Jacobi Elliptic Functions Research Articles

Chaotic pattern and traveling wave solution of the perturbed stochastic nonlinear Schrödinger equation with generalized anti-cubic law nonlinearity and spatio-temporal dispersion

The main object of this paper is to study the bifurcation, chaotic pattern and traveling wave solution of the perturbed stochastic nonlinear Schrödinger equation with generalized anti-cubic law nonlinearity and spatio-temporal dispersion. A traveling wave transformation is used to simplified the perturbed stochastic nonlinear Schrödinger equation into ordinary differential equation. The dynamic behavior of two-dimensional planar dynamical systems and their perturbed systems are studied, and bifurcation, phase portrait, and Poincaré section are presented. Furthermore, traveling wave solutions included Jacobian function solutions, trigonometric function solutions and hyperbolic function solutions are constructed.

Effects of Wiener process on analytical wave solutions for (3+1) dimensional nonlinear Schrödinger equation using modified extended mapping method

In this study, we investigate the soliton solutions for stochastic (3+1) dimensional nonlinear Schrödinger equation (NLSE). Our Study mainly depends on applying the modified extended mapping method to construct various and novel stochastic solutions for the proposed model. These solutions including (dark, bright, and singular) solitons. Moreover, singular periodic solutions, Weierstrass elliptic function solutions, and Jacobi elliptic function (JEF) solutions are raised. For a variety of nonlinear stochastic partial differential equations, this method offers a practical and effective method for determining exact stochastic solutions. The stochastic noise effects forcing the wave to behave differently from deterministic cases prognosticated. To illustrate the physical characteristics of the established stochastic solutions, some selected solutions are presented graphically. We used Mathematica (11.3) packages to find the coefficients and Matlab (R2015a) packages to plot the graphs.

The traveling wave solutions to a variant of the Boussinesq equation

In this study, we study a variant of the Boussinesq equation called as \(B( n+1,\,1,\,n )\) equation, and construct some traveling wave solutions by using an effective approach called the extended trial equation method. Thus, the soliton solutions, rational function solutions, elliptic function solutions and Jacobi elliptic function solutions, which show the existence of various mathematical and physical structures and events in the fundamental equation considered, have been constructured. In order to make a more detailed examination of the physical behavior of these solutions, two- and three-dimensional graphs of some solution functions were drawn with the help of the Mathematica package program. In the section of Discussion, we suggest a more general version of the trial equation method for nonlinear differential equations.

Free boundary minimal hypersurfaces outside of the ball

In this paper, we obtain two classification theorems for free boundary minimal hypersurfaces outside of the unit ball (exterior FBMH, for short) in Euclidean space. The first result states that the only exterior stable FBMH with parallel embedded regular ends are the catenoidal hypersurfaces. To achieve this, we prove a Bôcher-type result for positive Jacobi functions on regular minimal ends in \mathbb{R}^{n+1} that, after some calculations, implies the first theorem. The second theorem states that any exterior FBMH \Sigma with one regular end is a catenoidal hypersurface. Its proof is based on a symmetrization procedure similar to that of R. Schoen. We also give a complete description of the catenoidal hypersurfaces, including the calculation of their indices.

On some soliton structures for the perturbed nonlinear Schrödinger equation with Kerr law nonlinearity in mathematical physics

The most important physical structure that is assumed to illustrate the geometry of optical soliton replication in optical fiber theory is the nonlinear Schrödinger equation (NLSE). Optical soliton generation in nonlinear optical fibers is a topic of great contemporary interest because of the numerous applications of ultrafast signal routing systems and short light pulses in communications. This analysis's main goal is to create a large number of soliton solutions for the dynamical model using a variety of contemporary analytical methods. This paper studies different soliton solutions to the perturbed NLSE with Kerr law nonlinearity using two sets of two distinct integration strategies: the mapping approach and the unified auxiliary equation method. The majority of solutions have been found as Jacobi elliptic functions with limiting ellipticity modulus values. Solitons like dark, bright, optical, lonely, and others are also retrieved. We were able to create various single‐type solutions with the help of these strategies. As a result, there is a variety of optical, bell‐shaped, single periodic, and multi‐periodic solutions. In order to validate the computations, the stability of the acquired findings must also be proven. The study provides a highly stunning and suitable strategy for combining numerous exciting wave demonstrations for more advanced models of the present era. Furthermore, we can assert that the outcomes reported here are unique and novel.

