Jacobi Elliptic Functions Research Articles

Diverse general solitary wave solutions and conserved currents of a generalized geophysical Korteweg–de Vries model with nonlinear power law in ocean science

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 ). 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.

Time-fractional of cubic-quartic Schrödinger and cubic-quartic resonant Schrödinger equations with parabolic law: various optical solutions

Abstract Schrödinger's nonlinear equation is a fundamental model in fiber optics and many other areas of science. Using the Jacobi elliptic expansion function method, the time-fractional cubic-quartic nonlinear Schrödinger equation and cubic-quartic resonant nonlinear Schrödinger equation are investigated. By applying the effective Jacobi elliptic expansion function method, optical soliton solutions such as bright, dark, singular, periodic singular, exponential, and Jacobi elliptic function solutions have been obtained. The effect of the time-fractional derivative on the solutions is also revealed. Graphical representations are illustrated to showcase the physical properties of raised solutions, providing a comprehensive understanding of the solutions’ functionality.

Orbits of particles with magnetic dipole moment around magnetized Schwarzschild black holes: Applications to the S2 star orbit

This study provides a comprehensive analytical investigation of the bound and unbound motion of magnetized particles orbiting a Schwarzschild black hole immersed in an external asymptotically uniform magnetic field, which includes all conceivable types of bounded and unbounded orbits. In particular, for planetary orbits, we perform a comparative analysis of our findings with the observed position of the S2 star carrying magnetic dipole moment around Sagittarius A*. We found maximum and minimum values for the parameter of magnetic interaction between the magnetic dipole of the star and the external magnetic field, as well as the energy and angular momentum of the S2 star. As a result, we obtain estimations of the magnetic dipole of the star in the order of 106 G·cm3. Additionally, we explore deflecting trajectories akin to gravitational Rutherford scattering. In obtaining the solutions for the orbital equations, we articulate the elliptic integrals and Jacobi elliptic functions, and illustrative figures and simulations augment our study. Published by the American Physical Society 2024

Impact of relativistic positron beam on ion-acoustic solitary, periodic and breather waves in Earths' ionospheric region through the framework of KdV and modified KdV equation

Abstract This study explores the propagation characteristics of ion-acoustic periodic, soliton, and breather waves in electron-positron-ion (EPI) plasma with a relativistic positron beam. The Korteweg-de Vries (KdV) equation is obtained by applying the traditional reductive perturbation method (RPM) to the fundamental set of fluid equations. When the KdV model is unable to accurately represent the nonlinear system's evolution, a modified Korteweg-de Vries (mKdV) equation is constructed. In both models, Jacobi elliptic functions are used to derive periodic solutions, and a connection between periodic waves and soliton solutions is established. Hirota's bilinear method is used to generate breathers directly from the KdV type framework without utilizing the modified Schrodinger framework inferred from the KdV type framework, which is a prevalent method in studies of nonlinear waves. Numerical knowledge of various physical factors in the ionospheric region is incorporated into the model to elucidate wave propagation in the Earth's upper atmosphere.

Analytical Solutions and Instability Analysis of truncated $\mathrm{M}$-Fractional Coupled Dispersionless Equations

Abstract This paper systematically investigates the truncated M-fractional coupled dispersionless equations, a nonlinear system of partial differential equations characterized by its M-fractional derivative. We employ the Jacobi elliptic function expansion method to derive analytical solutions for the coupled system. Additionally, the modulation instability of the solutions is thoroughly explored, providing a detailed exposition of the mathematical framework governing the system. The analytical solutions are graphically illustrated and analyzed to highlight their physical significance. These findings hold significant applications in nonlinear optics, offering new insights into wave propagation and stability within such systems.

Yukawa theory in non-perturbative regimes: towards confinement, exact β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document}-function and conformal phase

We study possible hints towards confinement in a Z2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {Z}_2$$\\end{document}-invariant Yukawa system with massless fermions and a real scalar field in the strongly-coupled regime. Using the tools developed for studying non-perturbative physics via Jacobi elliptical functions, for a given but not unique choice of the vacuum state, we find the exact Green’s function for the scalar sector so that, after integrating out the scalar degrees of freedom, we are able to recover the low-energy limit of the theory that is a fully non-local Nambu–Jona–Lasinio (NJL) model. We provide an analytical result for the renormalization group (RG) running of the self-interaction coupling in the scalar sector in the strongly-coupled regime. In the fermion sector, we provide some clues towards confinement, after deriving the gap equation with the non-local NJL model, a property which is well-known to not emerge in the local limit of this model. We conclude that, for the scalar-Yukawa theory in the non-perturbative domain with our choice of the vacuum state, the fundamental fermions of the theory form bound states and cannot be observed as asymptotic states.

