Introduction

The study of quiescent optical solitons has gained momentum since the past few years [1,2,3,4,5,6,7,8,9,10,11,12,13,14]. This study comes from the fact that when chromatic dispersion (CD) is rendered to be nonlinear, such solitons are formed and this leads to a catastrophic disaster in the soliton communications through a variety of waveguides across intercontinental distances [15,16,17,18,19,20,21,22,23,24,25]. These stalled solitons are thus a serious hindrance and is an important technological glitch that must be mitigated at all costs [26,27,28,29,30,31,32,33,34,35,36]. Such quiescent solitons can form for any kind of self-phase modulation (SPM) structure [37,38,39,40,41,42,43,44,45,46,47]. The mathematical know–how of quiescent soliton formation has been studied earlier for a few models such as Lakshmanan–Kudu–Lakshmanan equation, Sasa–Satsuma equation as well as Radhakrishnan–Kundu–Lakshmanan equation [48,49,50,51,52,53,54,55,56,57,58].

The current paper, which is on another model known as the complex Ginzburg–Landau equation (CGLE), is an improvement of the results that have been previously reported in 2022 [10]. The previous results are all in terms of quadratures. The current work is an improved version of the previously reported results. Moreover, this work, which is with generalized temporal evolution, is a sequel to the previously reported results that was with linear temporal evolution [13]. The mathematical principle of Lie symmetry analysis retrieves the quiescent optical solitons to the CGLE with various SPM structures in the paper. Additionally, the saturating law as well as the exponential law of SPM are covered in this paper for the first time. The details of the derivation are jotted and exhibited in the rest of the paper after a few introductory words.

Governing model

The governing equation for the CGL equation with general non-Kerr law nonlinearity and generalized temporal evolution having nonlinear CD is given as

$$\begin{aligned}{} & {} i\left( q^l \right) _t + a \left( \left| q \right| ^n q^l \right) _{xx} + F(\left| q \right| ^2) q^l = \alpha \frac{\left| q_x \right| ^2}{\left( q^l\right) ^{*}}\nonumber \\{} & {} +\frac{\beta }{4\left| q \right| ^2 \left( q^l \right) ^{*}} \left[ 2\left| q \right| ^2 \left( \left| q \right| ^2 \right) _{xx} - \left\{ \left( \left| q \right| ^2 \right) _{x}\right\} ^2 \right] + \gamma q^l. \end{aligned}$$
(1)

Here, the first term is the generalized temporal evolution with l being the parameter for generalized temporal evolution. If \(l=1\), the model collapses to the special case of linear temporal evolution that has been studied [13]. The parameter a is the coefficient of nonlinear CD with n being the parameter of nonlinearity. For \(n=0\), the CD is rendered to be linear in which case, mobile soliton emerges from the model with linear temporal evolution for any feasible form of SPM. The third term with the functional F accounts for the SPM structure that stems from the intensity-dependent refractive index change. On the right hand side the terms with \(\alpha\), \(\beta\) and \(\gamma\) account for the nonlinear effects that emerge from the CGL structure. For non-zero n, the model leads to quiescent optical solitons whose structures will be derived for eleven different forms of SPM. For each of these cases, as will be seen, the soliton solutions that will emerge, by Lie symmetry analysis, are all implicit. The detailed analysis and the derivation of such solutions are sequentially exhibited in the rest of the paper for each of the eleven structural forms of SPM.

Mathematical analysis

To start with the search for the solution process for (1), the starting hypothesis is picked as

$$\begin{aligned} q \left( x,t \right) =\phi \left( x \right) {\textrm{e}^{i\lambda \,t}}. \end{aligned}$$
(2)

Upon substituting (2) into (1), one recovers the ordinary diferential equation (ODE) for \(\phi (x)\) as:

$$\begin{aligned}{} & {} a (l+n) \phi ^{2l+n-2}(x) \left[ (l+n-1) \left\{ \phi ^{\prime }(x) \right\} ^2 + \phi (x) \phi ^{\prime \prime }(x)\right] \nonumber \\{} & {} \quad - \left[ \phi ^{2l}(x) \left\{ \gamma + \lambda l - F\left( \phi ^2\right) \right\} \right] \nonumber \\{} & {} \quad - \alpha \left\{ \phi ^{\prime }(x) \right\} ^2 - \beta \phi (x) \phi ^{\prime \prime }(x) = 0. \end{aligned}$$
(3)

