Soliton solutions of fractional extended nonlinear Schrödinger equation arising in plasma physics and nonlinear optical fiber

In this research, we study traveling wave solutions to the fractional extended nonlinear SchrÖdinger equation (NLSE), and the effects of the third-order dispersion parameter. This equation is used to simulate the propagation of femtosecond, plasma physic and in nonlinear optical fiber. To accomplish this goal, we use the extended simple equation approach and the improved F-expansion method to secure a variety of distinct solutions in the form of dark, singular, periodic, rational, and exponential waves. Also, the stability of the outcomes is effectively examined. Several graphs have been sketched under appropriate parametric values to reinforce some reported findings. Computational work along with a graphical demonstration confirms the exactness of the proposed methods. The issue has not previously been investigated by taking into account the impact of the third order dispersion parameter. The main objective of this study is to obtain the different kinds of traveling wave solutions of fractional extended NLSE which are absent in the literature which justify the novelty of this study. We believe that these novel solutions hold a prominent place in the fields of nonlinear sciences and optical engineering because these solutions will enables a through understanding of the development and dynamic nature of such models. The obtained results indicate the reliability, efficiency, and capability of the implemented technique to determine wide-spectral stable traveling wave solutions to nonlinear equations emerging in various branches of scientific, technological, and engineering domains.

www.nature.com/scientificreports/ The femtosecond pulse propagation in monomode optical fibre is expressed by the fractional extended NLSE [61][62][63][64] as follows: where v = v(x, t) denotes the electric field, γ 1 shows the inverse of speed, γ 2 represents the second-order dispersion unknown, γ 3 denotes the third-order dispersion variable (TOD), ψ 1 is the coefficient of the derivative cubic term, ψ 2 is for soliton self-frequency shift and β is the effective nonlinear coefficient. In this study, we apply the extended simple equation approach and the improved F-expansion method to obtain the wave solutions for Eq. (1). The strength of the entire optical framework is determined by the TOD parameter and well-known effects, which also serve as conditional constraints. Securing results for this model that control these occurrences is crucial for a thorough understanding of these physical processes.
The remainder of the paper is summarized after the introduction section. In "Conformable fractional derivative", we discuss conformable fractional derivative. The improved F-expansion scheme and extended simple equation method are discussed in "Algorithm of the methods" of this article. In "Extraction of wave solutions", we use the aforementioned techniques to extract the periodic type, dark type, singular, and rational soliton solutions of the Eq .(1). In "Stability analysis", we discuss the analysis of solutions. Graphical representations and results of a few of the generated results are explained in "Results and discussions". Lastly, we present conclusion in "Conclusion".

Conformable fractional derivative
Assume that D σ η is a differential operator of any order, such as 0 < α ≤ 1 . Then conformable fractional derivative of V (η) is given by Following are some characteristics of this definition: Theorem 1 Suppose that function v(η) and w(η) are σ −differentiable at η > 0 with σ ∈ (0, 1] , therefore Theorem 2 Assume that v(η) is both differentiable and sigma-differentiable in the range σ ∈ (0, 1] . Furthermore, let v(η) be a differentiable function with the same range v(η),

Algorithm of the methods
The fractional order nonlinear equation with the spatial variable x and time variable t is as follows: where v(x, t) represents the unknown function , and J is a polynomial with fractional partial derivatives in v(x, t). The wave variable should be changed as follows: Here n, α and β are non-zero free unknowns to be evaluated. By switching (4) into (3), we get The improved F-expansion scheme. The circumstances for the improved F-expansion scheme are described in the following phases. Ist Step: The solution of Eq. (1) is presumable in the form that follows the improved F-expansion method.
. where m represents the real part of the equation and the prime represents derivatives with respect to χ . The three general solutions of the Riccati equation Eq.  (5), it is possible to calculate the polynomial in F(χ) . An algebraic system of equations is therefore produced when the same index of F(χ) is equal to zero. By using Mathematica to solve these equations, we can get the values of the unknowns p i , q i , m, and c, which will be utilized to obtain the answer to Eq. (3).
The extended simple equation method. We suppose the trial solution of the partial differential equation (PDE) of Eq. (1) that can be expanded in series as follows in order to achieve various results.
where b i = 0 is a constant to be determined later and N is a positive integer that can be calculated by applying the balancing principle to Eq. The general solution of Eq. (14) is We equal to zero all the accumulated factors of powers of R(χ) after switching Eqs. (13) and (14) into Eq. (5) and performing various calculations. We obtain an algebraic system of equations. It satisfies the requirement that the final solutions of Eq. (3) be determined.