Periodic, quasi-periodic, chaotic waves and solitonic structures of coupled Benjamin-Bona-Mahony-KdV system

Abstract In this paper, we mainly focus on studying the dynamical behaviour and soliton solution of the coupled Benjamin-Bona-Mahony-Korteweg-de Vries (BBM-KdV) system, which characterizes the propagation of long waves in weakly nonlinear dispersive media. The paper utilizes different tools to detect chaos, such as time series analysis, bifurcation diagrams, power spectra, phase portraits, Poincare maps, and Lyapunov exponents. This analysis helps in more accurate predictive modeling of the systems. This understanding can aid in the design of control strategies, resulting in enhancements in prediction, control, optimization, and design. Additionally, we construct the system's solitary wave structures using the Jacobi elliptic function (JEF) method. We identify periodic wave solutions expressed in terms of rational, hyperbolic, and trigonometric functions. Certain parameter values can lead to periodic wave solutions, solitary waves (bell-shaped solitons), shock wave solutions (kink-shaped soliton solutions), and double periodic wave solutions.

Impact of fractional and integer order derivatives on the [formula omitted]-dimensional fractional Davey–Stewartson–Kadomtsev–Petviashvili equation

In this study, the closed-form wave solutions of the (4+1)-dimensional fractional Davey–Stewartson–Kadomtsev–Petviashvili equation are investigated using the modified auxiliary equation method and the Jacobi elliptic function method. In the analysis, two fractional derivatives known as M-truncated, beta and integer order derivative are used. The fractional-order partial differential equation is transformed into an integer-order ordinary differential equation by using the wave transformation, fractional derivatives, and integer-order derivatives. As a result, wave function solutions are found, including bell shape, W-shaped, composite dark-bright and periodic wave. The effects of free parameters on the amplitudes and wave behaviors are illustrated. It is demonstrated extensively that changes in the free parameters lead to changes in the wave amplitude. A comparison of solutions using the two types of fractional derivatives and the integer-order derivatives is included. The effects of the beta derivative, the M-truncated derivative and integer order derivative on the considered model are presented using 2D and 3D figures.

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.

Nonlinear analysis of compound pendulum model

Abstract This paper investigates the relationships among various nonlinear physi cal equations, with a particular focus on the standard form of the ϕ^4 equation. Based on the theoretical framework of the Jacobi elliptic function, an exact solution for the ϕ^4 equation is derived. A key innovation of this work is the discovery of the consistency between the ϕ^4 equation and the motion equation of the compound pendulum. By utilizing this correspondence, an exact solution for the compound pendulum equation is obtained, grounded in the Jacobi elliptic function theory. Compared to numerical methods, this solution provides higher accuracy and has the potential to be applied to
more complex nonlinear physical systems. This model can also be applied in areas such as vibration damping in building materials, mechanical system analysis, and spacecraft control.

Stationary scattering for the nonlinear Schrödinger equation with point-like obstacles: exact solutions

AbstractWe solve the Nonlinear Schrödinger Equation (NLSE) in 1D in presence of one, two and several Dirac delta potentials. With the help of an equivalent central force problem we obtain the analytical solutions in terms of a biparametric family containing the Jacobi functions. Elliptic Jacobi functions are already reported in the literature but they have not been used in the context of a scattering problem under causal boundary conditions. In the simplest examples of one or two Dirac deltas we analyze how the nonlinear term of the equation affects the modulus and phase profiles of the wave function. We also study the transmission curves under the nonlinear modification of the tunneling behavior for the first time. For a Fabri-Perot configuration made of two deltas, we obtain the effect of nonlinear coupling in the positions of the local maxima (resonances). We lay the foundations for nonlinear Anderson localization of 1D BECs in a speckle field. Upon redefinition of parameters these novel results describe the dynamics of a stationary Higgs field in 1D. Finally, we discuss the conditions for soliton formation under the influence of a Dirac comb potential, giving rise to fully correlated locations and intensities of the defects.

