Jacobi Elliptic Function Expansion Method Research Articles

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.

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.

Rogue waves for the (2+1)-dimensional Myrzakulov–Lakshmanan-IV equation on a periodic background

In this paper, the rogue wave solutions of the (2+1)-dimensional Myrzakulov–Lakshmanan (ML)-IV equation, which is described by five component nonlinear evolution equations, are studied on a periodic background. By using the Jacobian elliptic function expansion method, the Darboux transformation (DT) method and the nonlinearization of the Lax pair, two kinds of rogue wave solutions which are expressed by Jacobian elliptic functions dn and cn, are obtained. The relationship between these five kinds of potential is summarized systematically. Firstly, the periodic rogue wave solution of one potential is obtained, and then the periodic rogue wave solutions of the other four potentials are obtained directly. The solutions we find present the dynamic phenomena of higher-order nonlinear wave equations.

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.

Exact solutions for the improved mKdv equation with conformable derivative by using the Jacobi elliptic function expansion method

Exact solutions for the improved mKdv equation with conformable derivative by using the Jacobi elliptic function expansion method

On the superpropagating nonlinear waveform in critical superthermal fluid

The superthermality indices of positron (electron) contributions to the new nonlinear wave profiles characterized by the model as supersoliton, cnoidal, shocklike, and super-periodic wave propagation have been obtained in the superthermal ionosphere plasma pair. These profiles that exhibit critical behaviors are described by the modified KP equation. Furthermore, the novel solutions have been exposed and the super-solitary and period solutions are derived and explained using the Jacobi elliptic function expansion method for the Modified Korteweg–De Vries (MKdV) equation. These solutions may be useful in the space of ionosphere fluids. Indeed, the MKdV model is potent for the dynamics of fluid flow. The super propagating waves obtained under the influence of critical densities represent a source of forcing or collapsing energy waves in space plasma fluids. To check the impacts of superthermality on the progress of nonlinear formations, all the given solutions are deliberated by the superthermal factor in ionosphere fluid observations and applications.

Optimal system and dynamics of optical soliton solutions for the Schamel KdV equation

In this research, we investigate the integrability properties of the Schamel–Korteweg–de Vries (S-KdV) equation, which is important for understanding the effect of electron trapping in the nonlinear interaction of ion-acoustic waves. Using the optimal system, we come over reduced ordinary differential equations (ODEs). To deal with reduced ODEs for this problem, Lie symmetry analysis is combined with the modified auxiliary equation (MAE) procedure and the generalized Jacobi elliptic function expansion (JEF) method. The analytical solutions reported here are novel and have a wide range of applications in mathematical physics.

Solitary and Periodic Wave Solutions of the Space-Time Fractional Extended Kawahara Equation

The extended Kawahara (Gardner Kawahara) equation is the improved form of the Korteweg–de Vries (KdV) equation, which is one of the most significant nonlinear evolution equations in mathematical physics. In that research, the analytical solutions of the conformable fractional extended Kawahara equation were acquired by utilizing the Jacobi elliptic function expansion method. The given expansion method was applied to different fractional forms of the extended Kawahara equation, such as the fraction that occurs in time, space, or both time and space by suitably changing the variables. In addition, various types of fractional problems are exhibited to expose the realistic application of the given method, and some of the obtained solutions were illustrated in two- or three-dimensional graphics as proof of the visualization.

Optical solitons with DNA dynamics arising in oscillator-chain of Peyrard–Bishop model

In this paper, we introduce the Noether approach for studying the Peyrard–Bishop (PB) DNA dynamic model equation. The Jacobi elliptic function expansion (JEFE) method is used to analyze this model. The obtained soliton solutions are hyperbolic, trigonometric, and rational functions. The dynamics of the obtained solutions conform to the properties of shock waves, periodic waves, solitary waves, double periodic waves, and rational solutions. A comparison between the results obtained by the JEFE method and the results existing in the literature shows that our results are more general, and novel, and have never been reported earlier. The wave profiles are investigated by Wolfram Mathematica simulations for the physical wave behavior.

