Perturbed Gerdjikov Ivanov Equation Research Articles

The dynamical behavior analysis of the fractional perturbed Gerdjikov–Ivanov equation

In this article, the fractional perturbed Gerdjikov–Ivanov equation is investigated. Firstly, the fractional perturbed Gerdjikov–Ivanov equation is transformed into an ordinary differential equation through traveling wave transformation. Secondly, using the trial method of rank homogeneous equation polynomials and the principle of homogeneous equilibrium, a two-dimensional planar dynamic system is presented and its bifurcation behavior is studied. Then, its two-dimensional phase portraits are drawn by using Maple software. Finally, disturbance factors are introduced into the planar dynamical system to study its chaotic behavior, and some two-dimensional phase portraits, three-dimensional phase portraits, Poincaré sections, and sensitivity analysis graphs of the perturbed system are plotted by using Maple software. The novelty lies in studying the dynamic behavior of the objective equation without the need for solving.

On the optical soliton solutions to the fractional complex structured (1+1)-dimensional perturbed gerdjikov-ivanov equation

In this work, we examine the complex structured Fractional Perturbed Gerdjikov-Ivanov equation (FPGIE), which describes the propagation of optical pulses with perturbation effects. This model finds applications in optical fibers, especially in photonic crystal fibers. We are discovered novel and unique optical soliton solutions using the modified Extended Direct Algebraic Method (mEDAM), which has never been used with this model previously. As a result, a hierarchy of traveling wave solutions including singular kink, periodic, solitary kink, and rogue-shaped soliton solutions, etc., are derived. Some obtained solutions are discussed graphically based on numerical values of some parameters related to the solution. The results add new and unique soliton types to the model and demonstrate how they interact and impact the system’s overall dynamics.

New chirp soliton solutions for the space-time fractional perturbed Gerdjikov–Ivanov equation with conformable derivative

The space–time perturbed fractional Gerdjikov–Ivanov equation is the main topic of this work, together with quintic nonlinearity and self-steepening, as it involves several intricate physical phenomena including nonlinearity, self-steepening and fractional calculus, where the fractional derivative is described by employing a conformable derivative. In addition, the governing equation is transformed into an integer-order ordinary differential equation by using an appropriate fractional complex transformation. Under certain restrictions, a direct algebraic method is employed to investigate the structures of chirp soliton solutions enfolding hyperbolic functional terms. The dynamic behaviour and bifurcation of equilibria of the system are thoroughly examined; chirp soliton solutions under specified constraints are investigated and the evolving profiles of the obtained solutions are visualized. Moreover, this research offers valuable perspectives on the behaviour of chirp solitons under specific conditions, which have practical applications in nonlinear physical systems and optical communication systems. The significant contribution of this work is the investigation and obtaining of novel chirp soliton solutions with hyperbolic functional terms under particular limitations using a novel approach. It further extends the prior approaches by treating difficult fractional differential equations from a fresh angle, offering new tools, and closely examining soliton solutions.

Dynamical behavior of perturbed Gerdjikov–Ivanov equation through different techniques

The objective of this work is to investigate the perturbed Gerdjikov–Ivanov (GI) equation along spatio-temporal dispersion which explains the dynamics of soliton dispersion and evolution of propagation distance in optical fibers, photonic crystal fibers (PCF), and metamaterials. The algorithms, namely hyperbolic extended function method and generalized Kudryashov’s method, are constructed to obtain the new soliton solutions. The dark, bright, periodic, and singular solitons are derived of the considered equation with the appropriate choice of parameters. These results are novel, confirm the stability of optical solitons, and have not been studied earlier. The explanation of evaluated results is given by sketching the various graphs in 3D, contour and 2D plots by using Maple 18. Graphical simulations divulge that varying the wave velocity affects the dynamical behaviors of the model. In summary, this research adds to our knowledge on how the perturbed GI equation with spatio-temporal dispersion behaves. The obtained soliton solutions and the methods offer computational tools for further analysis in this field. This work represents an advancement in our understanding of soliton dynamics and their applications in photonic systems.

Rhombus-shaped, M-shaped- solitons, 2D-lumps vector and tunneling in perturbed Gerdjikov–Ivanov equation with fourth-order dispersion: Modulation instability and global bifurcation

The Perturbed Gerdjikov–Ivanov Equation (PGIE) describes optical solitons in nonlinear fiber optics and photonic crystal fibers. It was currently studied in the literature. Here, we consider a new complex field equation, the PGIE with fourth-order dispersion. The exact solutions of the aforementioned equation are obtained by implementing the unified method. These solutions are evaluated numerically and they are shown in graphs. Our objectives, here, are to inspect the onset of self- steepening and self-phase modulation and investigate the structures of solitons produced. It is found that these phenomena are launched for high values of the relevant coefficients, and they may be overlapped. Multiple solitons shapes are observed; M-shaped solitons, rhombus shapes, solitons with tunneling, 2D-lumps vector, and solitons- cascade. Further, the analysis of modulation instability (MI) and global bifurcation are carried out. It is established that the MI triggers when the coefficient of Raman scattering exceeds a critical value. It is worth mentioning that two kinds of bifurcations can be studied, namely local and global bifurcations. Here, we are concerned with studying global bifurcation. This is based on constructing the phase portrait via Hamiltonian function.

