Distinct exact solutions for the conformable fractional derivative Gerdjikov-Ivanov equation via three credible methods

This paper reveals the conformable fractional space-time perturbed Gerdjikov-Ivanov (GI) equation, which is applied to nonlinear fibre optics together with photonic crystal fibres. The foremost intent of our operation is adopting the unified method, the modified F-expansion method and the modified Kudryashov method to hunt for definite solutions to the equation. In addition, by selecting fairish values, fluctuation behaviours of the solutions are drafted.

In this paper, on the basis of extant results, we will apply dissimilar approaches to dissect the (1 + 1)-dimensional conformable fractional space-time perturbed GI equation [39,40] which delineates, where D α t and D α x are the conformable fractional derivative terms, u = u(x, t) is a complex function that stands for some point of view the amplitudes of circularlypolarized waves in a nonlinear optical fibre, u * is the conjugate function of u, δ is the inter-modal dispersion, λ indicates self-steepening for short pulses, μ is the higher-order dispersion effect, α ∈ (0, 1), a, b, c are nonzero constant functions and i = √ −1.The fundamental objective of this work is to generate new exact solutions of Equation ( 1) on the strength of the unified method, the modified F-expansion method and the modified Kudryashov method.What is more, we test the validity of acquired solutions and compare the qualities of solutions under the aforementioned three different methods through graphical portrayal.We expect these new analytical solutions that belong to distinct classes are capable of applying in diversified branches of science and our results will go a long way in helping the exploration of the complicated physical world.
The outline of this paper is layout as follows: Section 2, it not only comply the alteration of Equation (1), but also makes the real and imaginary parts of the equation deviate.Section 3, tring to seek precise solutions of Equation (1) via the unified method.Section 4, in line with the modified F-expansion method, we can deal with Equation (1).Section 5, applying the modified Kudryashov method to reap accurate solutions and draw graphs for analysis.Finally, Section 6 is devoted to presenting the conclusion.

Mathematical analysis
First of all, we give the elementary definition of a conformable derivative [41].Definition 2.1: Let f : (0, ∞) → R be a function, then the conformable derivative (CD) of f is defined as, for all time variable t > 0 and fractional order α ∈ (0, 1]. Notably, if f in some (0, a), a > 0 is α-differentiable and lim t→0 + D α t f (t) exists, then the following relation is extended, It is worth to be noticing that we ought to convert Equation ( 1) into an ordinary differential equation of integer order in the first place.It can be described as follows: (1) Take into account the general structure of nonlinear temporal fractional differential equations in a mode given by, where f is a function in its parameters.(2) Utilize the transformation u(x, t) = U(ξ ) in the governing model, leads to the following ordinary differential equation, where U = U(ξ ) and ξ is a single wave variable.

The unified method
Here, we attempt to find the polynomial function solutions for Equation (1) via the unified method.We assume, (10) where b i and p i are arbitrary functions to be determined later.
(1) Equate the coefficients of ϕ(ξ ) and its exponential terms to zero.(2) Solve the auxiliary equations, namely, the formulas in the second row of Equation ( 10).(3) Receive the exact solutions supposed by Equation (10).
It is worthwhile referring to that the relationship of n, k is ascertained by UM in accordance with the balancing principle between the highest order derivative and nonlinear terms.For Equation (9), we balance the terms VV and V 4 , namely the terms V and V 3 , gives n = k−1.Analogously, the consistency condition ensures that the arbitrary coefficients that appeared in Equation ( 10) are capable of being consistently determined.

Solitary solution
In view of the unified method, when k = 2, the solitary solution of Equation ( 9) is able to depicted as, After that, inserting Equation ( 11) into Equation ( 9) and allowing every coefficient of φ i (ξ ) to zero, we can receive, where ab < 0.

Soliton solution
As said in the front of the subsection, in the case of k = 2, the soliton solution of Equation ( 9) is expressed as, Later, putting Equation ( 17) into Equation ( 9) in the same pattern, we have the following results, where abb 2 < 0. Through Equations ( 17), ( 18) and ( 9), the solution of Equation ( 1) is written as, where In the case of k = 3, the soliton solution of Equation ( 9) is presumed as, On the basic of the algorithm, we have, Putting these values in Equation ( 9) along with Equation (20).Next, the solution of Equation ( 1) can be derived as,
Using Equations ( 23) and ( 24), computes, where When k = 3, let us now suppose another solution for Equation (9) in the form, As we did in the former section, by symbolic calculation we obtain, where abb 6 < 0.
From what has been stated above, we figure out the solution of Equation ( 1), where 1 depicts the solitary wave solutions of Equation ( 1) under certain circumstances.(a) -(d) are rogue waves, and when t → 0, the value of u tends to be stable.In (e), when x > 0, we have u ≥ 0. Figure 2 portrays the soliton solutions of Equation ( 1).At t = 3, it is a dark soliton solution in (a) -(c).It is a rogue wave formed by multiple bright and dark solitons in (d)-(e).It can be seen from the figure that its minimum value is greater than 0, and the fluctuating values are mainly distributed in −3 < t < 3. Figure 3 denotes the elliptic solutions of Equation (1).In (a) -(d), when t = 0 and x = 0, it is a dark soliton solution.(e) state that the value of u is mainly concentrated at x > 5 and t > 0.

The modified F-expansion method
In what follows, we will adopt the modified F-expansion method to cope with Equation ( 9).The particulars have appeared in the following paragraph.As we showed in the unified method, we will also talk about the solutions in two categories.

