Implicit Quiescent Optical Soliton Perturbation with Nonlinear Chromatic Dispersion and Kudryashov’s Self-Phase Modulation Structures for the Complex Ginzburg–Landau Equation Using Lie Symmetry: Linear Temporal Evolution

Abstract

This paper investigates quiescent solitons in optical fibers and crystals, modeled by the complicated Ginzburg–Landau equation incorporating nonlinear chromatic dispersion and six self-phase modulation structures introduced by Kudryashov. The model is formulated with linear temporal evolution and analyzed using Lie symmetry methods. The study also identified parameter constraints under which solutions exist.

1. Introduction

The propagation dynamics of solitons in optical fibers, photonic crystal fibers (PCFs), and other waveguides are governed by two fundamental characteristics: chromatic dispersion (CD) and self-phase modulation (SPM) structures. The fragile equilibrium ensures the propagation of solitons via these waveguides across transcontinental and transoceanic distances [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]. At times, this equilibrium may be disrupted owing to various neglected characteristics of optical waveguides or improper treatment of these waveguides. This would result in the CD acting nonlinearly, causing the solitons to become impeded in their propagation. Such potential catastrophic consequences are prevalent and must be avoided at all costs. This study specifically addresses this type of problem. By analyzing the fundamental mechanisms that cause soliton instability, we aim to develop methods to mitigate these problems. We want to improve the reliability of soliton propagation in optical waveguides through thorough research and modeling, thereby providing effective communication systems across long distances.

Multiple models exist that examine the dynamics of optical solitons within these waveguides. One such model is the complex Ginzburg–Landau equation (CGLE). This study examines the formation of quiescent optical solitons in the context of a nonlinear CD in CGLE, influenced by Hamiltonian perturbation terms. It must be noted that the unperturbed version of the model has been previously addressed for CGLE with several additional forms of SPM structures [1,2]. The SPM structures that are going to be taken into account here were proposed by Kudryashov in the past decade. There are six such forms of SPM structures that will be considered in this work with nonlinear CD. These six forms of SPM structures have all been proposed by Kudryashov over the years. However, they have not yet been verified in an oscilloscope nor have such fiber materials been proposed as yet [16,17,18,19,20,21,22,23,24,25]. The next factor is the temporal evolution of solitons, which we have taken to be linear in the current work. The governing model would be subsequently integrated using Lie symmetry to recover the implicit quiescent optical solitons. A few of the results have been provided in terms of a special function. Additional integration schemes have previously been implemented to recover quiescent optical solitons, utilizing both linear and generalized temporal evolution [4]. The conditions required for the existence of such solitons are also outlined. The subsequent sections of the paper present details following a review of the model. This revisiting highlights the significance of the parameters influencing soliton stability and the impact of various perturbations.

Addressing quiescent optical solitons for nonlinear evolution equations, applicable to quantum optics, is not new. Stepping back, this concept has been considered for various models from Quantum Optics, such as the nonlinear Schrödinger’s equation, complex Ginzburg–Landau equation, Sasa–Satsuma equation, Lakshmanan–Porsezian–Daniel equation, Radhakrishnan–Kundu–Lakshmanan equation, Fokas–Lennells equation, Triki–Biswas model, and several others. In addition to Lie symmetry analysis, there are several integration schemes to recover quiescent optical solitons that have been successfully implemented in the past. These are the sine-Gordon equation approach, F-expansion scheme, Riccati equation method, extended Jacoboi’s elliptic function expansion, and extended trial function approach [16,17,18,19,20,21,22,23,24,25,26,27]. Additionally, in the past, quiescent gap solitons have been studied to a certain extent [28,29,30,31,32,33,34,35,36,37]. These schemes have revealed a plethora of solutions structures in the past. However, the Lie symmetry approach is the only one that provides a uniquely rich category of solutions of implicit type involving special functions, and the solutions also carry non-local features. This makes the solutions obtained with Lie symmetry rich and diverse, as will be exhibited in the current paper, which addresses the perturbed complex Ginzburg–Landau equation with arbitrary intensity.

2. Governing Mathematical Model

The non-dimensional representation of the perturbed CGLE is written as follows:

$$\begin{array}{ccc}& & i{\mathrm{\Psi }}_t+a{\left({\left|\mathrm{\Psi }\right|}^n\mathrm{\Psi }\right)}_{xx}+F\left({\left|\mathrm{\Psi }\right|}^2\right)\mathrm{\Psi }\hfill \\ & =& \alpha {\displaystyle \frac{{\left|{\mathrm{\Psi }}_x\right|}^2}{{\mathrm{\Psi }}^{*}}}+{\displaystyle \frac{\beta }{4{\left|\mathrm{\Psi }\right|}^2{\mathrm{\Psi }}^{*}}}\left[2{\left|\mathrm{\Psi }\right|}^2{\left({\left|\mathrm{\Psi }\right|}^2\right)}_{xx}-{\left\{{\left({\left|\mathrm{\Psi }\right|}^2\right)}_x\right\}}^2\right]\hfill \\ & +& \gamma \mathrm{\Psi }+i\left[\lambda {\left({\left|\mathrm{\Psi }\right|}^{2m}\mathrm{\Psi }\right)}_x+{\theta }_1{\left({\left|\mathrm{\Psi }\right|}^{2m}\right)}_x\mathrm{\Psi }+{\theta }_2{\left|\mathrm{\Psi }\right|}^{2m}{\mathrm{\Psi }}_x\right].\hfill \end{array}$$

(1)

Here, in Equation (1), $\mathrm{\Psi }(x,t)$ is a complex-valued function that represents the wave envelope, with x and t being the independent variables that represent the spatial and temporal variables, respectively, while $i=\sqrt{-1}$. The first term represents the temporal evolution of the pulses. The real-valued constant a is the coefficient of CD. The parameter n is its nonlinearity factor. For $n=0$, CD is linear, in which case the model (1) supports mobile solitons [3]. The third term is the SPM term where the functional F is the intensity-dependent nonlinear refractive index. Here, $\alpha$ and $\beta$ are real-valued functions while $\gamma$ stands for the detuning effect. From the perturbation terms, $\lambda$ is the effect of self-steepening to control the evolution of shock waves, while ${\theta }_j$ for $j=1,2$ are the coefficients of the soliton self-frequency shift.

3. Mathematical Preliminaries

Equation (1) does not support any mobile solitons unless $n=0$ [3]. Therefore, the assumption for the quiescent optical solitons is taken to be

$$\begin{array}{c}\hfill \mathrm{\Psi }\left(x,t\right)=\varphi \left(x\right){\mathrm{e}}^{i\omega t}.\end{array}$$

(2)

Substituting Equation (2) into Equation (1) and separating into real and imaginary components produces a pair of equations. The real component gives the ordinary differential equation (ODE) in $\varphi \left(x\right)$ as

$$\begin{array}{c}\hfill a(n+1){\varphi }^n\left(x\right)\left[n{\left\{{\varphi }^{\prime }\left(x\right)\right\}}^2+\varphi \left(x\right){\varphi }^{″}\left(x\right)\right]-{\varphi }^2\left(x\right)\left[\omega +\gamma -F\left\{{\varphi }^2\left(x\right)\right\}\right]\\ \hfill -\alpha {\left\{{\varphi }^{\prime }\left(x\right)\right\}}^2-\beta \varphi \left(x\right){\varphi }^{″}\left(x\right)=0,\end{array}$$

(3)

while the imaginary part leads to the parameter constraint

$$\begin{array}{c}\hfill {\theta }_2+2m{\theta }_1+(2m+1)\lambda =0.\end{array}$$

(4)

By virtue of Equation (4), the governing model given by Equation (1) modifies to

$$\begin{array}{ccc}& & i{\mathrm{\Psi }}_t+a{\left({\left|\mathrm{\Psi }\right|}^n\mathrm{\Psi }\right)}_{xx}+F\left({\left|\mathrm{\Psi }\right|}^2\right)\mathrm{\Psi }\hfill \\ & =& \alpha {\displaystyle \frac{{\left|{\mathrm{\Psi }}_x\right|}^2}{{\mathrm{\Psi }}^{*}}}+{\displaystyle \frac{\beta }{4{\left|\mathrm{\Psi }\right|}^2{\mathrm{\Psi }}^{*}}}\left[2{\left|\mathrm{\Psi }\right|}^2{\left({\left|\mathrm{\Psi }\right|}^2\right)}_{xx}-{\left\{{\left({\left|\mathrm{\Psi }\right|}^2\right)}_x\right\}}^2\right]+\gamma \mathrm{\Psi }\hfill \\ & +& i\left[\lambda {\left({\left|\mathrm{\Psi }\right|}^{2m}\mathrm{\Psi }\right)}_x+{\theta }_1{\left({\left|\mathrm{\Psi }\right|}^{2m}\right)}_x\mathrm{\Psi }-\left\{2{\theta }_{1}m+\lambda (2m+1)\right\}{\left|\mathrm{\Psi }\right|}^{2m}{\mathrm{\Psi }}_x\right].\hfill \end{array}$$

(5)

It is this Equation (3) that will be analyzed by Lie symmetry to retrieve its solutions to paint a complete picture to the model for six forms of the proposed SPM structures.