Exact chirped solutions of the perturbed Gerdjikov–Ivanov equation with spatio-temporal dispersion

Abstract In this paper, the perturbed Gerdjikov–Ivanov equation with spatio-temporal dispersion is investigated. The trial equation method, the complex envelope travelling wave transformation, and the complete discriminant system for polynomial method, respectively, are all utilized. As a consequence, twenty-eight exact chirped solutions are obtained, and some representative two-dimensional patterns under concrete parameters are presented. The results acquired are conducive to studying the dynamic behavior of solitons.

Bright, dark and kink exact soliton solutions for perturbed Gerdjikov–Ivanov equation with full nonlinearity

In this paper, the perturbed Gerdjikov-Ivanov equation (PGI) which describes the dynamics of the soliton in optical fiber is investigated. Using traveling wave transformation, the nonlinear PGI equation is transformed into two nonlinear ordinary differential equations. Consequently, these equations are reduced to a first -order ordinary differential equation that has a well-known exact solution. The exact solutions for the PGI equation will be introduced including Weierstrass, Jacobi elliptic, and hyperbolic functions. Various types of optical soliton solutions were raised such as bright, dark, and kink soliton solution. The obtained optical soliton solutions are presented graphically to clarify some of its physical parameters.

Investigation of optical soliton solutions for the perturbed Gerdjikov-Ivanov equation with full-nonlinearity

We have discussed the perturbed Gerdjikov-Ivanov (pGI) equation describing optical pulse propagation (PP) with perturbation effects, which has various applications in optical fibers, especially in photonic crystal fibers. According to our literature review, we have discovered new and original soliton types using the Sardar sub-equation and the modified Kudryashov methods, which have not been applied to this model before. We obtained dark, bright, periodic-singular and periodic-M-shaped soliton solutions, respectively. The analytical forms of the obtained solutions are represented by 3D, 2D and contour graphics. In addition, the physical effects of the solution parameters on the wave envelope have been described and clearly interpreted by presenting their 2D graphics.

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.

The analysis of solitonic, supernonlinear, periodic, quasiperiodic, bifurcation and chaotic patterns of perturbed Gerdjikov–Ivanov model with full nonlinearity

In this manuscript, the perturbed Gerdjikov–Ivanov equation is considered for the investigating model. It is studied to govern the dynamics of soliton propagation through optical fibers, metamaterials or PCF. The observed model is subjected to an extended simple equation technique, which disclosed an abundant of exact solutions including trigonometric function, singular, kink, peakon and compacton solutions. These solutions are made available with their essential conditions, which guarantee the persistence of such optical solitons and portraits using appropriate physical parameters in 3D, 2D and density plots. After that, the traveling wave transformation is used to turn the mth-order nonlinear perturbed Gerdjikov–Ivanov equation into a planar dynamical system. The dynamical and chaotic behaviors of the considered equation are also discussed. The qualitative analysis of the dynamical system and the chaotic behaviors of the perturbed system are investigated using the theory of plane dynamic systems. In addition, we utilize the RK4 method to discover patterns in the dynamical system that seem to be super nonlinear, periodic, and quasiperiodic. The reported results are new and have not been investigated. They can be used in the explanation of the physical phenomena modeled and will give information about the long-term dynamic behavior. Numerical simulations reveal that by changing the frequency and amplitude parameters have an impact on the dynamic behaviors of the system. It is established that the extended simple equation method and dynamical observations offer further influential mathematical tools for constructing exact solutions and their qualitative analysis in NLEEs in mathematical physics.

Computational extracting solutions for the perturbed Gerdjikov-Ivanov equation by using improved modified extended analytical approach

In this work, the improved modified extended tanh function method will be applied to study the perturbed Gerdjikov-Ivanov equation (PGI) which describe the dynamics of the soliton in optical fibers. New exact solutions for the PGI equation will be introduced including Weierstrass elliptic, Jacobi elliptic, dark and bright solitons, singular periodic and rational type solutions. Moreover, for the physical illustration of the obtained solutions, 3D and 2D graphs are presented.

Solitons and modulation instability of the perturbed Gerdjikov–Ivanov equation with spatio-temporal dispersion

This paper is exploring the optical solitons to perturbed Gerdjikov–Ivanov (GI) equation in optical fibers. Using traveling wave transformation, the complex form of perturbed GI equation is reduced to a nonlinear ordinary differential equation (ODE). Then, the improved projective Riccati equations method has been implemented to solve the ODE analytically. Different types of solutions such as bright, kink and singular optical soliton structures are obtained. In addition, W-shaped, kink-dark and singular-kink waves are obtained under specific values for the physical parameters in the model. The existence conditions of all optical solitons are given. Also, the behaviors of some optical solitons are illustrated graphically in this paper by selecting appropriate values for the physical parameters. Besides, the modulation instability of the perturbed GI equation is reported.

