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1. Introduction
Optical solitons are the basic component of fiber-optic telecommunication technology. Several models have been developed to investigate this mechanism, including the nonlinear Schrödinger’s equation (NLSE). There are different¨ forms of waveguides such as optical metamaterials, optical fibers and photonic crystal fibers, among others, that send a large amount of data across intercontinental distances [1,2]. This paper considers a particular type of optical waveguides with an artificially generated magnetic field, known as magneto-optic waveguides. The benefit of such waveguides is that they reduce the soliton clutter effect ensuring smooth information propagation [3-5].
In the field of nonlinear science, the NLSE is a wellknown model that can be used in a variety of physical instances, including nonlinear optics, nuclear physics, quantum mechanics, condensed matter physics, and plasma physics, etc. [6-12]. The fractal model is gaining significance in nonlinear evolution equations (NLEE) of physics and mathematics for its many attractive properties that traditional systems fail to provide. One form of fractal NLEE is coupled NLSE in nonlinear optics. This system can handle soliton solutions having applications in optical communications, logic gate devices, ultrafast soliton switches, and soliton lasers [13].
The optical soliton solutions of NLSE with various forms of nonlinearity possess a significant part in resolving realworld problems. In optics, a soliton is the wave that is unaltered during propagation due to a delicate balance between nonlinear and dispersive effects in the medium [14-16]. Solutions for various NLSE have been sought to investigate nonlinear phenomena with the solitons being either bright or dark depending on the details provided by the governing NLSE [17-20]. Researchers have been studying these solitons with quadratic-cubic nonlinearity since this form of nonlinearity was first suggested in 2011 [21]. The study of soliton dynamics in magneto-optic waveguides is crucial.
Bright solitons can be formed from a state of attraction to a state of separation from each other by magneto-optic components. This allows us to manage the so-called soliton clutter. This article explores the soliton solutions of coupled NLSE with quadratic-cubic nonlinearity by implementing He’s semi-inverse variational method and the Painlevé approach that may be conducive for engineers and physicist to physically comprehend this model.
The semi-inverse approach is an effective tool for finding different variational principles of physical problems [22,23]. He suggested the semi-inverse variational theorem as an efficient and direct algebraic approach for computing soliton solutions [24]. Many authors went on to expand this approach and contributed to the analysis of fractal models in distinct fields of science [25-28]. Another method adopted here to obtain soliton solutions of the governing model is the Painlevé approach, which is the generalization of well-known algorithms: simplest equation method, tanh-function method, and the G’/G-expansion method [29]. This is a powerful and reliable scheme to find exact solutions of NLSE by avoiding the meromorphic solutions.
The article is organized as: Section 2 is devoted to the mathematical description. Section 3 covers the study of soliton solutions of the FLE along with their graphics. Discussion of the results is presented in Sec. 4 and 5 gives the conclusion.
1.1 Governing system
The coupled model of NLSE with quadratic-cubic nonlinearity in magneto-optic waveguides is given as:
where l
i
,m
i
,n
i
,p
i
,s
i√
,R
i
,β
i
,µ
i
,υ
i
and η
i
for i = 1,2 are constants, while
2. Mathematical analysis
To continue, the initial assumptions are as follows:
Where
Here α, h, ν and η 0 are speed, frequency, wave number, and phase constant of the wave, respectively. F i (x,t) for i = 1,2 denote the amplitude of the pulses, whereas χ(x,t) represents the phase component of the pulses.
Substituting Eqs. (3) and (4) into Eqs. (1) and (2). So, the real parts become
while the imaginary parts are given as:
Integrating Eqs. (7) and (8) and setting the integration constants to zero yields
Equating the coefficients of linearly independent functions to zero in Eqs. (9) and (10), provides the constraints:
and
It can be deduced from Eqs. (11) and (13) that the soliton frequency is
provided l 1 ≠ l 2 and β 1 ≠ β 2. Furthermore, we set
where ε ≠ 0,1. As a consequence, Eqs. (5) and (6) become
Equations (17) and (18) are equivalent by taking the constraint conditions:
From the constraint Eq. (21), the wave number ν appears to be
Next, Eq. (17) can be rewritten as
where
provided l 1 ≠ 0.
In the view of [30], the fractal form of Eq. (1) and Eq. (2) can be written as:
where α is the fractal dimension value and dF 1 /dξ α is the fractal derivative represented as follows:
3. Extraction of solitons by proposed methods
3.1. Semi-inverse method
By He’s variational principle [22] we can derive the following variational formulation for Eq. (26) as:
where
be the Lagrangian, K = 1/2(dF 1 /dξ α ) is the kinetic energy and
is the potential energy.
