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# Bifurcation Analysis and Bounded Optical Soliton Solutions of the Biswas-Arshed Model

1 Department of Mathematics, College of Science at Alkharj, Prince Sattam bin Abdulaziz University, Alkharj, 11942, Saudi Arabia

2 Department of Mathematics, Pabna University of Science and Technology, Pabna, 6600, Bangladesh

3 Faculty of Science, Yibin University, Yibin, 644000, China

* Corresponding Author: Muhammad Nadeem. Email:

(This article belongs to the Special Issue: Fractal-Fractional Models for Engineering & Sciences)

*Computer Modeling in Engineering & Sciences* **2023**, *135*(3), 2197-2217. https://doi.org/10.32604/cmes.2023.022301

**Received** 03 March 2022; **Accepted** 19 April 2022; **Issue published** 23 November 2022

## Abstract

We investigate the bounded travelling wave solutions of the Biswas-Arshed model (BAM) including the low group velocity dispersion and excluding the self-phase modulation. We integrate the nonlinear structure of the model to obtain bounded optical solitons which pass through the optical fibers in the non-Kerr media. The bifurcation technique of the dynamical system is used to achieve the parameter bifurcation sets and split the parameter space into various areas which correspond to different phase portraits. All bounded optical solitons and bounded periodic wave solutions are identified and derived conforming to each region of these phase portraits. We also apply the extended sinh-Gordon equation expansion and the generalized Kudryashov integral schemes to obtain additional bounded optical soliton solutions of the BAM nonlinearity. We present more bounded optical shock waves, the bright-dark solitary wave, and optical rogue waves for the structure model via these schemes in different aspects.## Keywords

The study of optical solitons and their applications in the account of optical fiber transmissions is a paramount topic in communication networks. The general concepts of transmission optical solitons in nonlinear optical fiber systems are fundamentally important in controlling optical continuum creation and transferring information over very long distances. The general nonlinear complex models (see [1–7]) are the best examples of the exploration and description of the effect on the picosecond vibration with the group velocity dispersion (GVD) as well as the self-phase modulation (SPM). Such models can be used to explicitly address the electro-magnetic waves with ultrashort pules that can cover the world within a nanosecond. Thus these varieties of waves have been highly studied in the communication technologies, especially, in the optical fibers, data transmissions, telecommunication sectors, transoceanic spaces and so on [6–10]. Recently, Biswas and Arshed introduced a model which is greatly interesting due to the consideration of minor GVD and neglecting the SPM [1]. A number of works have done on this model to investigate optical solitons by using various approaches such as the trail solution technique [12], modified simple equation technique [13], Kerr and power law nonlinearity [14,15], mapping method [16], the extended trial function method [17], parameter restriction approach [18] and the

The purpose of this paper is to show how we can present optical shock wave including both bright and dark, optical soliton solutions, and construct the periodic wave solutions via dynamical system method with bifurcation analysis for the model. We also use the extended sinh-Gordon equation expansion (EShGEE) and the generalized Kudryashov (GK) approaches [20,21] to get a more bounded wave solution. To our knowledge, these types of investigations for the BAM are the first step in the study of the dynamical system, EShGEE and the GK methods.

The Biswas-Arshed model (BAM) [11] is given as the following form:

where

The transformation variable [11]

where

and the imaginary part:

One can easily find the following conditions that satisfies the Eq. (4):

Thus, we have to analyze the Eq. (3) only. In the following subsection, we analyze bifurcation to acquire phase portraits to identify the number and types of solutions that exist for the model. Beside this, we apply three different techniques, namely, the dynamical, the EShGEE and the GK approaches to the corresponding model Eq. (3) in order to obtain the corresponding exact optical soliton solutions of Eq. (1).

2.2 Bifurcations Analysis of the BAM

We first consider the bifurcations of phase orbits of ordinary differential equation (ODE) Eq. (3). To proceed of the motive, we need to convert the second order ODE to a dynamical system, which is possible by setting

with the Hamiltonian

where

Setting the system of Eq. (5) to zero gives critical points

If

(ii)

On the other hand, if

Moreover, a different dynamical system can arise for

with the Hamiltonian

It has a unique critical point

2.3 Bounded Travelling Wave Solutions of BAM

This section will provide the explicit expressions of all bounded travelling wave solution of the system Eq. (5).

