Original research articleDark optical solitons to the Biswas–Arshed equation with high order dispersions and absence of the self-phase modulation
Introduction
There are a lot of mathematical models to describe different physical phenomena in nonlinear optics. Some of such mathematical models with diverse applications are Fokas–lenells equation, Ginzburg–Landau equation, Lakshmanan–Porsezian–Daniel equation, Kaup–Newell equation, Radhakrishnan–Kundu–Lakshmanan equation, and Kundu–Eckhaus equation. The Biswas–Arshed equation is another mathematical model in nonlinear optics that has been established by Biswas and Arshed in [1]. The BA equation includes some interesting properties which have distinguished it from other models, for example, the self-phase modulation has been ignored in it and both second-order and third-order dispersions have been considered. The Biswas–Arshed equation with high order dispersions and absence of the self-phase modulation is expressed by [1]where signifies the wave profile such that and are spatial and temporal variables. The parameters and are the coefficients of the group velocity dispersion and the spatio-temporal dispersion whereas the parameters and are the coefficients of the third-order dispersion and the spatio-temporal third-order dispersion. The parameter is the coefficient of the self-steepening effect while and present the coefficients of nonlinear dispersions.
In recent years, the Biswas–Arshed equation has taken into consideration by many researchers and many effective methods have been used to obtain exact solutions of it. For instance, Yildirim [2] adopted the trial equation method to derive optical solitons of Biswas–Arshed equation. Ekici and Sonmezoglu [3] employed the extended trial equation method to obtain bright, dark, and singular soliton solutions of Biswas–Arshed equation. Chen [4] exerted the modified simple equation method to gain singular solitons of Biswas–Arshed equation. Aouadi et al. [5] applied the complex amplitude ansatz to acquire -shaped, bright, and dark solitons of Biswas–Arshed equation. Recently, the conserved quantities of Biswas–Arshed equation with full nonlinearity has been determined from the corresponding densities by Khan in [6]. The reader can see [[7], [8], [9], [10], [11], [12]].
The general purpose of the present work is to retrieve a number of optical solitons including dark soliton solutions to the Biswas–Arshed equation by using the -function method [[13], [14], [15], [16], [17], [18], [19], [20], [21], [22]] and demonstrate their dynamical behavior. The -function method is a generalization of its conventional version [13] and is considered as one of the reliable techniques to derive exact solutions of nonlinear PDEs. More recently, Hosseini et al. [18] utilized successfully the -function method to acquire a series of exact solutions for a high-order dispersive cubic-quintic Schrödinger equation. Some interesting papers can be found in [[23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43]].
The rest of this work is arranged as follows: In Section 2, we review the basic ideas of the -function method to solve nonlinear partial differential equations. In Section 3, we retrieve a number of optical solitons to the Biswas–Arshed equation using the -function method. In the final section, we summarize the results of this study.
Section snippets
-function method
Consider a nonlinear partial differential equation in the form
Through the use of the assumption , we are able to write the above nonlinear PDE as
The starting point is to consider the solution of Eq. (3) as belowin which and are determined later and is a positive integer. By substituting Eq. (4) into Eq. (3), one can obtain the following nonlinear system
Biswas–Arshed equation and its optical solitons
To extract optical solitons of Biswas–Arshed equation, we first take the following ansatzwhere indicates the wave amplitude component and is the speed of the wave. The parameters , , and in the phase portion signify frequency, wave number, and phase constant, respectively.
After setting the ansatz (5) in the BA Eq. (1) and distinguishing the real and imaginary components, we find
Conclusion
A mathematical model in optical fibers, the Biswas–Arshed equation, with high order dispersions and absence of the self-phase modulation was studied, in this paper. For this aim, by making use of a complex ansatz and distinguishing the real and imaginary components, the resulting second-order nonlinear ODE in the real domain was solved by the -function method and the symbolic computation system. As a success, several dark optical solitons to the BA equation were officially retrieved and
Declaration of Competing Interest
No conflict of interest exits in the submission of this manuscript, and manuscript is approved by all authors for publication.
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