Using the two scale transformation b = ξ α , Eq. (28) takes the form
Using the Ritz’s approach, consider the solitary wave solution as follows
where unknown constants X and Y are to be computed further. Substituting Eq. (30) into Eq. (29) gives
Taking the corresponding derivatives of J with respect to X and Y gives
From Eq. (32) and Eq. (33) we have
Equation (30) becomes
The solitary wave solution for Eq. (26) is
provided ε ≠ 0,1.
Now, consider another possible soliton solution, this time of the form
where unknown constants W and Z are to be calculated later. Plugging Eq. (39) into Eq. (29) yields
Taking the corresponding derivatives of J with respect to W and Z leads to
From Eqs. (41) and (42) we have
with the help of which (39) takes the form
The solitary wave solution for Eq. (26) is given as:
provided ε ≠ 0,1.
3.2. Painlevé Approach
According to Paul Painlevé, the exact solution of Eq. (26) has the form:
and f(U) in Eq. (48) satisfies f U − AU 2 = 0, which is a Riccati-equation.
The solution to this equation is given as
Differentiating Eq. (48) with respect to ξ and using Riccati equation give:
Substituting F 1, F 1 ξ and F 1 ξξ in Eq. (26) and comparing the coefficients of like powers of e −cξ f(U) equal to zero, we obtain the system of equations as:
which implies the following four cases:
(i) If
(ii) If
(iii) If
(iv) If
TABLE I Comparison of the results following the Painlevé approach, φ6 expansion, and semi-inverse methods.
| Methods | NLSE | Fractal NLSE |
| Painlevé |
|
|
| ϕ6 expansion |
|
|
| Semi-inverse |
|
|
4.Results and discussion
The graphical interpretation of the obtained results and the effect of fractal parameter on them are discussed in this section. The semi-inverse variational method yields the bright soliton solutions given by Eqs. (37), (38), (46), and (47). The physical appearance of these solitons is shown in terms of |u|2 and |v|2 by assigning different parameteric values. In Figs.1 and 2, the 2D profiles are provided for fractal dimension values α = 0.2,0.5,0.7,0.9 while 3D plots are the standard solitary waves of Eqs. (37), (38), (46) and (47). Kink soliton solutions, i.e., Eq. (51-54) of a given model are obtained following the Painleve approach. In Figs. 3 and 4, the 3D plots of Eq. (51) and Eq. (52) are shown for distinct fractal dimension values α = 0.2,0.5,0.7,1. In Fig. 3, the oscillation spikes on the surface are due to the fractal effect. In Fig. 4, the fractal effect on the solution is shown by the irregularity in the surface. Equations (53) and (54) display the same graphical behavior with just reflection as in Figs. 3 and 4, respectively.
FIGURE 1 The 3D profile of a) Eq. (37) for |u|2 and b) Eq. (38) for |v|2 for the parameters: δ 1 = −0.12, δ 2 = 0.55, δ 3 = 1.2, π = 22/7, α = −3, α = 1, є = 1.5 2D plots of c) |u|2 and d) |v|2 against x at t = 0 for fractal dimension value α = 0.2,0.5,0.7,0.9.
FIGURE 2 The 3D profile of a) Eq. (46) for |u|2 and b) Eq. (47) for |v|2 the parameters: δ 1 = −0.3, δ 2 = 0.55, δ 3 = 1.2, α = −3, α = 1, є = 1.5, 2D plots of c) |u|2 and d) |v|2 against x at t = 0 for fractal dimension value α = 0.2,0.5,0.7,0.9.
FIGURE 3 Plots of Eq. (51) for the parameters: δ 1 = −0.9, δ 3 = −2.1, α = 0.8, e 1 = 1, U 0 = 0.5 and α = 0.2,0.5,0.7,1.
FIGURE 4 Plots of Eq. (52) for the parameters: δ 1 = −0.55, δ 3 = −2.2, α = −2, e 1 = 1, U 0 = 0.5 and α = 0.2,0.5,0.7,1.
Remark The obtained results are compared to those existing in the literature [3, 23] and found to be novel. Kink solitons of the governing system are obtained following the Painlevé approach, while for the semi-inverse variational method we considered the fractal model of NLSE.
5.Conclusion
In this article, we have obtained the optical solitons for fractal coupled NLSE in magneto-optic waveguides that have many applications to the propagation of data in optical fibers. Bright and kink solitons are retrieved by the implementation of He’s semi-inverse and Painlevé methods. The semi-inverse approach is a fascinating integration scheme to deduce variational principles for various differential models. On the other hand, the Painlevé technique is compelling to find exact solutions of non-integrable nonlinear differential equations by averting their meromorphic solutions. The suitable choice of parameters enables us to discuss the fractal behavior of the system. The outcomes could be helpful in the telecommunication industry to increase transmission system output capability. The impact of fractal dimension value on solutions of the coupled system has been shown graphically, facilitating the understanding of understand the dynamics of the model. The applied methodologies may be conducive to solve a variety of problems arising in engineering and applied physics.