For its integrability, one must choose

$$\begin{aligned} l + n = 0. \end{aligned}$$
(4)

This transforms the CGLE (1) to

$$\begin{aligned}{} & {} \frac{i}{\left( q^n \right) _t} + a \left( \frac{\left| q \right| ^n}{ q^n } \right) _{xx} + \frac{F \left( \left| q \right| ^2 \right) }{ q^n} = \alpha \left| q_x \right| ^2 \left( q^n \right) ^{*}\nonumber \\{} & {} \quad + \frac{\beta \left( q^n \right) ^{*}}{4\left| q \right| ^2} \left[ 2\left| q \right| ^2 \left( \left| q \right| ^2 \right) _{xx} - \left\{ \left( \left| q \right| ^2 \right) _{x}\right\} ^2 \right] + \frac{\gamma }{ q^n}, \end{aligned}$$
(5)

while the ODE in \(\phi (x)\) given by (3) shrinks to

$$\begin{aligned}{} & {} \gamma - \lambda n - F\left( \phi ^2\right) \nonumber \\{} & {} \quad + \phi ^{2n} (x) \left[ \alpha \left\{ \phi ^{\prime }(x) \right\} ^2 + \beta \phi (x) \phi ^{\prime \prime }(x) \right] = 0. \end{aligned}$$
(6)

It is this version of the CGLE, namely (5), and the reduced ODE, given by (6), that will be studied for the eleven different structures of the SPM in the upcoming section to recover the implicit quiescent solitons.

Implicit quiescent solitons

This section will recover the quiescent solitons for the CGLE wih nonlinear CD and having generalized temporal evolution for the following eleven forms of SPM [59,60,61,62,63,64,65,66,67,68].

Kerr law

For Kerr law of nonlinearity, the nonlinear functional takes the form

$$\begin{aligned} F(\left| q \right| ^2)= & {} b\left| q \right| ^2, \end{aligned}$$
(7)

for b being a real-valued parameter. With this form of F, eqs. (5) and (6) reduce to

$$\begin{aligned}{} & {} \frac{i}{\left( q^n \right) _t} + a \left( \frac{\left| q \right| ^n}{ q^n } \right) _{xx} + b \frac{\left| q \right| ^2}{ q^n} = \alpha \left| q_x \right| ^2 \left( q^n \right) ^{*}\nonumber \\{} & {} \quad + \frac{\beta \left( q^n \right) ^{*}}{4\left| q \right| ^2} \left[ 2\left| q \right| ^2 \left( \left| q \right| ^2 \right) _{xx} - \left\{ \left( \left| q \right| ^2 \right) _{x}\right\} ^2 \right] + \frac{\gamma }{ q^n}, \end{aligned}$$
(8)

and

$$\begin{aligned} \gamma - \lambda n - b \phi ^2 + \phi ^{2n} (x) \left[ \alpha \left\{ \phi ^{\prime }(x) \right\} ^2 + \beta \phi (x) \phi ^{\prime \prime }(x) \right] = 0, \end{aligned}$$
(9)

respectively. Equation (9) admits a single Lie point symmetry, namely \(\partial /\partial x\). The usage of this symmetry integrates (9) to

$$\begin{aligned}{} & {} x=\pm \frac{\phi ^{1+n}}{n+1} \sqrt{-\frac{\alpha -n \beta }{\gamma - n \lambda }}\nonumber \\{} & {} \, _2F_1\left( \frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\frac{b (\alpha -n \beta ) \phi ^2}{(\alpha +\beta -n \beta ) (\gamma -n \lambda )}\right) , \end{aligned}$$
(10)

where the Gauss’ hypergeometric function is defined as

$$\begin{aligned} _2F_1 \left( \alpha , \beta ; \gamma ; z \right) = \sum _{n=0}^{\infty } \frac{ \left( \alpha \right) _n \left( \beta \right) _n}{ \left( \gamma \right) _n} \frac{z^n}{n!}, \end{aligned}$$
(11)

with the Pochhammer symbol being

$$\begin{aligned} (p)_n = \left\{ \begin{array}{ll} 1 &{} \text{ n=0 },\\ p(p+1)\cdots (p+n-1) &{} \text{ n } > 0. \end{array} \right. \end{aligned}$$
(12)

