Highly Dispersive Optical Solitons in Fiber Bragg Gratings with Quadratic-Cubic Nonlinearity

Abstract

Highly dispersive solitons in fiber Bragg gratings with quadratic-cubic law of nonlinear refractive index are studied in this paper. The $\left(G^{\prime }/G\right)$-expansion approach and the enhanced Kudryashov’s scheme have made this retrieval possible. A deluge of solitons, that emerge from the two integration schemes, are presented.

1. Introduction

One of the inherent drawbacks of soliton transmission is the compromise of the delicate balance between nonlinearity and chromatic dispersion (CD). There are several countermeasures that have been adopted to this extent. One such adopted modification is the introduction of dispersive optical solitons to compensate for the low count of CD. Some such well-known models that are studied in this context are Radhakrishnan–Kundu–Lakshmanan equation, Fokas–Lenells equation, Schrödinger–Hirota equation, and others. Later, additional dispersion terms were presented to address the model with surplus dispersive effects that led to the concept of highly dispersive (HD) optical solitons [1,2,3,4,5,6]. In this context, sixth-order dispersion (6OD), fifth-order dispersion (5OD), fourth-order dispersion (4OD), third-order dispersion (3OD), and intermodal dispersion (IMD) terms are considered, in addition to the pre-existing CD, which together makeup the HD solitons to provide the necessary delicate balance between self-phase modulation (SPM) and CD for the solitons to sustain the intercontinental distance propagation.

Another mechanism for the balance to sustain is the introduction of a grating structure along the internal walls of the fiber core. Such an engineering marvel was first introduced by the engineer Bragg; henceforth, this mechanism is referred to as fiber Bragg gratings [7,8,9,10]. The current paper is a combination of two such countermeasures to ensure a smooth transmission of solitons through the fibers for intercontinental distances. The SPM effect comes from the quadratic-cubic (QC) form of nonlinear refractive index structure. This leads to a model that is represented by the coupled nonlinear Schrödinger’s equation (NLSE), with dispersive reflectivity, which will be addressed using two integration schemes that will extract a phenomenal variety of solitons with the model. While there are several integration schemes to take care of the wide variety of optoelectronic phenomena [11,12,13,14,15,16], the current work implements two familiar integration technologies to handle the model. These schemes are the enhanced Kudryashov’s approach and $G^{\prime }/G$-expansion. The findings are derived from the two integration approaches and are collectively indicated after the model is presented.

The current paper is organized as indicated. Section 2 starts off with the introduction to the governing model with a physical interpretation of the involved parameters including the perturbation terms that are incorporated. Subsequently, Section 3 displays the preliminary mathematical analysis that is recovered by first decomposing the governing complex-valued coupled nonlinear evolution equations into the phase-amplitude format. This would give the velocity of the solitons, which would indicate the slowdown of the solitons. The parameter constraints are also revealed with this decomposition. Next, Section 4 yields a family of soliton and other solutions that stem from the application of G’/G-expansion scheme. Additional parameter constraints naturally emerge from the solution structures that guarantee the existence of the solitons and other forms of waves. Section 5 also reveals a few forms of soliton solutions, including the so-called straddled solitons that sit in between bright and singular solitons, depending on the parameter variations. The surface plots of bright and dark soliton pairs are also exhibited at the end of this section. Finally, a few conclusive statements are made with potential future avenues to venture into.

2. Governing Model

Here, the coupled mode Equations (1) and (2) describe wave propagation through nonlinear grating structures, where $q(x,t)$ and $r(x,t)$ represent the forward and backward propagating waves, respectively. ${\theta }_j$ and ${\mu }_j$$(j=1,2)$ give the nonlinear dispersions, while $c_j$ and $e_j$ yield the cross-phase modulation. ${\gamma }_j$ come from the self-steepening, while the first terms stand for the linear temporal evolution, where $i=\sqrt{-1}$. ${\alpha }_j$ arise from the IMD, while $q(x,t)$ and $r(x,t)$ stem from the wave profiles. ${\beta }_j$ yield the detuning parameters, while $d_j$ emerge from the SPM. ${\sigma }_j$ stem from the four-wave mixing. Finally, $a_l$ and $b_l$, $\left(l=1\cdots 6\right)$ are called the coefficients of IMD, CD, 3OD, 4OD, 5OD, and 6OD in sequence.

$$\begin{array}{c}i{q}_t+i{a}_1{r}_x+a_2{r}_{xx}+i{a}_3{r}_{xxx}+a_4{r}_{xxxx}+i{a}_5{r}_{xxxxx}+a_6{r}_{xxxxxx}+c_{1}q\sqrt{{\left|q\right|}^2+{\left|r\right|}^2+q^{*}r+q{r}^{*}}\\ \\ +\left(d_1{\left|q\right|}^2+e_1{\left|r\right|}^2\right)q+i{\alpha }_1{q}_x+{\beta }_{1}r+{\sigma }_1{q}^{*}{r}^2=i\left[{\gamma }_1{\left({\left|q\right|}^{2}q\right)}_x+{\theta }_1{\left({\left|q\right|}^2\right)}_{x}q+{\mu }_1{\left|q\right|}^2{q}_x\right],\end{array}$$

