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.

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 [14]. 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 [59]. 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 [1014]. 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 [1922] 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 [2327]. 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 [3032]. 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 [3338], authors have been used such derivatives for describing a wide range of physical processes. For instance, Uddin et al. [3336] 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].

where the complex–valued function is measuring the optical wave propagation in polarization preserving fibers, () is the spatial (temporal) variable, is the Kerr law of refractive index, respectively. The coefficients and 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):

Whereas the STF-CQNLSE is described in terms of CFD by

In the above coupled systems, (), indicated the coefficients of self-phase modulation (cross-phase modulation)and the effect of four–wave mixing is discarded. Many researchers [4047] 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 . 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. Let and be a function such that . Then the beta fractional derivative of order of is exhibited as and defined by

Properties 2. Let and suppose that , are two beta-differentiable functions of order . Then It is noted that if one introduce , when , in Equation (4), one of the most important property of BFD is defined as 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 that be a function such that . Then the conformable fractional derivative of order of is denoted as and given by

Properties 4. Suppose that and assuming , are two functions conformable differentiable of order . Then

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: where . The real functions and are representing the amplitude portion and the phase component of the pulse wave propagation, respectively. It is noted here that and are considered for BFD and defined by whereas in sense of CFD, , and are defined by

Here , , , and 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: where the superscripts , , , and are denoted the fourth, third, second, and first order derivatives of , respectively.

Imaginary parts:

Applying the linearly independent principle to imaginary parts for recovering:

This two equations gives the frequency of waves as

This suggests that and .

Setting , where is any nonzero constant so that . The real parts implies to the following:

Both these equations have the similar form under the following constraint conditions:

Now, considering

Equation (16) reduces to the following form

4. Analytical Solutions of STF-CQNLSE via EDAM

Applying the homogeneous balance principle to and 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 :

where are constants with . is satisfied the following ODE:

Note that Equation (22) offers a hung amount of solutions by depending on , , and as set out in Reference [39]. By considering Equation (21) and Equation (22), a polynomial in is obtained from Equation (20). As a result, a set of algebraic equations (ignored for convenience) for , and 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: where and .

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. When and ,

Case 2. When and ,

Case 3. When and ,

Case 4. When and ,

Case 5. When and ,

Case 6. When and ,

Case 7. When and

Case 8. When ,

Case 9. When and ,

Case 10. When and ,

It is noted that all the obtained solutions are valid whenever . Here, the formula for hyperbolic and trigonometric functions are defined as

where is the independent variable and 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 Figures 16) for demonstrating the effectiveness of fractional parameter along with relevant physical discussions. In our analysis, the parametric values of are assumed as and the other free parameters are considered based on the conditions of EDAM.