Optical Solitons for the Fokas-Lenells Equation with Beta and M-Truncated Derivatives

The Fokas-Lenells equation (FLE) including the M-truncated derivative or beta derivative is examined. Using the modi ﬁ ed mapping method, new elliptic, hyperbolic, rational, and trigonometric solutions are created. Also, we extend some previous results. Since the FLE has various applications in telecommunication modes, quantum ﬁ eld theory, quantum mechanics, and complex system theory, the solutions produced may be used to interpret a broad variety of important physical process. We present some of 3D and 2D diagrams to illustrate how M-truncated derivative and the beta derivative in ﬂ uence the exact solutions of the FLE. We demonstrate that when the derivative order decreases, the beta derivative pushes the surface to the left, whereas the M-truncated derivative pushes the surface to the right.

In this study, we consider Eq. ( 1) with two different time derivative operators as follows: With beta derivative operator, Eq. ( 1) takes the form where D α t is the beta derivative (BD) operator.And with M-truncated derivative operator, Eq. ( 1) takes the form where D α,σ m,t is the M-truncated derivative (MTD) operator.The novelty of this study is to find of the exact solutions of FLE ( 2) and (3).In order to reach these solutions, we use a modified mapping method (MM-method).We extend some previous results such as [24,27].Using the BD in Eq. ( 2) and the MTD in Eq. ( 3), the solutions would be very helpful to physicists in characterizing a wide variety of important physical processes.To further explore the effect of the BD and MTD on the acquired solution of FLE ( 2) and (3), we present some figures constructed in MATLAB.
The study's structure is as follows: In Section 2, we define the BD and MTD and state their prosperities.In Section 3, we explain the modified mapping method, while the wave equation of FLE-MTD (2) is obtained in Section 4. In Section 5, we get the exact solutions of the FLE-MTD (2).In Section 6, we can observe how the BD and MTD affect the obtained solutions of FLE-MTD (2).Lastly, the findings of the study are given.

Preliminaries
Recently, the fractional NEEs have increased in popularity owing to their broad variety of applications in domains such as biological population, signal processing, plasma physics, electrical networks, fluid flow, solid state, finance, chemical kinematics, optical fiber, and control theory physics.Various types of fractional derivatives were introduced by different mathematicians.The most prominent are those suggested by Caputo, Grunwald-Letnikov, Hadamard, Erdelyi, Riemann-Liouville, Marchaud, and Riesz [31][32][33][34].The bulk of fractional derivatives does not involve the standard derivative rules including the product rule, chain rule, and quotient rule.
2.1.Beta Derivative.Atangana et al. [35] developed a novel operator derivative known as BD.The BD [35] is defined for u 0,∞ ⟶ ℝ as Moreover, for any constants a and b, the BD has the following features [35]: 2. M-Truncated Derivative.Sousa and de Oliveira [36] proposed another derivative known as the MTD.The MTD of order α ∈ 0, 1 is defined as where , for σ > 0 and y ∈ ℂ 7 The MTD satisfies the following characteristics [36]:

The Clarification of MM-Method
Here, we implement the MM-method from [37].Let the solutions to Eq. ( 19) take the form where ℓ j and ℏ j are the undetermined constants for j = 1, 2, , M and φ solves where the constants r, q, and p are real numbers.Equation (10) has different solutions for r, p, and q as follows: sn η = sn η, κ ,dn η, κ = dn η, κ , and cn η = cn η, κ are the Jacobi elliptic functions (JEFs) for 0 < κ < 1 When 2 Journal of Function Spaces κ ⟶ 1, the following hyperbolic functions are produced from JEFs: Moreover, when κ ⟶ 0, the following trigonometric functions are produced from JEFs:

Traveling Wave Equation for FLE
To get the wave equation for FLE (2)/(3), we use where Ψ is a real function and η α and μ α are defined as follows: (i) In terms of beta derivative where μ 1 , μ 2 , η 1 , and η 2 are the nondefined constants.Putting Eq. ( 13) into Eq.( 2)/(3), we have the following system: From imaginary part (17), we obtained while the real part is given by where

Exact Solutions of FLE
To determine the value of M defined in Eq. ( 9), we balance Ψ 3 with Ψ′′ in Eq. (19) as Table 1: All solutions for Eq. ( 10) for different r, p, and q.

Effects of the Beta and M-Truncated Derivatives
Now, we examine the effect of the BD and MTD on the obtained solutions of the FLE (2)/(3).A number of diagrams are presented to illustrate how these solutions behave.For specific achieved solutions including ( 29), (40), and (43), let us fix the parameters t ∈ 0, 2 , and x ∈ 0, 4 to plot these graphs.
6.1.The Effect of Beta Derivative.From Figures 1-3, we infer that all solution curves differ from one another.Furthermore, the surface shifts to the right as the order of the derivative decreases.
6.2.The Effect of M-Truncated Derivative.Finally, from Figures 4-6, we deduce that all solution curves are distinct from one another.Furthermore, the surface shifts to the left when the derivative's order decreases.

Results and Discussion
The As a result, we obtained the exact solution for FLE including the M-truncated derivative or beta derivative.Utilizing the modified mapping method, new elliptic, hyperbolic, rational, and trigonometric solutions are acquired.For some fixed parameters and for various order of fractional   Journal of Function Spaces derivatives, we plotted many graphs to display the impacts of fractional derivatives on the solutions.We deduced that when the derivative order decreases, the beta derivative pushes the surface to the left as shown in Figures 1-3, whereas the M-truncated derivative pushes the surface to the right as shown in Figures 4-6.

Conclusions
We looked at the Fokas-Lenells equation (FLE) with the beta and M-truncated derivatives.The exact solutions of FLE were acquired through the implementation of a modified mapping method.These results are vital in clarifying a broad variety of interesting and difficult physical processes.Furthermore, we extended some previous results such as the results stated in [24,27].In addition, the beta and M-truncated derivative impacts on the exact solution of FLE (2)/(3) were addressed by using the MATLAB program.Finally, we deducedthatwhenthederivativeorderdecreases,thebeta derivative pushes the surface to the left, whereas the M-truncated derivative pushes the surface to the right.In future work, we look at Eq. ( 1) with additive noise.
Fokas-Lenells equation (FLE) is a nonlinear partial differential equation that arises in various fields of mathematical physics.It has several applications in fluid dynamics, quantum mechanics, and nonlinear optics, among others.