Extended Jacobi Elliptic Function Expansion Research Articles

Conserved vectors and symmetry solutions of the Landau–Ginzburg–Higgs equation of theoretical physics

This paper is devoted to the investigation of the Landau–Ginzburg–Higgs equation (LGHe), which serves as a mathematical model to understand phenomena such as superconductivity and cyclotron waves. The LGHe finds applications in various scientific fields, including fluid dynamics, plasma physics, biological systems, and electricity-electronics. The study adopts Lie symmetry analysis as the primary framework for exploration. This analysis involves the identification of Lie point symmetries that are admitted by the differential equation. By leveraging these Lie point symmetries, symmetry reductions are performed, leading to the discovery of group invariant solutions. To obtain explicit solutions, several mathematical methods are applied, including Kudryashov’s method, the extended Jacobi elliptic function expansion method, the power series method, and the simplest equation method. These methods yield solutions characterized by exponential, hyperbolic, and elliptic functions. The obtained solutions are visually represented through 3D, 2D, and density plots, which effectively illustrate the nature of the solutions. These plots depict various patterns, such as kink-shaped, singular kink-shaped, bell-shaped, and periodic solutions. Finally, the paper employs the multiplier method and the conservation theorem introduced by Ibragimov to derive conserved vectors. These conserved vectors play a crucial role in the study of physical quantities, such as the conservation of energy and momentum, and contribute to the understanding of the underlying physics of the system.

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.

Solitary Wave Solutions to a Fractional Model Using the Improved Modified Extended Tanh-Function Method

Nonlinear fractional partial differential equations (NLFPDEs) are widely used in simulating a variety of phenomena arisen in several disciplines such as applied mathematics, engineering, physics, and a wide range of other applications. Solitary wave solutions of NLFPDEs have become a significant tool in understanding the long-term dynamics of these events. This article primarily focuses on using the improved modified extended tanh-function algorithm to determine certain traveling wave solutions to the space-time fractional symmetric regularized long wave (SRLW) equation, which is used to discuss space-charge waves, shallow water waves, etc. The Jumarie’s modified Riemann-Liouville derivative is successfully used to deal with the fractional derivatives, which appear in the SRLW problem. We find many traveling wave solutions on the form of trigonometric, hyperbolic, complex, and rational functions. Furthermore, the performance of the employed technique is investigated in comparison to other techniques such as the Oncoming exp(−Θ(q))-expansion method and the extended Jacobi elliptic function expansion strategy. Some obtained results are graphically displayed to show their physical features. The findings of this article demonstrate that the used approach enables us to handle more NLFPDEs that emerge in mathematical physics.

New Complex Wave Solutions and Diverse Wave Structures of the (2+1)-Dimensional Asymmetric Nizhnik–Novikov–Veselov Equation

In this paper, we use a new, extended Jacobian elliptic function expansion method to explore the exact solutions of the (2+1)-dimensional asymmetric Nizhnik–Novikov–Veselov (aNNV) equation, which is a nonlinear physical model to describe an incompressible fluid. Combined with the mapping method, many new types of exact Jacobian elliptic function solutions are obtained. As we use two new forms of transformation, most of the solutions obtained are not found in previous studies. To show the complex nonlinear wave phenomena, we also provide various wave structures of a group of solutions, including periodic wave and solitary wave structures of ordinary traveling wave solutions, horseshoe-type wave, s-type wave and breaker-wave structures superposed by two kinds of waves: chaotic wave structures with chaotic behavior and spiral wave structures. The results show that this method is effective and powerful and can be used to construct various exact solutions for a wide range of nonlinear models and complex nonlinear wave phenomena in mathematical and physical research.