4. Application to the Six Forms of SPM by Kudryashov

4.1. Form-I

This form of refractive index is given as

$$\begin{array}{c}\hfill F\left(s\right)={\displaystyle \frac{b_1}{s^m}}+{\displaystyle \frac{b_2}{s^{\frac{m}{2}}}}+b_3{s}^{{\displaystyle \frac{m}{2}}}+b_4{s}^m,\end{array}$$

(6)

where, $b_j$ is the non-zero real-valued constant for $1\le j\le 4$ and m is the arbitrary light intensity parameter. Therefore, the governing model with nonlinear CD is

$$\begin{array}{ccc}& & i{\mathrm{\Psi }}_t+a{\left({\left|\mathrm{\Psi }\right|}^n\mathrm{\Psi }\right)}_{xx}+\left({\displaystyle \frac{b_1}{{\left|\mathrm{\Psi }\right|}^{2m}}}+{\displaystyle \frac{b_2}{{\left|\mathrm{\Psi }\right|}^m}}+b_3{\left|\mathrm{\Psi }\right|}^m+b_4{\left|\mathrm{\Psi }\right|}^{2m}\right)\mathrm{\Psi }\hfill \\ & =& \alpha {\displaystyle \frac{{\left|{\mathrm{\Psi }}_x\right|}^2}{{\mathrm{\Psi }}^{*}}}+{\displaystyle \frac{\beta }{4{\left|\mathrm{\Psi }\right|}^2{\mathrm{\Psi }}^{*}}}\left[2{\left|\mathrm{\Psi }\right|}^2{\left({\left|\mathrm{\Psi }\right|}^2\right)}_{xx}-{\left\{{\left({\left|\mathrm{\Psi }\right|}^2\right)}_x\right\}}^2\right]\hfill \\ & +& \gamma \mathrm{\Psi }+i\left[\lambda {\left({\left|\mathrm{\Psi }\right|}^{2m}\mathrm{\Psi }\right)}_x+{\theta }_1{\left({\left|\mathrm{\Psi }\right|}^{2m}\right)}_x\mathrm{\Psi }+{\theta }_2{\left|\mathrm{\Psi }\right|}^{2m}{\mathrm{\Psi }}_x\right].\hfill \end{array}$$

(7)

By replacing Equation (2) into Equation (7) and separating into real and imaginary components, the ordinary differential equation for $\varphi \left(x\right)$ can be extracted from the real part as

$$\begin{array}{c}\hfill a(n+1){\varphi }^n\left(x\right)\left[n{\left\{{\varphi }^{\prime }\left(x\right)\right\}}^2+\varphi \left(x\right){\varphi }^{″}\left(x\right)\right]\\ \hfill -{\varphi }^2\left(x\right)\left[\omega +\gamma -{\displaystyle \frac{b_1}{{\varphi }^m\left(x\right)}}-{\displaystyle \frac{b_2}{{\varphi }^{\frac{m}{2}}\left(x\right)}}-b_3{\varphi }^{{\displaystyle \frac{m}{2}}}\left(x\right)-b_4{\varphi }^m\left(x\right)\right]\\ \hfill -\alpha {\left\{{\varphi }^{\prime }\left(x\right)\right\}}^2-\beta \varphi \left(x\right){\varphi }^{″}\left(x\right)=0.\end{array}$$

(8)

Next, using the constraint condition given by Equation (4), Equation (7) reduces to

$$\begin{array}{ccc}& & i{\mathrm{\Psi }}_t+a{\left({\left|\mathrm{\Psi }\right|}^n\mathrm{\Psi }\right)}_{xx}+\left({\displaystyle \frac{b_1}{{\left|\mathrm{\Psi }\right|}^{2m}}}+{\displaystyle \frac{b_2}{{\left|\mathrm{\Psi }\right|}^m}}+b_3{\left|\mathrm{\Psi }\right|}^m+b_4{\left|\mathrm{\Psi }\right|}^{2m}\right)\mathrm{\Psi }\hfill \\ & =& \alpha {\displaystyle \frac{{\left|{\mathrm{\Psi }}_x\right|}^2}{{\mathrm{\Psi }}^{*}}}+{\displaystyle \frac{\beta }{4{\left|\mathrm{\Psi }\right|}^2{\mathrm{\Psi }}^{*}}}\left[2{\left|\mathrm{\Psi }\right|}^2{\left({\left|\mathrm{\Psi }\right|}^2\right)}_{xx}-{\left\{{\left({\left|\mathrm{\Psi }\right|}^2\right)}_x\right\}}^2\right]+\gamma \mathrm{\Psi }\hfill \\ & +& i\left[\lambda {\left({\left|\mathrm{\Psi }\right|}^{2m}\mathrm{\Psi }\right)}_x+{\theta }_1{\left({\left|\mathrm{\Psi }\right|}^{2m}\right)}_x\mathrm{\Psi }-\left\{2{\theta }_{1}m+\lambda (2m+1)\right\}{\left|\mathrm{\Psi }\right|}^{2m}{\mathrm{\Psi }}_x\right].\hfill \end{array}$$

(9)

Equation (8) supports translational Lie symmetry $\partial /\partial x$, which when applied integrates to

$$\begin{array}{c}\hfill x=\int \sqrt{{\displaystyle \frac{A}{B}}}d\varphi ,\end{array}$$

(10)

where

$$\begin{array}{c}\hfill A=-{\displaystyle \frac{{\left\{\beta -a(n+1){\varphi }^n\right\}}^2{\left[1-\frac{a(n+1){\varphi }^n}{\beta }\right]}^{-\frac{2\alpha }{\beta n}}}{\beta {\varphi }^2}},\end{array}$$

(11)

and

$$\begin{array}{c}\hfill B=-{\displaystyle \frac{\beta b_1{L}_2{\varphi }^{-2m}}{\alpha +\beta -\beta m}}+{\displaystyle \frac{2\beta b_2{L}_3{\varphi }^{-m}}{\beta (m-2)-2\alpha }}-{\displaystyle \frac{\beta b_4{L}_5{\varphi }^{2m}}{\alpha +\beta +\beta m}}-{\displaystyle \frac{2b_3{L}_4{\varphi }^m}{\frac{2\alpha }{\beta }+m+2}}+{\displaystyle \frac{\beta L_1(\gamma +\omega )}{\alpha +\beta }},\end{array}$$

(12)

with

$$\begin{array}{c}\hfill L_1={}_{2}F_1\left({\displaystyle \frac{2\alpha }{n\beta }}-1,{\displaystyle \frac{2(\alpha +\beta )}{n\beta }};{\displaystyle \frac{2(\alpha +\beta )}{n\beta }}+1;{\displaystyle \frac{a(n+1){\varphi }^n}{\beta }}\right),\end{array}$$

(13)

$$\begin{array}{c}\hfill L_2={}_{2}F_1\left({\displaystyle \frac{2\left(-m+\frac{\alpha }{\beta }+1\right)}{n}},{\displaystyle \frac{2\alpha }{n\beta }}-1;{\displaystyle \frac{-2m+\frac{2\alpha }{\beta }+2}{n}}+1;{\displaystyle \frac{a(n+1){\varphi }^n}{\beta }}\right),\end{array}$$

(14)

$$\begin{array}{c}\hfill L_3={}_{2}F_1\left({\displaystyle \frac{-m+\frac{2\alpha }{\beta }+2}{n}},{\displaystyle \frac{2\alpha }{n\beta }}-1;{\displaystyle \frac{-m+\frac{2\alpha }{\beta }+2}{n}}+1;{\displaystyle \frac{a(n+1){\varphi }^n}{\beta }}\right),\end{array}$$

(15)

$$\begin{array}{c}\hfill L_4={}_{2}F_1\left({\displaystyle \frac{m+\frac{2\alpha }{\beta }+2}{n}},{\displaystyle \frac{2\alpha }{n\beta }}-1;{\displaystyle \frac{m+\frac{2\alpha }{\beta }+2}{n}}+1;{\displaystyle \frac{a(n+1){\varphi }^n}{\beta }}\right),\end{array}$$

(16)

$$\begin{array}{c}\hfill L_5={}_{2}F_1\left({\displaystyle \frac{2\alpha }{n\beta }}-1,{\displaystyle \frac{2(\alpha +m\beta +\beta )}{n\beta }};{\displaystyle \frac{2(\alpha +m\beta +\beta )}{n\beta }}+1;{\displaystyle \frac{a(n+1){\varphi }^n}{\beta }}\right).\end{array}$$

(17)

The Gauss hypergeometric function is defined as follows:

$$\begin{array}{c}\hfill {}_{2}F_1\left(\alpha ,\beta ;\gamma ;z\right)=\sum _{n=0}^{\infty }{\displaystyle \frac{{\left(\alpha \right)}_n{\left(\beta \right)}_n}{{\left(\gamma \right)}_n}}{\displaystyle \frac{z^n}{n!}},\end{array}$$

(18)

where the Pochhammer symbol is given by

$$\begin{array}{c}\hfill {\left(p\right)}_n=\left\{\begin{array}{cc}1\hfill & n=0,\hfill \\ p(p+1)\cdots (p+n-1)\hfill & n>0.\hfill \end{array}\right.\end{array}$$

(19)

The criterion for the convergence of the hypergeometric series is

$$\begin{array}{c}\hfill \left|z\right|<1,\end{array}$$

(20)

which for Equations (13)–(17) implies

$$\begin{array}{c}\hfill -{\left|{\displaystyle \frac{\beta }{a(n+1)}}\right|}^{{\displaystyle \frac{1}{n}}}<\varphi \left(x\right)<{\left|{\displaystyle \frac{\beta }{a(n+1)}}\right|}^{{\displaystyle \frac{1}{n}}}.\end{array}$$

(21)

Next, Raabe’s test implies that for the series convergence

$$\begin{array}{c}\hfill n<{\displaystyle \frac{\alpha }{\beta }}\mathrm{for}\beta >0\mathrm{or}n>{\displaystyle \frac{\alpha }{\beta }}\mathrm{for}\beta <0.\end{array}$$

(22)

Another constraint that is dictated by Equation (10) for the solutions to exist is

$$\begin{array}{c}\hfill AB>0.\end{array}$$

(23)

4.2. Form-II

The law of the refractive index for the second form is structured as follows:

$$\begin{array}{c}\hfill F\left(s\right)={\displaystyle \frac{b_1}{s^{2m}}}+{\displaystyle \frac{b_2}{s^{\frac{3m}{2}}}}+{\displaystyle \frac{b_3}{s^m}}+{\displaystyle \frac{b_4}{s^{\frac{m}{2}}}}+b_5{s}^{{\displaystyle \frac{m}{2}}}+b_6{s}^m+b_7{s}^{{\displaystyle \frac{3m}{2}}}+b_8{s}^{2m},\end{array}$$

(24)

where $b_j$ is non-zero real-valued constants for $1\le j\le 8$ and m is the arbitrary light intensity parameter. Therefore, the governing NLSE with nonlinear CD is

$$\begin{array}{cc}\hfill & i{\mathrm{\Psi }}_t+a{\left({\left|\mathrm{\Psi }\right|}^n\mathrm{\Psi }\right)}_{xx}\hfill \\ \hfill & +\left({\displaystyle \frac{b_1}{{\left|\mathrm{\Psi }\right|}^{4m}}}+{\displaystyle \frac{b_2}{{\left|\mathrm{\Psi }\right|}^{3m}}}+{\displaystyle \frac{b_3}{{\left|\mathrm{\Psi }\right|}^{2m}}}+{\displaystyle \frac{b_4}{{\left|\mathrm{\Psi }\right|}^m}}+b_5{\left|\mathrm{\Psi }\right|}^m+b_6{\left|\mathrm{\Psi }\right|}^{2m}+b_7{\left|\mathrm{\Psi }\right|}^{3m}+b_8{\left|\mathrm{\Psi }\right|}^{4m}\right)\mathrm{\Psi }\hfill \\ \hfill & =\alpha {\displaystyle \frac{{\left|{\mathrm{\Psi }}_x\right|}^2}{{\mathrm{\Psi }}^{*}}}+{\displaystyle \frac{\beta }{4{\left|\mathrm{\Psi }\right|}^2{\mathrm{\Psi }}^{*}}}\left[2{\left|\mathrm{\Psi }\right|}^2{\left({\left|\mathrm{\Psi }\right|}^2\right)}_{xx}-{\left\{{\left({\left|\mathrm{\Psi }\right|}^2\right)}_x\right\}}^2\right]\hfill \\ & + \gamma \mathrm{\Psi }+i\left[\lambda {\left({\left|\mathrm{\Psi }\right|}^{2m}\mathrm{\Psi }\right)}_x+{\theta }_1{\left({\left|\mathrm{\Psi }\right|}^{2m}\right)}_x\mathrm{\Psi }+{\theta }_2{\left|\mathrm{\Psi }\right|}^{2m}{\mathrm{\Psi }}_x\right].\hfill \end{array}$$