(1)

and

$$\begin{array}{c}i{r}_t+i{b}_1{q}_x+b_2{q}_{xx}+i{b}_3{q}_{xxx}+b_4{q}_{xxxx}+i{b}_5{q}_{xxxxx}+b_6{q}_{xxxxxx}+c_{2}r\sqrt{{\left|r\right|}^2+{\left|q\right|}^2+r^{*}q+r{q}^{*}}\\ \\ +\left(d_2{\left|r\right|}^2+e_2{\left|q\right|}^2\right)r+i{\alpha }_2{r}_x+{\beta }_{2}q+{\sigma }_2{r}^{*}{q}^2=i\left[{\gamma }_2{\left({\left|r\right|}^{2}r\right)}_x+{\theta }_2{\left({\left|r\right|}^2\right)}_{x}r+{\mu }_2{\left|r\right|}^2{r}_x\right].\end{array}$$

(2)

The mathematical model for Bragg gratings in highly dispersive fibers having quadratic-cubic form of self-phase modulation is an extension/generalization of the model that was first introduced by Winful et al. [17]. Bragg gratings in optical fibers can be uniform or nonuniform. Lately, there has been great demand for nonuniform gratings, including chirped gratings, phase-shifted gratings, and apodized gratings, in tunable lasers and signal processing applications. Models that consider nonuniform gratings, often known as dispersive reflectivity models [10], have been extensively studied. Grating nonuniformities usually result from nonuniform variations in grating period or strength along the propagation direction. Thus, the amplitude modulation produces apodization of gratings, while a small variation in phase causes chirping of gratings. Both apodization and chirping of gratings cause variations in photonic bandgap. In addition, experimental and theoretical studies have demonstrated that the apodization of Bragg gratings induces reduction of soliton velocity. Additionally, variations in the intensity of ultraviolet laser beams employed to create grating cause random variations in the local refractive index of the medium. Standard coupled mode equations are unable to describe these effects accurately.

Atai and Malomed introduced a dispersive reflectivity model to explain the effects of nonuniformities in grating systems [10]. Other models describing the influence of nonstandard gratings have also been reported, and the bandgap spectrum of these models is typically wider and/or nonhomogeneous. For the practical application of nonuniform grating, the modified coupled model equations, including the dispersive reflectivity parameter, are displayed here. It is presumed that spatial variations in the amplitude of coupled modes assist in explaining the spatial alterations of the local refractive index.

3. Mathematical Analysis

The wave profiles are considered as

$$\begin{array}{c}q(x,t)={\phi }_1\left(\xi \right)exp\left[i\psi (x,t)\right],\\ \\ r(x,t)={\phi }_2\left(\xi \right)exp\left[i\psi (x,t)\right],\end{array}$$

(3)

and

$$\xi =x-vt,\psi (x,t)=-\kappa x+\omega t+{\theta }_0,$$

(4)

where v, $\kappa$, $\omega$, ${\theta }_0$, $\psi (x,t)$, and ${\phi }_j\left(\xi \right)$ stand for the velocity, frequency, wave number, phase constant, phase component, and amplitude components of the soliton in sequence. Substituting (3) and (4) into (1) and (2) yields the real parts

$$\begin{array}{c}{a}_6{\phi }_2^{\left(6\right)}+(a_4-5a_5\kappa -15a_6{\kappa }^2){\phi }_2^{\left(4\right)}+(a_2+3a_3\kappa -6a_4{\kappa }^2-10a_5{\kappa }^3+15a_6{\kappa }^4){\phi }_2^{″}\hfill \\ \\ +\left({\alpha }_1\kappa -\omega \right){\phi }_1+(-a_3{\kappa }^3-a_6{\kappa }^6+{\beta }_1+a_5{\kappa }^5+a_1\kappa +a_4{\kappa }^4-a_2{\kappa }^2){\phi }_2\hfill \\ \\ c_1{\phi }_1\left({\phi }_1+{\phi }_2\right)+\left[d_1-\kappa \left({\gamma }_1+{\mu }_1\right)\right]{\phi }_1^3+\left(e_1+{\sigma }_1\right){\phi }_1{\phi }_2^2=0,\hfill \end{array}$$