Extracting solitary solutions of the nonlinear Kaup–Kupershmidt (KK) equation by analytical method

Abstract Finding analytical solutions for nonlinear partial differential equations is physically meaningful. The Kaup-Kupershmidt (KK) equation is studied in this article. The KK equation is of fifth order, such that several solitary solutions are obtained. In this article, however, the modified auxiliary function approach is applied to this model to find solitary solutions. These solutions are written in terms of Jacobi functions. Therefore, the obtained solutions can be implemented graphically to show different patterns for appropriate parameters.

Multi-integral representations for Jacobi functions of the first and second kind

One may consider the generalization of Jacobi polynomials and the Jacobi function of the second kind to a general function where the degree is allowed to be a complex number instead of a non-negative integer. These functions are referred to as Jacobi functions. In a similar fashion as associated Legendre functions, these break into two categories, functions which are analytically continued from the real line segment ( − 1 , 1 ) and those analytically continued from the real ray ( 1 , ∞ ) . Using properties of Gauss hypergeometric functions, we derive multi-derivative and multi-integral representations for the Jacobi functions of the first and second kind.

Investigating the albedo effects on the dynamics of infinitesimal mass in the elliptic Sitnikov five-body problem

In this paper, we aim to showcase the significant impacts of albedo and eccentricity of the primaries on the infinitesimal mass within the elliptic case of the Sitnikov five-body problem. The Van der Pol transformation and averaging technique were employed to formulate the averaged equations of motion for the infinitesimal mass. Subsequently, the Hamiltonian equations of motion were derived using the action angle variables I and θ. Additionally, we obtained θ in terms of the Jacobi elliptic function sn through a canonical transformation. The investigation utilized the first return map to explore various orbits such as regular, periodic, quasi-periodic, chaotic, or stochastic in the presence of albedo and eccentricity. It was observed that chaotic and stochastic regions emerged as the albedo effect increased, disrupting the periodic tubes and islands. Moreover, by escalating the albedo effect and eccentricity of the primaries, several families of periodic orbits were revealed.

Dynamical characteristics and physical structure of cusp-like singular solitons in birefringent fibers

In this paper, we explore optical nonlinear waves in a birefringent fiber by investigating the dynamics of a coupled nonlinear Schrödinger system with four-wave mixing factors. Utilizing the Jacobi elliptic function approach, we achieve this goal by building cusp-like singular soliton solutions in this optical system. Depending on the characteristics of the focusing and defocusing optical medium, we investigate four different physical possibilities. We identify numerous cusp-type singular soliton profiles in these settings, namely cusp-like bright, dark, kink, and antikink solitons. Group velocity dispersion, spatio-temporal dispersion, and four-wave mixing effects are further evaluated in terms of their influence on the constructed singular soliton solutions. We show that the intensity of the singular soliton profiles can be suppressed or enhanced by altering the dispersion values. We also witness interesting transitions: when four-wave mixing parameters are tuned, cusp-like kink-type solitons can become cusp-like antikink solitons and vice versa. These novel findings about singular solitons in birefringent fibers have great promise for experimental uses in the realm of optics, as well as for advancing our knowledge of four wave mixing and nonlinear propagation.

Analytical solution of an Ill-posed system of nonlinear ODE’s

In this paper, we discuss a solution strategy for an overdetermined ODE system that uses elliptic functions. In particular, we show how an apparently ill-posed ODE system can be made amenable to treatment by Jacobian elliptic functions by introducing additional free parameters and still obtain a solution that corresponds well to the physically expected behaviour.