The condition that guarantees convergence of the hypergeometric series is

$$\begin{aligned} \left| z \right| < 1, \end{aligned}$$
(13)

which for (10) implies

$$\begin{aligned}{} & {} - \left| \frac{ \left( \alpha + \beta - n\beta \right) \left( \gamma - n \lambda \right) }{ b \left( \alpha - n \beta \right) } \right| ^{\frac{1}{2}}< \phi (x)\nonumber \\{} & {} \quad < \left| \frac{ \left( \alpha + \beta - n\beta \right) \left( \gamma - n \lambda \right) }{ b \left( \alpha - n \beta \right) } \right| ^{\frac{1}{2}}. \end{aligned}$$
(14)

Finally (10) compels the parameter constraint:

$$\begin{aligned} \left( \alpha - n \beta \right) \left( \gamma - n \lambda \right) < 0. \end{aligned}$$
(15)

Power law

For power-law of SPM, the functional F is

$$\begin{aligned} F(\left| q \right| ^2)= & {} b \left| q \right| ^{2m}, \end{aligned}$$
(16)

where m is the power-law nonlinearity parameter. Then, eqs (5) and (6) transfer to

$$\begin{aligned}{} & {} \frac{i}{\left( q^n \right) _t} + a \left( \frac{\left| q \right| ^n}{ q^n } \right) _{xx} + b \frac{\left| q \right| ^{2m}}{ q^n} = \alpha \left| q_x \right| ^2 \left( q^n \right) ^{*}\nonumber \\{} & {} \quad + \frac{\beta \left( q^n \right) ^{*}}{4\left| q \right| ^2} \left[ 2\left| q \right| ^2 \left( \left| q \right| ^2 \right) _{xx} - \left\{ \left( \left| q \right| ^2 \right) _{x}\right\} ^2 \right] + \frac{\gamma }{ q^n}, \end{aligned}$$
(17)

and

$$\begin{aligned} \gamma - \lambda n - b \phi ^{2m} + \phi ^{2n} (x) \left[ \alpha \left\{ \phi ^{\prime }(x) \right\} ^2 + \beta \phi (x) \phi ^{\prime \prime }(x) \right] = 0, \end{aligned}$$
(18)

respectively. This ODE admits a single Lie point symmetry, namely \(\partial /\partial x\) which when implemented leads to the following implicit solution in terms of Gauss’ hypergeometric functions as

$$\begin{aligned}{} & {} x=\pm \frac{\phi ^{n+1}}{n+1} \sqrt{-\frac{\alpha -n \beta }{\gamma - n \lambda }}\nonumber \\{} & {} \, _2F_1\left( \frac{1}{2},\frac{n+1}{2m}; \frac{n+2m+1}{2m}; \frac{b (\alpha -n \beta ) \phi ^{2 m}}{ \left\{ \alpha +(m-n) \beta \right\} (\gamma - n\lambda )}\right) . \end{aligned}$$
(19)

The convergence criteria (13) implies

$$\begin{aligned}{} & {} - \left| \frac{ \left( \left\{ \alpha + (m-n)\beta \right\} \right) \left( \gamma - n \lambda \right) }{ b \left( \alpha - n \beta \right) } \right| ^{\frac{1}{2m}}< \phi (x)\nonumber \\{} & {} < \left| \frac{ \left( \left\{ \alpha + (m- n)\beta \right\} \right) \left( \gamma - n \lambda \right) }{ b \left( \alpha - n \beta \right) } \right| ^{\frac{1}{2m}}, \end{aligned}$$
(20)

while the parameter domain restriction (15) holds for power-law as well.

