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Original Research ARTICLE

Front. Phys., 04 September 2019 | https://doi.org/10.3389/fphy.2019.00126

Optical Solitons With M-Truncated and Beta Derivatives in Nonlinear Optics

  • 1Department of Mathematics, Science Faculty, Firat University, Elazig, Turkey
  • 2Department of Mathematics, Science Faculty, Federal University Dutse, Jigawa, Nigeria
  • 3Cankaya University, Department of Mathematics, Ankara, Turkey
  • 4Institute of Space Sciences, Bucharest, Romania

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 [738].

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 [3941]. Many researchers have presented that, one can get to know more on the memory effect through non-integer operators [4245]. 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

Eiβ(z)=k=0izkΓ(βk+1).    (1)

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 exhibited TiMγ,β is given by

TiMγ,βf(τ)=limϵ0f(τ+iEβ(ϵτγ))f(τ)ϵ,    (2)

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 limτ0+(iTMγ,βf(τ)). Then we attain

TiMγ,βf(0)=limτ0+(TiMγ,βf(τ))    (3)

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

TiMγ,β(af+bg)=aiTMγ,β(f)+biTMγ,β(f), a, b ∈ ℝ

TiMγ,β(tμ)=μτμ-γ, μ ∈ ℝ

TiMγ,β(fg)=fiTMγ,β(g)+giTMγ,β(f),

TiMγ,β(fg)=giTMγ,β(f)-fiTMγ,β(g)g2,

If f is differentiable, then TiMγ,β(f)(τ)=τ1-γΓ(β+1)dfdτ,

TiMγ,β(fog)(τ)=f(g(τ))iTMγ,βg(τ), for f differentiable at g.

1.2. Beta Derivative

The beta derivative can be stated by [49]

T0Aηγ(F(η))=limϵ0F(η+ϵ(η+1Γ(γ)))F(η)ϵ.    (4)

along with the properties as comes next

1.

T0Aηγ(aF(η)+bG(η))=a0ATηγF(η)+b0AFηγG(η)0A    (5)

2.

Tηγ(c)=0,    (6)

for any c depicting a constant,

3.

T0Aηγ(F(η).G(η))=G(η)0ATηγF(η)+F(η)0ATηγG(η)    (7)

4.

T0Aηγ(F(η)G(η))=G(η)0ATηγF(η)F(η)0ATηγG(η)G2(η).    (8)

Considering ϵ=(η+1Γ(γ))γ-1h, h → 0 when ϵ → 0, therefore we have

T0AηγF(η)=(η+1Γ(γ))1γdF(η)dη,    (9)

with

ξ=lγ(η+1Γ(γ))γ    (10)

where l is a constant.

5.

T0Aηγ(F(τ)G(η))=ldF(τ)dτ.    (11)

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:

i0EDM,τγ,βu+a0EDM,η2γ,βu+bF(|u|2)u=1|u|2u*{δ0EDM,η2γ,β(|u|2)|u|2B(0EDM,ηγ,βu)2}+Au,    (12)

whereas in the sense of beta derivative is given by

i0EDtγu+a0EDη2γu+bF(|u|2)u=1|u|2u*{δ0EDη2γ(|u|2)|u|2B(0EDηγu)2}+Au,    (13)

where DoEM,τγ,β, DoEM,ηγ,β, and DoEτγ, DoEτγ 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

F(|u|2)um,n=1ck((-n,n)×(-m,m);2).

2.1. Mathematical Analysis

To solve Equations (12) and (13), the beginning step is as come next

u(η,τ)=u(ξ)eiϕ(η,τ),    (14)

the shape of the pulse is represented by u(η, τ) so that in the sense of M-truncated derivatives we have

ξ=Γ(β+1)γ(ηγ-υτγ)    (15)

and

ϕ(η,τ)=-Γ(β+1)γ(kηγ-wυτγ)+θ0(ξ),    (16)

and in the sense of beta derivative we have

ξ=1γ(η+1Γ(γ))γ-υγ(τ+1Γ(γ))γ    (17)

and

ϕ(η,τ)=-kγ(η+1Γ(γ))γ+wγ(τ+1Γ(γ))γ+θ0(ξ),    (18)

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

wu+a(u-k2u)+bF(u2)u=2(δ-2B)u2u+2δu+Au,    (19)

and

υ=-2ak.    (20)

Equation (20) denotes the soliton velocity. Setting δ = 2B in Equation (19) yields.

(a-4B)u-(w+ak2+A)u+bF(u2)u=0.    (21)

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 F(u)=u, therefore Equation (21) becomes.

(a-4B)u-(w+ak2+A)u+bu3=0.    (22)

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

u(η,τ)=a0+a1Φ(ξ),    (23)

with a0 and a1 depicting an unknown constants and Φ(ξ) holds for the Ricatti equation

Φ(ξ)=C+Φ(ξ)2,    (24)

with μ a non-zero constant. Plugging Equation (23) together with Equation (24) in Equation (22), one reaches

a1AΦ(ξ)+a0A+a03b-8a1BCΦ(ξ)+2aa1CΦ(ξ)+aa1k2Φ(ξ)+aa0k2+a0wa13bΦ(ξ)3+3a0a12bΦ(ξ)2+3a02a1bΦ(ξ)-8a1BΦ(ξ)3+a1+wΦ(ξ)+2aa1Φ(ξ)3=0    (25)