Extraction of wave solutions
In this section we tackle new traveling-wave solutions for the extended NLSE equation which profoundly relates to superconductivity, plasma physics and non-linear optics. From Eq. (1) with the assistance of Eq. (4) is separated into the following imaginary and real parts: The Eq. (23) is the nonlinear ordinary differential equation of Eq. (1), and Eq. (22) is integrated with χ , with zero as the integration constant. So, the Eq. (22) becomes, The following proportion is written using the homogeneous balance between Eqs. (23) and (24): From the aforementioned proposition, one can arrive at the following constraints: Application of improved F-expansion method. Following the balancing principle of the terms V ′′ and V 3 in Eq. (23), we obtain N = 1. Now, by combining Eq. (6) and the solutions of Eq. (7) in Eq. (23) and then using Mathematica, we obtain
Application of extended simple equation method. Switching the Eq. (13) togather with the solutions of Eq. (14) in Eq. (23) and then using Mathematica, we get If we take c 1 = 0 , using Eqs. (40) and (13), we finally arrive at the solutions for Eq. (1) given by Case-II: When c 0 , c 1 , c 2 , the solutions of Eq. (1) are Family 5

Stability analysis
This section examines the stability of the computed solutions to demonstrate their suitability for model applications. Using the characteristics of the Hamiltonian system to handle Eq. (36) under the following conditions in momentum having the following form: Thus, N represents momentum and v represents the potential of the electric field. The prerequisite for soliton stability is where n is the wave velocity. As a result, the following formulation is used to investigate the stability property of Eq. (36).
Thus, we find n .

Results and discussions
In this section, a comprehensive comparison of the evaluated results is made with the existing computed outcomes, which highlights the novelty of the current study. It is noticed that Ozisik et al. 61 calculated only a few numbers of solutions by using the modified F-expansion method. But we have constructed an abundance of traveling wave solutions in this article by using the extended simple equation approach and the improved F-expansion method. Several of our outcomes diverge from those mentioned in 61 if we compare our achievement with their results. Even so. if we give various values to the components involved, we can obtain some similar outcomes. It is crucial to note that the achievements of this article are practical, compact, eloquent, and straightforward to understand when it comes to nonlinear wave applications. It could also be used in plasma physics, semiconductor materials, optical fiber communications, ultrashort pulses, and other nonlinear optical phenomena, etc. In order to illustrate the relationship between two or more variables in a data set using a graph, a plot is a graphical process. Graphical description is a crucial technique for accurately representing nonlinear events. Plots of the nonlinear equation solutions are crucial for revealing the internal dynamics of various nonlinear processes. The numerical simulations of a few solutions that were discovered are provided in this section by selecting appropriate values for arbitrary parameters. The soliton has the ability to keep its amplitude, velocity, and form constant throughout its propagation. These reported solutions have some physical meaning for instance dark soliton is a soliton whose intensity is lower than the background and which isn't produced by a typical pulse but rather is basically devoid of energy in a continuous time beam. There are further types of solitary waves called singular solitons that have singularities, typically infinite discontinuities. Singular solitons might be linked to solitary waves when the location of the center of the solitary wave is imaginary. Therefore, discussing the topic of singular solitons is relevant. This type of solution contains spikes and therefore may recommend a description for the development of rogue waves. Periodic wave solution describes a wave with repeating continuous pattern, which determines its wavelength and frequency, while period defines as time required to complete cycle of waveform and frequency is a number of cycles per second of time. Here, we plotted various wave profiles that were extracted from the solutions to Eq. (1) and are shown in 3D and 2D. The Fig. 1 shows the periodic behavior E q . The 2D wave profiles of the obtained solutions have been sketched for various values of σ and δ to demonstrate the effect of the fractional derivative on the dynamic behavior of the waves. From the figures, it is observed that the fractional order has a significant impact on the characteristics of the wave profiles via the memory effect phenomenon, which means that the signal takes into account its past evolution at any point; acting on this parameter allows having better and more complete information about the shape of a signal or a pulse.

Conclusion
In this paper, we study the improved F-expansion and extended simple equation techniques to obtain the soliton solutions of modified extended NLSE with conformable fractional derivatives arising in mono-mode optical fibers. The somatic perspective of the derived wave solutions is illustrated in Figs. 1. 2, 3, 4, 5, 6 and 7 which is useful to comprehend the visuals of solitons and the effects of the third-order dispersion component. Also we obtained solutions of Eq. (1) in the form of exponential, rational, periodic, hyperbolic, and trigonometric functions. The outcomes are a collection of new, extended NLSE solutions, where the suggested methods showed to be more reliable, accurate, and effective. The obtained results and figures Figs. 1, 2, 3, 4, 5, 6 and 7 conclude that, the fractional parameter σ and δ plays the main rule in the solutions. We believe that the travelling wave solutions obtained in this paper should have significant applications in the field of sciences such as plasma physics and compact astronomical phenomena such as femtosecond pulse propagation in monomode optical fiber and ultrashort pulses. Also, the resulting soliton solutions secured in this study are encouraging and will benefit the community of researchers. In future the investigated model will be solved by some others powerful techniques.

Data availability
We have provided all the data within the article.