Solitons of the Twin-Core Couplers with Fractional Beta Derivative Evolution in Optical Metamaterials via Two Distinct Methods

,


Introduction
Nonlinear models encountered as a consequence of mathematical modelling of real-life problems and the analysis of these models is signifcant concerns of applied mathematics.Many models in the feld of physics and engineering reach signifcance when their solutions are analysed and allow to interpret the event they represent.Recently, an issue that has been increasing in importance is metamaterials and their models [1].Metamaterials can be defned as a new class of artifcial materials that do not exist in naturally occurring materials and exhibit extraordinary properties.Metamaterials are used to control and manipulate light and sound and have applications in many areas such as solar energy management, public security, medical device manufacturing, remote aviation, and sensor identifcation.On the other hand, studies on metamaterials that can provide protection against radiation and seismic movements can be used in the feld of sustainable energy, and make data transfer faster are continuing.
In the realm of modern optics, the ability to control and manipulate light has emerged as a compelling and consequential pursuit.Te advent of metamaterials, engineered materials with extraordinary properties not found in nature, has revolutionized this feld, enabling unprecedented control over the behaviour of light at the nanoscale.By intricately designing the structure and composition of metamaterials, researchers have unlocked remarkable possibilities for light manipulation, leading to a myriad of applications ranging from advanced photonic devices to information processing.Optical metamaterials (OMMs) [2,3] with negative refractive index can be used in invisibility cloak, superresolution imaging and efcient energy generation applications.
Within this expansive landscape of metamaterial research, twin-core couplers (TCCs) with beta derivative evolution [4] have emerged as a captivating avenue for exploration.Tese couplers possess the unique ability to channel light along distinct pathways within metamaterials, enhancing the precision and versatility of light manipulation.By harnessing the principles of beta derivative evolution, these couplers ofer the potential for compact and efcient devices that can guide, switch, and modulate light with unprecedented control.
Te concept of fractional derivative is a mathematical tool that extends the concept of diferentiation to noninteger orders.It is a generalization of the classical derivative, which corresponds to the case when the order of diferentiation is an integer.Fractional calculus plays an important role in modelling because in classical dynamic systems, it is not suitable to express the efect of physical problems due to their long-term occurrence [5].In these stages, processes are expressed using fractional derivatives (e.g., Riesz derivative, Caputo derivative, conformable derivative, Hadamard derivative, Riemann-Liouville derivative, etc.).However, except for the beta derivative, most of these derivatives do not satisfy some fundamental theorems of calculus.Te beta derivative [6] has found applications in various scientifc disciplines, including physics, engineering, signal processing, and optimization.It allows for the characterization of nonlocal and memory-dependent efects, which cannot be captured by traditional integer-order derivatives.
In the context of OMMs and TCCs, the beta derivative evolution refers to the use of fractional order derivatives to describe and manipulate the propagation of light within the metamaterial structure.Te inclusion of the beta derivative term in the mathematical model accounts for the nonlocal and memory efects that arise in these complex systems, enabling a more accurate representation of the light-guiding characteristics and facilitating the design and optimization of twin-core couplers with enhanced functionalities.
Te study and application of the beta derivative in OMMs ofer a deeper understanding of the underlying physics and provide a mathematical framework to analyse the dynamics of light propagation.By incorporating the beta derivative evolution into the modelling and analysis of twincore couplers, researchers can explore novel avenues for manipulating light and realize advanced photonic devices with tailored properties and improved performance.
Soliton solutions are key elements in the felds of engineering and applied sciences because of the applications in optical tools such as metamaterials and couplers [7][8][9][10][11][12][13][14][15][16][17][18].Some recent studies focus on a property demonstrated by solitons in symmetric optical couplers and symmetrybreaking transitions in quiescent and moving solitons in fractional couplers [19].Likewise, nonlinear directional couplers (NLDCs), due to the applications in nonlinear optics and signal processing, are gaining much more interest.To cover soliton solutions of the NLDCs, several computational methods have been proposed and applied by researchers [20][21][22].Along with them, TCCs studied by authors [23,24] in terms of integer order derivatives.
As a result of the details above, the study is focused on the soliton solutions for TCCs with beta derivative evolution (BDE) in OMMs.By using two diferent methods, the Bernoulli method [25] and complete polynomial discriminant system method (CPDS) [26,27], soliton solutions of TCCs having Kerr law nonlinearity via BDE and OMMs have been reported.By deriving analytical solutions, which are bell-shaped, kink-shaped and wings, for these equations, we believe that we will provide valuable insights into the fundamental properties such as energy, momentum, etc., and limitations of twin-core couplers.

