Abstract
Here, we look at the Sasa-Satsuma equation with M-truncated derivative (SSE-MTD). The analytical solutions in the form of trigonometric, hyperbolic, elliptic, and rational functions are constructed using the Jacobi elliptic function and generalizing Riccati equation mapping methods. Because the Sasa–Satsuma equation is applied to explain the propagation of femtosecond pulses in optical fibers, the acquired solutions can be employed to explain a wide range of important physical phenomena. Moreover, we apply the MATLAB tool to generate a series of graphs to address the effect of the M-truncated derivative on the exact solution of the SSE-MTD.
1 Introduction
Many authors have centered their attention on fractional nonlinear differential equations (FNLDEs) in the noble age of technology and science to examine complex mathematical models that are used in research area and real life, such as neuroscience, robotics, fluid dynamics, quantum mechanics, plasma physics, optical fibers, and so on. A lot studies have been published about some aspects of fractional differential equations, such as finding exact and numerical solutions, the existence and uniqueness of solutions, and the stability of solutions [1–7]. Therefore, it is essential to discover the exact solutions to these equations in order to understand the physical phenomenon and overcome the resulting obstacles. Recently, acquiring soliton solutions to important equations has emerged as a major field of study. Numerous researchers defended numerous novel methods to evaluate soliton solutions including (G′/G, 1/G)-expansion method [8] (G′/G)-expansion [9, 10], generalized (G′/G)-expansion [11], exp-function method [12], Jacobi elliptic function expansion [13], sine-cosine procedure [14], auxiliary equation scheme [15], first-integral method [16], sine-Gordon expansion technique [17], generalized Kudryashov approach [18], exp(−ϕ(ς))-expansion method [19], homotopy perturbation method with Aboodh transform [20], He–Laplace method, He’s variational iteration method [21, 22], and others.
In contrast, a new differentiation operator has grown up, that includes the concepts of fractional differentiation and fractal derivative. Therefore, various kinds of fractional derivatives were proposed by several mathematicians. The most well-known ones are the ones proposed by Grunwald-Letnikov, He’s fractional derivative, Atangana-Baleanu’s derivative, Riemann-Liouville, Marchaud, Riesz, Caputo, Hadamard, Kober, and Erdelyi [23–26]. The bulk of fractional derivative types do not follow classic derivative equations like the chain rule, quotient rule, and product rule. Sousa et al. [27] have developed a new derivative known as the M-truncated derivative (MTD), which is a natural extension of the classical derivative. The MTD for of order δ ∈ (0, 1] is indicated aswhere , for and β > 0, is the truncated Mittag-Leffler function and is defined as:
There are many authors have considered some nonlinear partial differential equations with M-truncated derivative such as [28–31] and the references therein. In this article, we examine the Sasa-Satsuma equation with M-truncated derivative (SSE-MTD):where is the optical soliton profile,, for k = 1, 2, 3, 4, are real constants. defines the temporal evolution of optical soliton molecules, is the group velocity dispersion. represents the third-order dispersion, while provides the stimulated Raman scattering, is the self-steepening and is Kerr-law fiber nonlinearity.
If we set δ = 1 and β = 0, then we have the Sasa-Satsuma (SS) equation (32, 33):
The SS Eq. 2, which was found while studying the integrability of Schrödinger equation, reduced to nonlinear Schrödinger equation when α1 = α2 = α3 = 0 as follows:
In 1991, Sasa and Satsuma [34] created Eq. 2. This equation has additional components that explain third-order dispersion, self-steepening, and self-frequency shift, which are prevalent in many areas of physics, such as ultrashort pulse propagation in optical fibers [35, 36]. Due to the importance of SS Eq. 2, many authors have obtained its exact solutions by using various methods such as new auxiliary equation method [37], extended trial equation and generalized Kudryashov methods [38], inverse scattering transform [39], improved F-expansion methods and improved auxiliary [40], Riemann problem method [41], unified transform method [42], Bäcklund transformation [43], Darboux transformation [44].
Our main objective of this work is to find the exact solutions of the SSE-MTD (1). The solutions in the form of hyperbolic, trigonometric, elliptic, and rational functions are constructed by utilizing the Jacobi elliptic function method (JEF-method) and generalizing Riccati equation mapping method (GREM-method). Because the Sasa–Satsuma equation is applied to clarify the propagation of femtosecond pulses in optical fibers, the solutions obtained can be employed to study a wide range of important physical phenomena. Furthermore, we utilize the MATLAB tool to generate a series of graphs to examine the effect of the M-truncated derivative on the exact solution of the SSE-MTD (1).
