Abstract
This paper retrieves the vector-coupled version of the generalized anticubic nonlinearity model in fiber Bragg gratings. With the help of the trial equation approach and the complete discriminant system for polynomial, nine families of the optical solitons solutions and single traveling wave solutions for the fiber Bragg gratings with generalized anticubic nonlinearity are obtained. Under specific parameter values, three-dimensional diagrams, two-dimensional diagrams, density plots, and contour plots of the obtained solutions are displayed. Moreover, the solutions obtained in the paper further demonstrated their accurate physical behaviors.
1. Introduction
The concept of optical soliton was first proposed by Hasegawa and Tappert in 1973. Later, Mollenaure et al. observed temporal solitons in optical fibers in the laboratory for the first time. Since then, the research of optical soliton communication has started. In 2014, researchers first created optical soliton state in Kippenberg’s Laboratory and informed that optical solitons can be generated by nonlinear processes. Optical soliton is a special type of ultrashort pulse, which can keep its shape, amplitude, and velocity unchanged for long time and distance, such as across the globe for transcontinental and transoceanic distances, when it propagates in optical fiber. Manufacturability of optical solitons can not only bring surprising ultrahigh data rate but also greatly reduce energy consumption of light source in an optical communication system. Therefore, optical solitons have a very important value in the field of modern communication and nonlinear optics [1]. Also, the study of optical solitons has attracted considerable attention. While the fiber Bragg gratings (FBGs) have been considered the outstanding sensor components, which can be used in the optical fiber communication system, fiber laser, and other fields. The optical nonlinearity of FBGs have been studied by integral technique [2, 3], such as Kerr law, log-law, triple-power law, power-law, dual-power law, quadratic-cubic law, cubic-quintic-septic law, anticubic law, parabolic-law and parabolic-nonlocal law. Optical solitons can transmit stably in fiber grating throughout the proper selection of input power and initial pulse width. Therefore, the study is a very important topic.
The governing generalized anticubic (AC) nonlinearity is represented as [4–6]
Here, is imaginary unit and . is the complex-valued function that expresses pulses transmitting across the fibers. In Equation (1), the first term is the linear temporal evolution, whereas the coefficient represents chromatic dispersion (CD), the constant coefficients present self-phase modulation (SPM), and is the power-law nonlinearity parameter such that . According to the reported literature [7], the majority of results are from numerical. In [8], Zayed et al. obtained the optical solitons of Equation (1) with three cases at by implementing extended auxiliary equation method. Obviously, the solutions obtained in [8] mainly concentrate on the Jacobi elliptic functions solutions.
It is impossible to derive the model equations in birefringent fibers or Bragg gratings with a general value of ; therefore, we consider the case of as permitted by the stability regime.
Then, the generalized AC splits into two components, and the vector-coupled model is expressed as (see [8]) where , and for are constants. The coefficients , and are from SPM and the coefficients , and are associated with crossphase modulation effects.
The rest of this article is listed as follows. In Section 2, under the traveling wave transformations and the trial equation method, Equation (3) is changed into planar dynamic system. In Section 3, all optical soliton solutions of Equation (3) are acquired via the complete discrimination system technique. In Section 4, we summarize the results of the current work.
2. Mathematical Analysis
For Equation (3), the following traveling wave transformation should be considered first (see [8]): where , , , and are all nonzero constants. represents the velocity of soliton. represents the frequency of soliton. represents the wave number. represents phase constant. , and are real functions.
Substituting (4) into the above system (3), separating the real part from the imaginary part, we obtain the formula: respectively.
Let , where is a nonzero constant. Thus, Equations (5)–(8) can be rewritten as
Thus, we obtain the following constraint condition from (11) and (12)
Therefore, where . So, Equations (9) and (10) have the same form under the constraint conditions:
Also, Equation (9) can be converted to (16) where , , , and satisfying .
Taking the transformation , where is a new function of . Thus, Equation (16) can be rewritten as
According to the polynomial trial equation method of the rank homogeneous equation, we take the following trial equation [9]: where are constants.
Multiplying both sides of Equation (18) by and integrating it yields [10] where is the integral constant.
Plugging (18) and (19) into (17), we can get a polynomial about , denoted by . Using the balance principle, we get . Let all the coefficients of be zero, and a system of algebraic equations is obtained. We can calculate that is an arbitrary constant, , and .