(25)

Replacing Equation (2) into Equation (25), the real part yields the ODE for $\varphi \left(x\right)$ as

$$\begin{array}{cc}\hfill & a(n+1){\varphi }^n\left(x\right)\left[n{\left\{{\varphi }^{\prime }\left(x\right)\right\}}^2+\varphi \left(x\right){\varphi }^{″}\left(x\right)\right]\hfill \\ \hfill & - {\varphi }^2\left(x\right)\left[\omega -{\displaystyle \frac{b_1}{{\varphi }^{2m}\left(x\right)}}-{\displaystyle \frac{b_2}{{\varphi }^{\frac{3m}{2}}\left(x\right)}}-{\displaystyle \frac{b_3}{{\varphi }^m\left(x\right)}}-{\displaystyle \frac{b_4}{{\varphi }^{\frac{m}{2}}\left(x\right)}}\right]\hfill \\ \hfill & - {\varphi }^2\left(x\right)\left[\gamma -b_5{\varphi }^{{\displaystyle \frac{m}{2}}}\left(x\right)-b_6{\varphi }^m\left(x\right)-b_7{\varphi }^{{\displaystyle \frac{3m}{2}}}\left(x\right)-b_8{\varphi }^{2m}\left(x\right)\right]\hfill \\ \hfill & - \alpha {\left\{{\varphi }^{\prime }\left(x\right)\right\}}^2-\beta \varphi \left(x\right){\varphi }^{″}\left(x\right)=0.\hfill \end{array}$$

(26)

The constraint condition given by Equation (4), which stems from the imaginary part in the equation leads to the governing model to be rewritten as:

$$\begin{array}{c}\hfill \begin{array}{cc}& i{\mathrm{\Psi }}_t+a{\left({\left|\mathrm{\Psi }\right|}^n\mathrm{\Psi }\right)}_{xx}\hfill \\ & +\left({\displaystyle \frac{b_1}{{\left|\mathrm{\Psi }\right|}^{4m}}}+{\displaystyle \frac{b_2}{{\left|\mathrm{\Psi }\right|}^{3m}}}+{\displaystyle \frac{b_3}{{\left|\mathrm{\Psi }\right|}^{2m}}}+{\displaystyle \frac{b_4}{{\left|\mathrm{\Psi }\right|}^m}}+b_5{\left|\mathrm{\Psi }\right|}^m+b_6{\left|\mathrm{\Psi }\right|}^{2m}+b_7{\left|\mathrm{\Psi }\right|}^{3m}+b_8{\left|\mathrm{\Psi }\right|}^{4m}\right)\mathrm{\Psi }\hfill \\ & =\alpha {\displaystyle \frac{{\left|{\mathrm{\Psi }}_x\right|}^2}{{\mathrm{\Psi }}^{*}}}+{\displaystyle \frac{\beta }{4{\left|\mathrm{\Psi }\right|}^2{\mathrm{\Psi }}^{*}}}\left[2{\left|\mathrm{\Psi }\right|}^2{\left({\left|\mathrm{\Psi }\right|}^2\right)}_{xx}-{\left\{{\left({\left|\mathrm{\Psi }\right|}^2\right)}_x\right\}}^2\right]+\gamma \mathrm{\Psi }\hfill \\ & + i\left[\lambda {\left({\left|\mathrm{\Psi }\right|}^{2m}\mathrm{\Psi }\right)}_x+{\theta }_1{\left({\left|\mathrm{\Psi }\right|}^{2m}\right)}_x\mathrm{\Psi }-\left\{2{\theta }_{1}m+\lambda (2m+1)\right\}{\left|\mathrm{\Psi }\right|}^{2m}{\mathrm{\Psi }}_x\right].\hfill \end{array}\end{array}$$

(27)

By the aid of the translational Lie symmetry supported by Equation (26), the ODE for $\varphi \left(x\right)$ integrates to

$$\begin{array}{c}\hfill x=\int \sqrt{{\displaystyle \frac{A}{2B}}}d\varphi ,\end{array}$$

(28)

where

$$A=exp\left[-{\displaystyle \frac{2\left\{(\alpha -n\beta )ln\left|a(n+1){\varphi }^n-\beta \right|-\alpha ln\left|{\varphi }^n\right|\right\}}{\beta n}}\right],$$

(29)

and

$$\begin{array}{cc}\hfill & B=-{\displaystyle \frac{{\beta }^2\gamma \rho L_1{\varphi }^{\frac{2\alpha }{\beta }+2}}{2(\alpha +\beta )}}-{\displaystyle \frac{{\beta }^2\rho \omega L_1{\varphi }^{\frac{2\alpha }{\beta }+2}}{2(\alpha +\beta )}}+{\displaystyle \frac{{\beta }^2\rho b_1{L}_3{\varphi }^{-4m+\frac{2\alpha }{\beta }+2}}{2(\alpha -2m\beta +\beta )}}\hfill \\ & -{\displaystyle \frac{{\beta }^2\rho b_2{L}_5{\varphi }^{-3m+\frac{2\alpha }{\beta }+2}}{(3m-2)\beta -2\alpha }}+{\displaystyle \frac{{\beta }^2\rho b_3{L}_7{\varphi }^{-2m+\frac{2\alpha }{\beta }+2}}{2(\alpha -m\beta +\beta )}}-{\displaystyle \frac{{\beta }^2\rho b_4{L}_9{\varphi }^{-m+\frac{2\alpha }{\beta }+2}}{(m-2)\beta -2\alpha }}\hfill \\ & +{\displaystyle \frac{{\beta }^2\rho b_5{L}_{11}{\varphi }^{m+\frac{2\alpha }{\beta }+2}}{2\alpha +(m+2)\beta }}+{\displaystyle \frac{{\beta }^2\rho b_6{L}_{13}{\varphi }^{2m+\frac{2\alpha }{\beta }+2}}{2(\alpha +m\beta +\beta )}}+{\displaystyle \frac{{\beta }^2\rho b_7{L}_{15}{\varphi }^{3m+\frac{2\alpha }{\beta }+2}}{2\alpha +(3m+2)\beta }}\hfill \\ & +{\displaystyle \frac{{\beta }^2\rho b_8{L}_{17}{\varphi }^{4m+\frac{2\alpha }{\beta }+2}}{2(\alpha +2m\beta +\beta )}}+{\displaystyle \frac{a\beta \gamma \rho L_2{\varphi }^{n+\frac{2\alpha }{\beta }+2}}{2\alpha +(n+2)\beta }}+{\displaystyle \frac{an\beta \gamma \rho L_2{\varphi }^{n+\frac{2\alpha }{\beta }+2}}{2\alpha +(n+2)\beta }}\hfill \\ & +{\displaystyle \frac{a\beta \rho \omega L_2{\varphi }^{n+\frac{2\alpha }{\beta }+2}}{2\alpha +(n+2)\beta }}+{\displaystyle \frac{an\beta \rho \omega L_2{\varphi }^{n+\frac{2\alpha }{\beta }+2}}{2\alpha +(n+2)\beta }}-{\displaystyle \frac{a\beta \rho b_1{L}_4{\varphi }^{-4m+n+\frac{2\alpha }{\beta }+2}}{2\alpha +(-4m+n+2)\beta }}\hfill \\ & -{\displaystyle \frac{an\beta \rho b_1{L}_4{\varphi }^{-4m+n+\frac{2\alpha }{\beta }+2}}{2\alpha +(-4m+n+2)\beta }}-{\displaystyle \frac{a\beta \rho b_2{L}_6{\varphi }^{-3m+n+\frac{2\alpha }{\beta }+2}}{2\alpha +(-3m+n+2)\beta }}-{\displaystyle \frac{an\beta \rho b_2{L}_6{\varphi }^{-3m+n+\frac{2\alpha }{\beta }+2}}{2\alpha +(-3m+n+2)\beta }}\hfill \\ & -{\displaystyle \frac{a\beta \rho b_3{L}_8{\varphi }^{-2m+n+\frac{2\alpha }{\beta }+2}}{2\alpha +(-2m+n+2)\beta }}-{\displaystyle \frac{an\beta \rho b_3{L}_8{\varphi }^{-2m+n+\frac{2\alpha }{\beta }+2}}{2\alpha +(-2m+n+2)\beta }}-{\displaystyle \frac{a\beta \rho b_4{L}_{10}{\varphi }^{-m+n+\frac{2\alpha }{\beta }+2}}{2\alpha +(-m+n+2)\beta }}\hfill \\ & -{\displaystyle \frac{an\beta \rho b_4{L}_{10}{\varphi }^{-m+n+\frac{2\alpha }{\beta }+2}}{2\alpha +(-m+n+2)\beta }}-{\displaystyle \frac{a\beta \rho b_5{L}_{12}{\varphi }^{m+n+\frac{2\alpha }{\beta }+2}}{2\alpha +(m+n+2)\beta }}-{\displaystyle \frac{an\beta \rho b_5{L}_{12}{\varphi }^{m+n+\frac{2\alpha }{\beta }+2}}{2\alpha +(m+n+2)\beta }}\hfill \\ & -{\displaystyle \frac{a\beta \rho b_6{L}_{14}{\varphi }^{2m+n+\frac{2\alpha }{\beta }+2}}{2\alpha +(2m+n+2)\beta }}-{\displaystyle \frac{an\beta \rho b_6{L}_{14}{\varphi }^{2m+n+\frac{2\alpha }{\beta }+2}}{2\alpha +(2m+n+2)\beta }}-{\displaystyle \frac{a\beta \rho b_7{L}_{16}{\varphi }^{3m+n+\frac{2\alpha }{\beta }+2}}{2\alpha +(3m+n+2)\beta }}\hfill \\ & -{\displaystyle \frac{an\beta \rho b_7{L}_{16}{\varphi }^{3m+n+\frac{2\alpha }{\beta }+2}}{2\alpha +(3m+n+2)\beta }}-{\displaystyle \frac{a\beta \rho b_8{L}_{18}{\varphi }^{4m+n+\frac{2\alpha }{\beta }+2}}{2\alpha +(4m+n+2)\beta }}-{\displaystyle \frac{an\beta \rho b_8{L}_{18}{\varphi }^{4m+n+\frac{2\alpha }{\beta }+2}}{2\alpha +(4m+n+2)\beta }}\hfill \end{array}$$

