Jacobi Elliptic Function Solutions Research Articles

Real-World Applications of Analytic Travelling Wave Solutions of a (3+1)-Dimensional Hirota–Satsuma–Ito-Like System Via Polynomial Complete Discriminant System and Elementary Integral Technique

Models reveal the dynamic character of wave motion, delineated in shallow waters alongside fluid dynamics; an example is the Hirota–Satsuma–Ito model equation. Therefore, this paper showcases the detailed analytical investigations of a (3+1)-dimensional Hirota–Satsuma–Ito-like system. In order to reduce the equation to a nonlinear ordinary differential system of equations, a traveling plane wave transformation is engaged. Thereafter, the direct integration technique is adopted to solve the model, thus culminating in obtaining Jacobi elliptic integral function solutions. Moreover, to attain more various solitonic solutions of diverse structures, a standard approach called the polynomial complete discriminant system and elementary integral technique is engaged. This provides exact traveling wave solutions of diverse known functions in the form of periodic, trigonometric, dark, mixed bright, and topological kink, as well as singular soliton solutions. These are found to appear in the form of Jacobi elliptic, trigonometric, as well as hyperbolic functions. Furthermore, some of these solutions are further examined by investigating their wave nature via numerical simulations.

Fractional Solitons in Optical Twin-Core Couplers with Kerr Law Nonlinearity and Local M-Derivative Using Modified Extended Mapping Method

This study focuses on optical twin-core couplers, which facilitate light transmission between two closely aligned optical fibers. These couplers operate based on the principle of coupling, allowing signals in one core to interact with those in the other. The Kerr effect, which describes how a material’s refractive index changes in response to the intensity of light, induces the nonlinear behavior essential for generating solitons—self-sustaining wave packets that preserve their shape and speed. In our research, we employ fractional derivatives to investigate how fractional-order variations influence wave propagation and soliton dynamics. By utilizing the modified extended mapping method (MEMM), we derive solitary wave solutions for the equations governing the behavior of optical twin-core couplers under Kerr nonlinearity. This methodology produces novel fractional traveling wave solutions, including dark, bright, singular, and combined bright–dark solitons, as well as hyperbolic, Jacobi elliptic function (JEF), periodic, and singular periodic solutions. To enhance understanding, we present physical interpretations through contour plots and include both 2D and 3D graphical representations of the results.

Unveiling of Highly Dispersive Dual-Solitons and Modulation Instability Analysis for Dual-Mode Extension of a Non-Linear Schrödinger Equation

The two-mode equations are nonlinear models that describe the behavior of two-way waves moving simultaneously while being affected by confined phase velocity. This article expands a non-linear Schrödinger equation (NLSE) by constructing it as a dual-mode structure. Applying the modified extended direct algebraic method (MEDAM) yields exact and explicit solutions. The results of this investigation have significant implications for the propagation of solitons in nonlinear optics. There are multiple resulted solutions that comprise singular periodic solutions, Weierstrass elliptic doubly periodic solutions, Jacobi elliptic function (JEF), singular soliton, bright soliton, dark soliton, and rational solutions, moreover, hyperbolic wave solutions. We show our acquired traveling wave solutions’ uniqueness and significant addition to current research by contrasting them with the body of existing literature. The method’s effectiveness shows that it may be used to address a wide variety of nonlinear problems across multiple disciplines, particularly in the theory of soliton, as the studied model appears in many applications. Additionally, we display the outlines of some of these discovered solution behaviors in 3D and 2D graphs to help with comprehension. Finally, we analyze modulation instability to examine the stability of the discovered solutions.

Abundant families of Jacobi elliptic function solutions and peculiar dynamical behavior for a generalized derivative nonlinear Schrödinger equation employing extended F-expansion method

Abstract This study investigates a generalized derivative nonlinear Schrödinger (GDNLS) equation , demonstrating how ultrashort pulses propagate in a single-mode optical fiber. The extended F-expansion method, which is a modification of Kudryashov’s auxiliary equation approach, is applied in this investigation to generate Jacobi elliptic solutions for the GDNLS. Three distinct solution instances are examined, and a variety of explicit solutions, including breathers, solitary waves, bright/dark solitons, bright-dark interaction solitons, a soliton-like solution, and a rogue-like solution, are obtained. To demonstrate the complex dynamical behavior of GDNLS equation, several representative solutions are chosen and their moduli are shown in three-dimensional, two-dimensional, and contour plots using Maple software.