Parabolic (cubic-quintic) law

For parabolic law, the structure of the functional is given by

$$\begin{aligned} F(\left| q \right| ^2)= & {} b_1 \left| q \right| ^2 + b_2 \left| q \right| ^4, \end{aligned}$$
(21)

where \(b_j\) for \(j = 1, 2\) are real-valued constants. The CGLE and the ODE for this case from (5) and (6) are given by

$$\begin{aligned}{} & {} \frac{i}{\left( q^n \right) _t} + a \left( \frac{\left| q \right| ^n}{ q^n } \right) _{xx} + \frac{b_1 \left| q \right| ^2 + b_2 \left| q \right| ^4 }{ q^n} = \alpha \left| q_x \right| ^2 \left( q^n \right) ^{*}\nonumber \\{} & {} + \frac{\beta \left( q^n \right) ^{*}}{4\left| q \right| ^2} \left[ 2\left| q \right| ^2 \left( \left| q \right| ^2 \right) _{xx} - \left\{ \left( \left| q \right| ^2 \right) _{x}\right\} ^2 \right] + \frac{\gamma }{ q^n}, \end{aligned}$$
(22)

and

$$\begin{aligned}{} & {} \gamma - \lambda n - b_1 \phi ^2 - b_2 \phi ^4\nonumber \\{} & {} \quad + \phi ^{2n} (x) \left[ \alpha \left\{ \phi ^{\prime }(x) \right\} ^2 + \beta \phi (x) \phi ^{\prime \prime }(x) \right] = 0, \end{aligned}$$
(23)

respectively. The ODE for \(\phi (x)\) given by (23) permits a single Lie point symmetry given by \(\partial /\partial x\) which leads to its implicit solution in terms of Appell hypergeometric function as

$$\begin{aligned}{} & {} x=\pm \frac{\phi ^{n+1}}{n+1} \sqrt{- \frac{\alpha -n \beta }{\gamma - n \lambda }}\,\nonumber \\{} & {} \, F_1\left( \frac{n+1}{2};\frac{1}{2},\frac{1}{2};\frac{n+3}{2}; \frac{2 (\alpha -n \beta ) \left\{ \alpha - (n-1)\beta \right\} \phi ^2 b_2}{A - \left\{ \alpha -(n-2) \beta \right\} (\alpha -n \beta ) b_1},\right. \nonumber \\{} & {} \left. - \frac{2 (\alpha -n \beta ) \left\{ \alpha - (n-1)\beta \right\} \phi ^2 b_2}{A + \left\{ \alpha -(n-2) \beta \right\} (\alpha -n \beta ) b_1} \right) , \end{aligned}$$
(24)

where

$$\begin{aligned} A= \sqrt{ \left\{ \alpha - (n-2) \beta \right\} (\alpha -n \beta ) \left[ 4 b_2 \left\{ \alpha - (n-1)\beta \right\} ^2 (\gamma -\lambda n) + b_1^2 (\alpha -\beta n) \left\{ \alpha - (n- 2) \beta \right\} \right] }. \end{aligned}$$
(25)

The Appell hypergeometric function of two variables has a primary definition through the hypergeometric series as

$$\begin{aligned}{} & {} F_1\left( a;b_1,b_2;c;x,y\right) \nonumber \\{} & {} = x^m y^n \left( \sum _{m=0}^{\infty } \sum _{n=0}^{\infty } \frac{(a)_{m+n} \left( b_1\right) _m \left( b_2\right) _n}{(c)_{m+n} m! n!}\right) , \end{aligned}$$
(26)

which is convergent inside the region

$$\begin{aligned} \max \left( \left| x \right| , \left| y \right| \right) < 1, \end{aligned}$$
(27)

which for (24) translates to

$$\begin{aligned}{} & {} \max \left( \left| \frac{(\alpha -n \beta ) \left\{ \alpha - (n-1)\beta \right\} \phi ^2 b_2}{A - \left\{ \alpha -(n-2) \beta \right\} (\alpha -n \beta ) b_1} \right| ,\right. \nonumber \\{} & {} \left. \left| \frac{(\alpha -n \beta ) \left\{ \alpha - (n-1)\beta \right\} \phi ^2 b_2}{A + \left\{ \alpha -(n-2) \beta \right\} (\alpha -n \beta ) b_1} \right| \right) < 1. \end{aligned}$$
(28)

Moreover, (25) implies

$$\begin{aligned}{} & {} \left\{ \alpha - (n-2) \beta \right\} (\alpha -n \beta )\nonumber \\{} & {} \left[ 4 b_2 \left\{ \alpha - (n-1)\beta \right\} ^2 (\gamma -\lambda n) + b_1^2 (\alpha -\beta n) \left\{ \alpha - (n- 2) \beta \right\} \right] > 0, \end{aligned}$$
(29)

while (15) must hold here as well.