The following is how the paper is organized: In the next section, we describe the methods employed in this paper. The wave equation for the SSE-MTD (1) is developed in Section 3. In Section 4, we employ the JEF-method and the GREM-method to get the precise solutions of the SSE-MTD (1). In Section 5, we study the effect of the MTD on the solution of Eq. 1. Finally, the findings of the article are presented.
2 Description of the methods
In this section, we describe the methods employed in this paper.
2.1 GREM-method
It is useful to outline the essential steps of GREM-method mentioned in [
45] as follows:
1. We begin by looking at a general kind of PDEs with MTD as follows
2. We use Eq. 4 to obtain the traveling wave solution
3. Using the next changes
5. Putting the following Riccati-Bernoulli equation
where
s,
r,
pare constants, into Eq.
8. Then we balance each coefficient of
to zero to get a system of ODE. We solve this system to attain the value of
s,
rand
p. It is straightforward to obtain the non-traveling wave solutions to Eq.
4by solving Eq.
8and utilizing Eq.
5.
2.2 JEF-method
While, we summarize here the main steps of the JEF-method described by Fan et al. [
46] as follows.
1. We repeat the first four steps from the previous subsection in order to obtain Eq. 7.
2. Assuming the solution of Eq. 7 in this type
where
Nis a positive integer that will be determined and
or
or
for 0 <
m< 1. The Jacobi elliptic functions
,
are periodic and have features of triangular functions as follows:
,
,
,
,
.
If
m→ 1, then
,
sech
and
sech
.
3. Usually, to determine the parameter N, we balance the highest order linear terms in the resulting equation with the highest order nonlinear terms. To determine the order, we follow these steps: Firstly, we define the degree of as . Secondly, we calculated the highest order nonlinear terms and the highest order nonlinear terms as
and
4. After we determine N, we substitute (9) into the ODE (7) in order to attain an equation in powers of .
5. Equating each coefficients of powers of in the resulting equation to zero. This will provide a set of equations containing the ak (k = 0, 1, …N) and . We solve these equations to attain the values of ak (k = 0, 1, …N) and and substitute with these value into Eq. 9.
3 Traveling wave Eq. For SSE-MTD
To derive the wave equation for SSE-MTD (1), we usewhere is a real function, μ1, μ2, η1, and η2 are non-zero constants. We note that
Inserting Eq. 11 into Eq. 1, we have for real partand for imaginary part
Integrating (13) once, we getwhere C is the integral constant. If we compare the coefficients of Eqs (12) and (14), we haveand
Now, we can rewrite Eq. 12 aswhere
Balancing with in Eq. 15 to calculate the parameter N as
4 Exact solutions of SSE-MTD
Two various methods such as GREM-method and JEF-method are used to attain the solutions to Eq. 15. The solutions to the SSE-MTD (1) are then determined.
4.1 REM-method
Utilizing Eq. 8, we obtain
Substituting (17) into (15), we have
We put each coefficient of equal zero in order to get
Solving these equations, we haveandwhere ℓ1 and ℓ2 are stated in Eq. 16. There are different sets for the solution of Eq. 8 relying on p and s as follows:
Set I: When ps > 0, thus the solutions of Eq. 8 are:Then, SSE-MTD (1) has the trigonometric functions solution:where .
Family II: When ps < 0, thus the solutions of Eq. 8 are:Then, SSE-MTD (1) has the hyperbolic functions solution:where .
Family III: When p = 0, s ≠ 0, then the solution of Eq. 8 is
Then, we get the rational function solution of SSE-MTD (1) as
Remark 1If we Putβ = 0 andδ = 0 in Eqs.(21)and(26), then we get the solutions (13) and (14) that stated in [40].
4.2 JEF-method
We assume the solutions of Eq. 15, with N = 1, areFirst, let . Differentiate Eq. 32 two times, we have
Setting Eqs 32, 33 into Eq. 15, we obtain
Plugging each coefficient of equal zero, we attainand
We obtain when we solve these equations
Consequently, the solution of Eq. 15 is
As a result, the solution of the SSE-MTD (1), for ℓ2 < 0 and ℓ1 > 0, iswhere . When m → 1, the solution (34) tends to:
Similarly, we can replace in (32) with or to derive the solutions of Eq. 15 as follows:and
Consequently, the solutions of the SSE-MTD (1) are as follows:for , ℓ1 < 0, andfor ℓ2 > 0, ℓ1 < 0, respectively. If m → 1, then the solutions (36) and (37) turn to:for ℓ2 > 0, ℓ1 < 0.