Essential Concepts of β-Derivative
Te beta derivative is defned by Atangana et al. [6] as Te defnition satisfes all fundamental properties of the conventional derivative.Te various properties are as follows: where a, b, c are real numbers, H and G are diferentiable functions with 0 Te beta derivative captures the fractional order behaviour of a function, providing a powerful tool for modelling and analysing systems with complex dynamics.Te study and application of the beta derivative in optical metamaterials ofer a deeper understanding of the underlying physics and provide a mathematical framework to analyse the dynamics of light propagation.Because of the mentioned advantages, a model consists of beta derivative is preferred in this study.

Twin-Core Couplers with Fractional Beta Derivative Evolution
Te TCCs equations with spatial-temporal BD evolution is given by [4,28].
Within that case S(x, t) and T(x, t) are complex valued functions describing the optical problems in two cores, respectively.A 0 D β t and A 0 D β x represent the beta derivatives with respect to time and space, correspondingly.g 1 , g 2 , α 1 , and α 2 are coefcients of dispersion while m 1 , m 2 , l 1 , and l 2 are related to trapping in the phase hole.k 1 and k 2 are coupling coefcient of TCCs.
To transfer the equations ( 4) and ( 5) into nonlinear ODE the following benefcial transformations can be used: where Here, for p � 1,2 the Q p (δ) shows the amplitude component of the wave, υ is velocity, l is a constant, κ is the frequency, ϕ(x, t) is the phase component, ω represents the wave number, and φ 0 is the phase constant.Also, phases of TCCs are identical.Tis step-matching condition is useful for extracting soliton solutions for the integrability of TCCs.
Te consequence of letting two coefcients of linearly independent functions to zero is If one equates two values of the speed of soliton, it yields Ten equation ( 9) can be written as Additionally, real parts of the examined equations are extracted as where For twin-core couplers with Kerr law nonlinearity equations ( 4) and ( 5) can be reduced as below with H(h) � sh: Equation ( 14) can be transformed by the equations ( 15) and ( 16).
To attain traveling wave solutions, l p � g p � 0 is used.Terefore, equation ( 17) is now rewritten as which have Kerr-Law nonlinearity.In order to interpret physical phenomena in nonlinear optics such as the dimensionless types of optical felds in the corresponding cores of optical fbres, S and T are evaluated in view of distinct mathematical functions.

Construction of Solutions via Complete Polynomial Discriminant System Method
To summarize the CPDS method, consider a nonlinear fractional PDE which can be reduced to a nonlinear ODE by the wave transformation [26,27].To apply the method, one can consider the auxiliary equation where a j and n are constants.If one rewrite the equation (19) in integral form and classify the roots of the polynomial in the denominator with the help of CPDS, one can obtain the exact analytical solutions of the mentioned fractional PDE.By using the balancing principle in equation (19), n � 4. When one substitute this value in the auxiliary equation, and then taking the necessary derivatives to plug into the equation ( 18), the following system yields: Te solution of the system is If we rewrite the auxiliary equation with equation ( 21) and choose a 0 � 0 and then integrate this equation by using the CDS for polynomial, the following solutions are raised: where δ � l/β(x Figures 1 and 2 represent the exact traveling wave solutions ( 22) and ( 23) as soliton solution with the parameters Figure 3 represents the exact traveling wave solutions ( 24) and ( 25) with the parameters Figure 4 shows the efect of the beta derivative to the graph of the solutions ( 22) and (23).
Ten, the auxiliary equation is chosen as If we apply the balancing procedure, it yields with n − m � 2.