Dual-power law

$$\begin{aligned} F(\left| q \right| ^2)= & {} b_1 \left| q \right| ^{2m} + b_2 \left| q \right| ^{2m+2}. \end{aligned}$$
(30)

Here \(b_j\) for \(j= 1, 2\) are real-valued constants and m is the power-law parameter. With such a functional form, eqs (5) and (6) respectively transform to

$$\begin{aligned}{} & {} \frac{i}{\left( q^n \right) _t} + a \left( \frac{\left| q \right| ^n}{ q^n } \right) _{xx} + \frac{b_1 \left| q \right| ^{2m} + b_2 \left| q \right| ^{2m+2} }{ q^n} = \alpha \left| q_x \right| ^2 \left( q^n \right) ^{*}\nonumber \\{} & {} + \frac{\beta \left( q^n \right) ^{*}}{4\left| q \right| ^2} \left[ 2\left| q \right| ^2 \left( \left| q \right| ^2 \right) _{xx} - \left\{ \left( \left| q \right| ^2 \right) _{x}\right\} ^2 \right] + \frac{\gamma }{ q^n}, \end{aligned}$$
(31)

and

$$\begin{aligned}{} & {} \gamma - \lambda n - b_1 \phi ^{2m} - b_2 \phi ^{2m+2} + \phi ^{2n} (x) \left[ \alpha \left\{ \phi ^{\prime }(x) \right\} ^2 \right. \nonumber \\{} & {} \quad \left. + \beta \phi (x) \phi ^{\prime \prime }(x) \right] = 0. \end{aligned}$$
(32)

Another parameter constraint that reels out for integrability is:

$$\begin{aligned} \alpha + (m-n+1)\beta = 0. \end{aligned}$$
(33)

With this additional parameter constraint, the ODE for \(\phi (x)\) reduces to

$$\begin{aligned}{} & {} \gamma - \lambda n + \beta \phi ^{2n+1}(x) \phi ^{\prime \prime }(x) - \beta (m-n+1) \phi ^{2n}(x) \left\{ \phi ^{\prime }(x) \right\} ^2\nonumber \\{} & {} \quad - b_1 \phi ^{2m}(x) - b_2 \phi ^{2m+2}(x) = 0. \end{aligned}$$
(34)

Equation (34) has a Lie point symmetry namely \(\partial /\partial x\) which aids in its integration that yields the implicit solution in quadrature as:

$$\begin{aligned} x=\pm \int \phi ^n \sqrt{\frac{(m+1) \beta }{\gamma -n \lambda - (m+1) \left( b_1 - 2b_2 \phi ^2 \ln \phi \right) \phi ^{2m}}} \, d\phi . \end{aligned}$$
(35)

Log-law

In this form of SPM, the nonlinear functional takes the form

$$\begin{aligned} F(\left| q \right| ^2)= & {} b \ln \left| q \right| ^{2}, \end{aligned}$$
(36)

for real-valued constant b. This makes (5) and (6) transform to

$$\begin{aligned}{} & {} \frac{i}{\left( q^n \right) _t} + a \left( \frac{\left| q \right| ^n}{ q^n } \right) _{xx} + b \frac{ \ln \left| q \right| ^2}{ q^n} = \alpha \left| q_x \right| ^2 \left( q^n \right) ^{*}\nonumber \\{} & {} \quad + \frac{\beta \left( q^n \right) ^{*}}{4\left| q \right| ^2} \left[ 2\left| q \right| ^2 \left( \left| q \right| ^2 \right) _{xx} - \left\{ \left( \left| q \right| ^2 \right) _{x}\right\} ^2 \right] + \frac{\gamma }{ q^n}, \end{aligned}$$
(37)

and

$$\begin{aligned}{} & {} \gamma - \lambda n - 2b \ln \phi + \phi ^{2n} (x) \left[ \alpha \left\{ \phi ^{\prime }(x) \right\} ^2 \right. \nonumber \\{} & {} \quad \left. + \beta \phi (x) \phi ^{\prime \prime }(x) \right] = 0. \end{aligned}$$
(38)

Equation (38) admits a single Lie point symmetry, namely \(\partial /\partial x\). This symmetry integrates the ODE implicitly in terms of Dawson integral as

$$\begin{aligned} x=\pm \sqrt{\frac{2 (\alpha - n \beta ) }{b(n+1)}} \phi ^{n+1} D \left( \sqrt{-\frac{(n+1) \left\{ b \beta +(\alpha -n \beta ) (\gamma -n \lambda )-2 b (\alpha -n \beta ) \ln \phi \right\} }{2 b (\alpha -n \beta )}}\right) . \end{aligned}$$
(39)

