# Optical Wave Phenomena in Birefringent Fibers Described by Space-Time Fractional Cubic-Quartic Nonlinear Schrödinger Equation with the Sense of Beta and Conformable Derivative

Authors: M. F. Uddin; M. G. Hafez
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

The work explores the optical wave solutions along with their graphical representations by proposing the coupled spatial-temporal fractional cubic-quartic nonlinear Schrödinger equation with the sense of two fractal derivatives (beta and conformable derivative) and Kerr law nonlinearity for birefringent fibers. The new extended direct algebraic method for the first time is implemented to achieve this goal. Many optical solutions are listed along with their existence criteria. Based on the existence criteria, the cubic-quartic bright, and singular optical soliton, periodic pulse, and rouge wave profiles are supported in birefringent fibers with the influence of both beta and conformable derivative parameter.


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## Body

## 1. Introduction

It is confirmed that the most idyllic information transferors, have the most captivating phenomena in fiber optics due to their expanding applications in optoelectronics, optical switching, optical metamaterials, telecommunications, and ultrafast signal processing systems [1–4]. As a result, one may study several nonlinear phenomena occurring on the aforementioned applications, namely nonlinear phase modulation, optical solitons, parametric and stimulated scattering processes, and supercontinuum generation. Such phenomena involving of different functions are very useful to understand the dynamical wave propagation in the area of all optical signal processing, namely optical amplification, multiwavelength sources, pulse generation, optical regeneration, wavelength conversion, and optical switching. The nonlinear Schrödinger equation (NLSE) is only the mathematical physics equation that characterizes the propagation of optical wave phenomena with different types of nonlinear media [5–9]. Over the past couple of decades, the majority of extant research has mostly focused on optical solitons in polarization-preserving fibers. However, solitons receive far less attention in birefringent fibers where the dynamics of soliton propagation are dictated by coupled NLSE [10–14]. As a result, the study of optical solitons in birefringent fibers is currently an important topic of nonlinear optics research.Actually, the optical pulses traveling through an optical fiber tend to become polarized due to fiber nonuniformities, random changes in fiber diameter, and other technical issues caused by fiber technologies such as fiber twists, bends, and external stress. Such causes yield various problems, including differential group delay of the pulses, which is known as the polarization mode dispersion [15]. As a result, the dispersion has a significant negative impact on the steady and stable transmission of pulses across transcontinental and transoceanic lengths via fiber. At this stage, a phenomenon referred to as birefringence, which is caused by the splitting of optical pulses into two orthogonally polarized pulses. Thus, the birefringence is basically one kind of physical phenomenon in optical fibers with variable propagation constants and group velocities. It is also generated due to have delicate circular symmetry for optical fibers [16]. In such an undesirable physical state, the controlling NLSE in the birefringent fibers (BFs) breaks into vector coupled NLSE.Furthermore, the notion of cubic-quartic (CQ) solitons is occurred due to the delicate balance between chromatic dispersion (CD) and self-phase modulation becomes fragile with the CD becomes low enough. In order to restore the much-needed equilibrium in this crisis situation for the compensation of low CD count, the third-order dispersion (3OD) and fourth-order dispersion (4OD) terms are needed. This phenomenon is known as the CQ dispersive effects and it was firstly introduced in 2017 [17, 18]. Consequently, this study has been garnered a lot of interest in the research community from 2017 until the present. Recently, a group of researchers [19–22] have made a momentous effort to recover a slew of remarkable results for CQ optical solitons in Bragg-gratings fiber, magneto-optic wave–guides described by NLSEs and the newly proposed Biswas-Milovic equation. Besides, a few results have been reported on CQ solitons in birefringent fibres with different laws of refractive index using various efficient mathematical methodologies [23–27]. Therefore, it is imperative to focus on the previous studies along with their limitations as a follow-up. One of the significant flaws of previous studies are illustrated that they have solely considered the physical models for localization.However, the fractional calculus has advanced over the last several decades. It is now well established that the influence of memory has been of major importance in the localization of modeling for quite some time. Profoundly, the integer-order physical models do not handle this memory issue efficiently [28, 29]. Many scholars have demonstrated that the noninteger operators can provide more information about the memory effect [30–32]. Consequently, the fractal derivatives, that is beta fractional derivative (BFD) and conformable fractional derivative (CFD) have been established recently as the extensions of integer order derivatives. Such derivatives play a dynamic role in modeling to study the local/nonlocal and conservative/nonconservative physical systems. The development of BFD and CFD meet many features of fundamental calculus, whereas the earlier fractional derivatives does not support some features of fundamental calculus. In References [33–38], authors have been used such derivatives for describing a wide range of physical processes. For instance, Uddin et al. [33–36] have reported that the effect of BFD parameter on nonlinear wave phenomena by considering different kinds of environments. Hafez et al. [37] and Iqbal et al. [38] have reported that the effect of CFD parameter on wave phenomena with their dynamical features in optical bullets and one-dimensional nonlinear electrical transmission line, respectively. It is now required to investigate nonlocal and nonconservative physical issues in optical fibers due to imperfections/nonuniformities in birefringent fibres along with some technical challenges resulting from fiber technologies discussed previously.Thus, this work explores some new solutions with their better visualization on the topic of optical wave propagation in BFs described by the model of the governing coupled space-time fractional cubic-quartic NLSE (STF-CQNLSE) via the new extended direct algebraic method (EDAM) [39]. It is noted that the EDAM technique is first time implemented to achieve this goal for the considered model STF-CQNLSE along with the Kerr law of nonlinear refractive index in the presence of BFD and CFD. In addition, this work reveals mysteries of nonlocal behavior in birefringent fibers for the governing STF-CQNLSE by securing bright, singular, periodic singular, combo, and pure periodic optical soliton solutions with constraint criteria that must remain valid for these optical wave propagation to exist. The rest of this paper is organized as follows. The governing model equations are presented in the next subsections. Section 2 discusses the essential mathematical preliminaries for deriving the converted ordinary differential equation (ODE) from the leading STF-CQNLSE, which is described in Section 3. In Section 4, use the EDAM approach to recover the desired optical solitons. Section 5 discusses the graphical depiction of certain acquired results along with the comparison of previously reported outcomes. Finally, in Section 6, some closing observations are presented.