Abundant fractional soliton solutions of a space-time fractional perturbed Gerdjikov-Ivanov equation by a fractional mapping method

A new fractional mapping method based on a generalized fractional auxiliary equation is proposed and applied to solve the space-time fractional perturbed Gerdjikov-Ivanov equation. The main feature of this approach is to obtain more accurate solutions by means of an auxiliary equation. Some exact fractional nonlinear wave solutions, including bright soliton, periodical wave and singularity soliton solutions are constructed by Mittag–Leffler function. Some deformations appear in those fractional nonlinear wave solutions, and those deformations become more obvious with the increase of the fractional order parameter. In addition, the coefficient of group velocity dispersion and the self-steepening for short pulses also affect the intensity of the soliton when the fractional order parameter remains unchanged. The effect of fractional order is explained by the graphical representation of a series of solutions and their physical meanings.

Classification of All Single Traveling Wave Solutions of Fractional Perturbed Gerdjikov–Ivanov Equation

The fractional perturbed Gerdjikov–Ivanov (pGI) equation plays a momentous role in nonlinear fiber optics, especially in the application of photonic crystal fibers. Constructing traveling wave solutions to this equation is a very challenging task in physics and mathematics. In the current article, our main purpose is to give the classifications of traveling wave solutions of the fractional pGI equation. These results can help physicists to further explain the complex fractional pGI equation.

Optical solitons to a perturbed Gerdjikov-Ivanov equation using two different techniques

In this article the perturbed Gerdjikov-Ivanov (GI)-equation which acts for the dynamics of propagation of solitons is employed. The balanced modified extended tanh-function and the non-balanced Riccati-Bernoulli Sub-ODE methods are used for the first time to obtain the new optical solitons of this equation. The obtained results give an accuracy interpretation of the propagation of solitons. We held a comparison between our results and those are in the previous work. The efficiency of these methods for constructing the exact solutions has been demonstrated. It is shown that these different technique's reduces the large volume of calculations.

Abundant wave solutions of the perturbed Gerdjikov–Ivanov equation in telecommunication industry

In this paper, the solitary wave solutions of the complex perturbed Gerdjikov–Ivanov (CPGI) equation are investigated by employing the extended simplest equation (ESE) and modified Khater (MKhat) methods. The studied equation describes the physical characterization of the optical soliton waves to mitigate internet bottlenecks with many different applications in the telecommunication industry. The evaluated solutions are explained through various sketches in two-, three-dimensional, and contour plots of their real, imaginary, and absolute values. The novelty of the obtained results is demonstrated by comparing them with the previously obtained solutions. The computational applied schemes’ performance is tested to illustrate their powerful and effective handling of many nonlinear evolutions equations.

Novel optical solitons to the perturbed Gerdjikov–Ivanov equation via collective variables

The objective of this manuscript is to study the collective variable (CV) technique to explore an important form of Schrödinger equation known as the Gerdjikov–Ivanov (GI) equation which expresses the dynamics of solitons for optical fibers in terms of pulse parameters. These parameters are temporal position, amplitude, width, chirp, phase, and frequency known as CVs. This is an effective and dynamic mathematical gadget to obtain soliton solutions of non-dimensional as well as perturbed GI equations. Moreover, an established numerical scheme that is the fourth-order Runge–Kutta method is exerted for the numerical simulation of the revealing coupled system of six ordinary differential equations which represent all the CVs included in the pulse ansatz. The CV approach is used to determine the evolution of pulse parameters with the propagation distance and illustrated them graphically. Furthermore, figures show the compelling periodic oscillations of pulse chirp, width, frequency and amplitude of soliton. For various values of super-Gaussian pulse parameters, the numerical behavior of solitons to illustrate variations in CVs is provided. Other significant aspects with regards to the current investigation are also inferred.

Exact single travelling wave solutions to the fractional perturbed Gerdjikov–Ivanov equation in nonlinear optics

In this paper, exact single travelling wave solutions to the nonlinear fractional perturbed Gerdjikov–Ivanov equation are captured by the complete discrimination system for polynomial method and the trial equation method. In the classification, we can find out the original equation has rational function solutions, solitary wave solutions, triangular function periodic solutions, and elliptic function periodic solutions, which are normally very difficult to be obtained by other methods. In particular, the concrete parameters are set to show that the solutions in the classification can be realized in almost all cases.

Closed Form Solutions of the Perturbed GerdjikovIvanov Equation With Variable Coefficients

Closed Form Solutions of the Perturbed GerdjikovIvanov Equation With Variable Coefficients

Generation of wing-shaped dark soliton for perturbed Gerdjikov–Ivanov equation in optical fibre

This paper investigates the dark soliton solutions to the perturbed Gerdjikov–Ivanov (G–I) equation describing the effects of ultra-short (femto second) optical soliton propagation in non-kerr media. The ordinary differential equations (ODE) are solved analytically by implementing the methods, such as, extended rational sinh-cosh and sin-cos method. Consequently, various types of dark solitons such as wing shape, double wing shape, broadening and steepening of dark optical solitons are obtained.