(30)

with

$$\begin{array}{c}\hfill L_1={}_{2}F_1\left({\displaystyle \frac{2\alpha }{n\beta }},{\displaystyle \frac{2(\alpha +\beta )}{n\beta }};{\displaystyle \frac{2(\alpha +\beta )}{n\beta }}+1;{\displaystyle \frac{a(n+1){\varphi }^n}{\beta }}\right),\end{array}$$

(31)

$$\begin{array}{c}\hfill L_2={}_{2}F_1\left({\displaystyle \frac{2\alpha }{n\beta }},{\displaystyle \frac{2\alpha +(n+2)\beta }{n\beta }};{\displaystyle \frac{2(\alpha +n\beta +\beta )}{n\beta }};{\displaystyle \frac{a(n+1){\varphi }^n}{\beta }}\right),\end{array}$$

(32)

$$\begin{array}{c}\hfill L_3={}_{2}F_1\left({\displaystyle \frac{2\alpha }{n\beta }},{\displaystyle \frac{2(\alpha -2m\beta +\beta )}{n\beta }};{\displaystyle \frac{2\alpha +(-4m+n+2)\beta }{n\beta }};{\displaystyle \frac{a(n+1){\varphi }^n}{\beta }}\right),\end{array}$$

(33)

$$\begin{array}{c}\hfill L_4={}_{2}F_1\left({\displaystyle \frac{2\alpha }{n\beta }},{\displaystyle \frac{2\alpha +(-4m+n+2)\beta }{n\beta }};{\displaystyle \frac{2(\alpha +(-2m+n+1\left)\beta \right)}{n\beta }};{\displaystyle \frac{a(n+1){\varphi }^n}{\beta }}\right),\end{array}$$

(34)

$$\begin{array}{c}\hfill L_5={}_{2}F_1\left({\displaystyle \frac{2\alpha }{n\beta }},{\displaystyle \frac{2\alpha -3m\beta +2\beta }{n\beta }};{\displaystyle \frac{2\alpha +(-3m+n+2)\beta }{n\beta }};{\displaystyle \frac{a(n+1){\varphi }^n}{\beta }}\right),\end{array}$$

(35)

$$\begin{array}{c}\hfill L_6={}_{2}F_1\left({\displaystyle \frac{2\alpha }{n\beta }},{\displaystyle \frac{2\alpha +(-3m+n+2)\beta }{n\beta }};{\displaystyle \frac{2\alpha +(-3m+2n+2)\beta }{n\beta }};{\displaystyle \frac{a(n+1){\varphi }^n}{\beta }}\right),\end{array}$$

(36)

$$\begin{array}{c}\hfill L_7={}_{2}F_1\left({\displaystyle \frac{2\alpha }{n\beta }},{\displaystyle \frac{2(\alpha -m\beta +\beta )}{n\beta }};{\displaystyle \frac{2\alpha +(-2m+n+2)\beta }{n\beta }};{\displaystyle \frac{a(n+1){\varphi }^n}{\beta }}\right),\end{array}$$

(37)

$$\begin{array}{c}\hfill L_8={}_{2}F_1\left({\displaystyle \frac{2\alpha }{n\beta }},{\displaystyle \frac{2\alpha +(-2m+n+2)\beta }{n\beta }};{\displaystyle \frac{2(\alpha +(-m+n+1\left)\beta \right)}{n\beta }};{\displaystyle \frac{a(n+1){\varphi }^n}{\beta }}\right),\end{array}$$

(38)

$$\begin{array}{c}\hfill L_9={}_{2}F_1\left({\displaystyle \frac{2\alpha }{n\beta }},{\displaystyle \frac{2\alpha -m\beta +2\beta }{n\beta }};{\displaystyle \frac{2\alpha +(-m+n+2)\beta }{n\beta }};{\displaystyle \frac{a(n+1){\varphi }^n}{\beta }}\right),\end{array}$$

(39)

$$\begin{array}{c}\hfill L_{10}={}_{2}F_1\left({\displaystyle \frac{2\alpha }{n\beta }},{\displaystyle \frac{2\alpha +(-m+n+2)\beta }{n\beta }};{\displaystyle \frac{2\alpha +(-m+2n+2)\beta }{n\beta }};{\displaystyle \frac{a(n+1){\varphi }^n}{\beta }}\right),\end{array}$$

(40)

$$\begin{array}{c}\hfill L_{11}={}_{2}F_1\left({\displaystyle \frac{2\alpha }{n\beta }},{\displaystyle \frac{2\alpha +(m+2)\beta }{n\beta }};{\displaystyle \frac{2\alpha +(m+n+2)\beta }{n\beta }};{\displaystyle \frac{a(n+1){\varphi }^n}{\beta }}\right),\end{array}$$

(41)

$$\begin{array}{c}\hfill L_{12}={}_{2}F_1\left({\displaystyle \frac{2\alpha }{n\beta }},{\displaystyle \frac{2\alpha +(m+n+2)\beta }{n\beta }};{\displaystyle \frac{2\alpha +(m+2n+2)\beta }{n\beta }};{\displaystyle \frac{a(n+1){\varphi }^n}{\beta }}\right),\end{array}$$

(42)

$$\begin{array}{c}\hfill L_{13}={}_{2}F_1\left({\displaystyle \frac{2\alpha }{n\beta }},{\displaystyle \frac{2(\alpha +m\beta +\beta )}{n\beta }};{\displaystyle \frac{2(\alpha +m\beta +\beta )}{n\beta }}+1;{\displaystyle \frac{a(n+1){\varphi }^n}{\beta }}\right),\end{array}$$

(43)

$$\begin{array}{c}\hfill L_{14}={}_{2}F_1\left({\displaystyle \frac{2\alpha }{n\beta }},{\displaystyle \frac{2\alpha +(2m+n+2)\beta }{n\beta }};{\displaystyle \frac{2(\alpha +(m+n+1\left)\beta \right)}{n\beta }};{\displaystyle \frac{a(n+1){\varphi }^n}{\beta }}\right),\end{array}$$

(44)

$$\begin{array}{c}\hfill L_{15}={}_{2}F_1\left({\displaystyle \frac{2\alpha }{n\beta }},{\displaystyle \frac{2\alpha +3m\beta +2\beta }{n\beta }};{\displaystyle \frac{2\alpha +(3m+n+2)\beta }{n\beta }};{\displaystyle \frac{a(n+1){\varphi }^n}{\beta }}\right),\end{array}$$

(45)

$$\begin{array}{c}\hfill L_{16}={}_{2}F_1\left({\displaystyle \frac{2\alpha }{n\beta }},{\displaystyle \frac{2\alpha +(3m+n+2)\beta }{n\beta }};{\displaystyle \frac{2\alpha +(3m+2n+2)\beta }{n\beta }};{\displaystyle \frac{a(n+1){\varphi }^n}{\beta }}\right),\end{array}$$

(46)

$$\begin{array}{c}\hfill L_{17}={}_{2}F_1\left({\displaystyle \frac{2\alpha }{n\beta }},{\displaystyle \frac{2(\alpha +2m\beta +\beta )}{n\beta }};{\displaystyle \frac{2(\alpha +2m\beta +\beta )}{n\beta }}+1;{\displaystyle \frac{a(n+1){\varphi }^n}{\beta }}\right),\end{array}$$

(47)

$$\begin{array}{c}\hfill L_{18}={}_{2}F_1\left({\displaystyle \frac{2\alpha }{n\beta }},{\displaystyle \frac{2\alpha +(4m+n+2)\beta }{n\beta }};{\displaystyle \frac{2(\alpha +(2m+n+1\left)\beta \right)}{n\beta }};{\displaystyle \frac{a(n+1){\varphi }^n}{\beta }}\right),\end{array}$$

(48)

$$\begin{array}{c}\hfill \rho ={\left\{a(n+1){\varphi }^n-\beta \right\}}^{-{\displaystyle \frac{2\alpha }{n\beta }}}{\left[1-{\displaystyle \frac{a(n+1){\varphi }^n}{\beta }}\right]}^{{\displaystyle \frac{2\alpha }{n\beta }}}\end{array}$$

(49)

The same parameter constraints given by Equations (21) and (22) must also remain valid for the solution to exist. Moreover, the Gauss’ hypergeometric function in Equations (31)–(48) was already introduced in Equation (18).