### 1.1. Governing Coupled Equation Having BFD and CFD Evolution

For polarization preserving fibers, the wave propagation through optical fibers with Kerr law of nonlinear refractive index is examined by the following cubic-quartic NLSE [23, 24].
(1)iϕt+ipϕxxx+qϕxxxx+rϕ2ϕ=0,where the complex–valued functionϕx,t is measuring the optical wave propagation in polarization preserving fibers, x (t) is the spatial (temporal) variable, r is the Kerr law of refractive index, respectively. The coefficients p and q are real constants represent the 3OD and 4OD, respectively. Note that most optical fibers are supported Kerr law nonlinearity. Recently, the fractional NLSEs (FNLSEs) are modeled and introduced to describe the physical process rather than NLSEs in the actual case especially in the field of nonlinear optical environments. To illustrate it, in References [40, 41], authors have reported the transmission dynamics of the beam describing by the FNLSE by considering fractional diffraction effect and self-defocusing saturable nonlinear media. They have studied the existence and dynamical evolution of fundamental solitons and multipole solitons supported by PT-symmetric potential. Wang and Wang [42] have studied the propagation of optical signal in fibers by considering the variable-coefficient wick-type stochastic FNLSE and found some analytical white noise functional solutions. They have shown that the FNLSEs can support a variety of new fractional optical solitons, such as multipole solitons, chirped and chirp-free fractional bright and dark soliton, and symmetric and antisymmetric solitons. Uddin et al. [33] have reported that one can derive nonlinear evolution equations of fractional order from nonlinear evolution equations of integer order using Agrawal’s variational principle when the complexity arises in a certain regimes of space or time for the impact of nonlocality as well as nonconservative energies arising in the physical systems. For the sake of simplicity, instead of deriving the fractional order equation, the following coupled STF-CQNLSE which incorporates BFD into consideration for the nonlocal nonlinear media and draws inspiration from the aforementioned literatures, is presented in this study to characterize the nonlocal behavior in BFs when the pulses are split into two direction frames from Equation (1):
(2)i0ADtμΨ1+ip10ADx3μΨ1+q10ADx4μΨ1+r1Ψ12+s1Ψ22Ψ1=0,i0ADtμΨ2+ip20ADx3μΨ2+q20ADx4μΨ2+r2Ψ22+s2Ψ12Ψ2=0.Whereas the STF-CQNLSE is described in terms of CFD by(3)iDtμΨ1+ip1Dx3μΨ1+q1Dx4μΨ1+r1Ψ12+s1Ψ22Ψ1=0,iDtμΨ2+ip2Dx3μΨ2+q2Dx4μΨ2+r2Ψ22+s2Ψ12Ψ2=0.In the above coupled systems,rj (sj), j=1,2 indicated the coefficients of self-phase modulation (cross-phase modulation)and the effect of four–wave mixing is discarded. Many researchers [40–47] have already implemented several kinds of mathematical procedures to solve the evolution equations involving of local and nonlocal operators. The solutions of Equations (2) and (3) have already been determined in the earlier investigations [23, 24] via the Riccati function, Sine-Gordon function, F–expansion, and exp-function expansion methods by considering only the classical models i.e. when μ=1. However, EDAM will be implemented to determine the traveling wave solutions of Equations (2) and (3) with the presence of nonlocal operators.


