Abstract
This paper studies optical solitons with M-truncated and beta derivatives (BD) for the Complex Ginzburg-Landau equation (CGLE) with Kerr Law nonlinearity. Two well-known integration schemes which are generalized tanh method (GTM) and generalized Bernoulli sub-ODE method (GBM) are utilized to extract such optical soliton solutions. For the successful existence of the solutions, the constraints conditions have been presented. The discussion for the physical features of the obtained solutions is reported.
1. Introduction
Nonlinearity has been potent field of research and its vitality is thought of through a sweer-amplitude wave oscillation analyzed in numerous fields from plasmas and fluids to biological and chemical phenomenon, solid state, to mention a few. Therefore, the most captivating viewpoint in nonlinear physical phenomenon are solitons. The availability of solitonic concepts are due to the philosophical balance of dispersion and nonlinearity [1]. A lot of researches on solitons and associated aspects of solitary wave (SW) solutions for example monopulse water wave which depict the first soliton can be found in Miller and Ross [2], Podlubny [3], Oldham [4], Kiryakova [5], and El-Sayed and Gaber [6]. Moreover, various mathematical insight and modeling can be interpreted through optical solitons for their numerical and analytical structures of the numerous mechanism. These stimulated many engineers and scientists to focus on the establishments of solitons with optical structures with the help of various integration schemes [7–38].
For quite a long time, the effect of memory is an idea that has been of great concern in the locality of modeling. Genuinely, the integer systems are not conveniently addressing this memory effect [39–41]. Many researchers have presented that, one can get to know more on the memory effect through non-integer operators [42–45]. An extension to integer order systems such as conformable [46], beta [47], and M-derivatives [48] have also been introduced and they play a vital role in modeling physical systems. These extension to integer order systems satisfy a lot of characteristics that were not satisfied before and it can be employed to model several physical phenomenon. In this study, we establish new optical solitons for the governing equation with M-truncated and beta derivatives with the aid of two well-known method. The M-truncated and beta derivatives are defined in the following subsections, respectively.
1.1. Truncated M-Fractional Derivative
We define the truncated Mittag-Leffler function of one parameter by
Truncated M-fractional derivative (TMD) is a fractional derivative that has been introduced in Sousa and de Oliveira [48]. This derivative has expunged the obstacles with the existing derivatives. It is defined in the following definition.
Definition 1.1. Assume that f : (0, ∞) → ℝ, the TMD of f with order γ exhibitedis given by
for τ > 0, and iEβγ ∈ (0, 1), β > 0 is a truncated Mittag-Leffler function of one parameter, as defined in (1). Note that, if f is γ-differentiable in some open interval (0, a), a > 0, and . Then we attain
Theorem 1.1. Surmise that f : (0, ∞) → ℝ is γ−differentiable for τ0 > 0, with γ ∈ (0, 1], β > 0, then f is continuous at τ0.
Theorem 1.2. Let 0 < γ ≤ 1, β > 0, a, b ∈ ℝ, f, g, γ-differentiable, at a point τ > 0. Then
, a, b ∈ ℝ
, μ ∈ ℝ
,
,
If f is differentiable, then,
, for f differentiable at g.
1.2. Beta Derivative
The beta derivative can be stated by [49]
along with the properties as comes next
1.
2.
for any c depicting a constant,
3.
4.
Considering , h → 0 when ϵ → 0, therefore we have
with
where l is a constant.
5.
The arrangements of the paper is as follows: In section 2 the governing equation has been presented. In section 3, applications have been reported, whereas section 4 provides the discussion of the obtained results along with their physical features. Finally, concluding remark is given in section 5.
2. Governing Equation
The CGLE equation [50, 51] in the sense of M-truncated derivative is given by:
whereas in the sense of beta derivative is given by
where , , and , depicts M-truncated and beta derivatives, respectively. 0 < γ ≤ 1, describing the order of the fractional derivatives and a, b, δ, B, and A are real constants.
In Equations (12) and (13), 𝔽 ∈ ℝ, and the complex function and its smoothness is necessary to be possessed 𝔽(|u|2)u : ℂ → ℂ. Consider ℂ to be a two-dimensional linear space ℝ2, and that 𝔽(|u|2)u is k times continuously differentiable, so that
2.1. Mathematical Analysis
To solve Equations (12) and (13), the beginning step is as come next
the shape of the pulse is represented by u(η, τ) so that in the sense of M-truncated derivatives we have
and
and in the sense of beta derivative we have
and
where w is the wave number of the soliton, k denotes the soliton frequency, υ indicates the speed of the soliton, ϕ(η, τ) is the phase component, θ0(ξ) depicts an additional phase function depending on ξ. Plugging (15) and (17) into (12) and (13), respectively, and decomposing the real and imaginary parts, one attains
and
Equation (20) denotes the soliton velocity. Setting δ = 2B in Equation (19) yields.