When k = 2
Advancing as usual, in the case of k = 2, setting the solution of Equation ( 9) to the following situation, Hereafter, substituting Equation ( 29) into Equation ( 9) and making the coefficients of the acquired algebraic equations 0, we figure out, Taking Equation (30) with Equation ( 29) into considering leads us to the following results,

When k = 3
According to the algorithm, in the case of k = 3, the assumption of the solution is, ( Proceeding as always, placing Equation (32) into Equation (9) and ascertaining all the coefficients ahead of φ i (ξ ) are equal to 0 by making use of symbolic calculation.We will achieve the three distinct solutions as shown below.
Case 1: b 0 = 0, we have, Accordingly, employing Equations ( 32) and ( 33) add to Equation ( 9), which admits to, where As a consequence, the solution of Equation ( 1) corresponds to, where Generally speaking, for the mentioned above solutions appear, we receive, 4 presents the rogue wave.when t = 0, the generated wave is antisymmetric on the x-u plane.Figure 5 reveals another solution of Equation ( 1).(a) -(c) are produced alternately by bright and dark solitons.In (d) and (e), they arise from the interaction of periodic waves and solitons.In (f), when t = 0, the value of u near the origin of x changes rapidly.

The modified kudryashov method
In this section, we will use another approach, that is the modified Kudryashov method to deal with Equation (9).In the light of this method, we can write, (39) By an analogous way as we expressed in the preceding parts, inserting Equation (39) into Equation ( 9) and collecting all of the terms with the same order of φ i (ξ ), we have the following three diverse results,  Family 1: Later on, which calculates, where Family 2: And we have, where T = which yields another solution of Equation ( 9) is revealed legibly as, where ξ = k( x β β − υ t α α ) and η = −κ x β β + ω t α α + θ .Figure 6 demonstrates the miscellaneous solutions of Equation (1).From the drawing above, It is not difficult for us to find that they are all antisymmetric waves in the x-u plane when t = 0.Among them, (a)-(c) are formed by the combination of periodic waves and bright soliton waves.(d)-(f) show the rogue waves, when x > 0, we have u > 0. (g) -(i) suggest that they are produced by the interaction of bright and dark solitons.

Results and discussion
Osman, et al applied the generalized Riccati equation together with the basic simplest equation method to obtain novel optical solitons solutions for the Perturbed Gerdjikov-Ivanov equation with truncated M-fractional derivative in 2020 [42].S Saha Ray dealt with the solutions of the Gerdjikov-Ivanov equation with the Riesz fractional derivative by means of the time-splitting spectral approach.Furthermore, an implicit finite difference method is utilized here to compare the results with the above-mentioned method [43].M Li, R Ye and Y Lou used the Darboux transformation to obtain the brightdark soliton, breather, rogue wave, kink, w-shaped soliton and periodic solutions of the nonlocal GI equation by constructing its 2n-fold Darboux transformation [44].
In this paper, we use three methods to obtain different solutions to the GI equation.The bright soliton solution exhibits a convex isolated waveform with a peak in the middle, and the attenuation on both sides is relatively smooth.The dark soliton solution is a concave isolated waveform.The singular wave solution is a finite duration pulse shape in time, usually consisting of a single or multiple wave packets.Periodic wave solutions exhibit recurring characteristics in both space and time.The shape of these solutions is determined by the specific form and parameters of the nonlinear equation, so there may be some changes in specific situations.
In optical systems, the propagation of bright soliton solutions is achieved by balancing the nonlinear self focussing effect and linear dispersion effect.When the nonlinear effect and dispersion effect cancel each other out, the bright soliton solution can maintain its amplitude and shape to propagate in the medium without diffusion.The propagation mechanism of dark soliton solutions is opposite to that of bright solitons.Periodic solutions are soliton solutions that recur in a periodic manner.Its formation is usually achieved by balancing nonlinear effects and other modulation mechanisms such as periodic potential fields.The propagation of singular solutions will be more complex, which usually requires more in-depth mathematical analysis.These solutions may have unbounded amplitudes and gradients at some positions or time points, and may involve multiple factors such as nonlinear effects, phase modulation, and interactions with other solutions.
This paper uses three different methods to solve Equation (1).A unified method is a method of categorizing different methods into a unified form.Its advantages are simplicity, intuition, and a wide range of applications.However, this method may require complex mathematical transformations and variable substitution, and in some cases it may not be possible to find a suitable unified form.The advantage of the modified F-expansion method is that it can improve convergence and accuracy by adjusting the parameters in the F-expansion method.However, this method may require complex calculations and iterative processes, and for some nonlinear equations, analytical expression may not be obtained.The advantage of the modified Kudryashov method is that more accurate analytical expression can be obtained, and it can be applied to some problems that are difficult to be solved by other methods.However, this method may not be applicable to general nonlinear differential equations, and in some cases, complex mathematical derivation and calculation may be required.

Conclusion
Here, we have effectively constructed a chain of exact solutions for the (1 + 1)-dimensional conformable fractional space-time perturbed GI equation by employing the unified method, the modified F-expansion method and the modified Kudryashov method.Furthermore, forcing apposite values in the solutions obtained, we can receive explicit graphs of nonidentical solutions.The solutions of the GI equation can play an important role in the field of optics, such as optical fibre beam transmission, optical fibre sensing and measurement, design and application of fibre gratings, design and optimization of fibre lasers, and so on.The issue of solving NLPDEs is an intriguing and practical one that could be validly probed in further investigation.