Collecting the terms in Φi(i = 0, 1, 2, 3), one attains

a0(a02b+ak2+A+w)=0,a1(3a02b+2aC+ak2+A-8BC+w)=0,3a0a12b=0,a1(a12b+2a-8B)=0.    (26)

Solving Equation (26), we obtain

Result 1. C=-k22, A = −4Bk2w, a0 = 0, b ≠ 0, a1=±2(4B-a)b. If C < 0, we attain

u(η,τ)=-2(4B-a)b-Ctanh(-Cξ)×eiϕ(η,τ),    (27)
u(η,τ)=-2(4B-a)b-Ccoth(-Cξ)×eiϕ(η,τ).    (28)

If C > 0, we acquire

u(η,τ)=±2(4B-a)bCtan(Cξ)×eiϕ(η,τ),    (29)
u(η,τ)=-2(4B-a)bCcot(Cξ)×eiϕ(η,τ).    (30)

Result 2. 2C + k2 ≠ 0, a=-A+8BC-w2C+k2, a0 = 0, b ≠ 0, a1=±2(A+4Bk2+w)2bC+bk2. If C < 0, we have

u(η,τ)=-2(A+4Bk2+w)2bC+bk2-Ctanh(-Cξ)×eiϕ(η,τ),    (31)
u(η,τ)=-2(A+4Bk2+w)2bC+bk2-Ccoth(-Cξ)×eiϕ(η,τ).    (32)

If C > 0, we attain

u(η,τ)=2(A+4Bk2+w)2bC+bk2Ctan(Cξ)×eiϕ(η,τ),    (33)
u(η,τ)=-2(A+4Bk2+w)2bC+bk2Ccot(Cξ)×eiϕ(η,τ),    (34)

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

u(η,τ)=a0+a1Φ(ξ),    (35)

with a0 and a1 representing an unknown constants and Φ(ξ) holds for the Ricatti equation

Φ(ξ)+λΦ(ξ)=μΦ(ξ)2,    (36)

with μ a non-zero constant. Plugging Equation (36) together with Equation (35) in Equation (22), one attains

a1AΦ(ξ)+a0A+a03b+aa1k2Φ(ξ)+aa1λ2Φ(ξ)+a1wΦ(ξ)+aa0k2+a0w4a1Bλ2Φ(ξ)+12a1BλμΦ(ξ)28a1Bμ2Φ(ξ)33aa1λμΦ(ξ)2+2aa1μ2Φ(ξ)3 a13bΦ(ξ)3+3a0a12bΦ(ξ)2+3a02a1bΦ(ξ)=0.    (37)

Collecting the terms in Φi(i = 0, 1, 2, 3), one obtains

a0(a02b+ak2+A+w)=0,a1(3a02b+ak2+aλ2+A-4Bλ2+w)=0,3a1(a0a1b-λμ(a-4B))=0,a1(a12b+2μ2(a-4B))=0.    (38)

Solving Equation (38), we reaches

Result 1. k=±λ2, A = −22w, b ≠ 0, a0=±4Bλ2-aλ22b, λ(a − 4B) ≠ 0, a1=μ2(4B-a)b. We obtain

u(η,τ)=(4Bλ2-aλ22b-λ22(4B-a)b(tanh(λ2ξ)-1))×eiϕ(η,τ),    (39)

or

u(η,τ)=(4Bλ2-aλ22b-λ22(4B-a)b(coth(λ2ξ)-1))×eiϕ(η,τ),    (40)

Result 2. 2k2 − λ2 ≠ 0, a=-2(A+2Bλ2+w)2k2-λ2, b ≠ 0, a0=±λ2(A+4Bk2+w)b(2k2-λ2), a1=-2μλ(A+4Bk2+w)b(2k2-λ2). We acquire

u(η,τ)=(λ2(A+4Bk2+w)b(2k2-λ2)+λλ(A+4Bk2+w)b(2k2-λ2)×(tanh(λ2ξ)-1))eiϕ(η,τ),    (41)

or

u(η,τ)=(λ2(A+4Bk2+w)b(2k2-λ2)+λλ(A+4Bk2+w)b(2k2-λ2)×(coth(λ2ξ)-1))eiϕ(η,τ),    (42)

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 13. 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
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Figure 1. Physical features with suitable values of the parameters. (A) M-truncated with γ = 0.5, β = 0.9 for (27). (B) Beta with γ = 0.5 for (27). (C) M-truncated with γ = 0.5, β = 0.9 for (28). (D) Beta with γ = 0.5, β = 0.9 for (27).

FIGURE 2
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Figure 2. Physical features with suitable values of the parameters. (A) M-truncated with γ = 0.5, β = 0.9 for (29). (B) Beta with γ = 0.5 for (29). (C) M-truncated with γ = 0.5, β = 0.9 for (30). (D) Beta with γ = 0.5, β = 0.9 for (30). (E) M-truncated with γ = 0.85, β = 0.76 for (39). (F) Beta with γ = 0.85 for (39).

FIGURE 3
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Figure 3. Physical features with suitable values of the parameters. (A) M-truncated with γ = 0.85, β = 0.76 for (40). (B) Beta with γ = 0.85, β = 0.9 for (40). (C) M-truncated with γ = 0.85, β = 0.76 for (41). (D) Beta with γ = 0.85 for (41). (E) M-truncated with γ = 0.85, β = 0.76 for (42). (F) Beta with γ = 0.85, β = 0.9 for (42).

Data Availability

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 Statement

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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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.

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

Copyright © 2019 Yusuf, Inc and Baleanu. 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: Abdullahi Yusuf, yusufabdullahi@fud.edu.ng