## 1.1. Governing Coupled Equation Having BFD and CFD Evolution

For polarization preserving fibers, the wave propagation through optical fibers with Kerr law of nonlinear refractive index is examined by the following cubic-quartic NLSE [23, 24].
(1)iϕt+ipϕxxx+qϕxxxx+rϕ2ϕ=0,where the complex–valued functionϕx,t is measuring the optical wave propagation in polarization preserving fibers, x (t) is the spatial (temporal) variable, r is the Kerr law of refractive index, respectively. The coefficients p and q are real constants represent the 3OD and 4OD, respectively. Note that most optical fibers are supported Kerr law nonlinearity. Recently, the fractional NLSEs (FNLSEs) are modeled and introduced to describe the physical process rather than NLSEs in the actual case especially in the field of nonlinear optical environments. To illustrate it, in References [40, 41], authors have reported the transmission dynamics of the beam describing by the FNLSE by considering fractional diffraction effect and self-defocusing saturable nonlinear media. They have studied the existence and dynamical evolution of fundamental solitons and multipole solitons supported by PT-symmetric potential. Wang and Wang [42] have studied the propagation of optical signal in fibers by considering the variable-coefficient wick-type stochastic FNLSE and found some analytical white noise functional solutions. They have shown that the FNLSEs can support a variety of new fractional optical solitons, such as multipole solitons, chirped and chirp-free fractional bright and dark soliton, and symmetric and antisymmetric solitons. Uddin et al. [33] have reported that one can derive nonlinear evolution equations of fractional order from nonlinear evolution equations of integer order using Agrawal’s variational principle when the complexity arises in a certain regimes of space or time for the impact of nonlocality as well as nonconservative energies arising in the physical systems. For the sake of simplicity, instead of deriving the fractional order equation, the following coupled STF-CQNLSE which incorporates BFD into consideration for the nonlocal nonlinear media and draws inspiration from the aforementioned literatures, is presented in this study to characterize the nonlocal behavior in BFs when the pulses are split into two direction frames from Equation (1):
(2)i0ADtμΨ1+ip10ADx3μΨ1+q10ADx4μΨ1+r1Ψ12+s1Ψ22Ψ1=0,i0ADtμΨ2+ip20ADx3μΨ2+q20ADx4μΨ2+r2Ψ22+s2Ψ12Ψ2=0.Whereas the STF-CQNLSE is described in terms of CFD by(3)iDtμΨ1+ip1Dx3μΨ1+q1Dx4μΨ1+r1Ψ12+s1Ψ22Ψ1=0,iDtμΨ2+ip2Dx3μΨ2+q2Dx4μΨ2+r2Ψ22+s2Ψ12Ψ2=0.In the above coupled systems,rj (sj), j=1,2 indicated the coefficients of self-phase modulation (cross-phase modulation)and the effect of four–wave mixing is discarded. Many researchers [40–47] have already implemented several kinds of mathematical procedures to solve the evolution equations involving of local and nonlocal operators. The solutions of Equations (2) and (3) have already been determined in the earlier investigations [23, 24] via the Riccati function, Sine-Gordon function, F–expansion, and exp-function expansion methods by considering only the classical models i.e. when μ=1. However, EDAM will be implemented to determine the traveling wave solutions of Equations (2) and (3) with the presence of nonlocal operators.


## 2. Mathematical Preliminaries

### 2.1. Beta Fractional Derivative (BFD)

The useful definition of BFD is recently introduced by the scholar group Atangana et al. [48].Definition 1.
Leta∈R and G be a function such that G:a,∞⟶R. Then the beta fractional derivative of order μ of G is exhibited as D0AxμGx and defined by
(4)D0AxμGx=limς⟶0Gx+ςx+1/Γμ1−μ−Gxς;0<μ≤1.Properties 2.
Let0<μ≤1 and suppose that α,α1,α2∈R, F,G≠0 are two beta-differentiable functions of order μ. Then
(5)D0Axμα1Fx+α2Gx=α1D0AxμFx+α2D0AxμGx,D0Axμα=0,D0AxμFx·Gx=Fx·D0AxμGx+Gx·D0AxμFx,D0AxμFx/Gx=Gx·D0AxμFx−Fx·D0AxμGxG2x.
It is noted that if one introduceς=x+1/Γμμ−1h, when ς⟶0, h⟶0 in Equation (4), one of the most important property of BFD is defined as
(6)D0AxμGx=x+1Γμ1−μdGxdx.which is indicated that the BFD can not only be treated as fractional derivative but also as a natural extension of the integer order derivative.


### 2.2. Conformable Fractional Derivative (CFD)

Khalil et al. [49] are introduced in details about the CFD with the following useful definition.Definition 3.
Assume thatG be a function such that G:0,∞⟶R. Then the conformable fractional derivative of order μ of G is denoted as DxμGx and given by
(7)DxμGx=limς⟶0Gx+ςx1−μ−Gxς;0<μ≤1.Properties 4.
Suppose that0<μ≤1 and assuming α,a,b,c∈R, F,G≠0 are two functions conformable differentiable of order μ. Then
(8)Dxμxα=αxα−μ,Dxμc=0,DxμaFx+bGx=aDxμFx+bDxμGx,DxμFx·Gx=Fx·DxμGx+Gx·DxμFx,DxμFx/Gx=Gx·DxμFx−Fx·DxμGxG2x,DxμGx=x1−μdGxdx.


## 2.1. Beta Fractional Derivative (BFD)

The useful definition of BFD is recently introduced by the scholar group Atangana et al. [48].Definition 1.
Leta∈R and G be a function such that G:a,∞⟶R. Then the beta fractional derivative of order μ of G is exhibited as D0AxμGx and defined by
(4)D0AxμGx=limς⟶0Gx+ςx+1/Γμ1−μ−Gxς;0<μ≤1.Properties 2.
Let0<μ≤1 and suppose that α,α1,α2∈R, F,G≠0 are two beta-differentiable functions of order μ. Then
(5)D0Axμα1Fx+α2Gx=α1D0AxμFx+α2D0AxμGx,D0Axμα=0,D0AxμFx·Gx=Fx·D0AxμGx+Gx·D0AxμFx,D0AxμFx/Gx=Gx·D0AxμFx−Fx·D0AxμGxG2x.
It is noted that if one introduceς=x+1/Γμμ−1h, when ς⟶0, h⟶0 in Equation (4), one of the most important property of BFD is defined as
(6)D0AxμGx=x+1Γμ1−μdGxdx.which is indicated that the BFD can not only be treated as fractional derivative but also as a natural extension of the integer order derivative.