2.2. Kerr Law
This law has got its origin through the reality that a light wave in an optical fiber heads to responses by a nonlinear patterns from non-harmonic motion of electrons bound in molecules, brought externally by an electric field. Although the responses by nonlinear terms are seriously low, over a long distance of propagation, the effects standstill in numerous patterns measuring in terms of light wavelength. This law is given by , therefore Equation (21) becomes.
3. Applications
This section will utilize the GT and GB sub-ODE methods to provide optical solitons for the governing equation with beta-derivative.
3.1. Application for GTM
According to GTM [52], Equation (22) has possessed the solution as comes next
with a0 and a1 depicting an unknown constants and Φ(ξ) holds for the Ricatti equation
with μ a non-zero constant. Plugging Equation (23) together with Equation (24) in Equation (22), one reaches
Collecting the terms in Φi(i = 0, 1, 2, 3), one attains
Solving Equation (26), we obtain
Result 1. , A = −4Bk2 − w, a0 = 0, b ≠ 0, . If C < 0, we attain
If C > 0, we acquire
Result 2. 2C + k2 ≠ 0, , a0 = 0, b ≠ 0, . If C < 0, we have
If C > 0, we attain
where ξ and ϕ(η, τ) are defined by (15) and (16) for M-truncated derivative solutions and by (17) and (18) for beta derivative solutions.
3.2. Application for GBM
This section will apply GBM for Equations (12) and (13). According to GB sub-ODE method [53], Equation (22) has possessed the solution as comes next
with a0 and a1 representing an unknown constants and Φ(ξ) holds for the Ricatti equation
with μ a non-zero constant. Plugging Equation (36) together with Equation (35) in Equation (22), one attains
Collecting the terms in Φi(i = 0, 1, 2, 3), one obtains
Solving Equation (38), we reaches
Result 1. , A = −2Bλ2 − w, b ≠ 0, , λ(a − 4B) ≠ 0, . We obtain
or
Result 2. 2k2 − λ2 ≠ 0, , b ≠ 0, , . We acquire
or
where ξ and ϕ(x, t) are defined by (15) and (16) for M-truncated derivative solutions and by (17) and (18) for beta derivative solutions.
4. Discussion
The M-truncated and beta-derivatives have been successfully utilized to reach optical solitons for the underlying equation. This has been achieved by utilizing two potent integration schemes which are GTM and GBM. Singular-dark, dark and singular-periodic solutions have been reported. The GTM has provided dark optical soliton (DOS) (27) and (31), singular optical soliton (28) and (32), optical singular periodic (29), (30), (33), and (34). The GBM has provided optical dark solitons reported in (39) and (41), optical singular solitons reported in (40) and (42).
Solitary waves (SW) with mitigating intensity than the background can be interpreted by DOS [49]. SW with discontinuous derivative can be depicted by singular solitons [54, 55]. These sorts of SW are potent as a results of efficiency and applicability they possessed optical communications of a long distance. Optical fibers can be considered as a thin lengthy strands of pure-ultra glass so that an electromagnetic radiations can be communicated without any mitigation from one point to the next [56].
5. Conclusion
In this research, we have applied the well-known M-truncated and beta derivatives to reach the optical solitons for the governing equation with Kerr Law nonlinearity. Two techniques which GTM and GBM have been used to attain such solutions. For the successful existence of the solutions, the constraints conditions have been presented. The discussion for the physical features of the obtained solutions are have been reported. The explicit behavior for the obtained results by suitable choice of the parameter values have been presented in the presented Figures 1–3. The effects of the γ, β-M-truncated derivative and γ-beta derivative have influenced the behavior of the solutions. The obtained solutions are new and novel and can be of great potent in explaining physical systems in nonlinear optics.
Figure 1
Figure 2
Figure 3
Statements
Data availability statement
All datasets generated for this study are included in the manuscript and the supplementary files.
Author contributions
All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.
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. The handling editor is currently organizing a Research Topic with one of the authors DB, and confirms the absence of any other collaboration.
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Summary
Keywords
complex Ginzburg-Landau equation, generalized tanh method, generalized Bernoulli sub-ODE method, beta derivative, optical solitons
Citation
Yusuf A, Inc M and Baleanu D (2019) Optical Solitons With M-Truncated and Beta Derivatives in Nonlinear Optics. Front. Phys. 7:126. doi: 10.3389/fphy.2019.00126
Received
15 April 2019
Accepted
21 August 2019
Published
04 September 2019
Volume
7 - 2019
Edited by
Alex Hansen, Norwegian University of Science and Technology, Norway
Reviewed by
Mostafa Eslami, University of Mazandaran, Iran; Haci Mehmet Baskonus, Harran University, Turkey
Updates
Copyright
© 2019 Yusuf, Inc and Baleanu.
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*Correspondence: Abdullahi Yusuf yusufabdullahi@fud.edu.ng
This article was submitted to Mathematical Physics, a section of the journal Frontiers in Physics
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