The Dawson integral is defined as

$$\begin{aligned} D(x)=e^{-x^2} \int _0^x e^{y^2} \, dy. \end{aligned}$$
(40)

The solution (39) is valid for

$$\begin{aligned} b > 0. \end{aligned}$$
(41)

Anti-cubic law

The structure of the nonlinear functional that stems from the nonlinear refractive index change reads

$$\begin{aligned} F(\left| q \right| ^2)= & {} \frac{b_1}{\left| q \right| ^{4}} +b_2 \left| q \right| ^{2} +b_3 \left| q \right| ^{4}, \end{aligned}$$
(42)

for real-valued constants \(b_j\) for \(j = 1, 2, 3\). Thus, eqs (5) and (6) takes the form:

$$\begin{aligned}{} & {} \frac{i}{\left( q^n \right) _t} + a \left( \frac{ \left| q \right| ^n}{ q^n } \right) _{xx} + \left( \frac{b_1}{\left| q \right| ^4} + b_2 \left| q \right| ^2 + b_3 \left| q \right| ^4 \right) q\nonumber \\{} & {} \quad = \alpha \left| q_x \right| ^2 \left( q^n \right) ^{*} + \frac{\beta \left( q^n \right) ^{*}}{4\left| q \right| ^2} \left[ 2\left| q \right| ^2 \left( \left| q \right| ^2 \right) _{xx}\right. \nonumber \\{} & {} \quad \left. -\left\{ \left( \left| q \right| ^2 \right) _{x}\right\} ^2 \right] + \frac{\gamma }{ q^n}, \end{aligned}$$
(43)

and

$$\begin{aligned}{} & {} \gamma - \lambda n - \frac{b_1}{\phi ^2} - b_2 \phi - b_3 \phi ^2 \nonumber \\{} & {} \quad + \phi ^{2n} (x) \left[ \alpha \left\{ \phi ^{\prime }(x) \right\} ^2 + \beta \phi (x) \phi ^{\prime \prime }(x) \right] = 0. \end{aligned}$$
(44)

Implementing the additional parameter constraint

$$\begin{aligned} \alpha - (n-2) \beta = 0, \end{aligned}$$
(45)

one secures the integral to (44) as

$$\begin{aligned} x=\pm 2 \int \phi ^{n+2} \sqrt{\frac{\beta }{b_1 - 2 \phi ^4 \left( \gamma -n \lambda - 2b_2 \phi ^2 + 4b_3 \phi ^4 \log \phi \right) }} \, d\phi . \end{aligned}$$
(46)

Generalized anti-cubic law

For the generalized anti-cubic law, the functional F takes the form:

$$\begin{aligned} F(\left| q \right| ^2)= & {} \frac{b_1}{ \left| q \right| ^{2(m+1)}} +b_2 \left| q \right| ^{2m} + b_3 \left| q \right| ^{2(m+1)}, \end{aligned}$$
(47)

where \(b_j\) for \(j = 1, 2, 3\) are real-valued parameters. Then, equations (5) and (6) turn out to be

$$\begin{aligned}{} & {} \frac{i}{\left( q^n \right) _t} + a \left( \frac{\left| q \right| ^n}{ q^n } \right) _{xx} + \left( \frac{b_1}{\left| q \right| ^{2m+2}} + b_2 \left| q \right| ^{2m} + b_3 \left| q \right| ^{2m+2} \right) q \nonumber \\{} & {} \quad = \alpha \left| q_x \right| ^2 \left( q^n \right) ^{*} + \frac{\beta \left( q^n \right) ^{*}}{4\left| q \right| ^2} \left[ 2\left| q \right| ^2 \left( \left| q \right| ^2 \right) _{xx} - \left\{ \left( \left| q \right| ^2 \right) _{x}\right\} ^2\right] \nonumber \\{} & {} \quad + \frac{\gamma }{ q^n}, \end{aligned}$$
(48)

and

$$\begin{aligned}{} & {} \gamma - \lambda n - \frac{b_1}{\phi ^{2(m+1)}} - b_2 \phi ^{2m} - b_3 \phi ^{2(m+1)} \nonumber \\{} & {} \quad + \phi ^{2n} (x) \left[ \alpha \left\{ \phi ^{\prime }(x) \right\} ^2 + \beta \phi (x) \phi ^{\prime \prime }(x) \right] = 0, \end{aligned}$$
(49)

respectively. For its integrability, the additional constrant parameter choice must be

$$\begin{aligned} 2m+1 = 0. \end{aligned}$$
(50)