Energy optimal path planning for wheeled mobile robots with circular obstacle

AbstractThe energy optimal motion planning algorithm for a differential drive robot with a circular obstacle in the maneuver space is introduced in this article. Rather than the traditional unicycle model, a non‐canonical state space model of the kinematics of a wheeled mobile robot is used to formulate the optimal control problem. The necessary conditions for optimality are formally derived using calculus of variations, resulting in constraints for the costates when the inequality constraint become active, revealing that the controls are required to be continuous. The optimality conditions permit decomposing the motion planning problem in two independent segments: prior to and after activation of the inequality constraints. Analytical expressions for the evolving states and control are found to be a function of Jacobi elliptic functions, permitting reducing the optimal control problem to a root‐finding problem. A comprehensive parametric study is included to demonstrate the impact on the optimal trajectories by varying the radius of the obstacle. The study shows that there are three distinct maneuver strategies with respect to the size of the obstacle. Furthermore, by including entering and exit time as optimization variables, a methodology for generalizing the development to any given motion scenario is developed and numerically illustrated.

Investigation of optical soliton solutions for the cubic-quartic derivative nonlinear Schrödinger equation using advanced integration techniques

This paper aims to investigate optical soliton solutions in the context of the cubic-quartic derivative nonlinear Schrödinger equation with differential group delay, incorporating perturbation terms for the first time. Motivated by the need to better understand soliton dynamics in advanced optical communication systems, we employ three integration techniques: the direct algebraic approach, Kudryashov’s method with an addendum, and the unified Riccati equation expansion method. Our study reveals that, by appropriately selecting parameter values, the resulting solutions include Jacobi elliptic functions that describe straddle solitons, bright, dark, and singular solitons. We also identify the conditions under which these soliton pulses can exist. Furthermore, we provide numerical simulations to illustrate these solutions under specific parameter settings, highlighting their potential applications in optical fiber systems.

Integrated Jacobi elliptic function solutions to the [formula omitted]-dimensional generalized Kadomtsev–Petviashvili equation by utilizing new solutions of the elliptic equation of order six

In this research, the generalized (3+1)-dimensional Kadomtsev–Petviashvili equation ((3+1)-GKPE) that expresses various nonlinear phenomena was studied. An extended Jacobi elliptic function expansion method (JEFEM) was developed by considering new solutions for the Jacobi elliptic equation of order six. Then the extended method was used to (3+1)-GKPE, where new exact Jacobi elliptic function solutions were obtained. This equation is of particular interest as it required a special transformation in order to apply the JEFEM. Moreover, some of the solutions are shown graphically.

The bifurcation, chaotic behavior and exact solutions of the fractional stochastic Jimbo–Miwa equations

The Jimbo–Miwa equation is the second equation in the KP hierarchy of integrable systems. In this paper, this equation is extended and introduced with the stochastic process and fractional derivatives. Firstly, the phase portrait of the Hamiltonian system generated by it is studied to understand its bifurcation behavior. Additionally, non-periodic and periodic perturbation terms are added to this system. Different values are assigned to the parameters in the perturbation terms to analyze its sensitivity and the resulting chaos is obtained. Finally, through integration techniques, the expression of the solution of this equation is obtained. These solutions are related to rational functions, trigonometric functions, exponential functions and Jacobi elliptic functions. To observe the form of the solutions more intuitively, 3D and 2D numerical simulations are conducted on the solutions and the solution images of the stochastic fractional differential equation are given by Matlab software. Compared with the existing literature, the research on the stochastic fractional equation of this equation is relatively rare and the analysis of the phase portrait is even scarcer. Our solution method is quite different from that in the previous literature. Therefore, this paper is novel. The conclusion of this paper will be of great help for the practical application of this equation.

Investigating innovative optical solitons for a (3+1)- dimensional nonlinear Schrödinger’s equation under the influences of 4th-order dispersive and parabolic law of nonlinearities

This study tackles a nonlinear Schrödinger’s equation in three dimensions, using three dispersive components of fourth order, often resulting in pure-quartic solitons. Pure-quartic bullets are known to deviate from conventional solitons due to their balance of fourth-order dispersion and nonlinearity. As bright and dark optically modulated bullets, we derive many solutions. Experiments on bullet transmission using optical nanofibers can benefit from the solutions found. Our creative solutions are produced by implementing the well-known scheme which is the improved modified extended tanh-function scheme. We find novel kinds of solutions (dark, bright, singular solitons, exponential, singular periodic, Weierstrass elliptic doubly periodic solutions, and Jacobi elliptic functions) that make their originality for the problem at hand evident by using the previously described approach. Contour plots and 2D and 3D visualizations in Wolfram Mathematica software are used to show how the well-furnished results propagate for various values of the necessary free parameters. The results demonstrate the computational processes’ precise, well-informed, and efficient nature. Through their integration with representational computations, they may be applied to increasingly complex phenomena. This work constitutes a major advancement in our comprehension of the intricate and erratic behavior of this mathematical model.