(5)

$$\begin{array}{c}{b}_6{\phi }_1^{\left(6\right)}+(b_4-5b_5\kappa -15b_6{\kappa }^2){\phi }_1^{\left(4\right)}+(b_2+3b_3\kappa -6b_4{\kappa }^2-10b_5{\kappa }^3+15b_6{\kappa }^4){\phi }_1^{″}\hfill \\ \\ +\left({\alpha }_2\kappa -\omega \right){\phi }_2+(-b_6{\kappa }^6+{\beta }_2+b_5{\kappa }^5-b_3{\kappa }^3+b_1\kappa +b_4{\kappa }^4-b_2{\kappa }^2){\phi }_1\hfill \\ \\ c_2{\phi }_2\left({\phi }_2+{\phi }_1\right)+\left[d_2-\kappa \left({\gamma }_2+{\mu }_2\right)\right]{\phi }_2^3+\left(e_2+{\sigma }_2\right){\phi }_1^2{\phi }_2=0,\hfill \end{array}$$

(6)

and the imaginary parts

$$\begin{array}{c}(a_5-6a_6\kappa ){\phi }_2^{\left(5\right)}+(a_3-4a_4\kappa -10a_5{\kappa }^2+20a_6{\kappa }^3){\phi }_2^{‴}-\left[3{\gamma }_1+2{\beta }_1+{\mu }_1\right]{\phi }_1^2{\phi }_1^{\prime }\hfill \\ \\ +({\alpha }_1-v){\phi }_1^{\prime }+(a_1-2a_2\kappa -3a_3{\kappa }^2+4a_4{\kappa }^3+5a_5{\kappa }^4-6a_6{\kappa }^5){\phi }_2^{\prime }=0,\hfill \end{array}$$

(7)

$$\begin{array}{c}(b_5-6b_6\kappa ){\phi }_1^{\left(5\right)}+(b_3-4b_4\kappa -10b_5{\kappa }^2+20b_6{\kappa }^3){\phi }_1^{‴}-\left[3{\gamma }_2+2{\beta }_2+{\mu }_2\right]{\phi }_2^2{\phi }_2^{\prime }\hfill \\ \\ +({\alpha }_2-v){\phi }_2^{\prime }+(-6b_6{\kappa }^5+b_1+5b_5{\kappa }^4-2b_2\kappa +4b_4{\kappa }^3-3b_3{\kappa }^2){\phi }_1^{\prime }=0.\hfill \end{array}$$

(8)

Set

$${\phi }_2\left(\xi \right)=\mathrm{\Pi }{\phi }_1\left(\xi \right),\mathrm{\Pi }\ne 0\mathrm{or}1.$$

(9)

Equations (5) and (6) are thus presented as below:

$$\begin{array}{c}{a}_6\mathrm{\Pi }{\phi }_1^{\left(6\right)}+(a_4-5a_5\kappa -15a_6{\kappa }^2)\mathrm{\Pi }{\phi }_1^{\left(4\right)}+(a_2+3a_3\kappa -6a_4{\kappa }^2-10a_5{\kappa }^3+15a_6{\kappa }^4)\mathrm{\Pi }{\phi }_1^{″}\hfill \\ \\ +\left[{\alpha }_1\kappa -\omega +({\beta }_1+a_1\kappa -a_6{\kappa }^6+a_5{\kappa }^5-a_2{\kappa }^2+a_4{\kappa }^4-a_3{\kappa }^3)\mathrm{\Pi }\right]{\phi }_1\hfill \\ \\ c_1\left(1+\mathrm{\Pi }\right){\phi }_1^2+\left[d_1-\kappa \left({\gamma }_1+{\mu }_1\right)+\left(e_1+{\sigma }_1\right){\mathrm{\Pi }}^2\right]{\phi }_1^3=0,\hfill \end{array}$$

(10)

and

$$\begin{array}{c}{b}_2{\phi }_1^{\left(6\right)}+(b_4-5b_5\kappa -15b_6{\kappa }^2){\phi }_1^{\left(4\right)}+(b_2+3b_3\kappa -6b_4{\kappa }^2-10b_5{\kappa }^3+15b_6{\kappa }^4){\phi }_1^{″}\hfill \\ \\ +\left[\left({\alpha }_2\kappa -\omega \right)\mathrm{\Pi }+{\beta }_2-b_6{\kappa }^6+b_1\kappa +b_5{\kappa }^5-b_2{\kappa }^2+b_4{\kappa }^4-b_3{\kappa }^3\right]{\phi }_1\hfill \\ \\ c_2\mathrm{\Pi }\left(1+\mathrm{\Pi }\right){\phi }_1^2+\left[d_2{\mathrm{\Pi }}^2-\kappa \left({\gamma }_2+{\mu }_2\right){\mathrm{\Pi }}^2+e_2+{\sigma }_2\right]\mathrm{\Pi }{\phi }_1^3=0,\hfill \end{array}$$

(11)

while Equations (7) and (8) become

$$\begin{array}{c}(a_5-6a_6\kappa )\mathrm{\Pi }{\phi }_1^{\left(5\right)}+(a_3-4a_4\kappa -10a_5{\kappa }^2+20a_6{\kappa }^3)\mathrm{\Pi }{\phi }_1^{‴}-\left[3{\gamma }_1+2{\beta }_1+{\mu }_1\right]{\phi }_1^2{\phi }_1^{\prime }\hfill \\ \\ +\left[{\alpha }_1-v+\mathrm{\Pi }(a_1-2a_2\kappa -3a_3{\kappa }^2+4a_4{\kappa }^3+5a_5{\kappa }^4-6a_6{\kappa }^5)\right]{\phi }_1^{\prime }=0,\hfill \end{array}$$

