Abstract
In this paper, the coupled Kundu-Mukherjee-Naskar (KMN) model in Bragg grating fibers is considered to retrieve some new optical soliton solutions in () dimensions. Plenty of new exact solutions, including rational function solutions and triangle function solutions, in addition to the Jacobian elliptic function solutions, are obtained by using a complete discrimination system method. The 3D-surface plots, 2D-shape plots, and corresponding 2D contour plots of some obtained solutions are drawn, which provides a visualized structures and propagation of solitons. The novel results are new and show the effectiveness of the proposed method.
1. Introduction
The Kundu-Mukherjee-Naskar (KMN) equation was first proposed in 2014 to address rogue waves (RWs) in the ocean [1], which is usually denotes as
In recent years, many mathematicians and physicists have devoted themselves to the research and discussion of this equation. In [2], Qiu et al. considered this model to obtain first-order oceanic rogue wave solution via the Darboux transformation. In [3], Singh et al. obtained higher-dimensional nonlinear wave solutions and study the dynamics of deep water oceanic RWs. In [4], Ekici et al. indicated that the KMN equation is applicable to study the dynamics of soliton propagation through optical fibers in () dimensions on the bases of the fact that RWs are observed in a crystal fiber [5]. Note that the use of special methods constructing exact solutions of nonlinear differential models (see [6–13]) is a main research area of nonlinear optical science, the optical solitons in KMN equation have been addressed by broad researchers to recover the exact solutions by applying many effective methods including Kudryashov’s approach method [14, 15], the function method [16], the new auxiliary equation method [17], the new extended direct algebraic method [18, 19], the method of undetermined coefficients and Lie symmetry [20, 21], the trial equation technique [22], the functional variable method [23], and the modified simple equation approach technique [24]. It is worth mentioning that Yldrm [25] also used the modified simple equation approach technique to discuss a new model of coupled KMN equations in birefringent fibers. Then, Zayed et al. [26] first proposed the coupled KMN model in the Bragg gratings fibers.
In this study, we consider the nonlinear coupled ()-dimensional KMN model in the Bragg grating fibers [26]: where and are complex-valued functions that represent the wave profiles, while , , , , , , , and are real-valued constants. The parameters are the coefficients of dispersion terms. The parameters , , , and are the coefficients of nonlinearity. The parameters , , and give the intermodal dispersions, the detuning parameters, and the four wave mixing parameters, respectively. In the same work [26], the modified and addendum Kudryashov’s method are used to obtain optical solitons for model (2).
Recently, the complete discrimination system for the polynomial method was first proposed by Liu [27] as it is modified form of the discrimination system for high degree polynomial and provides a bit wider range of exact solutions than the famous methods mentioned above. Many authors have solved a lot of models in mathematics, physics, engineering, fluid mechanics, plasma, optical fibers, and other areas of science (see [28–33]). The aim of this paper is to extract the new exact optical solutions of the model (2) after utilizing Liu’s method.
The rest of the paper is organized as follows: In Section 2, a glance of discrimination system for polynomial method is given. In Section 3, after taking travelling wave transformation upon (2), the model is reduced to ordinary differential equations (ODEs). Implementing the complete discrimination system method to the ODEs, the exact solutions are obtained. In Section 4, the focus is on the graphical representation for the obtained solutions. We fix parameters to draw the 3D-surface plots, 2D-shape plots, and the corresponding 2D contour plots of some obtained solutions. Finally, the comparison of the obtained results is discussed and conclusions are illustrated in Section 5.
2. Outline of the Complete Discrimination System Method
We consider a nonlinear differential equation takes the form as follows: where is an unknown function and is a polynomial of all the derivatives of .
By applying the classical complex traveling wave transformation , we can convert Equation (3) into an ordinary differential equation (ODE), which can be written as where is a polynomial of and its derivatives, notation “” denotes the derivative with respect to .
Next, Equation (4) can be reduced to where is an integral constant and denotes a degree polynomial. According to the complete discriminant system method, we can retrieve all types of single wave solutions of Equation (2) from Equation (5) by classifying the roots for polynomial .