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.

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 applied to the (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.

Classical and nonclassical Lie symmetries, bifurcation analysis, and Jacobi elliptic function solutions to a 3D-modified nonlinear wave equation in liquid involving gas bubbles

The current paper undertakes an in-depth exploration of the dynamics of nonlinear waves governed by a 3D-modified nonlinear wave equation, a significant model in the study of complex wave phenomena. To this end, the study employs both classical and nonclassical Lie symmetries for rigorously deriving invariant solutions of the governing equation. These symmetries enable the formal construction of exact solutions, which are crucial for understanding the complex behavior of the model. Furthermore, the research extends into the realm of bifurcation analysis through the application of planar dynamical system theory. Such an analysis reveals the conditions under which the 3D-modified nonlinear wave equation admits Jacobi elliptic function solutions. The study also delves into the impact of the nonlinear parameter on the physical characteristics of bright and kink solitary waves as well as continuous periodic waves using Maple. Overall, the comprehensive analysis presented not only enhances the understanding of complex nonlinear wave dynamics but also sets the stage for future advancements in vast areas of fluid dynamics and plasma physics.

The analysis of traveling wave structures and chaos of the cubic–quartic perturbed Biswas–Milovic equation with Kudryashov's nonlinear form and two generalized nonlocal laws

The cubic–quartic perturbed Biswas–Milovic equation, which contains Kudryashov's nonlinear form and two generalized nonlocal laws, has been explored qualitatively and quantitatively, as demonstrated in the present work. The research methods used include the complete discrimination system for polynomial method and the trial equation method. The results show that the Hamiltonian has the conservation property, and the global phase diagrams obtained via the bifurcation method reveal the existence of periodic and soliton solutions. Furthermore, we fully classify all the single traveling wave solutions to substantiate our findings, covering singular solutions, solitons, and Jacobian elliptic function solutions. We analyze their topological stabilities and present two‐dimensional graphs of solutions. We also delve deeper into the dynamic system by incorporating the perturbation item to explore the chaotic phenomena associated with the equation. These outcomes are valuable for studying the propagation of high‐order dispersive optical solitons and have potential applications in optimizing optical communication systems to improve efficiency.

The chaotic behavior and traveling wave solutions of the conformable extended Korteweg–de-Vries model

Abstract In this article, the phase portraits, chaotic patterns, and traveling wave solutions of the conformable extended Korteweg–de-Vries (KdV) model are investigated. First, the conformal fractional order extended KdV model is transformed into ordinary differential equation through traveling wave transformation. Second, two-dimensional (2D) planar dynamical system is presented and its chaotic behavior is studied by using the planar dynamical system method. Moreover, some three-dimensional (3D), 2D phase portraits and the Lyapunov exponent diagram are drawn. Finally, many meaningful solutions are constructed by using the complete discriminant system method, which include rational, trigonometric, hyperbolic, and Jacobi elliptic function solutions. In order to facilitate readers to see the impact of fractional order changes more intuitively, Maple software is used to draw 2D graphics, 3D graphics, density plots, contour plots, and comparison charts of some obtained solutions.

Diverse exact solutions to Davey–Stewartson model using modified extended mapping method

In this study, we obtain solitary wave solutions and other exact wave solutions for Davey–Stewartson equation (DSE), which explains how waves move through water with a finite depth while being affected by gravity and surface tension. The study is conducted with the aid of the modified extended mapping method (MEMM). A variety of distinct traveling wave solutions are furnished. The obtained solutions comprise dark, bright, and singular solitary wave solutions. Additionally, Jacobi elliptic function solutions, exponential wave solutions, singular periodic wave solutions, rational wave solutions, and periodic wave solutions are also offered. To help readers physically grasp the acquired solutions, graphical representations of some of the extracted solutions are provided.