(12)

and

$$\begin{array}{c}(b_5-6b_6\kappa ){\phi }_1^{\left(5\right)}+(b_3-4b_4\kappa -10b_5{\kappa }^2+20b_6{\kappa }^3){\phi }_1^{‴}-\left[3{\gamma }_2+2{\beta }_2+{\mu }_2\right]{\mathrm{\Pi }}^3{\phi }_1^2{\phi }_1^{\prime }\hfill \\ \\ +\left[({\alpha }_2-v)\mathrm{\Pi }+(-6b_6{\kappa }^5+b_1+5b_5{\kappa }^4-2b_2\kappa +4b_4{\kappa }^3-3b_3{\kappa }^2)\right]{\phi }_1^{\prime }=0.\hfill \end{array}$$

(13)

From Equations (12) and (13), one obtains the frequency of the soliton as

$$\kappa =\frac{a_5}{6a_6}=\frac{b_5}{6b_6},\mathrm{thus}{a}_5{b}_6=a_6{b}_5,$$

(14)

the constraint conditions

$$a_3-4a_4\kappa -10a_5{\kappa }^2+20a_6{\kappa }^3=0,$$

(15)

$$b_3-4b_4\kappa -10b_5{\kappa }^2+20b_6{\kappa }^3=0,$$

(16)

$${\mu }_j+2{\beta }_j+3{\gamma }_j=0,$$

(17)

and the velocity of the soliton as

$$\begin{array}{c}v={\alpha }_1+\mathrm{\Pi }(a_1-2a_2\kappa -3a_3{\kappa }^2+4a_4{\kappa }^3+5a_5{\kappa }^4-6a_6{\kappa }^5),\\ \\ v={\alpha }_2+\frac{1}{\mathrm{\Pi }}(-6b_6{\kappa }^5+b_1+5b_5{\kappa }^4-2b_2\kappa +4b_4{\kappa }^3-3b_3{\kappa }^2).\end{array}$$

(18)

From (18), we have the constraint condition

$$\left({\alpha }_2-{\alpha }_1\right)\mathrm{\Pi }=(a_1-2a_2\kappa -3a_3{\kappa }^2+4a_4{\kappa }^3+5a_5{\kappa }^4-6a_6{\kappa }^5){\mathrm{\Pi }}^2-(-6b_6{\kappa }^5+5b_5{\kappa }^4+4b_4{\kappa }^3+b_1-3b_3{\kappa }^2-2b_2\kappa ).$$

(19)

One of Equations (10) and (11) can be addressed by the aid of the restrictions

$$\begin{array}{c}\frac{a_6\mathrm{\Pi }}{b_6}=\frac{(a_4-5a_5\kappa -15a_6{\kappa }^2)\mathrm{\Pi }}{b_4-5b_5\kappa -15b_6{\kappa }^2}=\frac{(a_2+3a_3\kappa -6a_4{\kappa }^2-10a_5{\kappa }^3+15a_6{\kappa }^4)\mathrm{\Pi }}{b_2+3b_3\kappa -6b_4{\kappa }^2-10b_5{\kappa }^3+15b_6{\kappa }^4}=\frac{c_1}{c_2\mathrm{\Pi }}\hfill \\ \\ =\frac{{\alpha }_1\kappa -\omega +({\beta }_1+a_1\kappa -a_2{\kappa }^2-a_3{\kappa }^3+a_4{\kappa }^4+a_5{\kappa }^5-a_6{\kappa }^6)\mathrm{\Pi }}{\left({\alpha }_2\kappa -\omega \right)\mathrm{\Pi }+{\beta }_2+b_1\kappa -b_2{\kappa }^2-b_3{\kappa }^3+b_4{\kappa }^4+b_5{\kappa }^5-b_6{\kappa }^6}=\frac{d_1-\kappa \left({\gamma }_1+{\mu }_1\right)+\left(e_1+{\sigma }_1\right){\mathrm{\Pi }}^2}{\left[d_2{\mathrm{\Pi }}^2-\kappa \left({\gamma }_2+{\mu }_2\right){\mathrm{\Pi }}^2+e_2+{\sigma }_2\right]\mathrm{\Pi }}.\hfill \end{array}$$

(20)

Thus, Equation (10) falls out as

$${\phi }_1^{\left(6\right)}+{\Delta }_4{\phi }_1^{\left(4\right)}+{\Delta }_2{\phi }_1^{\prime \prime }+W_1{\phi }_1+W_2{\phi }_1^2+W_3{\phi }_1^3=0,$$

