Abstract
The Fokas-Lenells equation (FLE) including the M-truncated derivative or beta derivative is examined. Using the modified 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 field 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 influence 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.
1. Introduction
Nonlinear evolution equations (NEEs) have a broad application in engineering and scientific areas, such as chemical kinematics, heat flow, fluid mechanics, optical fibers, wave propagation phenomena, solid-state physics, shallow water wave propagation, and quantum mechanics. Obtaining traveling wave solutions is one of the crucial physical problems for NEEs. Consequently, in nonlinear sciences, the search for mathematical approaches for creating the analytical solutions to NEEs is currently a crucial and vital task. In recent years, numerous techniques for addressing NEEs have been developed, including F-expansion technique [1], spectral methods [2], -expansion [3], tanh-sech method [4], Hirota’s method [5], extended trial equation [6], extended tanh-coth method [7, 8], Lie’s symmetry analysis [9–11], He’s semi-inverse method [12], generalized exponential rational function [13], generalized Riccati simplest equation [14, 15], new auxiliary equation approach [16], perturbation method [17], -expansion [18, 19], improved Sardar subequation [20], Jacobi elliptic function [21, 22], modified simple equation method [23], and more recent techniques.
One of the most important of NEEs is the Fokas-Lenells equation (FLE) [24–26]: where describes the complex field, , , and , , and are the positive constants. Equation (1) has various applications in complex system theory, quantum field theory, telecommunication, and quantum mechanics models. Moreover, it occurs as a pattern that indicates the propagation of nonlinear pulses in optical fibers. Many researchers have obtained the exact solutions of Eq. (1) by employing several methods such as the generalized Kudryashov and extended trial equation methods [24], Riccati’s equation method [27], ()-expansion method [28], complex envelope function ansatz [29], and mapping method [30].
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 is the beta derivative (BD) operator.
And with M-truncated derivative operator, Eq. (1) takes the form where 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.
2. 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 [31–34]. 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 as
Moreover, for any constants and , the BD has the following features [35]:
2.2. M-Truncated Derivative
Sousa and de Oliveira [36] proposed another derivative known as the MTD. The MTD of order is defined as where
The MTD satisfies the following characteristics [36]:
3. The Clarification of MM-Method
Here, we implement the MM-method from [37]. Let the solutions to Eq. (19) take the form where and are the undetermined constants for and solves where the constants , , and are real numbers. Equation (10) has different solutions for , , and as follows:
, and are the Jacobi elliptic functions (JEFs) for When , the following hyperbolic functions are produced from JEFs:
Moreover, when the following trigonometric functions are produced from JEFs:
4. Traveling Wave Equation for FLE
To get the wave equation for FLE (2)/(3), we use where is a real function and and are defined as follows: (i)In terms of beta derivative(ii)In terms of M-truncated derivativewhere , , , and 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
5. Exact Solutions of FLE
To determine the value of defined in Eq. (9), we balance with in Eq. (19) as
With , Eq. (9) becomes
Putting Eq. (22) into Eq. (19), we have
Setting all coefficient of and in Eq. (23) be zero for we get
There are three sets derived from these equations:
Set 1
Set 2
Set 3
Set 1. The solution of Eq. (19), utilizing Eqs. (22) and (25), takes the form
Therefore, the solutions of FLE (2)/(3), by using Table 1, are as follows:
Elliptic solutions
Rational solutions
Hyperbolic solutions (if in (29)-(43))
Trigonometric solutions (if in (29)-(43))
where and are defined in (14) or (15) in the sense of BD or MTD, respectively
Set 2. The solution of Eq. (19), utilizing Eqs. (22) and (26), is
Therefore, the solutions of FLE (2)/(3), by using Table 1, are as follows:
Elliptic solutions
Hyperbolic solutions (if in (50)-(59) for )
Trigonometric solutions (if in (50)-(59)for )
where and are defined in (14) or (15) in the sense of BD or MTD, respectively
Set 3. The solution of Eq. (19), utilizing Eqs. (22) and (27), is
Therefore, the solutions of FLE (2)/(3), by using Table 1, are as follows:
Elliptic solutions
Hyperbolic solutions (if in (65)-(69))
Trigonometric solutions (if in (65)-(69) for )
where and are defined in (14) or (15) in the sense of BD or MTD, respectively
Remark 1. If we put (or and ) in Eq. (40), then we obtain the same results (71), respectively, reported in [24].
Remark 2. If we put (or and ) in Eqs. (60), (70), (44), and (41), then we get the same results (34), (35), (36), and (37), respectively, reported in [27].
6. 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 , , , , and 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.
(a)
(b)
(c)
(d) , 0.7, and 1
(a)
(b)
(c)
(d) , 0.7, and 1
(a)
(b)
(c)
(d) , 0.7, and 1
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.
(a)
(b)
(c)
(d) , 0.7, and 1
(a)
(b)
(c)
(d) , 0.7, and 1
(a)
(b)
(c)
(d) , 0.7, and 1
7. Results and Discussion
The 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. Obtaining exact solutions for the FLE is of utmost importance. These solutions deepen our understanding of the physical phenomena, validate approximations and numerical methods, foster the development of new mathematical techniques, and facilitate education and training. The pursuit of exact solutions drives progress in mathematical physics and contributes to a broader scientific and technological advancement.
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 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.
8. 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 deduced 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 future work, we look at Eq. (1) with additive noise.
Data Availability
All data are available in this paper.
Conflicts of Interest
The author declares that they have no competing interests.
Acknowledgments
This work is supported by the Princess Nourah bint Abdulrahman University Researcher Supporting Project (number PNURSP2023R 273), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.