3. Applications to the Model
Firstly, we assume Equation (2) have solutions in the form: where and are the functions stand for the profile for optical pulse, and describes the phase portion. Here, the parameters and in the amplitude component stand for the direct cosines of the solitons along the - and -directions, respectively, and denotes the soltion velocity. The parameters and in the amplitude component signify the frequencies of the solitons along the - and -directions, respectively, while is the wave number and is the phase constant.
Inserting (6) into (2) and decomposing into the real and imaginary parts, we get the real parts for the and as while the imaginary parts are
In the view of physical reality, we suppose that where is a constant.
Then, implementing the condition (11) in (7)–(8), the real parts are rewritten as
Making Equations (12) and (13) equal for convenient, the coefficients satisfy the following results:
Putting (11) into (9) and (10), the imaginary parts become
Taking into account the linear independence of (17) and (18) and making the coefficients to zero, we obtain the constrain conditions:
Implementing (14) in (19), the velocity of the soliton is obtained as
Multiplying on both sides of (12) and integrating it on , we get where is an arbitrary constant.
For obtaining exact solutions, the following transformations are selected:
Inserting (23) into (22), we yield where is the an arbitrary constant and coefficient has the forms:
It is easy to reformulate Equation (24) with an integral representation as where is the integration constant.
Note that the sign and comprise the complete discriminant system for the polynomial , and we retrieve the exact solutions of (24) by applying the complete discrimination system method in the paragraph below.
Case 1. . Then, we have . When , the exact traveling wave solutions for Equation (2) are recovered as below:
If , the exact solutions for Equation (2) are
If , the solution is
If , we get
Case 2. , . Then, we have . When , the exact traveling wave solutions for Equation (2) are obtained as follows.
If , we have
If , we have
Case 3. , , that is, . We note that, in this case, .
If , when , the exact solution for Equation (2) takes the form
when , the exact solution for Equation (2) takes the form
where .
If , when , the exact solution for Equation (2) takes the form when , the exact solution for Equation (2) takes the form where .
If , when , the exact solution for Equation (2) takes the form when , the exact solution for Equation (2) takes the form where .
Case 4. . When , we obtain the exact traveling wave solutions for Equation (2) as where .
4. Graphical Representation
In this section, we graphically illustrate the amplitude profile of some retrieved solutions of four types by using Maple software. Graphs 1-4 show the 2D, 3D, and corresponding contour plots for the solutions , , , and , respectively, by taking suitable choice of free parameters satisfying all the constrict conditions (14)–(16) and (19)–(20). Figure 1 shows the 2D and 3D views and contour plot of kink soliton solution of (27) by taking suitable parameters. Figures 2 and 3 elaborate both cases of singular periodic solutions of (31) and (38) in 2D and 3D shapes and contour plot. The 2D and 3D views and contour plot of the bright soliton solution of (34) is depicted in Figure 4. Figures 1–4 express an insight view of the structures for the solutions of (2) which give us a better understanding of the propagation and mechanism for the origin system.
5. Conclusion
The governing model in Bragg grating fibers for the coupled Kundu-Mukherjee-Naskar model was examined on purpose of obtaining new optical soliton solutions. The complete discrimination system for the polynomial method was adopted to uncover the rational function solutions, triangle function solutions, and the Jacobian elliptic function solutions of this model. All of the derived solutions are guaranteed by putting them back into the original equation. Many other structures can be obtained by choosing free parameters. Comparing the results with the works done in [26], some of our obtained solutions are in good agreement with the published results which is presented in Table 1. The solutions (31) and (32) and (38) and (39) were not obtained by Zayed et al. [26]. The listed solutions in Case 3 and 4 are reported for the first time and can hardly be obtained by other methods. As a rather newly proposed system, there is a lot of scope to extend the horizon in this context. For this reason, additional techniques such as the Lie group, the new extended direct algebraic method, and the new auxiliary equation method will be considered on this model. These details shall be revealed, respectively, with time.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Authors’ Contributions
Chen Peng provided resources; did the investigation, methodology, writing—review and editing, and acquired the software. Furong Zhang did the validation. Hongwei Zhao acquired the software and did the data curation and formal analysis. Zhao Li did the project administration, supervision, and methodology. All authors read and agreed to the published version of the manuscript.
Acknowledgments
This work was supported by the Scientific Research Funds of Chengdu University under grant No. 2081920034.