## 2.2. Conformable Fractional Derivative (CFD)

Khalil et al. [49] are introduced in details about the CFD with the following useful definition.Definition 3.
Assume thatG be a function such that G:0,∞⟶R. Then the conformable fractional derivative of order μ of G is denoted as DxμGx and given by
(7)DxμGx=limς⟶0Gx+ςx1−μ−Gxς;0<μ≤1.Properties 4.
Suppose that0<μ≤1 and assuming α,a,b,c∈R, F,G≠0 are two functions conformable differentiable of order μ. Then
(8)Dxμxα=αxα−μ,Dxμc=0,DxμaFx+bGx=aDxμFx+bDxμGx,DxμFx·Gx=Fx·DxμGx+Gx·DxμFx,DxμFx/Gx=Gx·DxμFx−Fx·DxμGxG2x,DxμGx=x1−μdGxdx.


## 3. Converted ODE for STF-CQNLSE

To start with the following traveling wave variables transform to convert the systems of Equations(2) and (3) into its corresponding ODEs:
(9)Ψ1x,t=E1ΞjeiTjx,t,Ψ2x,t=E2ΞjeiTjx,t.where j=B,C. The real functions EiΞj and Tjx,t are representing the amplitude portion and the phase component of the pulse wave propagation, respectively. It is noted here that ΞB and TB are considered for BFD and defined by
(10)ΞB=1μx+1Γμμ−νμt+1Γμμ,TB=−κμx+1Γμμ+ωμt+1Γμμ+θ0.whereas in sense of CFD, ΞC, and TC are defined by
(11)ΞC=1μxμ−νμtμ,TC=−κμxμ+ωμtμ+θ0.Hereν, κ, ω, and θ0 are representing the speed, frequency, number, and phase constant of wave, respectively. The coupled system of Equation (2) and Equation (3) are then converted to the same ODEs by substituting separately the pair of equations, either Equation (9) and Equation (10) for BFD or Equation (9) and Equation (11) for CFD along with their suitable properties. Finally, for the sake of simplicity of this report, after decomposing the obtained ODEs into real and imaginary parts, one attains the following:Real parts:(12)q1E1iv+3κp1−6κ2q1E1′′+κ4q1−κ3p1−ωE1+r1E13+s1E1E22=0,q2E2iv+3κp2−6κ2q2E2′′+κ4q2−κ3p2−ωE2+r2E23+s2E12E2=0,where the superscripts E1iv, E1′′′, E1′′, and E1′ are denoted the fourth, third, second, and first order derivatives of E1, respectively.Imaginary parts:(13)p1−4κq1E1′′′−ν+3κ2p1−4κ3q1E1′=0,p2−4κq2E2′′′−ν+3κ2p2−4κ3q2E2′=0.Applying the linearly independent principle to imaginary parts for recovering:(14)ν=4κ3q1−3κ2p1,p1−4κq1=0.ν=4κ3q2−3κ2p2,p2−4κq2=0.This two equations gives the frequency of waves as(15)κ=3p1−p24q1−q2.This suggests thatp1≠p2 and q1≠q2.SettingE2=ΠE1, where Π is any nonzero constant so that Π≠1. The real parts implies to the following:
(16)q1E1iv+3κp1−6κ2q1E1′′+κ4q1−κ3p1−ωE1+r1+Π2s1E13=0,(17)Πq2E1iv+Π3κp2−6κ2q2E1′′+Πκ4q2−κ3p2−ωE1+Π3r2+Πs2E13=0.Both these equations have the similar form under the following constraint conditions:(18)q1=Πq2,3κp1−6κ2q1=Π3κp2−6κ2q2,κ4q1−κ3p1−ω=Πκ4q2−κ3p2−ω,r1+Π2s1=ΠΠ2r2+s2.Now, considering(19)Y0=3κp1−6κ2q1,Y1=κ4q1−κ3p1,Y2=r1+Π2s1,Equation (16) reduces to the following form
(20)q1E1iv+Y0E1′′+Y1−ωE1+Y2E13=0.


