Elsevier

Optik

Volume 243, October 2021, 167406
Optik

Original research article
Solitons in optical fiber Bragg gratings for perturbed NLSE having cubic–quartic dispersive reflectivity with parabolic-nonlocal combo law of refractive index

https://doi.org/10.1016/j.ijleo.2021.167406Get rights and content

Abstract

This paper retrieves optical fiber Bragg gratings for perturbed NLSE having cubic–quartic dispersive reflectivity with parabolic-nonlocal combo law of refractive index. The Jacobi elliptic function solutions are found using the unified auxiliary equation methodDark solitons, singular solitons, bright solitons, combo singular solitons and combo dark-singular solitons are obtained.

Introduction

The study of optical solitons in optical fibers, PCF, metamaterials, optical couplers, fibers with Bragg gratings is going on strong and steady for the past few decades [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31]. Lately, the concept of cubic–quartic (CQ) solitons have emerged when it was realized that the delicate balance between chromatic dispersion (CD) and self-phase modulation (SPM) grows feeble when CD gets low. In that case, to boost up the much-needed balance, it is necessary to compensate for the low count of CD by introducing higher order dispersive effects. These are third-order dispersion (3OD) and fourth-order (4OD) dispersion terms. Together, these constitute CQ dispersive effects. For fiber Bragg gratings, it is the dispersive reflectivity, which originally stems from CD, now comes from CQ dispersion. In this paper, the coupled system of cubic–quartic nonlinear Schrödinger equation (CQ-NLSE) in fibers Bragg gratings with parabolic-nonlocal combo law of refractive index will be studied for the first time by the aid of the method mentioned in the abstract. The Jacobi elliptic function solutions are found using the unified auxiliary equation method. Singular solitons, bright solitons, combo singular solitons, dark solitons and combo dark-singular solitons of the governing model are recovered. The details of the paper are exposed in the subsequent sections.

The dimensionless form of the coupled cubic–quartic perturbed NLSE in fiber Bragg gratings with parabolic-nonlocal combo law nonlinearity is written, for the first time, as iqt+ia1rxxx+b1rxxxx+c1|q|2+d1|r|2q+ξ1|q|4+η1|q|2|r|2+ζ1|r|4q+f1|q|2xx+g1|r|2xxq+iα1qx+β1r+σ1qr2=iγ1|q|2qx+θ1|q|2xq+ρ1|q|2qx,and irt+ia2qxxx+b2qxxxx+c2|r|2+d2|q|2r+ξ2|r|4+η2|r|2|q|2+ζ2|q|4r+f2|r|2xx+g2|q|2xxr+iα2rx+β2q+σ2rq2=iγ2|r|2rx+θ2|r|2xr+ρ2|r|2rx,where al,bl,cl,dl,ξl,ηl,ζl,fl,gl,αl,βl,σl,γl,θl and ρl,(l=1,2) are real parameters. Here, qx,t, r(x,t) are the complex wave profiles. The coefficients of al, bl are third order dispersion(3OD) and fourth order dispersion (4OD), respectively. The parameters cl, ξl, fl represent the self-phase modulation (SPM) coefficients, while the cross-phase modulation (XPM) effect comes from the coefficients dl,ηl, ζl and gl. The parameters αl, βl and σl are the coefficients of inter-modal dispersion (IMD), detuning parameter and four-wave mixing effect (4WM) for Kerr part of the nonlinearity, respectively, while, γl is the coefficient of  self-steepening (SS) term. Lastly, the parameters θl, ρl are the coefficients of nonlinear dispersion terms. The system (1), (2) is a manifested version of the standard model. That is the well-known NLSE model in fiber Bragg gratings with chromatic dispersion (CD), that is structured as [23]: iqt+E1rxx+c1|q|2+d1|r|2q+ξ1|q|4+η1|q|2|r|2+ζ1|r|4q+f1|q|2xx+g1|r|2xxq+iα1qx+β1r+σ1qr2=0,and irt+E2qxx+c2|r|2+d2|q|2r+ξ2|r|4+η2|q|2|r|2+ζ2|q|4r+f2|r|2xx+g2|q|2xxr+iα2rx+β2q+σ2rq2=0,i=1,where E1 and E2 are the coefficients of chromatic dispersion (CD). In the system (1), (2), it is this CD that is replaced by 3OD and 4OD, which formulate the dispersion effects.

The objective of this article is to solve the system (1), (2) using the unified auxiliary equation approach to find the Jacobi elliptic function solutions, dark solitons, singular solitons, bright solitons, combo singular solitons and combo dark-singular solitons.

The organization of this article can be written as: the mathematical analysis is discussed in Section 2, the unified auxiliary equation algorithm is applied to the coupled system (1), (2) in Section 3 and the numerical simulations for some solutions are designed in Section 4. Lastly, conclusions are given in Section 5.

Section snippets

Mathematical analysis

In order to recover solitons of the CQ-NLSE in fiber Bragg gratings with parabolic non-local law nonlinearity, we set q(x,t)=H1(ξ)expiϕ(x,t),r(x,t)=H2(ξ)expiϕ(x,t),and ξ=xυt,ϕ(x,t)=κx+ωt+θ0,where υ,κ,ω and θ0 are all non zero parameters. Here, υ is the velocity of soliton, κ is the frequency of soliton, ω is the wave number of the soliton and finally, θ0 is the phase parameter, while H1ξ,H2ξ and ϕx,t are real functions representing the amplitude portion of the soliton and the phase component

Unified auxiliary equation method

According to this method, we assume that Eq. (31) has the formal solution H1(ξ)=A0+s=1Nfs1(ξ)Asf(ξ)+Bsg(ξ),where A0, As, Bs (s=1,.., N) are constants to be determined later, such that AN0 or BN0, while f(ξ) and g(ξ) satisfy the auxiliary ODEs : f(ξ)=f(ξ)g(ξ),g(ξ)=q1+g2(ξ)+r1f2(ξ),g2(ξ)=q1+r12f2(ξ)+cf2(ξ), where q1, r1, c are constants. Balancing H1(4)with H15 in (31), yields the balance number N=1, we have H1ξ=A0+A1fξ+B1gξ,where A0, A1 and B1 are constants and A10 or B10 [22].

Numerical simulations

In this section, we introduce the graphs of some solutions for Eqs. (59), (60), Eqs. (64), (65) and Eqs. (93), (94). Let us now examine Fig. 1, Fig. 2, Fig. 3. as it illustrates some of our solutions obtained in this paper. To this purpose, we choose some special values of the obtained parameters.

Fig. 1 Shows the profile of the dark soliton solutions (59), (60).

Fig. 2 Shows the profile of the bright soliton solutions (64), (65).

Fig. 3 Shows the profile of the combo singular soliton solutions

Conclusions

Many exact solutions have been obtained for the coupled system of cubic–quartic nonlinear Schrödinger equation (CQ-NLSE) in fiber Bragg gratings with parabolic-nonlocal combo law of refractive index. The unified auxiliary equation algorithm is applied to the coupled system (1), (2). The Jacobi elliptic function solutions have been found using the unified auxiliary equation method. The limiting process applied with modulus of ellipticity enabled this retrieval. The results are thus

Declaration of Competing Interest

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 paper.

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