Investigating the potential of optical metamaterials with highly dispersive solitons in twin couplers with stochastic perturbations and white noise effects

This study emphasizes a unique contribution as we investigate the dimensionless version of the perturbed stochastic NLSE within an optical metamaterial-designed coupler. We employ Kudryashov's sextic-power law nonlinearity to present this formulation, and, for the first time, we introduce multiplicative white noise using the Ito interpretation. This study aims to bridge the gap in understanding the influence of stochastic effects on optical metamaterials and offers valuable insights for future research in this field. The methods introduced include enhancements to Kudryashov's approach and extensions to the modified sub-ODE technique. These methodologies are leveraged to derive a wide array of solutions, such as bright, dark, singular, and combined solitons, with some solutions presented using Jacobi and Weierstrass elliptic functions. The effectiveness of these methods in addressing various nonlinear partial differential equations is demonstrated through the results. The study further delves into how random noise impacts optical solitons and analyzes soliton behavior when subjected to different conditions. It is observed that white noise plays a crucial role in the formation of optical solitons within the coupler and that these solitons demonstrate highly dispersive properties under specific scenarios. These insights are pivotal for designing and innovating optical metamaterials for diverse technological uses.

Optical soliton solutions of the stochastic resonant nonlinear Schrödinger equation with spatio temporal and inter-modal dispersion under generalized Kudryashov's law non-linearity

This article investigates optical soliton solutions within the stochastic resonant nonlinear Schrödinger equation (SRNLSE). This equation incorporates both spatio-temporal dispersion (STD) and inter-modal dispersion (IMD), along with multiplicative white noise and generalized Kudryashov's law non-linearity. By applying the extended auxiliary equation method, we identify a range of soliton solutions, including bright, dark, and singular solitons. Additionally, we derive solutions that take the form of Jacobi elliptic functions, Weierstrass elliptic functions, and periodic wave functions. The study provides significant insights into soliton dynamics within nonlinear optical systems affected by stochastic influences and complex dispersion interactions. Specifically, it highlights how the interplay of STD and IMD, coupled with the presence of multiplicative noise, shapes the behavior of solitons. Moreover, we delve into the effects of multiplicative noise on the exact solutions of the NLSE using the Maple software. Our analysis reveals that multiplicative noise, interpreted in the Ito sense, plays a crucial role in stabilizing the soliton solutions, particularly maintaining their stability around the zero state. This finding underscores the importance of noise in influencing the stability and dynamics of solitons in optical systems.

Rogue waves on the periodic background for a higher-order nonlinear Schrödinger–Maxwell–Bloch system

In this paper, we construct the rogue wave solutions on the background of the Jacobian elliptic functions for a higher-order nonlinear Schrödinger–Maxwell–Bloch system. The Jacobian elliptic function traveling wave solutions as the seed solutions are considered. Through the approach of the nonlinearization of the Lax pair and Darboux transformation method, the rogue waves and the line rogue waves on the Jacobian elliptic functions dn and cn background are obtained, respectively.

Optical soliton solutions for the nonlinear Schrödinger equation with higher-order dispersion arise in nonlinear optics

Optical solitons and traveling wave solutions for the higher-order dispersive extended nonlinear Schrödinger equation are studied. Ultrashort pulse propagation in optical communication networks is described by this equation. To find exact solutions to the model, the unified Riccati equation expansion method and the Jacobi elliptic function expansion method are successfully applied. The optical solutions includes various solitary wave solutions, such as dark, bright, combined dark-bright, singular, combined periodic, periodic, Jacobian elliptic, and rational functions. Three-dimensional and two-dimensional graphs of solutions are presented. Also, the dynamical behavior of waves and the impact of time on solutions by selecting appropriate parameters are illustrated.