## 4. Analytical Solutions of STF-CQNLSE via EDAM

Applying the homogeneous balance principle toE1iv and E13 in Equation (20) and according to EDAM (see details in Reference [39]), the formal analytical solutions of Equation (20) can be represented as a polynomial in SΞj:
(21)E1Ξj=η0+∑L=12ηLSLΞj,whereηLL=0,1,2 are constants with η2≠0. SΞj is satisfied the following ODE:
(22)dSΞjdΞj=lnAα+λSΞj+σS2Ξj;A≠0,1.Note that Equation (22) offers a hung amount of solutions by depending on α, λ, and σ as set out in Reference [39]. By considering Equation (21) and Equation (22), a polynomial in SΞj is obtained from Equation (20). As a result, a set of algebraic equations (ignored for convenience) for η0,η1,η2,α,λ,σ, and ω are formulated by setting the coefficients of this polynomial equal to zero. By simplifying such equations via the computational package Maple-18, the following relations are determined:
(23)η0=±6ασlnA2−10κp1Y2H1,η1=±6λσlnA2−10κp1Y2H1,η2=±6σ2lnA2−10κp1Y2H1,ω=H1Y1−12lnA4Φ2κp1H1,q1=3κp1H1.where H1=20ασlnA2−5λ2lnA2+6κ2 and Φ=λ2−4ασ.By considering the above relations together with the solutions of Equation (22) and the assistance of Equation (21), the following solitary wave solutions for the considered STF-CQNLSE as in Equation (2) and Equation (3) in sense of BFD and CFD, respectively, are determined:Case 1.
WhenΦ<0 and σ≠0,
(24)Ψ11x,t=∓3Φ2lnA2−10κp1Y2H11+tanA−Φ2Ξj2eiTj,(25)Ψ21x,t=∓3ΠΦ2lnA2−10κp1Y2H11+tanA−Φ2Ξj2eiTj,(26)Ψ12x,t=∓3Φ2lnA2−10κp1Y2H11+cotA−Φ2Ξj2eiTj,(27)Ψ22x,t=∓3ΠΦ2lnA2−10κp1Y2H11+cotA−Φ2Ξj2eiTj,(28)Ψ13x,t=∓3Φ2lnA2−10κp1Y2H11+tanA−ΦΞj±MNsecA−ΦΞj2eiTj,(29)Ψ23x,t=∓3ΠΦ2lnA2−10κp1Y2H11+tanA−ΦΞj±MNsecA−ΦΞj2eiTj,(30)Ψ14x,t=∓3Φ2lnA2−10κp1Y2H11+cotA−ΦΞj±MNcscA−ΦΞj2eiTj,(31)Ψ24x,t=∓3ΠΦ2lnA2−10κp1Y2H11+cotA−ΦΞj±MNcscA−ΦΞj2eiTj,(32)Ψ15x,t=∓3Φ8lnA2−10κp1Y2H14+tanA−Φ4Ξj−cotA−Φ4Ξj2eiTj,(33)Ψ25x,t=∓3ΠΦ8lnA2−10κp1Y2H14+tanA−Φ4Ξj−cotA−Φ4Ξj2eiTj.Case 2.
WhenΦ>0 and σ≠0,
(34)Ψ16x,t=∓3Φ2lnA2−10κp1Y2H11−tanhAΦ2Ξj2eiTj,(35)Ψ26x,t=∓3ΠΦ2lnA2−10κp1Y2H11−tanhAΦ2Ξj2eiTj,(36)Ψ17x,t=∓3Φ2lnA2−10κp1Y2H11−cothAΦ2Ξj2eiTj,(37)Ψ27x,t=∓3ΠΦ2lnA2−10κp1Y2H11−cothAΦ2Ξj2eiTj,(38)Ψ18x,t=∓3Φ2lnA2−10κp1Y2H11−tanhAΦΞj∓iMNsechAΦΞj2eiTj,(39)Ψ28x,t=∓3ΠΦ2lnA2−10κp1Y2H11−tanhAΦΞj∓iMNsechAΦΞj2eiTj,(40)Ψ19x,t=∓3Φ2lnA2−10κp1Y2H11−cothAΦΞj∓MNcschAΦΞj2eiTj,(41)Ψ29x,t=∓3ΠΦ2lnA2−10κp1Y2H11−cothAΦΞj∓MNcschAΦΞj2eiTj,(42)Ψ110x,t=∓3Φ8lnA2−10κp1Y2H14−tanhAΦ4Ξj+cothAΦ4Ξj2eiTj,(43)Ψ210x,t=∓3ΠΦ8lnA2−10κp1Y2H14−tanhAΦ4Ξj+cothAΦ4Ξj2eiTj.Case 3.
Whenασ>0 and λ=0,
(44)Ψ111x,t=±6ασlnA2−10κp1Y2H11+tanAασΞj2eiTj,Ψ211x,t=±6ΠασlnA2−10κp1Y2H11+tanAασΞj2eiTj,Ψ112x,t=±6ασlnA2−10κp1Y2H11+cotAασΞj2eiTj,Ψ212x,t=±6ΠασlnA2−10κp1Y2H11+cotAασΞj2eiTj,Ψ113x,t=±6ασlnA2−10κp1Y2H11+tanA2ασΞj±MNsecA2ασΞj2eiTj,Ψ213x,t=±6ΠασlnA2−10κp1Y2H11+tanA2ασΞj±MNsecA2ασΞj2eiTj,Ψ114x,t=±6ασlnA2−10κp1Y2H11+cotA2ασΞj∓MNcscA2ασΞj2eiTj,Ψ214x,t=±6ΠασlnA2−10κp1Y2H11+cotA2ασΞj∓MNcscA2ασΞj2eiTj,Ψ115x,t=±32ασlnA2−10κp1Y2H14+tanAασ2Ξj−cotAασ2Ξj2eiTj,Ψ215x,t=±3Π2ασlnA2−10κp1Y2H14+tanAασ2Ξj−cotAασ2Ξj2eiTj.Case 4.