Diverse analytical wave solutions and dynamical behaviors of the new (2 + 1)-dimensional Sakovich equation emerging in fluid dynamics

The primary goal of this article is to extend previous results and obtain a more comprehensive variety of analytical wave solutions to the (2 + 1)-dimensional Sakovich equation, a nonlinear evolution equation that plays a remarkable role in condensed physics, fiber optics, and fluid dynamics. By employing two relatively new techniques, Lie symmetry analysis, and the extended Jacobian elliptic function expansion method, some standard form of new and unique closed-form solutions are established in terms of trigonometric, hyperbolic, and Jacobi elliptic functions. An optimal system of the one-dimensional subalgebras is also constructed using Lie vectors. These generated solutions contain several ascendant parameters that play a crucial role in describing the inner mechanism of a given physical model. Therefore, the obtained wave solutions are demonstrated graphically by three- and two-dimensional graphics using Mathematica. In addition, a couple of varieties of solutions, including multi soliton, periodic soliton, Bell-shaped, and parabolic profile, are depicted for the suitable values of the included parameters. Finally, it must be noted that the variation in obtained wave profiles by the change in subsidiary involved parameters reveals the parameter effects on the wave. The derived solutions are more generalized than previous established results (Özkan and Yaşar in Alex Eng J 59:(6):5285–5293, 2020), showing the novelty and significance of our solutions. Also, the solutions obtained in the form of Jacobi elliptic functions have never been reported in the literature. This study ensures that the forgoing techniques are practical and may be used to seek the solitary solitons to a diversity of nonlinear evolution equations.

On distinctive solitons type solutions for some important nonlinear Schrödinger equations

The extended Jacobi elliptic function expansion (EJEFE) method is used to retrieve several types of optical solitons of two nonlinear Schrodinger equations, namely the Heisenberg ferromagnetic spin chains and Alfven envelop equations. The obtained traveling wave solutions and the corresponding plots are analysed by means of the symbolic package Mathematica. The solutions show that the proposed strategy is effective and reliable for solving different types of nonlinear differential equations.

Soliton solutions for system of ion sound and Langmuir waves

We study on constructing new traveling wave solution of a system known as ion sound and Langmuir waves by using method of extended Jacobian elliptic function expansion and He’s semi-inverse technique. Some new solutions will be beneficial for researchers concerning with nonlinear physical phenomena. The proposed methods may serve as the framework for solutions of various equations in applied science. Graphical simulations are provided to illustrate the behavior of these solutions.

Stochastic treatment of the solutions for the resonant nonlinear Schrödinger equation with spatio-temporal dispersions and inter-modal using beta distribution

In this paper, the extended Jacobian elliptic function expansion method is implemented in order to construct some new traveling wave solutions for the resonant nonlinear Schrodinger equation with both spatio-temporal dispersion and inter-modal dispersion. These new traveling wave solutions are obtained by the proposed method, which is easy to implement and computationally very attractive. Moreover, these solutions may be applicable for some physical fields, such as plasma physics. The main aim of this paper is the stochastic treatment of the solutions when the spatio-temporal coefficient or the wave transition is beta random variables. The priority of using beta statistical distribution for the spatio-temporal is discussed. Some graphical simulations are given to illustrate the behavior of these solutions in the deterministic and stochastic case studies. Indeed the proposed techniques are very powerful tool to solve other models in applied science.

Analytical and Numerical Investigation for the DMBBM Equation

The nonlinear dispersive modified Benjamin-Bona-Mahony (DMBBM) equation is solved numerically using adaptive moving mesh PDEs (MMPDEs) method. Indeed, the exact solution of the DMBBM equation is obtained by using the extended Jacobian elliptic function expansion method. The current methods give a wider applicability for handling nonlinear wave equations in engineering and mathematical physics. The adaptive moving mesh method is compared with exact solution by numerical examples, where the explicit solutions are known. The numerical results verify the accuracy of the proposed method.

Numerical investigation of the dispersive long wave equation using an adaptive moving mesh method and its stability

This paper investigates the exact and numerical solutions of the dispersive long wave equations (DLWE) using the extended Jacobian elliptic function expansion method (EJEFET) and the adaptive moving mesh method (AMMT). The uniform mesh method (UMM) is also applied to obtain some numerical results. The analysis of the stability of these solutions is also presented. The achieved solutions are stable. We compare both numerical and exact solutions and obtained that the numerical results are accurate. Some 3D figures are plotted to reinforce and ensure the integrity of the qualitative and quantitative results obtained in this work. Finally, L2 and L∞ errors are analyzed to discover the proper performance of the considered techniques in solving DLWE.