New solutions of the general elliptic equation and its applications to the new (3 + 1)-dimensional integrable Kadomtsev-Petviashvili equation

In this article, we introduce many new Jacobi elliptic function solutions to the general elliptic equation. Consequently, the Jacobi elliptic function expansion method is improved to accommodate the general elliptic equation and its new solutions for constructing exact traveling wave solutions of nonlinear partial differential equations (NLPDEs). Moreover, the improved method is used to obtain new explicit solutions for the (3+1)-dimensional integrable Kadomtsev-Petviashvili (KP) equation. This method can be applied to many other NLPDEs as well for obtaining new exact solutions.

Modulational instability, modulated wave, and optical solitons for a generalized highly dispersive cubic-quintic-septic-nonic medium with self-frequency shift and self-steepening nonlinear terms

In this research, we delve into a generalized highly dispersive (HD) nonlinear Schrödinger equation, enriched with cubic-quintic-septic-nonic (CQSN) nonlinearities. The core of our investigation revolves around the perturbation of plane waves, aiming to understand their stability characteristics in such a complex medium. We investigate the influence of various factors such as the amplitude of the plane wave, perturbed wave number, nonic nonlinear term, and fourth-order dispersion term. Our findings indicate that increasing the amplitude of the plane wave widens the modulation instability (MI) bands and amplifies the MI growth rate. In contrast, increasing the nonic nonlinear term has opposing effects, narrowing the MI bands and diminishing the amplitude of the MI growth rate. Increasing the fourth-order dispersion term does not affect the amplitude of the MI growth rate but narrows the MI bands. The observed pattern of increasing and then decreasing MI intensity with rising K can be attributed to the complex interplay among phase matching conditions, dispersion effects, and nonlinear saturation. Initially, higher K enhances phase matching and boosts MI growth. However, as K increases further, the combined influence of dispersion and nonlinear effects can diminish the effectiveness of phase matching, resulting in a reduction in MI intensity. A significant portion of our work is dedicated to identifying and analyzing modulated rational, polynomial Jacobi elliptic function solutions, and the emergence of optical solitons within this framework. These findings provide new insights into the nonlinear dynamics underpinning the generalized HDNLSE, enriched with CQSN nonlinearities, offering valuable contributions to the theoretical understanding of such phenomena.

Chaotic Pattern and Solitary Solutions for the (21)-Dimensional Beta-Fractional Double-Chain DNA System

The dynamical behavior of the double-chain deoxyribonucleic acid (DNA) system holds significant implications for advancing the understanding of DNA transmission laws in the realms of biology and medicine. This study delves into the investigation of chaos patterns and solitary wave solutions for the (2+1) Beta-fractional double-chain DNA system, employing the theory of planar dynamical systems and the method of complete discrimination system for polynomials (CDSP). The results demonstrate a diverse spectrum of solitary wave solutions, sensitivity to perturbations, and manifestations of chaotic behavior within the system. Through the utilization of the complete discrimination system for polynomials, a multitude of novel solitary wave solutions, encompassing periodic, solitary wave, and Jacobian elliptic function solutions, were systematically constructed. The influence of Beta derivatives on the solutions was elucidated through parameter comparison analysis, emphasizing the innovative nature of this study. These findings underscore the potential of this system in unraveling various biologically significant DNA transmission mechanisms.

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

Modulational instability and chirped modulated wave, chirped optical solitons for a generalized (3+1)-dimensional cubic-quintic medium with self-frequency shift and self-steepening nonlinear terms

We consider a generalized (3+1)-dimensional nonlinear Schrödinger with cubic-quintic nonlinear and self-frequency shift and self-steepening terms. We disrupt the plane wave to study the stability of the wave in this media. We examine how various factors—such as the amplitude of the plane wave, cubic and quintic nonlinear terms, self-steepening, and self-frequency shift—affect the modulational instability (MI). Our findings reveal that the amplitude of the plane wave and the quintic nonlinear term can expand the MI bands and increase the amplitude of the MI growth rate. Conversely, the self-steepening and self-frequency shift terms exert opposite effects, narrowing the MI bands and reducing the amplitude of the MI growth rate. We investigate the existence of modulated chirped rational and polynomial Jacobi elliptic function solutions and chirped optical solitons.