Travelling Wave Solutions of the Non-Linear Wave Equations

Abstract This article focuses on the exact periodic solutions of nonlinear wave equations using the well-known Jacobi elliptic function expansion method. This method is more general than the hyperbolic tangent function expansion method. The periodic solutions are found using this method which contains both solitary wave and shock wave solutions. In this paper, the new results are computed using the closed-form solution including solitary or shock wave solutions which are obtained using Jacobi elliptic function method. The corresponding solitary or shock wave solutions are compared with the actual results. The results are visualised and the periodic behaviour of the solution is described in detail. The shock waves are found to break with time, whereas, solitary waves are found to be improved continuously with time.

On a modified Korteweg–de Vries equation for electrostatic structures in relativistic degenerate electron–positron plasma

In this paper, the propagation of acoustic solitons in an electron–positron (e–p) degenerate plasma with the relativistically degenerate electrons and positrons is studied. Moreover, we considered stationary positive ions for neutralizing the plasma background. Using the reductive perturbation method, we obtained a modified Korteweg–de Vries equation. By applying a Jacobi elliptic function expansion method, we have proposed the exact analytical solitary, kink, and periodic solutions for the nonlinear wave structures. Also, using the phase plane analysis three equilibrium points have been perceived two of them should be centers and the last is a saddle for quantum acoustic waves. The numerical and computational results show that firstly some of them should be discarded by physical values and then these solutions have been influenced by the plasma parameters such as the electron degenerate density parameter (ne0) and the ion-to-electron density ratio (δ) which peaked out in white dwarf stars. It is found that both the amplitude and the width of the nonlinear structures are affected by the mentioned parameters. This investigation is useful for the relativistically degenerate plasma media in ultra-dense astrophysical environments (such as neutron stars, white dwarfs, etc.), inertia fusion science, experiments involving the interaction of laser and solid density plasma, and inertial confinement fusion in which the relativistic effects play a major role.

Quantitative and qualitative analyses of the mKdV equation and modeling nonlinear waves in plasma

In this paper, nonlinear electrostatic structures on the ion time scale in plasma consisting of two populations of electrons (cold and hot), positrons, and warm adiabatic ions are investigated. The multiple scale method is used to derive the modified Korteweg–de Vries equation (mKdVE). The Jacobi elliptic function expansion method (JEFEM) is employed to find some exact analytical solutions such as periodic, solitonic, and shock solutions. It is shown that the variation in the plasma parameters of interest, for our model, allows the existence of solitary and periodic structures and no shocks. It is also shown that the most important plasma parameters for the plasma model under consideration are positron concentration, α, and the percentage of cold and hot electrons, represented by the parameters μ and ν, respectively. Additionally, the qualitative behavior of the mKdVE is studied using dynamical system theory. The topological structure of the solution is discussed in the phase plane. In this work, the phase plane analysis, which is restricted to the discrete values of the parameter, is extended to the continuous range of the parameter using a bifurcation diagram. Bifurcation diagrams are drawn to forecast the behavior of the solution for specifically chosen essential plasma parameters. The analytical solution and the qualitative behavior of the solution presented in this paper are shown to be compatible with each other. The results presented here are general and can be gainfully employed to study a variety of nonlinear waves in space, laboratory plasmas, and astrophysical plasmas.