(21)

where

$$\left.\begin{array}{c}{\Delta }_4={\displaystyle \frac{a_4-5a_5\kappa -15a_6{\kappa }^2}{a_6\mathrm{\Pi }}},{\Delta }_2={\displaystyle \frac{a_2+3a_3\kappa -6a_4{\kappa }^2-10a_5{\kappa }^3+15a_6{\kappa }^4}{a_6}},\hfill \\ W_1={\displaystyle \frac{{\alpha }_1\kappa -\omega +(-a_2{\kappa }^2+a_5{\kappa }^5+{\beta }_1+a_1\kappa +a_4{\kappa }^4-a_3{\kappa }^3-a_6{\kappa }^6)\mathrm{\Pi }}{a_6\mathrm{\Pi }}},\hfill \\ W_2={\displaystyle \frac{c_1\left(1+\mathrm{\Pi }\right)}{a_6\mathrm{\Pi }}},W_3={\displaystyle \frac{d_1-\kappa \left({\gamma }_1+{\mu }_1\right)+\left(e_1+{\sigma }_1\right){\mathrm{\Pi }}^2}{a_6\mathrm{\Pi }}},a_6\ne 0.\hfill \end{array}\right\}$$

(22)

4. ($G^{\prime }/G$)-Expansion

Equation (21) admits the solution form

$${\phi }_1\left(\xi \right)=\sum _{J=0}^N{\delta }_J{Q}^J\left(\xi \right),{\delta }_N\ne 0,Q\left(\xi \right)=\frac{G^{\prime }\left(\xi \right)}{G\left(\xi \right)},$$

(23)

along with the ancillary equations

$$G^{″}+\lambda G^{\prime }+\mu G=0,$$

(24)

and

$$Q^{\prime }\left(\xi \right)=-\left[Q^2\left(\xi \right)+\lambda Q\left(\xi \right)+\mu \right],$$

(25)

which satisfy the combo soliton

$$Q\left(\xi \right)=-\frac{\lambda }{2}+\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\left[{\displaystyle \frac{r_{1}sinh\left(\frac{1}{2}\xi \sqrt{{\lambda }^2-4\mu }\right)+r_{2}cosh\left(\frac{1}{2}\xi \sqrt{{\lambda }^2-4\mu }\right)}{r_{1}cosh\left(\frac{1}{2}\xi \sqrt{{\lambda }^2-4\mu }\right)+r_{2}sinh\left(\frac{1}{2}\xi \sqrt{{\lambda }^2-4\mu }\right)}}\right],\mathrm{if}{\lambda }^2-4\mu >0,$$

(26)

the singular periodic wave

$$Q\left(\xi \right)=-\frac{\lambda }{2}+\frac{1}{2}\sqrt{-\left({\lambda }^2-4\mu \right)}\left[{\displaystyle \frac{r_{1}cos\left(\frac{1}{2}\xi \sqrt{-\left({\lambda }^2-4\mu \right)}\right)-r_{2}sin\left(\frac{1}{2}\xi \sqrt{-\left({\lambda }^2-4\mu \right)}\right)}{r_{1}sin\left(\frac{1}{2}\xi \sqrt{-\left({\lambda }^2-4\mu \right)}\right)+r_{2}cos\left(\frac{1}{2}\xi \sqrt{-\left({\lambda }^2-4\mu \right)}\right)}}\right],\mathrm{if}{\lambda }^2-4\mu <0,$$

(27)

and the rational wave

$$Q\left(\xi \right)=-\frac{\lambda }{2}+\frac{r_2}{r_1+r_2\xi },\mathrm{if}{\lambda }^2-4\mu =0,$$

(28)

where N comes from the balancing method, while $\lambda$, $r_1$, $\mu$, ${\delta }_J(J=0-N)$, and $r_2$ are constants. Balancing ${\phi }_1^{\left(6\right)}$ with ${\phi }_1^3$ in (21) simplifies (23) to

$${\phi }_1\left(\xi \right)={\delta }_0+{\delta }_{1}Q\left(\xi \right)+{\delta }_2{Q}^2\left(\xi \right)+{\delta }_3{Q}^3\left(\xi \right),{\delta }_3\ne 0.$$

(29)

Inserting (29) along with (25) into (21) leaves us with the results

$${\delta }_0=-\frac{6{\Delta }_4}{6889}\sqrt{\frac{2905{\Delta }_4}{W_3}},{\delta }_1=\frac{18{\Delta }_4}{83}\sqrt{-\frac{35}{W_3}},{\delta }_2=0,{\delta }_3=24\sqrt{-\frac{35}{W_3}},\lambda =0,\mu =\frac{{\Delta }_4}{332},$$

(30)

and

$${\Delta }_2=\frac{946}{6889}{\Delta }_4,W_1=\frac{2520}{571787}{\Delta }_4^3,W_2=\frac{18W_3{\Delta }_4}{6889}\sqrt{\frac{2905{\Delta }_4}{W_3}},W_3<0,{\Delta }_4<0.$$

