Abundant M-fractional optical solitons to the pertubed Gerdjikov-Ivanov equation treating the mathematical nonlinear optics

In this paper, the perturbed Gerdjikov-Ivanov (GI) equation using a truncated M-fractional derivative is studied in mathematical nonlinear optics. We explore its novel dark and other soliton solutions and compared them with the existing results. To obtain the objective, two particular methods, modiﬁed extended tanh expansion method and Exp a function method, are implemented. In this exert, a arrangement of exact solitons are received as well as veriﬁed by utilizing the MATHEMATICA software. The dynamical characteristics of the obtained results, along with a fractional parameter, are also discussed via two and three-dimensional graphs. These solutions suggest that the employed methods are impressive, determined and smooth as compared to many other methods. The work of this paper is of high importance regarding its applications in photonic crystal ﬁbers and mathematical physics.


Introduction
Many phenomenon are often described in the form of nonlinear Schrödinger (NLS) equations. Then those NLS equations are studied for different purposes with different approach. These have much importance in the different fields of science and engineering. In recent years, to solve the different NLS equations different analytical and numerical approaches have been developed [22][23][24][25]. The NLS equations that are mostly studied are those have cubic nonlinearity. For example, NLS equation along quintic non-linearity is the perturbed GI equation. Various approaches have been applied to find different exact solitons of this equation. For Instance, two types of bright wave solutions of perturbed GI equation have been obtained with the help of semi-inverse variational method [16]. Distinct solitons are investigated by applying the sine-Gordon equation method [17]. Biswas et al. have obtained the singular and bright solitons with the use of extended trial equation approach for the perturbed GI equation [18]. Different solitary wave solutions of the perturbed GI equation have been found by the implementation of the exp(ϕ(ξ))−expansion and the Kudryashov techniques [19]. Various wave solutions have obtained by applying the two different method, exp(ϕ(ξ))−expansion method and (G ′ /G 2 )−expansion technique [20]. Different optical solitons have been determined by using the exp a function method and the modified Kudryashov method [21]. Recently, some novel wave solutions were investigated by using the generalized exponential rational function method [26]. Besides of these methods, there are two methods that are more reliable, simple and useful named as: modified extended tanh expansion method (EThEM) and Exp a function method. The modified extended tanh expansion technique [1] has been used to discuss Biswas and Arshed model with non-linearity factor "n". The different solitons of (2 + 1)dimensional integrable nonlinear Schrödinger equation were explained in [2]. Various solitons of new coupled evolution equation were explained [4]. Optical soliton solutions of the travelling wave nonlinear equations have been determined in [5][6][7][8]. Explicit exact solitons of two nonlinear Schrödinger equations have investigated through two different techniques [13]. Similarly, this techniques have been applied to solve the other many NLS equations [9,11]. The main task in this research is to search some new dark and other optical soliton solutions of the perturbed GI equation with truncated M. fractional derivative. The modified extended tanh expansion method and Exp a function method are employed to acquire the aforesaid task.

The governing model and the mathematical analysis
The perturbed Gerdjikov-Ivanov (GI) equation can be read as [15][16][17][18][19][20][21]: here g = g(x, t) shows the complex-valued wave function, depends on independent variables x and t. In Eq. (1), τ 1 shows the coefficient of GVD, τ 2 represents the coefficient of the quintic non-linearity of the model and τ 3 indicates the coefficient of the nonlinear dispersion term. Moreover, the parameters θ 1 , θ 2 and ρ represent the perturbation effects. Finally, the term g * shows the complex conjugate of g. Eq. (1) with truncated M-fractional derivative is given as: where that E β (.) is a truncated Mittag-Leffler function of one parameter [14]. Now by using the complex constraints conditions given in the following: Here λ indicates the phase component, σ 1 shows the frequency of the wave solutions and σ 2 represents the wave number of solitons. By putting the Eq. (4) into the model Eq.
(2), we get: Real part: Imaginary part: From Eq.(6), we get: Using the terms G ′′ and G 5 and the homogenous balance approach, we observe m = 1/2. Therefore, we use the, we use the following transformation to get solution in retrieve form: By using the Eq. (8) into Eq. (5), yields

Application of the modified EThEM:
Here, a quick review of the said method and its implementation both are explained. Let's assume the below non-linear PDE of the form: here u. Let us assume the below wave transformation: Here ν shows the wave velocity. Putting the Eq. (11) into Eq. (10), taking the following nonlinear ODE: Here primes represents the derivatives w.r.t ζ.
Moreover, consider the solution of Eq. (12) is of the form: In Eq. (13), α 0 , α n , β n , (n = 1, 2, 3, ..., m) are undetermined and to be find later. Notice that α n and β n are not both zero at a time. By balancing nonlinear term and the highest derivative in Eq. (12), we get m.

Application of Exp a function approach
Here, we recall the main points of the aforesaid approach and then its demonstration has been exercised for required solutions. let's assume we have a NPDE in the following form: The above PDE given in the Eq.(41) may be obtained in the below form of NODE : by implementing the below wave transformations: Let us suppose a solution of Eq. (12) is of the below form [9,10,12]: here α i and β i (0 ≤ i ≤ m) are unknown constants and to be find later. Positive integer m is obtained by using the homogenous balance technique and balancing the highest derivative and nonlinear term in the Eq. (12). Putting Eq. (44) into non-linear Eq. (12), yields Putting ℓ i (0 ≤ i ≤ t) in Eq. (45) equal to zero, a set of algebraic equations is gained as follows.
By this obtained sets, we get the nontrivial solutions of the NPDE (41).

Conclusion:
In this paper, the perturbed Gerdjikov-Ivanov (GI) equation with conformable M-fractional derivative has been worked out and found its dark and other optical soliton solutions. For this, a fractional wave transformation was used for reducing the perturbed GI equation with truncated M -fractional derivative into a nonlinear ODE. Then, by applying modified extended tanh expansion and the Exp a function methods, the novel soliton solutions are obtained. The achieved results are verified by software computation as well as explain with the help of 2-dimensional and 3-dimensional graphs.