Journal of Mathematics
Te corresponding system is From the solution of the system above, one can obtain the set of coefcients and corresponding solutions as follows: Set 1: Set 2: For these two sets, analytical solutions of equations ( 4) and ( 5) are extracted as Figure 5 shows the behaviour of exact traveling wave solutions (35) and (36) with respect to the parameters From the solution of the system equation (32), set 3 is obtained as Set 3: Corresponding solutions are as follows: Journal of Mathematics Figure 6 shows the exact analytical solutions (44) and (45) with the parameters ω � 5,

Construction of Solutions via the Bernoulli Method
Bernoulli method [25,29] is one of the ansatz-based methods used to solve nonlinear evolution equations.To apply the method Bernoulli type diferential equation is considered as the auxiliary equation: where z � z(δ), P, and R are parameters.Additionally, the solution assumption is determined via balancing principle Q �  N�1 n�0 a ip z i � a 0p + a 1p z.Substituting Bernoulli type diferential equation and the assumption into equation (18) and equalling the coefcients of each power of z(δ) to zero, nonlinear algebraic system is hold.Te solution set of the system is obtained When the solution of the Bernoulli type diferential equation, the obtained parameters are substituted into the assumption of the solution, the solutions are obtained as Figure 7 represents the exact wave solution as soliton solution with the parameters Te 3D, 2D plots (at t � 0.5 for various values of β ) of solution (51) are shown in Figures 7(a

Journal of Mathematics 11
Te obtained results are for the equations ( 4) and ( 5) under some assumptions/restrictions.Now, the solutions are obtained without any assumptions/restrictions for the general case for equations (4) and (5).With the help of the benefcial transformations, equations (4) and ( 5) are reduced to two nonlinear ODEs as corresponding to imaginary and real parts, respectively.Applying the Bernoulli method procedure, following solution set for parameters is obtained When the solution of the Bernoulli type diferential equation, the obtained parameters is substituted into the assumption of the solution, the solutions are obtained Figure 8 represents the exact wave solution as soliton solution with the parameters α Diferent patterns, dynamic behaviours and types of solitary wave solutions can be seen from the graphs sketched for diferent parameters to understand the physical behaviours of the obtained solutions.One can also see that the beta fractional parameter has  a great impact on the wave structure and the amplitude is afected by the diferent values of the dispersion coefcient.

Conclusions
Te twin-core couplers with fractional BDE including Kerr nonlinearity have been considered.To obtain analytical solutions, two diferent methods, namely, the Bernoulli method and complete polynomial discriminant system method have been applied.Several types of soliton solutions have been successfully obtained by these methods.Several exact soliton solutions of the model have been attained by the CPDS method under some restrictions.Besides, by using the Bernoulli method the solutions are obtained without any assumptions/restrictions for the general case.For stability, three options are available: Lyapunov stability, orbital stability, and Vakhitov-Kolokolov criterion [19].In general, the obtained solution is considered as a soliton/travelling wave solution of a nonlinear partial diferential equation, and then the stability of the particular motion Q p (x − ct) has to be compared with a general class of motion.When compare our solutions with [4,28], one can see that some of the results are new.It is believed that some of the fndings in this study are not seen in the literature till now.From integrated photonic circuits to optical sensing and beyond, these couplers hold immense potential for advancing various felds of science and technology.By leveraging the analytical solutions presented in this paper, researchers can unravel the complex dynamics of light propagation within these structures and pave the way for transformative advancements in the feld of light manipulation.By using the analytical frameworks, researchers can gain deeper insights into the underlying physics, establish design guidelines, and open up avenues for further innovation and optimization of these couplers in OMMs.
7, respectively.Te solutions Journal of Mathematics occur as a bright solitary wave solution and β parameter has an efect on the solutions.