4.3. Form-III

The law of refractive index is:

$$\begin{array}{c}\hfill F\left(s\right)=b_1{s}^{{\displaystyle \frac{m}{2}}}+b_2{s}^m+b_3{\left(s^{{\displaystyle \frac{m}{2}}}\right)}^{″}.\end{array}$$

(50)

Therefore, the governing NLSE takes the form

$$\begin{array}{c}\hfill i{\mathrm{\Psi }}_t+a{\left({\left|\mathrm{\Psi }\right|}^n\mathrm{\Psi }\right)}_{xx}+\left[b_1{\left|\mathrm{\Psi }\right|}^m+b_2{\left|\mathrm{\Psi }\right|}^{2m}+b_3{\left({\left|\mathrm{\Psi }\right|}^m\right)}_{xx}\right]\mathrm{\Psi }\\ \hfill =\alpha {\displaystyle \frac{{\left|{\mathrm{\Psi }}_x\right|}^2}{{\mathrm{\Psi }}^{*}}}+{\displaystyle \frac{\beta }{4{\left|\mathrm{\Psi }\right|}^2{\mathrm{\Psi }}^{*}}}\left[2{\left|\mathrm{\Psi }\right|}^2{\left({\left|\mathrm{\Psi }\right|}^2\right)}_{xx}-{\left\{{\left({\left|\mathrm{\Psi }\right|}^2\right)}_x\right\}}^2\right]\\ \hfill +\gamma \mathrm{\Psi }+i\left[\lambda {\left({\left|\mathrm{\Psi }\right|}^{2m}\mathrm{\Psi }\right)}_x+{\theta }_1{\left({\left|\mathrm{\Psi }\right|}^{2m}\right)}_x\mathrm{\Psi }+{\theta }_2{\left|\mathrm{\Psi }\right|}^{2m}{\mathrm{\Psi }}_x\right].\end{array}$$

(51)

Replacing Equation (2) in Equation (51) results in the ordinary differential equation for $\varphi \left(x\right)$ as

$$\begin{array}{c}\hfill \begin{array}{c}\hfill a(n+1){\varphi }^n\left(x\right)\left[n{\left\{{\varphi }^{\prime }\left(x\right)\right\}}^2+\varphi \left(x\right){\varphi }^{″}\left(x\right)\right]\\ \hfill -{\varphi }^2\left(x\right)\left[\omega +\gamma -b_1{\varphi }^{{\displaystyle \frac{m}{2}}}\left(x\right)-b_2{\varphi }^m\left(x\right)-b_3{\left\{{\varphi }^{{\displaystyle \frac{m}{2}}}\left(x\right)\right\}}^{″}\right]\\ \hfill -\alpha {\left\{{\varphi }^{\prime }\left(x\right)\right\}}^2-\beta \varphi \left(x\right){\varphi }^{″}\left(x\right)=0.\end{array}\end{array}$$

(52)

The parameter constraint condition given by Equation (4) condenses the governing model given by Equation (51) to

$$\begin{array}{ccc}& & i{\mathrm{\Psi }}_t+a{\left({\left|\mathrm{\Psi }\right|}^n\mathrm{\Psi }\right)}_{xx}+\left[b_1{\left|\mathrm{\Psi }\right|}^m+b_2{\left|\mathrm{\Psi }\right|}^{2m}+b_3{\left({\left|\mathrm{\Psi }\right|}^m\right)}_{xx}\right]\mathrm{\Psi }\hfill \\ & =& \alpha {\displaystyle \frac{{\left|{\mathrm{\Psi }}_x\right|}^2}{{\mathrm{\Psi }}^{*}}}+{\displaystyle \frac{\beta }{4{\left|\mathrm{\Psi }\right|}^2{\mathrm{\Psi }}^{*}}}\left[2{\left|\mathrm{\Psi }\right|}^2{\left({\left|\mathrm{\Psi }\right|}^2\right)}_{xx}-{\left\{{\left({\left|\mathrm{\Psi }\right|}^2\right)}_x\right\}}^2\right]+\gamma \mathrm{\Psi }\hfill \\ & +& i\left[\lambda {\left({\left|\mathrm{\Psi }\right|}^{2m}\mathrm{\Psi }\right)}_x+{\theta }_1{\left({\left|\mathrm{\Psi }\right|}^{2m}\right)}_x\mathrm{\Psi }-\left\{2{\theta }_{1}m+\lambda (2m+1)\right\}{\left|\mathrm{\Psi }\right|}^{2m}{\mathrm{\Psi }}_x\right].\hfill \end{array}$$

(53)

The translational Lie symmetry integrates Equations (52)–(28) where

$$\begin{array}{c}\hfill A=exp\left(-2{\int }^{\varphi }{\displaystyle \frac{m^2{b}_3{\tau }_1^m-m{b}_3{\tau }_1^m+a{n}^2{\tau }_1^n+an{\tau }_1^n-\alpha }{{\tau }_1\left(-m{b}_3{\tau }_1^m-a{\tau }_1^n-an{\tau }_1^n+\beta \right)}}d{\tau }_1\right),\end{array}$$

(54)

and

$$\begin{array}{c}\hfill B={\int }^{\varphi }{\displaystyle \frac{\left(b_1{\tau }_2^{m+2}+b_2{\tau }_2^{2m+2}-\gamma {\tau }_2^2-{\tau }_2^2\omega \right)exp\left(-2{\int }^{{\tau }_2}\frac{m^2{b}_3{\tau }_1^m-m{b}_3{\tau }_1^m+a{n}^2{\tau }_1^n+an{\tau }_1^n-\alpha }{{\tau }_1\left(-m{b}_3{\tau }_1^m-a{\tau }_1^n-an{\tau }_1^n+\beta \right)}d{\tau }_1\right)}{{\tau }_2\left(-a{\tau }_2^n-an{\tau }_2^n-m{b}_3{\tau }_2^m+\beta \right)}}d{\tau }_2.\end{array}$$

(55)

In this case, the parameter constraint given by Equation (23) must also hold for the solution to exist.

4.4. Form-IV

The law of refractive index for this form would be

$$\begin{array}{c}\hfill F\left(s\right)=b_1{s}^{{\displaystyle \frac{m}{2}}}+b_2{s}^m+b_3{s}^{{\displaystyle \frac{3m}{2}}}+b_4{s}^{2m}+b_5{s}^{{\displaystyle \frac{5m}{2}}}+b_6{s}^{3m},\end{array}$$

(56)

where, $b_j$ is real-valued constants for $1\le j\le 6$ and m is the arbitrary light intensity parameter. Therefore, the governing model is written as

$$\begin{array}{ccc}& & i{\mathrm{\Psi }}_t+a{\left({\left|\mathrm{\Psi }\right|}^n\mathrm{\Psi }\right)}_{xx}+\left(b_1{\left|\mathrm{\Psi }\right|}^m+b_2{\left|\mathrm{\Psi }\right|}^{2m}+b_3{\left|\mathrm{\Psi }\right|}^{3m}+b_4{\left|\mathrm{\Psi }\right|}^{4m}+b_5{\left|\mathrm{\Psi }\right|}^{5m}+b_6{\left|\mathrm{\Psi }\right|}^{6m}\right)\mathrm{\Psi }\hfill \\ & =& \alpha {\displaystyle \frac{{\left|{\mathrm{\Psi }}_x\right|}^2}{{\mathrm{\Psi }}^{*}}}+{\displaystyle \frac{\beta }{4{\left|\mathrm{\Psi }\right|}^2{\mathrm{\Psi }}^{*}}}\left[2{\left|\mathrm{\Psi }\right|}^2{\left({\left|\mathrm{\Psi }\right|}^2\right)}_{xx}-{\left\{{\left({\left|\mathrm{\Psi }\right|}^2\right)}_x\right\}}^2\right]\hfill \\ & +& \gamma \mathrm{\Psi }+i\left[\lambda {\left({\left|\mathrm{\Psi }\right|}^{2m}\mathrm{\Psi }\right)}_x+{\theta }_1{\left({\left|\mathrm{\Psi }\right|}^{2m}\right)}_x\mathrm{\Psi }+{\theta }_2{\left|\mathrm{\Psi }\right|}^{2m}{\mathrm{\Psi }}_x\right]\hfill \end{array}$$

(57)

Substituting Equation (2) into Equation (57), the real part yields the ODE

$$\begin{array}{ccc}& & a(n+1){\varphi }^n\left(x\right)\left[n{\left\{{\varphi }^{\prime }\left(x\right)\right\}}^2+\varphi \left(x\right){\varphi }^{″}\left(x\right)\right]\hfill \\ & -& {\varphi }^2\left(x\right)\left[\omega +\gamma -b_1{\varphi }^{{\displaystyle \frac{m}{2}}}\left(x\right)-b_2{\varphi }^m\left(x\right)-b_3{\varphi }^{{\displaystyle \frac{3m}{2}}}\left(x\right)-b_4{\varphi }^{2m}\left(x\right)-b_5{\varphi }^{{\displaystyle \frac{5m}{2}}}\left(x\right)-b_6{\varphi }^{3m}\left(x\right)\right]\hfill \\ & -& \alpha {\left\{{\varphi }^{\prime }\left(x\right)\right\}}^2-\beta \varphi \left(x\right){\varphi }^{″}\left(x\right)=0.\hfill \end{array}$$

(58)

The parameter constraint given by Equation (4), which comes from the imaginary part, restructures the governing model given by Equation (57) to

$$\begin{array}{ccc}& & i{\mathrm{\Psi }}_t+a{\left({\left|\mathrm{\Psi }\right|}^n\mathrm{\Psi }\right)}_{xx}+\left(b_1{\left|\mathrm{\Psi }\right|}^m+b_2{\left|\mathrm{\Psi }\right|}^{2m}+b_3{\left|\mathrm{\Psi }\right|}^{3m}+b_4{\left|\mathrm{\Psi }\right|}^{4m}+b_5{\left|\mathrm{\Psi }\right|}^{5m}+b_6{\left|\mathrm{\Psi }\right|}^{6m}\right)\mathrm{\Psi }\hfill \\ & =& \alpha {\displaystyle \frac{{\left|{\mathrm{\Psi }}_x\right|}^2}{{\mathrm{\Psi }}^{*}}}+{\displaystyle \frac{\beta }{4{\left|\mathrm{\Psi }\right|}^2{\mathrm{\Psi }}^{*}}}\left[2{\left|\mathrm{\Psi }\right|}^2{\left({\left|\mathrm{\Psi }\right|}^2\right)}_{xx}-{\left\{{\left({\left|\mathrm{\Psi }\right|}^2\right)}_x\right\}}^2\right]+\gamma \mathrm{\Psi }\hfill \\ & +& i\left[\lambda {\left({\left|\mathrm{\Psi }\right|}^{2m}\mathrm{\Psi }\right)}_x+{\theta }_1{\left({\left|\mathrm{\Psi }\right|}^{2m}\right)}_x\mathrm{\Psi }-\left\{2{\theta }_{1}m+\lambda (2m+1)\right\}{\left|\mathrm{\Psi }\right|}^{2m}{\mathrm{\Psi }}_x\right].\hfill \end{array}$$

(59)

From the translational Lie symmetry supported by Equation (58), its implicit solution emerges as in Equation (28) where

$$A=exp\left[-{\displaystyle \frac{2\left\{(\alpha -n\beta )ln\left|a(n+1){\varphi }^n-\beta \right|-\alpha ln\left|{\varphi }^n\right|\right\}}{\beta n}}\right],$$

