This work was devoted to unearth W-chirped to the famous Chen–Lee–Liu equation (CLLE) in optical monomode fibres. The results obtained will be useful to explain wave propagating with the chirp component. To attempt the main goal, we have used the new sub-ordinary differential equation (ODE) technique which was upgraded recently by Zayed EME, Mohamed EMA. Application of newly proposed sub-ODE method to locate chirped optical solutions to the Triki–Biswas equation equation. Optik. 2020;207:164360. On the other hand, we have used the modulation analysis to study the steady state of the obtained chirped soliton solutions in optical monomode fibres.
Today, many authors focused their interest on treatment of the nonlinear physical system to unearth the wave called “soliton.” Solitons have been found in many different physical systems such as bright soliton, dark soliton, breather-like solitons, and abundant vector solitons [1,2,3, 4,5,6, 7,8,9, 10,11,12, 13,14]. The prototypical model for the study of these solitons is the nonlinear Schrödinger (NLS) equation with attractive or repulsive interactions . Many experiments suggested that the NLS model described well the evolution dynamics of many different nonlinear systems, such as water wave tank, optical fibre, and Bose–Einstein condensate . The propagation of chirped soliton in the nonlinear physical system has become nowadays a wide field of research. During the spread of the latter, the pulse is loosed in nonlinear physical system because of many kinds of perturbations and properties of materials . Today, with the event of chirped soliton which is able to remain in its form after perturbation or collision, it is possible to make pulse uniform in the nonlinear physical system . By exciting chirped soliton, it is possible to amplify or reduce a pulse which propagates in the nonlinear physical system . Otherwise, the W-chirped soliton as mentioned in the title is a chirped soliton with W-shape. In short, the solitons are full of many interests in various fields of applications such as communication, medicine, hydrodynamic just to name a few .
In addition, other models have been designed in recent years to illustrate the propagation of optical waves, among which the the Triki–Biswas equation , Fokas–Lenells equation  and so on. In this logic, the CLLE model was built to describe soliton in monomode optical fibres. It is given in the following form :
To turn to the ODE, the following transformation is assumed:
To arrive at an integrable form of equation (9), we conjecture , which makes it possible to obtain the following form:
From there we get the solutions of equation (11), which is written as follows:
Next, we employ equation (10) to point out the following result: , , , , .
, and assume that , the following case is obtained
According to the new type of algorithm set out in ref. , the following solutions should be revealed
i.1 , and , what follows from
Considering , , we have gained combined bright soliton and hyperbolic functions as solutions
Figure 1 plots ( ) the corresponding W-chirp bright soliton of equation (17) and the bright soliton equation (16) when the speed of the soliton is and the parameter of the sub-ODE method is given by . Furthermore, Figure 2 is the spatiotemporal plot 2D of equations (16) and (17), respectively, at the same constraint relation on the velocity and free parameter of the sub-ODE algorithm. Moreover, Figure 3 is the spatiotemporal plot of the dark soliton solutions of analytical solution equation (25).
Considering , and , the following bright soliton and hyperbolic functions as solutions is obtained
For , , , we obtain the following cases
iv.1 we gained hyperbolic function solutions
iv.2 we gained trigonometric function solutions
For , in what follows, Jacobian elliptic function solutions will be presented. It should be noted that these solutions can turn to soliton solutions such as bright, dark solitons and periodic or singular functions when or Here the parameter is the Jacobian modulus.
Having already obtained bright and dark soliton solutions, periodic and singular solutions above, we will limit ourselves strictly to the Jacobian elliptic solutions type. They look like
vi.1 For , , , it is revealed
vi.2 For , , , it is revealed
vi.3 For , , , hence
After obtaining abundant analytical solutions, it is judicious to seek the stability of these solutions by using the famous linear stability technique in equation (1). To get there, we project the perturbed solution of equation (1) in the following form:
Figure 4 depicts the MI gain spectrum with the effect of SPM, which is related to the kerr nonlinearity of the monomode optical fibres. Inspecting the curve, two regimes are setting out. In the first case, we assume the positive SPM and the second one with negative valued of SPM.
This article concerns the study of the W-chirped soliton and other solutions in optical monomode fibres by using the dimensionless CLLE. More recently, chirped soliton to the model with dual power law of nonlinearity has been studied. The steady state of the obtained results has not been made. In view of the importance of the chirped signal in the communication system, we undertook the modulation analysis of the steady state of the obtained results in order to be able to define their domain of existence through graphical representations of the gain spectrum. The perturbation effect to will be added next the model to better circumscribe the constraints linked to the propagation of the chirped signal with perturbation effects.
Declaration of competing interests: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this article.
Funding: This work was supported by the Natural Science Foundation of China (Grant Nos. 61673169, 11301127, 11701176, 11626101, and 11601485).
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