Whenασ<0 and λ=0,
(45)Ψ116x,t=±6ασlnA2−10κp1Y2H11−tanhA−ασΞj2eiTj,Ψ216x,t=±6ΠασlnA2−10κp1Y2H11−tanhA−ασΞj2eiTj,Ψ117x,t=±6ασlnA2−10κp1Y2H11−cothA−ασΞj2eiTj,Ψ217x,t=±6ΠασlnA2−10κp1Y2H11−cothA−ασΞj2eiTj,Ψ118x,t=±6ασlnA2−10κp1Y2H11−−tanhA2−ασΞj±iMNsechA2−ασΞj2eiTj,Ψ218x,t=±6ΠασlnA2−10κp1Y2H11−−tanhA2−ασΞj±iMNsechA2−ασΞj2eiTj,Ψ119x,t=±6ασlnA2−10κp1Y2H11−−cothA2−ασΞj±MNcschA2−ασΞj2eiTj,Ψ219x,t=±6ΠασlnA2−10κp1Y2H11−−cothA2−ασΞj±MNcschA2−ασΞj2eiTj,Ψ120x,t=±32ασlnA2−10κp1Y2H14−tanhA−ασ2Ξj+cothA−ασ2Ξj2eiTj,Ψ220x,t=±3Π2ασlnA2−10κp1Y2H14−tanhA−ασ2Ξj+cothA−ασ2Ξj2eiTj.Case 5.
Whenσ=α and λ=0,
(46)Ψ121x,t=±6α2lnA2−10κp1Y2H1tanAαΞj2+1eiTj,(47)Ψ221x,t=±6Πα2lnA2−10κp1Y2H1tanAαΞj2+1eiTj,(48)Ψ122x,t=±6α2lnA2−10κp1Y2H1cotAαΞj2+1eiTj,(49)Ψ222x,t=±6Πα2lnA2−10κp1Y2H1cotAαΞj2+1eiTj,(50)Ψ123x,t=±6α2lnA2−10κp1Y2H1tanA2αΞj±MNsecA2αΞj2+1eiTj,(51)Ψ223x,t=±6Πα2lnA2−10κp1Y2H1tanA2αΞj±MNsecA2αΞj2+1eiTj,(52)Ψ124x,t=±6α2lnA2−10κp1Y2H1−cotA2αΞj±MNcscA2αΞj2+1eiTj,(53)Ψ224x,t=±6Πα2lnA2−10κp1Y2H1−cotA2αΞj±MNcscA2αΞj2+1eiTj,(54)Ψ125x,t=±32α2lnA2−10κp1Y2H1tanAα2Ξj−cotAα2Ξj2+4eiTj,(55)Ψ225x,t=±3Π2α2lnA2−10κp1Y2H1tanAα2Ξj−cotAα2Ξj2+4eiTj.Case 6.
Whenσ=−α and λ=0,
(56)Ψ126x,t=±6α2lnA2−10κp1Y2H1tanhAαΞj2−1eiTj,Ψ226x,t=±6Πα2lnA2−10κp1Y2H1tanhAαΞj2−1eiTj,Ψ127x,t=±6α2lnA2−10κp1Y2H1cothAαΞj2−1eiTj,Ψ227x,t=±6Πα2lnA2−10κp1Y2H1cothAαΞj2−1eiTj,Ψ128x,t=±6α2lnA2−10κp1Y2H1−tanhA2αΞj±iMNsechA2αΞj2−1eiTj,Ψ228x,t=±6Πα2lnA2−10κp1Y2H1−tanhA2αΞj±iMNsechA2αΞj2−1eiTj,Ψ129x,t=±6α2lnA2−10κp1Y2H1−cothA2αΞj±MNcschA2αΞj2−1eiTj,Ψ229x,t=±6Πα2lnA2−10κp1Y2H1−cothA2αΞj±MNcschA2αΞj2−1eiTj,Ψ130x,t=±32α2lnA2−10κp1Y2H1tanhAα2Ξj+cothAα2Ξj2−4eiTj,Ψ230x,t=±3Π2α2lnA2−10κp1Y2H1tanhAα2Ξj+cothAα2Ξj2−4eiTj.Case 7.
Whenλ2=4ασ andCase 8.
Whenλ=α=0,
(57)Ψ131x,t=±6−10κp1Y2H11Ξj2eiTj,Ψ231x,t=±6Π−10κp1Y2H11Ξj2eiTj.Case 9.
Whenλ≠0 and α=0,
(58)Ψ132x,t=∓6λ2lnA2M−10κp1Y2H1coshAλΞj−sinhAλΞjcoshAλΞj−sinhAλΞj+M2eiTj,Ψ232x,t=∓6Πλ2lnA2M−10κp1Y2H1coshAλΞj−sinhAλΞjcoshAλΞj−sinhAλΞj+M2eiTj,Ψ133x,t=∓6λ2lnA2N−10κp1Y2H1sinhAλΞj+coshAλΞjsinhAλΞj+coshAλΞj+N2eiTj,Ψ233x,t=∓6Πλ2lnA2N−10κp1Y2H1sinhAλΞj+coshAλΞjsinhAλΞj+coshAλΞj+N2eiTj.Case 10.
Whenλ=K,σ=mKm≠0 and α=0,
(59)Ψ134x,t=±6K2mMlnA2−10κp1Y2H1mMA2KΞjM−mNAKΞj2+AKΞjM−mNAKΞjeiTj,Ψ234x,t=±6ΠK2mMlnA2−10κp1Y2H1mMA2KΞjM−mNAKΞj2+AKΞjM−mNAKΞjeiTj.It is noted that all the obtained solutions are valid wheneverκp1Y2H1<0. Here, the formula for hyperbolic and trigonometric functions are defined as
(60)sinhAΞ=MAΞ−NA−Ξ2,coshAΞ=MAΞ+NA−Ξ2,tanhAΞ=MAΞ−NA−ΞMAΞ+NA−Ξ,cothAΞ=MAΞ+NA−ΞMAΞ−NA−Ξ,sechAΞ=2MAΞ+NA−Ξ,cschAΞ=2MAΞ−NA−Ξ,sinAΞ=MAiΞ−NA−iΞ2i,cosAΞ=MAiΞ+NA−iΞ2,tanAΞ=−iMAiΞ−NA−iΞMAiΞ+NA−iΞ,cotAΞ=iMAiΞ+NA−iΞMAiΞ−NA−iΞ,secAΞ=2MAiΞ+NA−iΞ,cscAΞ=2iMAiΞ−NA−iΞ.whereΞ is the independent variable and M,N>0 are arbitrary constants called as deformation parameters.