(31)

Type-1: Placing (30) along with (26) into (29), the combo solitons come out as

$$\begin{array}{c}q(x,t)=\pm \frac{3{\Delta }_4}{83}\sqrt{\frac{35{\Delta }_4}{83W_3}}exp\left[i\left(-\kappa x+\omega t+{\theta }_0\right)\right]\hfill \\ \times \left\{\frac{r_{1}sinh\left(\frac{1}{2}\sqrt{-\frac{{\Delta }_4}{83}}\left(x-vt\right)\right)+r_{2}cosh\left(\frac{1}{2}\sqrt{-\frac{{\Delta }_4}{83}}\left(x-vt\right)\right)}{r_{1}cosh\left(\frac{1}{2}\sqrt{-\frac{{\Delta }_4}{83}}\left(x-vt\right)\right)+r_{2}sinh\left(\frac{1}{2}\sqrt{-\frac{{\Delta }_4}{83}}\left(x-vt\right)\right)}\left[3-{\left(\frac{r_{1}sinh\left(\frac{1}{2}\sqrt{-\frac{{\Delta }_4}{83}}\left(x-vt\right)\right)+r_{2}cosh\left(\frac{1}{2}\sqrt{-\frac{{\Delta }_4}{83}}\left(x-vt\right)\right)}{r_{1}cosh\left(\frac{1}{2}\sqrt{-\frac{{\Delta }_4}{83}}\left(x-vt\right)\right)+r_{2}sinh\left(\frac{1}{2}\sqrt{-\frac{{\Delta }_4}{83}}\left(x-vt\right)\right)}\right)}^2\right]-2\right\},\hfill \end{array}$$

(32)

and

$$\begin{array}{c}r(x,t)=\pm \frac{3\mathrm{\Pi }{\Delta }_4}{83}\sqrt{\frac{35{\Delta }_4}{83W_3}}exp\left[i\left(-\kappa x+\omega t+{\theta }_0\right)\right]\hfill \\ \times \left\{\frac{r_{1}sinh\left(\frac{1}{2}\sqrt{-\frac{{\Delta }_4}{83}}\left(x-vt\right)\right)+r_{2}cosh\left(\frac{1}{2}\sqrt{-\frac{{\Delta }_4}{83}}\left(x-vt\right)\right)}{r_{1}cosh\left(\frac{1}{2}\sqrt{-\frac{{\Delta }_4}{83}}\left(x-vt\right)\right)+r_{2}sinh\left(\frac{1}{2}\sqrt{-\frac{{\Delta }_4}{83}}\left(x-vt\right)\right)}\left[3-{\left(\frac{r_{1}sinh\left(\frac{1}{2}\sqrt{-\frac{{\Delta }_4}{83}}\left(x-vt\right)\right)+r_{2}cosh\left(\frac{1}{2}\sqrt{-\frac{{\Delta }_4}{83}}\left(x-vt\right)\right)}{r_{1}cosh\left(\frac{1}{2}\sqrt{-\frac{{\Delta }_4}{83}}\left(x-vt\right)\right)+r_{2}sinh\left(\frac{1}{2}\sqrt{-\frac{{\Delta }_4}{83}}\left(x-vt\right)\right)}\right)}^2\right]-2\right\}.\hfill \end{array}$$

(33)

In particular, if $r_1\ne 0$ and $r_2=0$, the dark solitons evolve as

$$q(x,t)=\pm \frac{3{\Delta }_4}{83}\sqrt{\frac{35{\Delta }_4}{83W_3}}\left\{tanh\left(\frac{1}{2}\sqrt{-\frac{{\Delta }_4}{83}}\left(x-vt\right)\right)\left[3-{tanh}^2\left(\frac{1}{2}\sqrt{-\frac{{\Delta }_4}{83}}\left(x-vt\right)\right)\right]-2\right\}exp\left[i\left(-\kappa x+\omega t+{\theta }_0\right)\right],$$

(34)

and

$$r(x,t)=\pm \frac{3\mathrm{\Pi }{\Delta }_4}{83}\sqrt{\frac{35{\Delta }_4}{83W_3}}\left\{tanh\left(\frac{1}{2}\sqrt{-\frac{{\Delta }_4}{83}}\left(x-vt\right)\right)\left[3-{tanh}^2\left(\frac{1}{2}\sqrt{-\frac{{\Delta }_4}{83}}\left(x-vt\right)\right)\right]-2\right\}exp\left[i\left(-\kappa x+\omega t+{\theta }_0\right)\right],$$