New exact solitary waves for the Sasa-Satsuma model with variable coefficients

In this paper, we investigate the variable coefficients Sasa-Satsuma model, which can describe the propagation of a light pulse in a cylindrical fiber. We study this model and obtain rich solutions using two separate methods. We obtain analytical Weierstrass elliptic function solutions using the Weierstrass elliptic function expansion method. Some Jacobi elliptic function solutions are obtained using the modified Jacobi elliptic function expansion method. When the Jacobi elliptic function degenerates, we obtain the corresponding trigonometric, hyperbolic function solutions and periodic solutions. We also try to take the coefficients of the equation as some functions and obtain some more complicated exact solutions, which have not appeared in previous studies. Finally, we simulate some waveform diagrams of the solutions using the computer software Mathematica and obtain periodic waves, bright and dark soliton, double solitons and some complex periodic waves. With these waveform diagrams, we can observe the dynamical behavior of the solutions more clearly.

Investigating (2+1)-dimensional dissipative long wave system in water waves using three innovative integration norms

This research article investigates the dissipative long wave system in (2+1) dimensions. To analyze this system, three separate integrating strategies were used: the single manifold approach, the unified method, and the generalized projective Ricatti equation method. Novel soliton solutions were effectively obtained by applying these strategies to the suggested model. The resulting solutions are various, including mixed soliton-like solutions, dark solitons, rational solutions, triangular periodic solutions, and combined Jacobi elliptic wave function solutions. Using the aforementioned approaches, an extensive variety of solution types has been discovered for the proposed model. The dynamic character of these closed-form solutions has been adequately demonstrated using both three dimensional and two dimensional graphical representations. This visualization helps to better understand the behavior of the acquired solutions.

Exact solutions and bifurcations for the (3+1)-dimensional generalized KdV-ZK equation

In this paper, a class of (3+1)-dimensional generalized Korteweg–de Vries-Zakharov-Kuznetsov (KdV-ZK) equation is studied by utilizing the bifurcation theory of the planar dynamical systems and the Fan sub-function method. This model can be used to explain the effects of magnetic fields on weakly nonlinear ion-acoustic waves investigated in plasma fields composed of cold and hot electrons. Under the different parameter conditions, the phase portraits and bifurcations are derived, and new exact solutions including soliton, periodic, kink and breaking wave solutions for the model are constructed. Moreover, some exact solutions, which contain soliton, kink, trigonometric function, hyperbolic function, Jacobi elliptic function solutions, are derived via the improved Fan sub-function method. The types of solutions obtained completely correspond to the types of the orbits acquired above, which verifies the validity of the method. Finally, the physical structures of some exact solutions are analyzed in graphical forms.

Novel analytic solutions of strain wave model in micro-structured solids

Abstract In this article, the modified extended direct algebraic method is implemented to investigate the strain wave model that governs the wave propagation in micro-structured solids. The proposed method provides many new exact traveling wave solutions with certain free parameters. Exact solutions are extremely important in interpreting the inner structures of the natural phenomena. Solitary and other wave solutions are provided for this model, such as bright solitary solutions, dark solitary solutions, singular solitary solutions, singular-dark combo solitary solutions. Also, periodic solutions and Jacobi elliptic function solutions are presented. To show the physical characteristics of the raised solutions, the graphical illustration of some solutions is presented.

Explicit and exact travelling wave solutions for Hirota equation and computerized mechanization.

By using the power-exponential function method and the extended hyperbolic auxiliary equation method, we present the explicit solutions of the subsidiary elliptic-like equation. With the aid of the subsidiary elliptic-like equation and a simple transformation, we obtain the exact solutions of Hirota equation which include bright solitary wave, dark solitary wave, bell profile solitary wave solutions and Jacobian elliptic function solutions. Some of these solutions are found for the first time, which may be useful for depicting nonlinear physical phenomena. This approach can also be applied to solve the other nonlinear partial differential equations.