(60)

and

$$\begin{array}{cc}\hfill & B={\displaystyle \frac{{\beta }^2\rho {\left({\varphi }^n\right)}^{\frac{2\alpha }{n\beta }}{\left(a(n+1){\varphi }^n-\beta \right)}^{-\frac{2\alpha }{n\beta }}{b}_1{L}_3{\varphi }^{m+2}}{2\alpha +m\beta +2\beta }}+{\displaystyle \frac{{\beta }^2\rho {\left({\varphi }^n\right)}^{\frac{2\alpha }{n\beta }}{\left(a(n+1){\varphi }^n-\beta \right)}^{-\frac{2\alpha }{n\beta }}{b}_2{L}_5{\varphi }^{2m+2}}{2(\alpha +m\beta +\beta )}}\hfill \\ & +{\displaystyle \frac{{\beta }^2\rho {\left({\varphi }^n\right)}^{\frac{2\alpha }{n\beta }}{\left(a(n+1){\varphi }^n-\beta \right)}^{-\frac{2\alpha }{n\beta }}{b}_3{L}_7{\varphi }^{3m+2}}{2\alpha +3m\beta +2\beta }}+{\displaystyle \frac{{\beta }^2\rho {\left({\varphi }^n\right)}^{\frac{2\alpha }{n\beta }}{\left(a(n+1){\varphi }^n-\beta \right)}^{-\frac{2\alpha }{n\beta }}{b}_4{L}_9{\varphi }^{4m+2}}{2(\alpha +2m\beta +\beta )}}\hfill \\ & +{\displaystyle \frac{{\beta }^2\rho {\left({\varphi }^n\right)}^{\frac{2\alpha }{n\beta }}{\left(a(n+1){\varphi }^n-\beta \right)}^{-\frac{2\alpha }{n\beta }}{b}_5{L}_{11}{\varphi }^{5m+2}}{2\alpha +5m\beta +2\beta }}+{\displaystyle \frac{{\beta }^2\rho {\left({\varphi }^n\right)}^{\frac{2\alpha }{n\beta }}{\left(a(n+1){\varphi }^n-\beta \right)}^{-\frac{2\alpha }{n\beta }}{b}_6{L}_{13}{\varphi }^{6m+2}}{2(\alpha +3m\beta +\beta )}}\hfill \\ & +{\displaystyle \frac{a\beta \gamma \rho {\left({\varphi }^n\right)}^{\frac{2\alpha }{n\beta }}{\left(a(n+1){\varphi }^n-\beta \right)}^{-\frac{2\alpha }{n\beta }}{L}_2{\varphi }^{n+2}}{2\alpha +n\beta +2\beta }}+{\displaystyle \frac{an\beta \gamma \rho {\left({\varphi }^n\right)}^{\frac{2\alpha }{n\beta }}{\left(a(n+1){\varphi }^n-\beta \right)}^{-\frac{2\alpha }{n\beta }}{L}_2{\varphi }^{n+2}}{2\alpha +n\beta +2\beta }}\hfill \\ & +{\displaystyle \frac{a\beta \rho {\left({\varphi }^n\right)}^{\frac{2\alpha }{n\beta }}{\left(a(n+1){\varphi }^n-\beta \right)}^{-\frac{2\alpha }{n\beta }}\omega L_2{\varphi }^{n+2}}{2\alpha +n\beta +2\beta }}+{\displaystyle \frac{an\beta \rho {\left({\varphi }^n\right)}^{\frac{2\alpha }{n\beta }}{\left(a(n+1){\varphi }^n-\beta \right)}^{-\frac{2\alpha }{n\beta }}\omega L_2{\varphi }^{n+2}}{2\alpha +n\beta +2\beta }}\hfill \\ & -{\displaystyle \frac{a\beta \rho {\left({\varphi }^n\right)}^{\frac{2\alpha }{n\beta }}{\left(a(n+1){\varphi }^n-\beta \right)}^{-\frac{2\alpha }{n\beta }}{b}_1{L}_4{\varphi }^{m+n+2}}{2\alpha +m\beta +n\beta +2\beta }}-{\displaystyle \frac{an\beta \rho {\left({\varphi }^n\right)}^{\frac{2\alpha }{n\beta }}{\left(a(n+1){\varphi }^n-\beta \right)}^{-\frac{2\alpha }{n\beta }}{b}_1{L}_4{\varphi }^{m+n+2}}{2\alpha +m\beta +n\beta +2\beta }}\hfill \\ & -{\displaystyle \frac{a\beta \rho {\left({\varphi }^n\right)}^{\frac{2\alpha }{n\beta }}{\left(a(n+1){\varphi }^n-\beta \right)}^{-\frac{2\alpha }{n\beta }}{b}_2{L}_6{\varphi }^{2m+n+2}}{2\alpha +2m\beta +n\beta +2\beta }}-{\displaystyle \frac{an\beta \rho {\left({\varphi }^n\right)}^{\frac{2\alpha }{n\beta }}{\left(a(n+1){\varphi }^n-\beta \right)}^{-\frac{2\alpha }{n\beta }}{b}_2{L}_6{\varphi }^{2m+n+2}}{2\alpha +2m\beta +n\beta +2\beta }}\hfill \\ & -{\displaystyle \frac{a\beta \rho {\left({\varphi }^n\right)}^{\frac{2\alpha }{n\beta }}{\left(a(n+1){\varphi }^n-\beta \right)}^{-\frac{2\alpha }{n\beta }}{b}_3{L}_8{\varphi }^{3m+n+2}}{2\alpha +3m\beta +n\beta +2\beta }}-{\displaystyle \frac{an\beta \rho {\left({\varphi }^n\right)}^{\frac{2\alpha }{n\beta }}{\left(a(n+1){\varphi }^n-\beta \right)}^{-\frac{2\alpha }{n\beta }}{b}_3{L}_8{\varphi }^{3m+n+2}}{2\alpha +3m\beta +n\beta +2\beta }}\hfill \\ & -{\displaystyle \frac{a\beta \rho {\left({\varphi }^n\right)}^{\frac{2\alpha }{n\beta }}{\left(a(n+1){\varphi }^n-\beta \right)}^{-\frac{2\alpha }{n\beta }}{b}_4{L}_{10}{\varphi }^{4m+n+2}}{2\alpha +4m\beta +n\beta +2\beta }}-{\displaystyle \frac{an\beta \rho {\left({\varphi }^n\right)}^{\frac{2\alpha }{n\beta }}{\left(a(n+1){\varphi }^n-\beta \right)}^{-\frac{2\alpha }{n\beta }}{b}_4{L}_{10}{\varphi }^{4m+n+2}}{2\alpha +4m\beta +n\beta +2\beta }}\hfill \\ & -{\displaystyle \frac{a\beta \rho {\left({\varphi }^n\right)}^{\frac{2\alpha }{n\beta }}{\left(a(n+1){\varphi }^n-\beta \right)}^{-\frac{2\alpha }{n\beta }}{b}_5{L}_{12}{\varphi }^{5m+n+2}}{2\alpha +5m\beta +n\beta +2\beta }}-{\displaystyle \frac{an\beta \rho {\left({\varphi }^n\right)}^{\frac{2\alpha }{n\beta }}{\left(a(n+1){\varphi }^n-\beta \right)}^{-\frac{2\alpha }{n\beta }}{b}_5{L}_{12}{\varphi }^{5m+n+2}}{2\alpha +5m\beta +n\beta +2\beta }}\hfill \\ & -{\displaystyle \frac{a\beta \rho {\left({\varphi }^n\right)}^{\frac{2\alpha }{n\beta }}{\left(a(n+1){\varphi }^n-\beta \right)}^{-\frac{2\alpha }{n\beta }}{b}_6{L}_{14}{\varphi }^{6m+n+2}}{2\alpha +6m\beta +n\beta +2\beta }}-{\displaystyle \frac{an\beta \rho {\left({\varphi }^n\right)}^{\frac{2\alpha }{n\beta }}{\left(a(n+1){\varphi }^n-\beta \right)}^{-\frac{2\alpha }{n\beta }}{b}_6{L}_{14}{\varphi }^{6m+n+2}}{2\alpha +6m\beta +n\beta +2\beta }}\hfill \\ & -{\displaystyle \frac{{\beta }^2\gamma \rho {\left({\varphi }^n\right)}^{\frac{2\alpha }{n\beta }}{\left(a(n+1){\varphi }^n-\beta \right)}^{-\frac{2\alpha }{n\beta }}{L}_1{\varphi }^2}{2(\alpha +\beta )}}-{\displaystyle \frac{{\beta }^2\rho {\left({\varphi }^n\right)}^{\frac{2\alpha }{n\beta }}{\left(a(n+1){\varphi }^n-\beta \right)}^{-\frac{2\alpha }{n\beta }}\omega L_1{\varphi }^2}{2(\alpha +\beta )}},\hfill \end{array}$$

(61)

with

$$\begin{array}{c}\hfill \rho ={\left[1-{\displaystyle \frac{a(n+1){\varphi }^n}{\beta }}\right]}^{{\displaystyle \frac{2\alpha }{n\beta }}},\end{array}$$

(62)

$$\begin{array}{c}\hfill L_1={}_{2}F_1\left({\displaystyle \frac{2\alpha }{n\beta }},{\displaystyle \frac{2(\alpha +\beta )}{n\beta }};{\displaystyle \frac{2(\alpha +\beta )}{n\beta }}+1;{\displaystyle \frac{a(n+1){\varphi }^n}{\beta }}\right),\end{array}$$

(63)

$$\begin{array}{c}\hfill L_2={}_{2}F_1\left({\displaystyle \frac{2\alpha }{n\beta }},{\displaystyle \frac{2\alpha +n\beta +2\beta }{n\beta }};{\displaystyle \frac{2\alpha +n\beta +2\beta }{n\beta }}+1;{\displaystyle \frac{a(n+1){\varphi }^n}{\beta }}\right),\end{array}$$

(64)

$$\begin{array}{c}\hfill L_3={}_{2}F_1\left({\displaystyle \frac{2\alpha }{n\beta }},{\displaystyle \frac{2\alpha +m\beta +2\beta }{n\beta }};{\displaystyle \frac{2\alpha +m\beta +2\beta }{n\beta }}+1;{\displaystyle \frac{a(n+1){\varphi }^n}{\beta }}\right),\end{array}$$

(65)

$$\begin{array}{c}\hfill L_4={}_{2}F_1\left({\displaystyle \frac{2\alpha }{n\beta }},{\displaystyle \frac{2\alpha +m\beta +n\beta +2\beta }{n\beta }};{\displaystyle \frac{2\alpha +m\beta +n\beta +2\beta }{n\beta }}+1;{\displaystyle \frac{a(n+1){\varphi }^n}{\beta }}\right),\end{array}$$

(66)

$$\begin{array}{c}\hfill L_5={}_{2}F_1\left({\displaystyle \frac{2\alpha }{n\beta }},{\displaystyle \frac{2(\alpha +m\beta +\beta )}{n\beta }};{\displaystyle \frac{2(\alpha +m\beta +\beta )}{n\beta }}+1;{\displaystyle \frac{a(n+1){\varphi }^n}{\beta }}\right),\end{array}$$