Extended Jacobi elliptic function solutions for general boussinesq systems

In this research paper, we have utilized the Jacobi elliptic function expansion method to obtain the exact solutions of (1+1)- dimensional Boussinesq System (GBQS). The most important difference that distinguishes this method from other methods is the parameters included in the auxiliary equation F’ (ξ) =  Ö P F4(ξ) + QF2(ξ) + R. As far as the authors know, there is no other study in which such a variety of solutions has been given. Depending on P, Q and R, nineteen the solitary wave and periodic wave solutions are obtained at their limit conditions. In addition, 3D and contour plot graphics for the constructed waves are investigated with the computer package program by giving special values to the parameters involved. The validity and reliability of the method is examined by its applications on a class of nonlinear evolution equations of special interest in nonlinear mathematical physics. The results were acquired to verify that the recommended method is applicable and reliable for the analytic treatment of a wide application of nonlinear phenomena

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.

The attributes of the dust-acoustic solitary and periodic structures in the Saturn's inner magnetosphere

Nonlinear characteristics of dust-acoustic (DA) structures including the localized and periodic waves in a plasma having Maxwellian ions and superthermal two-temperature electrons are investigated. The wave equations, including both Kadomtsev–Petviashvili (KP) and modified KP (mKP) equations, are derived using the reductive perturbation technique (RPT). The quantitative and qualitative characteristics of both compressive and rarefactive structures are studied. The Jacobi elliptic function expansion method (JEFEM) is employed for the purpose of quantitative analysis, while the qualitative behavior is studied by the dint of the dynamical system approach. The solutions to the mKP equation hold under a critical condition where the quadratic nonlinearity ceases to exist. It is noticed that the KP equation admits only rarefactive solitary waves (SWs), whereas the mKP equation admits both compressive and rarefactive SWs. It is found that the profile (amplitude and width) of both DA solitary and periodic structures are different at different radii of Saturn's inner magnetosphere. The effect of the kappa spectral index is studied, and it is found that when the population of energetic cold electrons is decreased, the solitary structure gets energized. Our study is applied to Saturn's inner magnetosphere where kappa distributed two-temperature electrons and dust grains with negative charge are observed by various satellite missions.

Optical solitons to the Perturbed Gerdjikov-Ivanov equation with quantic nonlinearity

In this article, we examine the Perturbed Gerdjikov-Ivanov equation by using the Jacobi elliptic function expansion method. This results in obtaining distinct solutions including dark, bright, singular solitons, periodic waves, singular periodic waves, and Jacobi elliptic function solutions. The 2- and 3-dimensional graphs of the reported solutions are presented. The reported results may be useful in explaining the physical features of the studied equation.

Exact Solutions for the Generalized Atangana-Baleanu-Riemann Fractional (3 + 1)-Dimensional Kadomtsev–Petviashvili Equation

In this article, the generalized Jacobi elliptic function expansion method with four new Jacobi elliptic functions was used to the generalized fractional (3 + 1)-dimensional Kadomtsev–Petviashvili (GFKP) equation with the Atangana-Baleanu-Riemann fractional derivative, and abundant new types of analytical solutions to the GFKP were obtained. It is well known that there is a tight connection between symmetry and travelling wave solutions. Most of the existing techniques to handle the PDEs for finding the exact solitary wave solutions are, in essence, a case of symmetry reduction, including nonclassical symmetry and Lie symmetries etc. Some 3D plots, 2D plots, and contour plots of these solutions were simulated to reveal the inner structure of the equation, which showed that the efficient method is sufficient to seek exact solutions of the nonlinear partial differential models arising in mathematical physics.

Analytical analysis of the fourth-order Boussinesq equation by traveling wave solutions

To find periodic solutions of nonlinear equations, the Jacobi elliptic function expansion method is used. This strategy is more extensive than expanding the hyperbolic tangent series. This approach can produce periodic shock wave solutions such as solitary wave solutions.

The exact solutions of Fokas-Lenells equation based on Jacobi elliptic function expansion method

The Fokas-Lenells (FL) equation, which is rich in physical property in soliton theory as well as optical fiber, is a generalization of the higher-order Schrödinger equation. We construct the periodic solutions of the FL equation based on the Jacobi elliptic function expansion method in this context. Moreover, the characteristics of the obtained solutions are visualized graphically by selecting appropriate parameters.

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.