Abstract
The rapid advancements in metamaterial research have brought forth a new era of possibilities for controlling and manipulating light at the nanoscale. In particular, the design and engineering of optical metamaterials have created advances in the field of photonics, enabling the development of advanced devices with unprecedented functionalities. Among the myriad of intriguing metamaterial structures, the nonlinear directional couplers with beta derivative evolution have emerged as a significant avenue of exploration, offering remarkable potential for light propagation and manipulation. This study obtains the solitary wave solutions for twin-core couplers having spatial-temporal fractional beta derivative evolution by using two different methods, the Bernoulli method and the complete polynomial discriminant system method. By graphing some of the obtained solutions, the effect of the beta derivative has been shown. The findings would be beneficial to understand physical behaviours in nonlinear optics, particularly twin-core couplers with optical metamaterials.
1. Introduction
Nonlinear models encountered as a consequence of mathematical modelling of real-life problems and the analysis of these models is significant concerns of applied mathematics. Many models in the field of physics and engineering reach significance 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 defined as a new class of artificial 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 identification. On the other hand, studies on metamaterials that can provide protection against radiation and seismic movements can be used in the field 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. The advent of metamaterials, engineered materials with extraordinary properties not found in nature, has revolutionized this field, 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, super-resolution imaging and efficient 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. These 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 offer the potential for compact and efficient devices that can guide, switch, and modulate light with unprecedented control.
The concept of fractional derivative is a mathematical tool that extends the concept of differentiation to noninteger orders. It is a generalization of the classical derivative, which corresponds to the case when the order of differentiation is an integer. Fractional calculus plays an important role in modelling because in classical dynamic systems, it is not suitable to express the effect 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. The beta derivative [6] has found applications in various scientific disciplines, including physics, engineering, signal processing, and optimization. It allows for the characterization of nonlocal and memory-dependent effects, 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. The inclusion of the beta derivative term in the mathematical model accounts for the nonlocal and memory effects 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.
The study and application of the beta derivative in OMMs offer 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 twin-core 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 fields of engineering and applied sciences because of the applications in optical tools such as metamaterials and couplers [7–18]. Some recent studies focus on a property demonstrated by solitons in symmetric optical couplers and symmetry-breaking 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 [20–22]. 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 different 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.
2. Essential Concepts of β-Derivative
The beta derivative is defined by Atangana et al. [6] as
The definition satisfies all fundamental properties of the conventional derivative. The various properties are as follows:where are real numbers, and are differentiable functions with . Letting , when , then one can obtain
The beta derivative captures the fractional order behaviour of a function, providing a powerful tool for modelling and analysing systems with complex dynamics. The study and application of the beta derivative in optical metamaterials offer 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.
3. Twin-Core Couplers with Fractional Beta Derivative Evolution
The TCCs equations with spatial-temporal BD evolution is given by [4, 28].where
Within that case and are complex valued functions describing the optical problems in two cores, respectively. and represent the beta derivatives with respect to time and space, correspondingly. , and are coefficients of dispersion while , and are related to trapping in the phase hole. and are coupling coefficient of TCCs.
To transfer the equations (4) and (5) into nonlinear ODE the following beneficial transformations can be used:where
Here, for the shows the amplitude component of the wave, is velocity, is a constant, is the frequency, is the phase component, represents the wave number, and is the phase constant. Also, phases of TCCs are identical. This step-matching condition is useful for extracting soliton solutions for the integrability of TCCs.
The consequence of letting two coefficients of linearly independent functions to zero is
If one equates two values of the speed of soliton, it yields
Then equation (9) can be written as
Additionally, real parts of the examined equations are extracted aswhere and Owing to balancing procedure with equation (13) is turned into
For twin-core couplers with Kerr law nonlinearity equations (4) and (5) can be reduced as below with :
Equation (14) can be transformed by the equations (15) and (16).
To attain traveling wave solutions, is used. Therefore, equation (17) is now rewritten aswhich have Kerr-Law nonlinearity. In order to interpret physical phenomena in nonlinear optics such as the dimensionless types of optical fields in the corresponding cores of optical fibres, and are evaluated in view of distinct mathematical functions.
4. 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 equationwhere and 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), 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:
The solution of the system is
If we rewrite the auxiliary equation with equation (21) and choose and then integrate this equation by using the CDS for polynomial, the following solutions are raised:where
Figures 1 and 2 represent the exact traveling wave solutions (22) and (23) as soliton solution with the parameters (a) and (b) respectively. The solutions occur as a bright solitary wave solution and parameter has an effect on the solutions.
(a)
(b)
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(b)
Figure 3 represents the exact traveling wave solutions (24) and (25) with the parameters , , and respectively.
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Figure 4 shows the effect of the beta derivative to the graph of the solutions (22) and (23).
Then, the auxiliary equation is chosen as
If we apply the balancing procedure, it yields with
For and
The corresponding system is
From the solution of the system above, one can obtain the set of coefficients 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 and respectively. From the solution of the system equation (32), set 3 is obtained as Set 3:
Corresponding solutions are as follows:
(a)
(b)
Figure 6 shows the exact analytical solutions (44) and (45) with the parameters . The 3D plot and 2D plot for of the solution are shown in Figures 6(a) and 6(b), respectively.
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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 differential equation is considered as the auxiliary equation:where , , and are parameters. Additionally, the solution assumption is determined via balancing principle . Substituting Bernoulli type differential equation and the assumption into equation (18) and equalling the coefficients of each power of to zero, nonlinear algebraic system is hold. The solution set of the system is obtained
When the solution of the Bernoulli type differential 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 , , , , , , , , , , . The 3D, 2D plots (at = 0.5 for various values of ) of solution (51) are shown in Figures 7(a) and 7(b), respectively.
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The 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 beneficial 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 differential 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 , , , , , , , , , . The 3D and 2D plots (at for various values of ) of are shown in Figures 8(a) and 8(b), respectively.
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(b)
Different patterns, dynamic behaviours and types of solitary wave solutions can be seen from the graphs sketched for different 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 affected by the different values of the dispersion coefficient.
6. Conclusions
The twin-core couplers with fractional BDE including Kerr nonlinearity have been considered. To obtain analytical solutions, two different 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 differential equation, and then the stability of the particular motion 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 findings 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 fields 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 field 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.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.