New extended Jacobi elliptic function expansion scheme for wave-wave interaction in ionic media

New extended Jacobi elliptic function expansion scheme for wave-wave interaction in ionic media

Optical solitons, nonlinear self-adjointness and conservation laws for Kundu–Eckhaus equation

In this article, Kundu–Eckhaus equation (KE) is studied from the perspective of modified tanh-coth method (MTC), extended Jacobi elliptic function expansion method (EJEF), Lie symmetry analysis, nonlinear self-adjointness and conservation laws (Cls). New soliton solutions like combined dark-bright, dark, periodic wave and singular soliton solutions are obtained. The equation is found to be a nonlinear self-adjoint, we construct the Cls using the new conservation theorem presented by Ibragimov. Physical interpretation for some of the obtained solutions are illustrated in Figures.

Optical soliton solutions for the higher-order dispersive cubic-quintic nonlinear Schrödinger equation

In this paper, we analyze new optical soliton solutions to the higher-order dispersive cubic-quintic nonlinear Schrödinger equation (NLSE) using three integration schemes. The schemes used in this paper are modified tanh-coth (MTC), extended Jacobi elliptic function expansion (EJEF), and two variable (G′∕G,1∕G)-expansion methods. We obtain new solutions that to the best of our knowledge do not exist previously. The obtained solutions includes bright, dark, combined bright-dark, singular as well as periodic waves solitons. The obtained solutions may be used to explain and understand the physical nature of the wave spreads in the most dispersive optical medium. Some interesting figures for the physical interpretation of the obtained solutions are also presented.

The investigation of soliton solutions of the coupled sine-Gordon equation in nonlinear optics

The coupled sine-Gordon equation, which can be used to describe the propagation of an optical pulse in fibre waveguide, has been investigated analytically. The hyperbolic and trigonometric function solutions are constructed by the extended expansion method, and then the Jacobi elliptic, hyperbolic and trigonometric function solutions are derived with the help of the extended Jacobi elliptic function expansion method.

Soliton solutions for Davydov solitons in α-helix proteins

The propagation equation for describing Davydov solitons in α-helix proteins has been investigated analytically. There are seven integration tools to extract analytical soliton solutions. They are the Ricatti equation expansion approach, ansatz scheme, improved extended tanh-equation method, the extend exp(−Ψ(τ))-expansion method, the extended Jacobi elliptic function expansion method, the extended trial equation method and the extended G'/G− expansion method.

Explicit solitons in the parabolic law nonlinear negative-index materials

This paper deals with the existence of explicit ultra-short solitons in the negative-index materials with third-order dispersion and higher-order nonlinearities. Four integration algorithms, that are the improved modified extended tanh-function method, the extended trial equation method, the extended Jacobi elliptic function expansion method and the \(\exp \left( -\Phi \left( \eta \right) \right) \)-expansion method, are employed to extract analytical traveling wave solutions. The presented results show that exact solitons can exist in this physical model.

Extended Jacobian Elliptic Function Expansion Method and Its Applications in Mathematical Physics

In this work, an extended Jacobian elliptic function expansion method is proposed for constructing the exact solutions of nonlinear evolution equations. The validity and reliability of the method are tested by its applications to some nonlinear evolution equations which play an important role inmathematical physics.

Traveling wave solutions for the Couple Boiti-Leon-Pempinelli System by using extended Jacobian elliptic function expansion method

In this work, an extended Jacobian elliptic function expansion method is pro-posed for constructing the exact solutions of nonlinear evolution equations. The validity and reliability of the method are tested by its applications to the Couple Boiti-Leon-Pempinelli System which plays an important role in mathematical physics.

Extended Jacobian Elliptic Function Expansion Method and its Applications for Solving some Nonlinear Evolution Equations in Mathematical Physics

Extended Jacobian elliptic function expansion method is employed to find the exact traveling wave solutions involving parameters for nonlinear evolution equations. When these parameters are taken to be special values, the solitary wave solutions are derived from the exact traveling wave solutions. It is shown that extended Jacobian elliptic function expansion method provides an effective and a more powerful mathematical tool for solving nonlinear evolution equations in mathematical physics. Comparison between our results and the well-known results will be presented.

Extended Jacobian Elliptic Function Expansion Method and Its Applications in Biology

In this work, an extended Jacobian elliptic function expansion method is proposed for constructing the exact solutions of nonlinear evolution equations. The validity and reliability of the method are tested by its applications to Dynamical system in a new Double-Chain Model of DNA and a diffusive predator-prey system which play an important role in biology.