Remark 2If we Putβ = 0 andδ = 0 in Eqs. (34) and (36), then we get the solutions (48) and (49) that stated in [40].
5 Discussion and effects of M-truncated derivative
Discussion: For the Sasa-Satsuma equation with a M-truncated derivative, we found the optical solutions in this paper. Two effective methods, the REM-method and JEF-method, were used to arrive at these results. The REM-method has provided optical singular periodic (21) and (22), singular optical solution (27), and dark optical solution (26). While JEF-method has provided elliptic solutions. Dark optical solution can interpret solitary waves (SW) with less intensity than the background [47]. SW with discontinuous derivatives can be illustrated using singular solitons [48, 49]. These kinds of SW are effective because of their efficacy and applicability in optical long-distance communications. Optical fibers can be thought of as thin, long strands of pure-ultra glass that allow electromagnetic radiations to travel unimpeded from one location to another.
Effects of M-truncated derivative: Now, we examine the influence of MTD on the exact solution of the SSE-MTD (1). Several graphical representations depict the behavior of some obtained solutions, including (26) (34) and (38). Let us fix the parameters and t ∈ [0, 2] to simulate these graphs.
Now, we deduce from Figures 1, 2, 3 that when the derivative order δ of M-truncated derivative increases, the surface moves into the right.
FIGURE 1
FIGURE 2
FIGURE 3
6 Conclusion
In this study, the Sasa-Satsuma equation with M-truncated derivative (SSE-MTD) was examined. We acquired the exact solutions by utilizing Jacobi elliptic function and generalizing Riccati equation mapping methods. Because of the application of the Sasa–Satsuma equation in explaining the propagation of femtosecond pulses in optical fibers, these solutions may explain a wide range of interesting and complex physical phenomena. Furthermore, using the MATLAB program, the M-truncated derivative effects on the exact solutions of SSE-MTD (1) were illustrated. We concluded that when the derivatives order increases the surface moves into the right. In the future work, we can consider Sasa-Satsuma equation with stochastic term.
Statements
Data availability statement
The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.
Author contributions
WM: software, data curation, formal analysis, investigation, methodology, writing—original draft. FA-A: data curation, investigation, formal analysis, writing—original draft. All authors contributed to the article and approved the submitted version.
Acknowledgments
Princess Nourah bint Abdulrahman University Researcher Supporting Project number (PNURSP2023R 273), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
Abbreviations
2D, Two dimension; 3D, Three dimension; FNLDEs, Fractional nonlinear differential equations; GREM-method, generalizing Riccati equation mapping method; JEF-method, Jacobi elliptic function method; MTD, M-truncated derivative; ODE, Ordinary differential equation; SSE, Sasa-Satsuma equation; SW, Solitary waves.
Nomenclature
| M-truncated derivative operator | |
| Truncated Mittag-Leffler function | |
| Γ(⋅) | Gamma function |
| β | Positive real number |
| δ | Fractional derivative order |
| aandb | Real constants |
| αk, for k= 1, 2, 3, 4 | Real constants |
| xandt | Independent variables |
| Solution of wave equation | |
| ηδ | The wave variable |
| μδ | The phase component |
| μ1 | The wave frequency |
| μ2 | The wave number |
| η1 | The wave frequency |
| η2 | The wave velocity |
| Sn | The elliptic sine |
| Cn | The elliptic cosine |
| Dn | The delta amplitude |
| M | Modulus |
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Summary
Keywords
Sasa-Satsuma equation, M-truncated derivative, optical solitons, generalizing Riccati equation mapping method, analytical solutions
Citation
Al-Askar FM and Mohammed WW (2023) Abundant optical solutions for the Sasa-Satsuma equation with M-truncated derivative. Front. Phys. 11:1216451. doi: 10.3389/fphy.2023.1216451
Received
03 May 2023
Accepted
10 July 2023
Published
21 July 2023
Volume
11 - 2023
Edited by
Hamid M. Sedighi, Shahid Chamran University of Ahvaz, Iran
Reviewed by
Muhammad Nadeem, Qujing Normal University, China
Naveed Anjum, Government College University, Faisalabad, Pakistan
Fernane Khaireddine, 8 May 1945 University of Guelma, Algeria
Updates
Copyright
© 2023 Al-Askar and Mohammed.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Wael W. Mohammed, wael.mohammed@mans.edu.eg
ORCID: Farah M. Al-Askar, orcid.org/0000-0002-2394-0041; Wael W. Mohammed, orcid.org/0000-0002-1402-7584
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.