(67)

$$\begin{array}{c}\hfill L_6={}_{2}F_1\left({\displaystyle \frac{2\alpha }{n\beta }},{\displaystyle \frac{2\alpha +2m\beta +n\beta +2\beta }{n\beta }};{\displaystyle \frac{2\alpha +2m\beta +n\beta +2\beta }{n\beta }}+1;{\displaystyle \frac{a(n+1){\varphi }^n}{\beta }}\right),\end{array}$$

(68)

$$\begin{array}{c}\hfill L_7={}_{2}F_1\left({\displaystyle \frac{2\alpha }{n\beta }},{\displaystyle \frac{2\alpha +3m\beta +2\beta }{n\beta }};{\displaystyle \frac{2\alpha +3m\beta +2\beta }{n\beta }}+1;{\displaystyle \frac{a(n+1){\varphi }^n}{\beta }}\right),\end{array}$$

(69)

$$\begin{array}{c}\hfill L_8={}_{2}F_1\left({\displaystyle \frac{2\alpha }{n\beta }},{\displaystyle \frac{2\alpha +3m\beta +n\beta +2\beta }{n\beta }};{\displaystyle \frac{2\alpha +3m\beta +n\beta +2\beta }{n\beta }}+1;{\displaystyle \frac{a(n+1){\varphi }^n}{\beta }}\right),\end{array}$$

(70)

$$\begin{array}{c}\hfill L_9={}_{2}F_1\left({\displaystyle \frac{2\alpha }{n\beta }},{\displaystyle \frac{2(\alpha +2m\beta +\beta )}{n\beta }};{\displaystyle \frac{2(\alpha +2m\beta +\beta )}{n\beta }}+1;{\displaystyle \frac{a(n+1){\varphi }^n}{\beta }}\right),\end{array}$$

(71)

$$\begin{array}{c}\hfill L_{10}={}_{2}F_1\left({\displaystyle \frac{2\alpha }{n\beta }},{\displaystyle \frac{2\alpha +4m\beta +n\beta +2\beta }{n\beta }};{\displaystyle \frac{2\alpha +4m\beta +n\beta +2\beta }{n\beta }}+1;{\displaystyle \frac{a(n+1){\varphi }^n}{\beta }}\right),\end{array}$$

(72)

$$\begin{array}{c}\hfill L_{11}={}_{2}F_1\left({\displaystyle \frac{2\alpha }{n\beta }},{\displaystyle \frac{2\alpha +5m\beta +2\beta }{n\beta }};{\displaystyle \frac{2\alpha +5m\beta +2\beta }{n\beta }}+1;{\displaystyle \frac{a(n+1){\varphi }^n}{\beta }}\right),\end{array}$$

(73)

$$\begin{array}{c}\hfill L_{12}={}_{2}F_1\left({\displaystyle \frac{2\alpha }{n\beta }},{\displaystyle \frac{2\alpha +5m\beta +n\beta +2\beta }{n\beta }};{\displaystyle \frac{2\alpha +5m\beta +n\beta +2\beta }{n\beta }}+1;{\displaystyle \frac{a(n+1){\varphi }^n}{\beta }}\right),\end{array}$$

(74)

$$\begin{array}{c}\hfill L_{13}={}_{2}F_1\left({\displaystyle \frac{2\alpha }{n\beta }},{\displaystyle \frac{2(\alpha +3m\beta +\beta )}{n\beta }};{\displaystyle \frac{2(\alpha +3m\beta +\beta )}{n\beta }}+1;{\displaystyle \frac{a(n+1){\varphi }^n}{\beta }}\right),\end{array}$$

(75)

$$\begin{array}{c}\hfill L_{14}={}_{2}F_1\left({\displaystyle \frac{2\alpha }{n\beta }},{\displaystyle \frac{2\alpha +6m\beta +n\beta +2\beta }{n\beta }};{\displaystyle \frac{2\alpha +6m\beta +n\beta +2\beta }{n\beta }}+1;{\displaystyle \frac{a(n+1){\varphi }^n}{\beta }}\right).\end{array}$$

(76)

Notably, the constraint given by Equation (23) still holds for the solution to exist while the Gauss’ hypergeometric functions was introduced earlier in Equation (18).

4.5. Form-V

For this form, the law of refractive index is structured as follows:

$$\begin{array}{c}\hfill F\left(s\right)=b_1{s}^{{\displaystyle \frac{m}{2}}}+b_2{s}^m+b_3{s}^{{\displaystyle \frac{3m}{2}}}+b_4{s}^{2m}+b_5{\left(s^{{\displaystyle \frac{m}{2}}}\right)}^{″}+b_6{\left(s^m\right)}^{″}.\end{array}$$

(77)

The governing CGLE is, therefore, written as

$$\begin{array}{cc}\hfill & i{\mathrm{\Psi }}_t+a{\left({\left|\mathrm{\Psi }\right|}^n\mathrm{\Psi }\right)}_{xx}+\left[b_1{\left|\mathrm{\Psi }\right|}^m+b_2{\left|\mathrm{\Psi }\right|}^{2m}+b_3{\left|\mathrm{\Psi }\right|}^{3m}+b_4{\left|\mathrm{\Psi }\right|}^{4m}+b_5{\left({\left|\mathrm{\Psi }\right|}^m\right)}_{xx}+b_6{\left({\left|\mathrm{\Psi }\right|}^{2m}\right)}_{xx}\right]\mathrm{\Psi }\hfill \\ \hfill & =\alpha {\displaystyle \frac{{\left|{\mathrm{\Psi }}_x\right|}^2}{{\mathrm{\Psi }}^{*}}}+{\displaystyle \frac{\beta }{4{\left|\mathrm{\Psi }\right|}^2{\mathrm{\Psi }}^{*}}}\left[2{\left|\mathrm{\Psi }\right|}^2{\left({\left|\mathrm{\Psi }\right|}^2\right)}_{xx}-{\left\{{\left({\left|\mathrm{\Psi }\right|}^2\right)}_x\right\}}^2\right]\hfill \\ & + \gamma \mathrm{\Psi }+i\left[\lambda {\left({\left|\mathrm{\Psi }\right|}^{2m}\mathrm{\Psi }\right)}_x+{\theta }_1{\left({\left|\mathrm{\Psi }\right|}^{2m}\right)}_x\mathrm{\Psi }+{\theta }_2{\left|\mathrm{\Psi }\right|}^{2m}{\mathrm{\Psi }}_x\right].\hfill \end{array}$$

(78)

Upon substituting Equation (2) into Equation (78), the ODE for $\varphi \left(x\right)$ is written as:

$$\begin{array}{ccc}& & a(n+1){\varphi }^n\left(x\right)\left[n{\left\{{\varphi }^{\prime }\left(x\right)\right\}}^2+\varphi \left(x\right){\varphi }^{″}\left(x\right)\right]\hfill \\ & -& {\varphi }^2\left(x\right)\left[\omega +\gamma -b_1{\varphi }^{{\displaystyle \frac{m}{2}}}\left(x\right)-b_2{\varphi }^m\left(x\right)-b_3{\varphi }^{{\displaystyle \frac{3m}{2}}}\left(x\right)-b_4{\varphi }^{2m}\left(x\right)-b_5{\left\{{\varphi }^{{\displaystyle \frac{m}{2}}}\left(x\right)\right\}}^{″}-b_6{\left\{{\varphi }^m\left(x\right)\right\}}^{″}\right]\hfill \\ & -& \alpha {\left\{{\varphi }^{\prime }\left(x\right)\right\}}^2-\beta \varphi \left(x\right){\varphi }^{″}\left(x\right)=0.\hfill \end{array}$$

(79)

Next, the governing model after implementing the parameter constraint given by Equation (4) is restructured as:

$$\begin{array}{cc}\hfill & i{\mathrm{\Psi }}_t+a{\left({\left|\mathrm{\Psi }\right|}^n\mathrm{\Psi }\right)}_{xx}\hfill \\ & +\left[b_1{\left|\mathrm{\Psi }\right|}^m+b_2{\left|\mathrm{\Psi }\right|}^{2m}+b_3{\left|\mathrm{\Psi }\right|}^{3m}+b_4{\left|\mathrm{\Psi }\right|}^{4m}+b_5{\left({\left|\mathrm{\Psi }\right|}^m\right)}_{xx}+b_6{\left({\left|\mathrm{\Psi }\right|}^{2m}\right)}_{xx}\right]\mathrm{\Psi }\hfill \\ \hfill & =\alpha {\displaystyle \frac{{\left|{\mathrm{\Psi }}_x\right|}^2}{{\mathrm{\Psi }}^{*}}}+{\displaystyle \frac{\beta }{4{\left|\mathrm{\Psi }\right|}^2{\mathrm{\Psi }}^{*}}}\left[2{\left|\mathrm{\Psi }\right|}^2{\left({\left|\mathrm{\Psi }\right|}^2\right)}_{xx}-{\left\{{\left({\left|\mathrm{\Psi }\right|}^2\right)}_x\right\}}^2\right]+\gamma \mathrm{\Psi }\hfill \\ & + i\left[\lambda {\left({\left|\mathrm{\Psi }\right|}^{2m}\mathrm{\Psi }\right)}_x+{\theta }_1{\left({\left|\mathrm{\Psi }\right|}^{2m}\right)}_x\mathrm{\Psi }-\left\{2{\theta }_{1}m+\lambda (2m+1)\right\}{\left|\mathrm{\Psi }\right|}^{2m}{\mathrm{\Psi }}_x\right].\hfill \end{array}$$

(80)

The translational Lie symmetry aids in the integration of Equation (79), which leads to the solution given by Equation (28) where

$$\begin{array}{c}\hfill A=exp\left(-2{\int }^{\varphi }{\displaystyle \frac{m^2{b}_5{\tau }_1^m-m{b}_5{\tau }_1^m+4m^2{b}_6{\tau }_1^{2m}-2m{b}_6{\tau }_1^{2m}+a{n}^2{\tau }_1^n+an{\tau }_1^n-\alpha }{{\tau }_1\left(-m{b}_5{\tau }_1^m-2m{b}_6{\tau }_1^{2m}-a{\tau }_1^n-an{\tau }_1^n+\beta \right)}}d{\tau }_1\right),\end{array}$$

(81)

$$\begin{array}{c}\hfill B={\int }^{\varphi }{\displaystyle \frac{\mathrm{\Gamma }exp\left(-2{\int }^{{\tau }_2}\frac{m^2{b}_5{\tau }_1^m-m{b}_5{\tau }_1^m+4m^2{b}_6{\tau }_1^{2m}-2m{b}_6{\tau }_1^{2m}+a{n}^2{\tau }_1^n+an{\tau }_1^n-\alpha }{{\tau }_1\left(-m{b}_5{\tau }_1^m-2m{b}_6{\tau }_1^{2m}-a{\tau }_1^n-an{\tau }_1^n+\beta \right)}d{\tau }_1\right)}{{\tau }_2\left(-a{\tau }_2^n-an{\tau }_2^n+b_5(-m){\tau }_2^m-2b_{6}m{\tau }_2^{2m}+\beta \right)}}d{\tau }_2,\end{array}$$

(82)

and

$$\begin{array}{c}\hfill \mathrm{\Gamma }=b_1{\tau }_2^{m+2}+b_2{\tau }_2^{2m+2}+b_3{\tau }_2^{3m+2}+b_4{\tau }_2^{4m+2}-\gamma {\tau }_2^2-{\tau }_2^2\omega .\end{array}$$

(83)

Once again, the condition given by Equation (23) must remain valid for the solution to exist.