(35)

while, if $r_1=0$ and $r_2\ne 0$, the singular solitons shape up as

$$q(x,t)=\pm \frac{3{\Delta }_4}{83}\sqrt{\frac{35{\Delta }_4}{83W_3}}\left\{coth\left(\frac{1}{2}\sqrt{-\frac{{\Delta }_4}{83}}\left(x-vt\right)\right)\left[3-{coth}^2\left(\frac{1}{2}\sqrt{-\frac{{\Delta }_4}{83}}\left(x-vt\right)\right)\right]-2\right\}exp\left[i\left(-\kappa x+\omega t+{\theta }_0\right)\right],$$

(36)

and

$$r(x,t)=\pm \frac{3\mathrm{\Pi }{\Delta }_4}{83}\sqrt{\frac{35{\Delta }_4}{83W_3}}\left\{coth\left(\frac{1}{2}\sqrt{-\frac{{\Delta }_4}{83}}\left(x-vt\right)\right)\left[3-{coth}^2\left(\frac{1}{2}\sqrt{-\frac{{\Delta }_4}{83}}\left(x-vt\right)\right)\right]-2\right\}exp\left[i\left(-\kappa x+\omega t+{\theta }_0\right)\right].$$

(37)

Type-2: Inserting $\mu =\frac{1}{4}{\lambda }^2$ into (30) and (31) provides us the coefficients

$${\delta }_0=0,{\delta }_1=0,{\delta }_2=0,{\delta }_3=24\sqrt{-\frac{35}{W_3}},\lambda =0,$$

(38)

and

$${\Delta }_2=0,{\Delta }_4=0,W_1=0,W_2=0,W_3=W_3,W_3<0.$$

(39)

Substituting (38) along with (28) into (29), the rational waves fall out as

$$q(x,t)=\pm 24\sqrt{-\frac{35}{W_3}}{\left[{\displaystyle \frac{r_2}{r_1+r_2\left(x-vt\right)}}\right]}^{3}exp\left[i\left(-\kappa x+\omega t+{\theta }_0\right)\right],$$

(40)

and

$$r(x,t)=\pm 24\mathrm{\Pi }\sqrt{-\frac{35}{W_3}}{\left[{\displaystyle \frac{r_2}{r_1+r_2\left(x-vt\right)}}\right]}^{3}exp\left[i\left(-\kappa x+\omega t+{\theta }_0\right)\right].$$

(41)

5. Enhanced Kudryashov’s Procedure

Equation (21) holds the solution structure

$${\phi }_1\left(\xi \right)=\sum _{j=0}^N{H}_j{R}^j\left(\xi \right),H_N\ne 0,$$

(42)

along with the auxiliary equation

$$R^{{}^{\prime }2}\left(\xi \right)=R^2\left(\xi \right)[1-B{R}^{2p}\left(\xi \right)]{ln}^{2}a,0<a\ne 1,$$

(43)

which satisfies the combo soliton

$$R\left(\xi \right)={\left[\frac{4A}{\left(4A^2+B\right)cosh\left(p\xi lna\right)+\left(4A^2-B\right)sinh\left(p\xi lna\right)}\right]}^{\frac{1}{p}},$$

(44)

where p is a natural number, N comes from the balancing method, while $H_j(j=0,1,2,\dots ,N)$, B, and A are constants. Balancing ${\phi }_1^{\left(6\right)}$ and ${\phi }_1^3$ in Equation (21) gives

$$N+6p=3N⟹N=3p.$$

(45)

Case-1: Setting $p=1$ simplifies (42) to

$${\phi }_1\left(\xi \right)=H_0+H_{1}R\left(\xi \right)+H_2{R}^2\left(\xi \right)+H_3{R}^3\left(\xi \right),H_3\ne 0.$$

(46)

Inserting (46) along with (43) into (21) yields the coefficients

$$H_0=\frac{105{ln}^{3}a}{\sqrt{W_3}},H_1=0,H_2=0,H_3=24B\sqrt{\frac{35B}{W_3}}{ln}^{3}a,$$

(47)

and

$${\Delta }_1=1891{ln}^{4}a,{\Delta }_4=-83{ln}^{2}a,W_1=22050{ln}^{6}a,W_2=-315\sqrt{W_3}{ln}^{3}a,W_3>0,B>0.$$

(48)

Plugging (47) along with (44) into (46), the combo solitons evolve as

$$q(x,t)=\frac{3{ln}^{3}a}{\sqrt{W_3}}\left\{35+8B\sqrt{35B}{\left(\frac{4A}{\left(4A^2+B\right)cosh\left[(x-vt)lna\right]+\left(4A^2-B\right)sinh\left[(x-vt)lna\right]}\right)}^3\right\}exp\left[i\left(-\kappa x+\omega t+{\theta }_0\right)\right],$$

(49)

and

$$r(x,t)=\frac{3\mathrm{\Pi }{ln}^{3}a}{\sqrt{W_3}}\left\{35+8B\sqrt{35B}{\left(\frac{4A}{\left(4A^2+B\right)cosh\left[(x-vt)lna\right]+\left(4A^2-B\right)sinh\left[(x-vt)lna\right]}\right)}^3\right\}exp\left[i\left(-\kappa x+\omega t+{\theta }_0\right)\right].$$