2.3.1 The Periodic Wave Solution

Recall the cases

where

Combining Eqs. (12) and (13), it leads to

By the direct elliptic integral and calculation, we arrive at the solution

where

where

Similarly, the periodic solution corresponding to

where

where

Again, recall the cases

where

Combining Eqs. (20) and (21), it leads to

Again by the elliptic integral and after some calculation, we arrive at the solution

where

where

2.3.2 The Solitary Wave Solutions

Using the cases

where

Applying the elliptic integral and some calculation, we arrive at the solution

The resulting solution

Similarly, let the homoclinic orbit and initial value

Through the elliptic integral and after some calculation, we arrive at the solution

The obtained solution

2.3.3 The Shock Wave Solutions

For the cases

where

and applying the elliptic integral and after some calculation, we arrive at the solution

Noting that

The resulting solution

2.4 Optical Soliton Solutions to BAM via the EShGEE

One considers the general form of the trail solution in the extended sinh-Gordon equation expansion approach (EShGEE) [23],

where

Due to balance principal, the value n of Eq. (35) can be obtained. The Eq. (36) has been obtained from the sinh-Gordon equation [23],

and they [23] obtained the solutions,

and

where

We now compute the balance number of Eq. (3), which leads to

Putting Eq. (40) into Eq. (3) along with Eq. (36), we obtain a polynomial of

where

Now for the Set 1, if we combine Eq. (41) with Eqs. (38)–(40), and substituting into Eq. (2), we obtain the exact soliton solutions of Eq. (1). These solutions of the model give us the optical shock wave and optical singular shock solitons,

where

Similarly, for the Set 2, if we combine Eq. (42) with Eqs. (38)–(40), and substituting into Eq. (2), we obtain the exact soliton solutions of Eq. (1). These solutions give us the combination of chirp-free bright and optical shock wave double solitons, and the combination of optical singular double solitons,

where

where

2.5 Optical Soliton Solutions to the Biswas-Arshed Model via the GK Method

The GK method [30] is a unique approach to obtaining the generalized solitons and periodic rogue waves of the nonlinear evolution equations (NLEEs) [20,21]. We now consider a rational series in this method as:

where

with the solution

where h is the integral constant.

The trail solution Eq. (50) to the BAM takes the following form for the balance numbers

Putting Eq. (53) into Eq. (3) along with Eq. (51), we obtain a polynomial of

where

where

where

where

Now for the Set 1, if we combine Eq. (54) with Eq. (53) and substituting into Eq. (2), we obtain the exact soliton solutions of Eq. (1). These solutions are able to give us optical rogue wave solitons,

where

The behavior of the solution

Similar, for the Set 2, if we combine Eq. (55) with Eq. (53) and substituting into Eq. (2), we obtain the exact soliton solutions of Eq. (1). These solutions lead to optical rogue wave solitons,

where

where

where

This section will provide some discussions of the physical importance of the acquired results. The results of the BAM structure presented in this research have richer physical structure than earlier outcomes in the literature [11–18]. The recorded solutions are significant in the context of nonlinear dynamics, physical science, mathematical physics and the optical communication through optical fiber. We studied BAM given by Eq. (1) using bifurcation analysis, dynamical system, EShGEE and GK schemes. The bifurcation scheme provided us the evidence of the existence of various periodic wave and optical soliton solution splitting parametric areas shown in different phase portraits in Figs. 1 and 3 of the BAM. We also illustrated the physical meaning of the obtained explicit solutions 3D and contour plots that appeared in Figs. 4 and 11.

The main results of this paper are on the application of the bifurcation analysis via a dynamical system scheme and the derivation of all bounded optical wave solutions of the Biswas-Arshed model. We obtained all types of phase portraits and corresponding bounded optical shock wave, bounded solitary wave, and the bounded periodic wave solutions of the BAM by using the dynamical scheme, the extended sinh-Gordon equation expansion, and the generalized Kudryashov integral schemes. To our knowledge, these types of solitons for the Biswas-Arshed model have not been explored before [13–16,19]. All the solutions are illustrated graphically. The model could be investigated to get multi-soliton and rogue wave solutions by the other existing methods, in particular, Hirota bilinear approach [24,25] and Darboux transformation [26]. The results would be interesting to use in social media, telecommunication industries, internet zone, and many other aspects.

Acknowledgement: The authors thank to Prof. Ji-Huan He for his valuable suggestions that helped us in improving the quality and presentation of this paper.

Funding Statement: This research was supported by the Deanship of Scientific Research, Prince Sattam bin Abdulaziz University, Alkharj, Saudi Arabia, under Grant No. 2021/01/19122.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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## Cite This Article

**APA Style**

*Computer Modeling in Engineering & Sciences*,

*135*

*(3)*, 2197-2217. https://doi.org/10.32604/cmes.2023.022301

**Vancouver Style**

**IEEE Style**

*Comput. Model. Eng. Sci.*, vol. 135, no. 3, pp. 2197-2217. 2023. https://doi.org/10.32604/cmes.2023.022301

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