4.6. Form-VI

For this form, the law of the refractive index is structured as follows:

$$\begin{array}{c}\hfill F\left(s\right)=b_1{s}^{{\displaystyle \frac{m}{2}}}+b_2{s}^m+b_3{s}^{{\displaystyle \frac{3m}{2}}}+b_4{s}^{2m}+b_5{s}^{{\displaystyle \frac{5m}{2}}}+b_6{s}^{3m}+b_7{\left(s^{{\displaystyle \frac{m}{2}}}\right)}^{″}+b_8{\left(s^m\right)}^{″}.\end{array}$$

(84)

Here, $b_j$ is the real-valued constant for $1\le j\le 8$ and m is the arbitrary light intensity parameter. This leads to the governing CGLE to be written as:

$$\begin{array}{cc}\hfill & i{\mathrm{\Psi }}_t+a{\left({\left|\mathrm{\Psi }\right|}^n\mathrm{\Psi }\right)}_{xx}\hfill \\ \hfill & +\left[b_1{\left|\mathrm{\Psi }\right|}^m+b_2{\left|\mathrm{\Psi }\right|}^{2m}+b_3{\left|\mathrm{\Psi }\right|}^{3m}+b_4{\left|\mathrm{\Psi }\right|}^{4m}+b_5{\left|\mathrm{\Psi }\right|}^{5m}+b_6{\left|\mathrm{\Psi }\right|}^{6m}+b_7{\left({\left|\mathrm{\Psi }\right|}^m\right)}_{xx}+b_8{\left({\left|\mathrm{\Psi }\right|}^{2m}\right)}_{xx}\right]\mathrm{\Psi }\hfill \\ \hfill & =\alpha {\displaystyle \frac{{\left|{\mathrm{\Psi }}_x\right|}^2}{{\mathrm{\Psi }}^{*}}}+{\displaystyle \frac{\beta }{4{\left|\mathrm{\Psi }\right|}^2{\mathrm{\Psi }}^{*}}}\left[2{\left|\mathrm{\Psi }\right|}^2{\left({\left|\mathrm{\Psi }\right|}^2\right)}_{xx}-{\left\{{\left({\left|\mathrm{\Psi }\right|}^2\right)}_x\right\}}^2\right]\hfill \\ & + \gamma \mathrm{\Psi }+i\left[\lambda {\left({\left|\mathrm{\Psi }\right|}^{2m}\mathrm{\Psi }\right)}_x+{\theta }_1{\left({\left|\mathrm{\Psi }\right|}^{2m}\right)}_x\mathrm{\Psi }+{\theta }_2{\left|\mathrm{\Psi }\right|}^{2m}{\mathrm{\Psi }}_x\right].\hfill \end{array}$$

(85)

Upon substituting Equation (2) into Equation (85), the governing ODE for $\varphi \left(x\right)$ is written as:

$$\begin{array}{cc}\hfill & a(n+1){\varphi }^n\left(x\right)\left[n{\left\{{\varphi }^{\prime }\left(x\right)\right\}}^2+\varphi \left(x\right){\varphi }^{″}\left(x\right)\right]\hfill \\ \hfill & -{\varphi }^2\left(x\right)\left[\omega +\gamma -b_1{\varphi }^{{\displaystyle \frac{m}{2}}}\left(x\right)-b_2{\varphi }^m\left(x\right)-b_3{\varphi }^{{\displaystyle \frac{3m}{2}}}\left(x\right)-b_4{\varphi }^{2m}\left(x\right)-b_5{\varphi }^{{\displaystyle \frac{5m}{2}}}\left(x\right)\right]\hfill \\ \hfill & -{\varphi }^2\left(x\right)\left[-b_6{\varphi }^{3m}\left(x\right)-b_7{\left\{{\varphi }^{{\displaystyle \frac{m}{2}}}\left(x\right)\right\}}^{″}-b_8{\left\{{\varphi }^m\left(x\right)\right\}}^{″}\right]\hfill \\ \hfill & -\alpha {\left\{{\varphi }^{\prime }\left(x\right)\right\}}^2-\beta \varphi \left(x\right){\varphi }^{″}\left(x\right)=0,\hfill \end{array}$$

(86)

while the governing model, by virtue of the constraint condition given by Equation (4), collapses to

$$\begin{array}{cc}\hfill & i{\mathrm{\Psi }}_t+a{\left({\left|\mathrm{\Psi }\right|}^n\mathrm{\Psi }\right)}_{xx}\hfill \\ & +\left[b_1{\left|\mathrm{\Psi }\right|}^m+b_2{\left|\mathrm{\Psi }\right|}^{2m}+b_3{\left|\mathrm{\Psi }\right|}^{3m}+b_4{\left|\mathrm{\Psi }\right|}^{4m}+b_5{\left|\mathrm{\Psi }\right|}^{5m}+b_6{\left|\mathrm{\Psi }\right|}^{6m}+b_7{\left({\left|\mathrm{\Psi }\right|}^m\right)}_{xx}+b_8{\left({\left|\mathrm{\Psi }\right|}^{2m}\right)}_{xx}\right]\mathrm{\Psi }\hfill \\ \hfill & =\alpha {\displaystyle \frac{{\left|{\mathrm{\Psi }}_x\right|}^2}{{\mathrm{\Psi }}^{*}}}+{\displaystyle \frac{\beta }{4{\left|\mathrm{\Psi }\right|}^2{\mathrm{\Psi }}^{*}}}\left[2{\left|\mathrm{\Psi }\right|}^2{\left({\left|\mathrm{\Psi }\right|}^2\right)}_{xx}-{\left\{{\left({\left|\mathrm{\Psi }\right|}^2\right)}_x\right\}}^2\right]+\gamma \mathrm{\Psi }\hfill \\ & + i\left[\lambda {\left({\left|\mathrm{\Psi }\right|}^{2m}\mathrm{\Psi }\right)}_x+{\theta }_1{\left({\left|\mathrm{\Psi }\right|}^{2m}\right)}_x\mathrm{\Psi }-\left\{2{\theta }_{1}m+\lambda (2m+1)\right\}{\left|\mathrm{\Psi }\right|}^{2m}{\mathrm{\Psi }}_x\right].\hfill \end{array}$$

(87)

Implementing the translational Lie symmetry in Equation (86), the implicit integral is given by Equation (28) where

$$\begin{array}{c}\hfill A=exp\left(-2{\int }^{\varphi }{\displaystyle \frac{m^2{b}_7{\tau }_1^m-m{b}_7{\tau }_1^m+4m^2{b}_8{\tau }_1^{2m}-2m{b}_8{\tau }_1^{2m}+a{n}^2{\tau }_1^n+an{\tau }_1^n-\alpha }{{\tau }_1\left(-m{b}_7{\tau }_1^m-2m{b}_8{\tau }_1^{2m}-a{\tau }_1^n-an{\tau }_1^n+\beta \right)}}d{\tau }_1\right),\end{array}$$

(88)

$$\begin{array}{c}\hfill B={\int }^{\varphi }{\displaystyle \frac{\mathrm{\Gamma }exp\left(-2{\int }^{{\tau }_2}\frac{m^2{b}_7{\tau }_1^m-m{b}_7{\tau }_1^m+4m^2{b}_8{\tau }_1^{2m}-2m{b}_8{\tau }_1^{2m}+a{n}^2{\tau }_1^n+an{\tau }_1^n-\alpha }{{\tau }_1\left(-m{b}_7{\tau }_1^m-2m{b}_8{\tau }_1^{2m}-a{\tau }_1^n-an{\tau }_1^n+\beta \right)}d{\tau }_1\right)}{{\tau }_2\left(-a{\tau }_2^n-an{\tau }_2^n+b_7(-m){\tau }_2^m-2b_{8}m{\tau }_2^{2m}+\beta \right)}}d{\tau }_2,\end{array}$$

(89)

and

$$\begin{array}{c}\hfill \mathrm{\Gamma }=b_1{\tau }_2^{m+2}+b_2{\tau }_2^{2m+2}+b_3{\tau }_2^{3m+2}+b_4{\tau }_2^{4m+2}+b_5{\tau }_2^{5m+2}+b_6{\tau }_2^{6m+2}-\gamma {\tau }_2^2-{\tau }_2^2\omega .\end{array}$$

(90)

The solution existence criteria, as given by Equation (23), must hold here as well.

5. Conclusions

The paper is a comprehensive study on the retrieval of quiescent solitons in optical fibers and crystals that is modeled by CGLE with nonlinear CD as well as six forms of SPM structures, together with linear temporal evolution. The integration methodology uses Lie symmetry analysis. The solutions appeared with several parameter constraints that are also listed in the work. Some of the solutions are in terms of Gauss’ hypergeometric functions, which provided its own parameter constraints for the existence of solutions.

Thus, the optoelectronics community clearly understands the message. If CD is rendered to be nonlinear due to rough handling of fibers and crystals or by any other accidental means, the solitons that travel across intercontinental distances would be stalled during their propagation, which would trigger catastrophic consequences. This paper should, therefore, convey a stern warning to telecommunication engineers. They must prioritize the integrity of optical components to ensure reliable communication channels. Implementing rigorous testing and maintenance protocols will be essential for preventing such disruptions in the future.

The current paper opens up a plethora of avenues to explore. An immediate thought would be to address the model with a generalized temporal evolution which would produce a bigger and better picture to this study of quiescent optical solitons with CGLE as its fundamental model. The results of such research are under way and could soon be visible. Moreover, these research endeavors could gradually extend to additional optoelectronic devices. This is just a foot in the door!