(50)

When $W_3>0$ and $B=4A^2$, the bright solitons read as

$$q(x,t)=\frac{3{ln}^{3}a}{\sqrt{W_3}}\left\{35+8\sqrt{35}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}{}^3\left[(x-vt)lna\right]\right\}exp\left[i\left(-\kappa x+\omega t+{\theta }_0\right)\right],$$

(51)

and

$$r(x,t)=\frac{3\mathrm{\Pi }{ln}^{3}a}{\sqrt{W_3}}\left\{35+8\sqrt{35}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}{}^3\left[(x-vt)lna\right]\right\}exp\left[i\left(-\kappa x+\omega t+{\theta }_0\right)\right].$$

(52)

Case-2: Setting $p=2$ transforms (42) to

$${\phi }_1\left(\xi \right)=\sum _{j=0}^6{H}_j{R}^j\left(\xi \right),H_6\ne 0.$$

(53)

Inserting (53) along with (43) into (21) gives the coefficients

$$H_0=\frac{840{ln}^{3}a}{\sqrt{W_3}},H_1=0,H_2=0,H_3=0,H_4=0,H_5=0,H_6=192B\sqrt{\frac{35B}{W_3}}{ln}^{3}a,$$

(54)

and

$${\Delta }_1=30256{ln}^{4}a,{\Delta }_4=-332{ln}^{2}a,W_1=1411200{ln}^{6}a,W_2=-2520\sqrt{W_3}{ln}^{3}a,W_3>0,B>0.$$

(55)

Substituting (54) along with (44) into (53), the combo solitons stand as

$$q(x,t)=\frac{24{ln}^{3}a}{\sqrt{W_3}}\left\{35+8B\sqrt{35B}{\left(\frac{4A}{\left(4A^2+B\right)cosh\left[2(x-vt)lna\right]+\left(4A^2-B\right)sinh\left[2(x-vt)lna\right]}\right)}^3\right\}exp\left[i\left(-\kappa x+\omega t+{\theta }_0\right)\right],$$

(56)

and

$$r(x,t)=\frac{24\mathrm{\Pi }{ln}^{3}a}{\sqrt{W_3}}\left\{35+8B\sqrt{35B}{\left(\frac{4A}{\left(4A^2+B\right)cosh\left[2(x-vt)lna\right]+\left(4A^2-B\right)sinh\left[2(x-vt)lna\right]}\right)}^3\right\}exp\left[i\left(-\kappa x+\omega t+{\theta }_0\right)\right].$$

(57)

If $W_3>0$ and $B=4A^2$, the bright solitons stick out as

$$q(x,t)=\frac{24{ln}^{3}a}{\sqrt{W_3}}\left\{35+8\sqrt{35}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}{}^3\left[2(x-vt)lna\right]\right\}exp\left[i\left(-\kappa x+\omega t+{\theta }_0\right)\right],$$

(58)

and

$$r(x,t)=\frac{24\mathrm{\Pi }{ln}^{3}a}{\sqrt{W_3}}\left\{35+8\sqrt{35}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}{}^3\left[2(x-vt)lna\right]\right\}exp\left[i\left(-\kappa x+\omega t+{\theta }_0\right)\right].$$

(59)

The Figure 1 and Figure 2 represent the surface plots of dark and bright solitons respectively. The selected parameter values are indicated in the captions.

Figure 1. Surface plots of dark solitons (34) and (35) with $\mathrm{\Pi }=2$, $a_1=b_1=1$, $a_2=b_2=1$, $a_3=b_3=1$, $a_4=b_4=-1$, $a_5=b_5=1$, $a_6=b_6=1$, $d_1=1$, ${\gamma }_1=1$, ${\mu }_1=1$, $e_1=1$, ${\sigma }_1=1$, and ${\alpha }_1={\alpha }_2=1$.

Figure 2. Surface plots of bright solitons (51) and (52) with $a=e$, $\mathrm{\Pi }=2$, $a_1=b_1=1$, $a_2=b_2=1$, $a_3=b_3=1$, $a_4=b_4=-1$, $a_5=b_5=1$, $a_6=b_6=1$, $d_1=1$, ${\gamma }_1=1$, ${\mu }_1=1$, $e_1=1$, ${\sigma }_1=1$, and ${\alpha }_1={\alpha }_2=1$.

6. Conclusions

The current paper revealed HD solitons with the model that is studied in fiber Bragg gratings having QC form of nonlinear refractive index. Two integration algorithms were employed to extract such solitons. These solitons are now going to be of greater value to move forward. The results will be utilized to compute the conservation laws when the conserved densities would come from multiplier approach or using the Lagrangian or Lie symmetry. These results would be later displayed. Moreover, the variational principle would also give the soliton parameter variations that would further provide additional perspective to the model. Such results and other such findings would be reported in future along the lines of the pre-existing results [5,6,7,8,9].