Consequently, Equation (19) can be rewritten as
In Equation (20), if , that is, , taking the transformation
So, we have where and .
If , that is, , then we take the transformation
So, we have where , and .
Denote
Then, Equation (20) can be written in the following integral form: where is the integral constant.
Since Yang et al. have introduced the algorithm [11] for distinguishing the roots of polynomials, many solitons have been constructed in recent years. The discriminant system of quartic polynomials of is
Integrating formula (26), we obtain optical solitons and exact traveling wave solutions [12–17] of Equation (3) under nine cases. Details will be given in the next section.
3. Optical Wave Patterns
Case 1. When , that is, . If , the solutions of Equation (3) are given in the following form:
When , we draw the grah of as shown in Figure 1.
Case 2. When , has a pair of conjugate complex roots of multiplicities two, that is, , where are real numbers and . If , the solutions of Equation (3) are given in the following form:
Case 3. When , has two real roots of multiplicities two, that is, , where are real numbers and . If .
Case 3.1. When or , the solutions of Equation (3) are given in the following form:
(a) 3D graph
(b) 2D graph
(c) Density plot
(d) Contour plt
When , we draw the graph of as shown in Figure 2.
Case 3.2. When , the solutions of Equation (3) are given in the following form:
Case 4. When , has two real roots and real roots with multiplicities two, that is, , where are real numbers satisfying .
Case 4.1. If
Case 4.1.1. When and , or and , we can calculate (26) and get the solutions of Equation (3):
Case 4.1.2. When , we can calculate (26) and get the solutions of Equation (3)
Case 4.2. If .
Case 4.2.1. When and , or and , we can calculate (26) and get the solutions of Equation (3)
Case 4.2.2. When , we can calculate (26) and get the solutions of Equation (3)
Case 5. When , has real roots of multiplicities three and real roots with multiplicities one, that is, , where are real numbers.
Case 5.1. If , when and , or and , the solutions of Equation (3) are given in the following form:
Case 5.2. If , when and , or and , the solutions of Equation (3) are given in the following form:
Case 6. When , has real roots of multiplicities two and a pair of conjugate complex roots, that is, . If , the solutions of Equation (3) are given in the following form: which is a solitary wave solution and where .
Case 7. When , has four distinct real roots, that is, , where , and are real numbers satisfying .
Case 7.1. If .
Case 7.1.1. When or , the solutions of Equation (3) are given in the following form:
Case 7.1.2. When , the solutions of Equation (3) are given in the following form:
Case 7.2. If .
Case 7.2.1. When , the solutions of Equation (3) are given in the following form:
Case 7.2.2. When , the solutions of Equation (3) are given in the following form: where .
Case 8. When , has two different real roots and a pair of conjugate complex roots, that is, , where are real numbers, , and .
Take the transformation , where , and .
Case 8.1. If , the solutions of Equation (3) are given in the following form:
Case 8.2. If , the solutions of Equation (3) are given in the following form which are elliptic double periodic function solutions and where .
Case 9. When , has two pairs of conjugate complex roots, that is, , where are real numbers and .
If , we take the transformation , where , the solutions of Equation (3) are given in the following form:
where .
(a) 3D graph
(b) 2D graph
(c) Density plot
(d) Contour plot
Compared with the reported in the existing literature [8], we have obtained exponential function solutions, trigonometric function solutions, rational function solutions, hyperbolic function solutions, and implicit function solutions(see Table 1).
4. Conclusion
This paper studied optical solitons with generalized anticubic nonlinearity in fiber Bragg gratings with one case at . The complete discriminant system for polynomial method combining with the trial equation method yielded a plethora of solutions. In addition to the Jacobian function solutions reported in the existing literature [7, 8], we have obtained exponential function solutions, trigonometric function solutions, rational function solutions, hyperbolic function solutions, and soliton solutions. All such studies will yield marvelous and novel results that will be gradually disclosed. In the future work, we will still focus on the study of optical soliton solutions of coupled nonlinear partial differential equations. Moreover, we will further analyze the dynamic behavior of such equations.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This project is supported by the Scientific Research Funds of Chengdu University under grant no. 2081920034.