Abstract
This research paper explores the Atangana conformable nonlinear fractional Schrödinger equation’s optical soliton wave solutions through three recently introduced computational schemes. The simplest expanded equation, the generalized Kudryashov method, and the sech-tanh expansion approaches are used for describing the structure of optical solitons by nonlinear optical fibers with the modern fractional operator. Several formulas such as hyperbolic, trigonometric, logical, dim, light, moon-bright hybrid, singular, combined singular, and regular wave solutions have been created. The employed methods are effective and worthy of being tested. The features of the Hamiltonian process were used to analyze the stability properties of the solutions obtained.
1. Introduction
The nonlinear partial differential equation is one of the most exciting science types, as it can describe several nonlinear phenomena as nonlinear partial distinctiveness equations (NLPD) or a form of NLPD [1–3], including diffusion, heat, electrostatics, fluid dynamics, electrodynamic, elasticity, and quantum mechanics. Some equations include the uncertain functions of multivariable and partial byproducts, including integral and partial derivatives [4, 5]. Several investigators in various fields have been studying the fractional order in the NLPD equations [6]. This analysis is focused on the nonlocal properties that only occur in such a derivative, as they are not present in an integer derivative [7–9]. This property is generally used to describe a location, quantity, and nonlocal Lagrangian behavior at a distance [10]. According to this property, various forms of concepts are employed to transform the NLPD equation into a standard differential equation with an integer order, including Riemann–Liouville, the two-scale fractal derivative, He fractional derivative, Caputo, and compliant fractional derivatives [11–13]. The compliant fractional derivative of Atangana is used here on the grounds of its supremacy over other concepts of fractional derivatives [14, 15]. This superiority is apparent in its capacity to extend all organizational characteristics, such as a chain law, quotient law, semigroup property, and element rule, to the traditional first derivative [16, 17].
Many analytical and semianalytical schemes are derived for the same purpose, such as Lie group method, -expansion method, tanh-expansion method, extended tanh-expansion method, improved Bernoulli subequation function, Riccati–Bernoulli sub-ODE method, sinh-cosh method, simplest equation method, generalized Kudryashov method, Adomian decomposition method, homotopy perturbation method, general homotopy analysis method, and so on [18–25].
The rest of the paper’s sections are ordered as follows: Section 2 studies the performance of the extended simplest equation [26, 27], the generalized Kudryashov [28, 29], and the sech-tanh expansion methods [30, 31] on the perturbed time-fractional nonlinear Schrödinger equation. Moreover, we study the stability property of obtained analytical wave solutions. Section 3 represents some of the obtained solutions in three & two-dimensional and contour plots. Section 4 gives the conclusion.
2. Application
In this section, we apply the Atangana conformable fractional operator to the perturbed time-fractional nonlinear Schrödinger equation that is given by [31–35].where are the arbitrary constants. Also, represents the dispersion term, is the nonlinear dispersion, is the nonlinear dispersion term and describes the propagation of optical solitons in optical fibers that exhibits a Kerr law nonlinearity. During the evolution age of communications, many fundamentals’ applications are related to equation (1) such as plasma physics and optical fiber communications. This fiber is used to transmit light between the two ends of the fiber and the widespread use in fiber-optical links. They provide a broader and higher range of communication (data) than electrical cables. Typically crafted from drawing glass (silica) or plastic of a much thicker diameter than a human scalp.
Handling equation (1) via the Atangana conformable fractional in the next wave transformation where is an arbitrary constant, yields
Separating the real and imaginary part of equation (2) and integrating the imaginary part once with zero constant of integration lead to
Comparing the coefficients of in equations (3) and (4) explains the equivalence between both equations under the following constraint
This relation leads to Applying the homogenous balance rule to equation (4) for determining the balance between the nonlinear and highest order derivative terms.
2.1. Extended Simplest Equation Method
Applying this scheme to equation (4) leads to the following general solutionwhere are arbitrary constants. Also, satisfies the next auxiliary equation [26, 27]where are arbitrary constants. Substituting equations (6) along (7) into equation (4) and collecting all terms with the same power of lead to a system of algebraic equations. Solving this system yields the following families.
2.1.1. Family I
.
2.1.2. Family II
Thus, the optical solitary wave solutions of the Atangana conformable fractional nonlinear Schrödinger equation are given based on family I as follows:
When
for
For
When
For
For
When
Additionally, the optical solitary wave solutions of the Atangana conformable fractional nonlinear Schrödinger equation are given based on family II by
When
for
For
When ,
2.2. Generalized Kudryashov Method
Applying this scheme to equation (4) leads to the following general solutionwhere are arbitrary constants. Also, satisfies the next auxiliary equation [26, 27]:
Substituting equation (21) along (22) into equation (4) and collecting all terms with the same power of lead to a system of algebraic equations. Solving this system yields the following families.
2.2.1. Family I
2.2.2. Family II
2.2.3. Family III
2.2.4. Family IV
Consequently, the optical solitary wave solutions of the Atangana conformable fractional nonlinear Schrödinger equation are given based on the above families respectively by
2.3. Sech-Tanh Expansion Method
Applying this scheme to equation (4) leads to the following general solution:where are arbitrary constants. Substituting equation (30) into equation (4) and collecting all terms of lead to a system of algebraic equations. Solving this system yields the following families.
2.3.1. Family I
2.3.2. Family II
Consequently, the optical solitary wave solutions of the Atangana conformable fractional nonlinear Schrödinger equation are given based on the above families respectively by
3. Stability Property of Solutions
This part studies the stability property of each one of the obtained solutions by using the Hamiltonian system property since the momentum in this system is given by [36, 37]where is the solution of the model. Consequently, the condition for stability of the solutions can be formulated aswhere c is the wave velocity. The momentum in the Hamiltonian system for equation (11) is given byand, thus
We conclude that this solution is unstable on the interval when . This result shows the ability of the solutions for their application. Applying the previous steps to other obtained solutions leads to studying the stability property of all solutions. This process makes choosing suitable stable solutions to use in model applications is very easy and interesting.
4. Representation of Obtained Solutions
This part studies the physical interpretation of obtained solutions under a suitable choice for the values of parameters. It represents each one of the selected solutions by three- and two-dimensional plots. For each kind, we mean the shape of the solution in real and imaginary plots to show the similarities and differences between them in these cases. We have plotted each of Figures 1–6 under the following conditions: Figure 1 illustrates the optical periodic solitary wave solution of in three-dimensional view when in the next interval to explain the perspective view of the solution. Figure 2 describes the optical periodic solitary wave solution of in two-dimensional view when in the next interval to represent the wave propagation pattern of the wave along the x-axis. Figure 3 illustrates the optical periodic solitary wave solution of in three-dimensional view when in the next interval to explain the perspective view of the solution. Figure 4 presents the optical periodic solitary wave solution of in two-dimensional view when in the next interval to represent the wave propagation pattern of the wave along the x-axis. Figure 5 illustrates the optical periodic solitary wave solution of in three-dimensional view when in the next interval to explain the perspective view of the solution. Figure 6 illustrates the optical periodic solitary wave solution of in two-dimensional view when in the next interval to represent the wave propagation pattern of the wave along the x-axis.
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5. Conclusion
The consistent fragmental derivative concept of Atangana successfully enables us to introduce three recent computational systems for the disturbed time-fractional nonlinear Schrödinger equation. This model has been given modern wave analytical solutions. We have researched the reliability of strategies to illustrate their potential to work in the sample implementations. The robust and efficient implementation of the computing schemes has been tested, demonstrating its capacity with an integer or fractional order to handle many nonlinear partial equations.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.