## 5. Graphical Representations and Discussions

Based on the attained hung amount of solitary wave solutions for the considered STF-CQNLSE in birefringent fibres with the sense of BFD and CFD, some of them are illustrated graphically (see Figures1–6) for demonstrating the effectiveness of fractional parameter μ along with relevant physical discussions. In our analysis, the parametric values of μ are assumed as 0<μ≤1 and the other free parameters are considered based on the conditions of EDAM.Figure 1
3D periodic wave profileΨ1x,t of Equation (26) for BFD parameter (a) μ=0.9, (b) μ=1 and CFD parameter (c) μ=0.9 and (d) μ=1 with other physical parameters by considering α=2, λ=1, σ=3, A=e, M=0.2, N=0.1, κ=0.1, Π=2, p1=r1=0.2, s1=−0.3, and θ0=1.
(a)(b)(c)(d)Figure 2
Contour plots ofΨ1x,t of Equation (26) for (a) μ=0.9 and (b) μ=1 for BFD, (c) μ=0.9 and (d) μ=1 for CFD, and 2D wave profile (Solid line = BFD; Dash Line = CFD) of the effect of (e) time for μ=0.75 and (f) fractional parameter μ for t=150. The remaining parameters are as in Figure 1.
(a)(b)(c)(d)(e)(f)Figure 3
Optical rouge wave profiles ofΨ1x,t and Ψ2x,t as in Equation (38) and Equation (39), respectively, for (a) BFD parameter μ=0.85, (b) CFD parameter μ=0.85, the effect (solid line = BFD; dash line = CFD) of time for (c) μ=0.98 and (d) μ=1, (e) the effect of fractional parameter for t=750. The remaining parameters are α=0.1, λ=0.5, σ=0.2, A=e, M=−0.1, N=0.5, κ=0.1, Π=2, p1=0.2, r1=−0.2, s1=0.3, and θ0=1.
(a)(b)(c)(d)(e)Figure 4
Optical bright soliton profiles ofΨ1x,t and Ψ2x,t as in Equation (40) and Equation (41), respectively, for (a) BFD parameter μ=0.95, (b) CFD parameter μ=0.95, the effect (Solid line = BFD; Dash Line = CFD) of time for (c) μ=0.95 and (d) μ=1, (e) the effect of fractional parameter for t=1200. The remaining parameters are α=0.1, λ=0.9, σ=0.5, A=e, M=N=1, κ=0.1, Π=2, p1=0.2, r1=−0.2, s1=0.3, and θ0=1.
(a)(b)(c)(d)(e)Figure 5
3D periodic wave profiles ofΨ1x,t and Ψ2x,t as in Equation (54) and Equation (55), respectively, for BFD parameter (a) μ=0.95, (b) μ=1 and CFD parameter (c) μ=0.95, (d) μ=1. The other parameters are considered as α=σ=0.5, λ=0, A=e, M=0.2, N=0.1, κ=0.1, Π=2, p1=r1=0.2, s1=−0.3, and θ0=1.
(a)(b)(c)(d)Figure 6
Contour plots ofΨ1x,t of Equation (54) for (a) BFD parameter μ=0.95 and (b) CFD parameter μ=0.95, and 2D wave profiles (solid line = BFD; dash line = CFD) with the effect of time for (c) μ=0.75, (d) μ=1 and (e, f) with the effect of fractional parameter for t=900. The remaining parameters are as in Figure 5.
(a)(b)(c)(d)(e)(f)Figures1(a)–1(d) display the 3D periodic wave profiles Ψ1x,t by considering Equation (26) along with the influence of BFD parameter (CFD parameter) and α=2, λ=1, σ=3, A=e, M=0.2, N=0.1, κ=0.1, Π=2, p1=r1=0.2, s1=−0.3, θ0=1. In order to properly understand the influences of both BFD and CFD parameters, Figure 2 displays the contour plots and 2D wave profiles Ψ1x,t by considering the same Equation (26) and the same typical values of other parameter as in Figure 1. It is depicted from Figures 1 and 2 that the wavelength (frequency) are shorter (higher) for conformable derivative rather than beta derivative. In addition, the wavelength (frequency) are shorter (increasing) with the increase of both BFD and CFD parameter.Figure3 represents the 3D and 2D resonance solitary wave (like killer wave, singular soliton) profiles by considering Equation (38) and Equation (39) with the variation of both BFD and CFD parameter for time. To display these figures, the other parametric values are considered as α=0.1, λ=0.5, σ=0.2, A=e, M=−0.1, N=0.5, κ=0.1, Π=2, p1=0.2, r1=−0.2, s1=0.3, and θ0=1. It is interesting to show from Figure 3 that for different choice of arbitrary parameters, one can produce unstable soliton, that is, rouge waves for fractional value of the parameter μ from Equation (38) and Equation (39), but it would always produce singular soliton for μ=1. The amplitudes of rouge waves are increasing with regard to increase of fractional parameter, but decreasing and behaves pulse like with the increasing values of time.Figure4 presents the 3D and 2D resonance bright soliton profiles with the variation of time for both of BFD and CFD parameter by considering Equation (40) and Equation (41). To display these figures, the other parametric values are considered as α=0.1, λ=0.9, σ=0.5, A=e, M=N=1, κ=0.1, Π=2, p1=0.2, r1=−0.2, s1=0.3, and θ0=1. It is depicted from Figure 4 that the amplitudes of bright solitons are decreasing with the increasing values of fractional parameter μ for any given time. For non-local value μ=0.95 of BFD (solid line) and CFD (dash line) parameter, the amplitudes of