Non-perturbative Lee-Wick gauge theory: Towards Confinement & RGE with strong couplings

We consider a non-Abelian Lee–Wick gauge theory and discuss Becchi-Rouet-Stora-Tyutin invariance. It contains fourth-order derivative as extensions of the kinetic term, leading to massive ghosts in the theory upon quantization. We particularly provide essential clues towards confinement conditions in strongly-coupled regimes, using the Kugo–Ojima approach, and obtain the functions in the non-perturbative regimes. This is achieved using a set of exact solutions of the corresponding local theory in terms of Jacobi elliptical functions. We obtain a similar function just as for the ordinary Yang-Mills theory but the main differences are that now, the cut-off arises naturally from the Lee–Wick heavy mass scales (M). We show that the fate of the ghosts are fixed in these regimes: they are no more the propagating degrees of freedom in the infrared-limit. As it also happens for the ordinary case, confinement is due to the non-Abelian nature of the theory. In the limit , one recovers the standard results for the local non-Abelian Yang-Mills theory.

Applications of complete discrimination system approach to analyze the dynamic characteristics of the cubic–quintic nonlinear Schrodinger equation with optical soliton and bifurcation analysis

In this research, the complete discriminant system (CDS) of polynomial method (CDSPM) will be applied to analyze the dynamic characteristics of the cubic–quintic nonlinear Schrodinger equation (CQNLSE) with an additional anti-cubic nonlinear term (CQNLSE-AC) with a particular emphasis on the introduction of various optical solitons and wave structures. Our analysis of the CQNLSE-AC illustrates the importance of CDSPM, and we give dynamic results, such as critical conditions and bifurcation points for solutions. Additionally, we determine several types of optical soliton solutions, including Jacobian elliptic function (JEF), hyperbolic function, and trigonometric function solutions, and also convert JEF into solitary wave (SW) solutions.

Phase portraits and new exact traveling wave solutions of the (2+1)-dimensional Hirota system

In this paper, we investigate the (2+1)-dimensional Hirota system of equations, which is essential for modeling physical events since it contains the NLS equation and the cmKdV equation. We apply the Jacobi elliptic function method to obtain new results. By using some ansatz in terms of finite Jacobi elliptic functions, we have obtained various exact solutions, such as dark solitons, bright solitons, and periodic solutions. Moreover, for some reductions as α=0,β=1 and α=1,β=0, we obtain the exact solutions for the (2+1)-dimensional NLS equation and the (2+1)-dimensional cmKdV equation. The dynamics of the obtained solutions are presented in the figures. We also analyze the phase portraits of the (2+1)-dimensional Hirota system.

Construction of chirped propagation with Jacobi elliptic functions for the nonlinear Schrödinger equations with quadratic nonlinearity with inter-modal and spatio-temporal dispersions

Construction of chirped propagation with Jacobi elliptic functions for the nonlinear Schrödinger equations with quadratic nonlinearity with inter-modal and spatio-temporal dispersions

Propagation of solitons in inhomogeneous birefringent nonlinear dispersive media

We study the existence and wave-speed management of nonlinear localized waves in an inhomogeneous birefringent nonlinear dispersive medium in the presence of external potential. The coupled nonlinear Schrödinger equation with varying coefficients which can be applied to model the interaction among the modes in nonlinear optics and in Bose–Einstein condensation is considered. By means of a complex transformation, we derive new types of periodic nonlinear waves for the present coupled system which are expressed in terms of Jacobi elliptic functions. Novel soliton pulse solutions including bright-dipole and bright–dark solutions are identified in the long-wave limit of these periodic solutions. As an application of these soliton pulses, we discuss their wave-speed management for the different choices of dispersion parameter. The results show that the varied dispersion parameter can be used to control effectively the wave speed of propagating localized waves.

