overall, this research both deepens our understanding of the Improved Modified Korteweg-de Vries equation and provides more evidence that theoretical mathematics can be useful in solving practical problems. Mathematical modeling and computational visualization have the potential to make substantial contributions to engineering and the sciences, and our results support a multidisciplinary approach to study.

Bell and Kink type, Weierstrass and Jacobi elliptic, multiwave, kinky breather, M-shaped and periodic-kink-cross rational solutions for Einstein’s vacuum field model

Bell and Kink type, Weierstrass and Jacobi elliptic, multiwave, kinky breather, M-shaped and periodic-kink-cross rational solutions for Einstein’s vacuum field model

Novel closed-form analytical solutions and modulation instability spectrum induced by the Salerno equation describing nonlinear discrete electrical lattice via symbolic computation

A nonlinear electrical lattice is an effective experimental tool for dealing with nonlinear dispersive media and creating possibilities for the realistic modeling of electrical soliton. In this work, we discuss the dynamical behavior of the governing (1+1)-dimensional time–space Salerno equation, which describes the discrete electrical lattice with nonlinear dispersion. The different kinds of solutions in the form of periodic, Jacobi elliptic, and exponential functions are obtained by the two analytical approaches, namely the modified generalized exponential rational function method (MGERFM) and the extended sinh-Gordon equation expansion method (shGEEM). The MGERF approach is applied to construct exact solitary wave solutions containing trigonometry, hyperbolic and exponential functions, while the extended shGEE technique is employed to obtain periodic wave solutions containing hyperbolic and Jacobi elliptic functions via performing the symbolic computation in the software package MATHEMATICA. Moreover, by means of standard linear-stability analysis, the modulation instability (MI) and the MI gain spectrum is analytically computed for the considered equation. By selecting suitable parametric values, solution profiles are portrayed in order to make the solutions physically relevant. These results yield periodic waves, kink waves, multi-soliton, and their interactions. By comparing the obtained solutions to the previous findings, we can reveal the novelty of the solutions. In addition, all the obtained solutions were verified by substituting them back into the governing equation using the Mathematica software. The acquired results are significant for the study of wave propagation, signal transmission, and applications of super-transmission phenomena.

New Jacobi Elliptic Solutions and Other Solutions in Optical Metamaterials Having Higher-Order Dispersion and Its Stability Analysis

New Jacobi Elliptic Solutions and Other Solutions in Optical Metamaterials Having Higher-Order Dispersion and Its Stability Analysis

Chirped pulses for Nematicons in liquid crystals with cubic-septic law nonlinearity

This manuscript concentrates the behaviour of nematic liquid crystals (NLC) incorporating cubic-septic law property. The resulting solutions of the model play a significant role in the energy transport of soliton molecules in liquid crystals. In this study, the Jacobi elliptic (JE) function technique, which is one of the efficient integration procedure, is used to investigate chirped elliptic and solitary wave solitons like bright, dark, kink, singular, hyperbolic with some constraint conditions while the hyperbolic type traveling wave solutions of the equation defining the NLC incorporating cubic-septic law property is also represented as dark and singular solitons. The linear section of the obtained pulse chirp has two intensity-dependent chirping components that change the chirp. The results of this work could help us understand how chirped solitary waves (CSW) propagate in a weakly nonlocal cubic-septic law medium. This study improves the understanding of nonlinear light-matter interactions in liquid crystals by providing new insights into the role of chirped pulses in forming and modulating spatial optical solitons. Furthermore, we provide successful results in 3D, 2D, and contour forms. It is emphasized that the constrained technique is more effective and powerful than other ways, and the conclusions drawn in this work can contribute in understanding of the soliton molecules found in liquid crystals.

New optical solitons for perturbed stochastic nonlinear Schrödinger equation by functional variable method

In this paper, the functional variable method is used to obtain new optical soliton solutions for the perturbed stochastic nonlinear Schrödinger equation with generalized anti-cubic nonlinearity and multiplicative white noise. Using some transformations, new rational, Jacobi elliptic, Weierstrass, and hyperbolic stochastic solutions are obtained. Several optical soliton solutions were proposed, including dark, bright, and compacton soliton solutions. Graphical presentations of the obtained optical soliton solutions are shown to illustrate some of its physical parameters.

Bright, dark and kink exact soliton solutions for perturbed Gerdjikov–Ivanov equation with full nonlinearity

In this paper, the perturbed Gerdjikov-Ivanov equation (PGI) which describes the dynamics of the soliton in optical fiber is investigated. Using traveling wave transformation, the nonlinear PGI equation is transformed into two nonlinear ordinary differential equations. Consequently, these equations are reduced to a first -order ordinary differential equation that has a well-known exact solution. The exact solutions for the PGI equation will be introduced including Weierstrass, Jacobi elliptic, and hyperbolic functions. Various types of optical soliton solutions were raised such as bright, dark, and kink soliton solution. The obtained optical soliton solutions are presented graphically to clarify some of its physical parameters.

Orbital Dynamics, Chaotic Orbits and Jacobi Elliptic Functions

Orbital Dynamics, Chaotic Orbits and Jacobi Elliptic Functions