the bright solitons are increasing with pulse like behavior due to the increase of time. But, the amplitudes of the bright solitons are remaining constant and the bright solitons behaves pulse with the variation of time only for the local value of μ=1.Finally, Figures5 and 6 demonstrate the 3D contour plot and 2D resonance periodic wave profiles by considering Equation (54) and Equation (55) with the variation of both BFD and CFD parameter as well as time. To display these figures, the other parametric values are considered as α=σ=0.5, λ=0, A=e, M=0.2, N=0.1, κ=0.1, Π=2, p1=r1=0.2, s1=−0.3, and θ0=1. It is depicted from Figures 5 and 6 that soliton types periodic waves are produced for any values of BFD and CFD parameter and the increasing values of time. But, only the wavelength of such waves are increasing along the space domain for a specific time with a fractional value of the BFD and CFD parameter. Further, the wavelength of these waves are remarkably decreased when the values of the fractional parameter are sufficiently increased. It is also observed that the frequency of these waves are increased dramatically for local value μ=1 rather than any nonlocal value of BFD and CFD parameter. Consequently, Figure 6(f) is clearly shown that the blue light (short wavelength) travels faster than red light (long wavelength).It is noted that Yildirim et al. [23] have only secured dark and singular type optical solitons by adopting exp-function expansion algorithm. In another report [24], they recovered bright, dark, and singular solitons by Riccati function approach, sine-Gordon function technique, and F-expansion scheme. Additionally, Zahran and Bekir [25] implemented extended simple equation method to recover some periodic and singular type solutions by taking four forms of nonlinearity. But in all these studies, their model equation is identical with this considered STF-CQNLSE when the parametric value μ=1 is taken only. Further, Dutta et al. [26] have also investigated similar types of solutions by considering the equation for parabolic law nonlinearity having conformable derivative by utilizing the extended sinh-Gordon expansion method. But the novelty of the current study is that we investigate more new solitonic structures, like periodic pulse, combo bright-dark, rogue structures, etc. along with their obtained solutions based on the existence criteria of the solutions as well as some parametric values which were not found in previous literatures. We also illustrated whether there is any effect of the fractional parameter in the equation considered and found that it has a considerable impact on the optical wave structures. This new EDAM technique is being implemented for the first time to derive several forms of optical solitons from the considered model equation, and some of the solutions return similar wave structures for other techniques suggested in the literature. Thus, the derived solutions take on an entirely new form because of the fractional order. As a result, this technique can be utilized to construct more general forms of solutions not only for the STF-CQNLSE under consideration, but also for any model equation in other disciplines.


## 6. Concluding Remarks

The classical derivative operators are sometimes unable to address the physical issues in the dynamical systems. This actually happens when the complexity may arise suddenly in a certain regime of space or time due to the impact of nonlocality or nonconservative energies or any other physical reasons in birefringent fibres. To overcome such difficulty, one may use the fractal, or so-called the BFD and CFD. Such derivatives are treated as the local or nonlocal derivatives based on the parametric value of fractional parameter. Thus, the coupled cubic-quartic nonlinear Schrödinger equations having Kerr-law nonlinearity and both of BFD and CFD space–time evolution have been considered to study the optical pulses in birefringent fibers. The effective integration scheme, that is EDAM has been successfully implemented to determine the optical wave solutions. It is observed that the fractal nonlocal operator parameter is remarkably changed the optical wave phenomena in birefringent fibres. In this way, one may study optical wave phenomena in birefringent fibres by considering the presented model having other nonlinearities. Note that some useful new solutions have been determined by considering the constraint conditions. However, one can study the stability analysis for traveling waves by forming the planar dynamical system, but beyond the scope of this study.



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*Source: 7265164-2022-12-16.xml*