Lie symmetries, bifurcation analysis, and Jacobi elliptic function solutions to the nonlinear Kodama equation

The present article formally studies the propagation of optical pulses in a nonlinear medium governed by the nonlinear Kodama equation. For this purpose, the Lie symmetries group is first adopted for similarity reduction and constructing some exact solutions of the governing equation. After deriving the dynamical system of the nonlinear Kodama equation, its bifurcation analysis is accomplished using the idea of the planar dynamical system. Through perturbing the resultant dynamical system using a trigonometric function, chaotic characteristics of the governing model are analyzed by serving several two- and three-dimensional phase portraits. A sensitivity analysis of the dynamical system is performed using the Runge–Kutta method to ensure that small changes in initial conditions have little impact on solution stability. Finally, using the technique of the planar dynamical system, a number of Jacobi elliptic function solutions (in special cases, bright and dark solitons) are constructed for the nonlinear Kodama equation. It has been shown that bright and dark solitons can be controlled for their width and height effectively by the achievements of the current paper.

New traveling wave solutions and dynamic behavior analysis of the nonlinear Rangwala–Rao model

This work investigates the nonlinear Rangwala–Rao equation, which stems from the mixed derivative nonlinear Schrödinger equation. For retrieving new exact solutions to the equation, the complete discriminant system for polynomial method is employed. In results, some novel traveling wave solutions, including solitary wave solutions, triangular function solutions, periodic solutions and Jacobi elliptic function solutions are obtained and demonstrated through numerical simulations. The bifurcations of phase velocity fields are depicted to reveal the dynamic behavior of the Rangwala-Rao equation using the qualitative theory of dynamical systems. Furthermore, considering external perturbation, the chaotic motions of the perturbed Rangwala-Rao equation are investigated.

Rogue periodic waves of the short pulse equation and the coupled integrable dispersionless equation

In this paper, we explore the topic of rogue waves on periodic backgrounds. Using the travelling wave reduction method, we derive two families of periodic solutions for the coupled integrable dispersionless (CD) equation expressed in Jacobi elliptic dnoidal and conoidal functions, respectively. We use these Jacobi elliptic functions as seed solutions to construct algebraic decaying solitons on the dnoidal background of the CD equation based on the one-fold Darboux transformation (DT). Furthermore, we utilize the two-fold DT to find rogue wave solutions on the conoidal background of the CD equation. To obtain solutions of the short pulse equation, we use the hodograph transformation. We analyse the dynamics of the explicit solutions obtained from this research through plots.

Optical soliton solution via complete discrimination system approach along with bifurcation and sensitivity analyses for the Gerjikov-Ivanov equation

The aim of this research intends to investigate the complex wave patterns and dynamic behavior of the Gerjikov-Ivanov equation (GIE), commonly known as the derivative nonlinear Schrödinger equation (DNLSE). The GIE model is the one that examines dynamics of optical soliton propagation for transmission technologies, including transcontinental and transoceanic lengths of optical fibers, data transfer, and the telecommunications sector. The GIE equation can be regarded as an extension of the NLSE when certain higher-order nonlinear effects are taken into account. We study various optical soliton solutions, including Jacobian elliptic, hyperbolic, and trigonometric functions, are identified using the polynomial method’s complete discrimination system (CDSPM). In addition, Jacobian elliptic functions can be transformed into solitary wave solutions. We also discuss bifurcation points, sensitivity analysis and critical solution conditions, offering a thorough grasp of the system’s dynamic features. The CDSPM plays an important role in studying equilibrium points, phase diagrams, bifurcation behavior, and performing quantitative and qualitative evaluations.

Steady-state currents of planar interdigitated electrode arrays in shallow cells

Analytical equations for the steady-state current at shallow electrochemical cells with embedded interdigitated electrode (IDE) arrays were found by solving the diffusion equation in steady state using conformal transformations. These equations consider the cases where the IDE is used with an external counter electrode (four-electrode configuration) and where one of its arrays is used as internal counter electrode (three- and two-electrode configuration). Since the expressions for the current depend on Jacobian elliptic functions, we found approximations which depend only on trigonometric and hyperbolic functions which can ease the calculations at the expense of an error below 5%. Additionally, we provide expressions for steady-state voltammograms for the case of Nernstian (reversible) electrode reactions. The obtained theoretical results agree with theoretical and simulation results from the literature and with our own experiments. These results can be useful as criteria for designing electrochemical cells based on IDEs, and as benchmarks for assessing their performance, especially for polymer- or paper-based microfluidics.