This changes eqs (48) and (49) respectively to:

$$\begin{aligned}{} & {} \frac{i}{\left( q^n \right) _t} + a \left( \frac{\left| q \right| ^n}{ q^n } \right) _{xx} + \left( \frac{b_1 + b_2}{ \left| q \right| {}} + b_3 \left| q \right| \right) \frac{1}{q^n} = \alpha \left| q_x \right| ^2 \left( q^n \right) ^{*}\nonumber \\{} & {} \quad + \frac{\beta \left( q^n \right) ^{*}}{4\left| q \right| ^2} \left[ 2\left| q \right| ^2 \left( \left| q \right| ^2 \right) _{xx} - \left\{ \left( \left| q \right| ^2 \right) _{x}\right\} ^2 \right] + \frac{\gamma }{ q^n}, \end{aligned}$$
(51)

and

$$\begin{aligned}{} & {} b_1 + b_2 + b_3 \phi ^2(x) - \left( \gamma - \lambda n \right) \phi (x) - \phi ^{2n+1}(x) \left[ \alpha \left\{ \phi ^{\prime }(x) \right\} ^2 \right. \nonumber \\{} & {} \quad \left. + \beta \phi (x) \phi ^{\prime \prime }(x) \right] =0, \end{aligned}$$
(52)

respectively. This equation admits a single Lie point symmetry, given by \(\partial /\partial x\) which leads to its implicit solution in terms of Appell hypergeometric function as

$$\begin{aligned} x= & {} \pm \frac{\phi ^{n+1}}{2n+3} \sqrt{\frac{2 \left\{ 2 \alpha - (2n+1)\beta \right\} \phi }{b_1+b_2}} F_1\nonumber \\{} & {} \quad \left( \frac{2n+3}{2};\frac{1}{2},\frac{1}{2};\frac{2n+5}{2};\frac{A_1}{A_2+\sqrt{A_3 A_4}},\frac{A_1}{A_2-\sqrt{A_3 A_4}}\right) , \end{aligned}$$
(53)

where

$$\begin{aligned} A_1= & {} 4 b_3 \phi (\alpha - n\beta ) \left\{ 2 \alpha - (2n+1)\beta \right\} ,\end{aligned}$$
(54)
$$\begin{aligned} A_2= & {} \left\{ 2 \alpha -(2n-1)\beta \right\} \left\{ 2\alpha - (2n+1)\beta \right\} (\gamma - n \lambda ),\end{aligned}$$
(55)
$$\begin{aligned} A_3= & {} \left\{ 2 \alpha -(2n-1)\beta \right\} \left\{ 2\alpha - (2n+1)\beta \right\} , \end{aligned}$$
(56)

and

$$\begin{aligned} A_4= & {} \left\{ 2 \alpha -(2n-1)\beta \right\} \left\{ 2\alpha - (2n+1)\beta \right\} (\gamma - n \lambda )^2 \nonumber \\{} & {} \quad - 16 b_3 \left( b_1 + b_2\right) (\alpha - n \beta )^2. \end{aligned}$$
(57)

From (53), one needs to have

$$\begin{aligned} A_3 A_4 > 0. \end{aligned}$$
(58)

The convergence criteria (27) in this case leads to

$$\begin{aligned} \max \left( \left| \frac{A_1}{A_2 + \sqrt{A_3 A_4}} \right| , \left| \frac{A_1}{A_2 - \sqrt{A_3 A_4}} \right| \right) < 1. \end{aligned}$$
(59)

Quadratic-cubic law

For quadratic-cubic law, the functional F for the nonlinear refractive index change is structured as

$$\begin{aligned} F(\left| q \right| ^2)= & {} b_1 \left| q \right| +b_2 \left| q \right| ^{2}, \end{aligned}$$
(60)

where \(b_j\) for \(j = 1, 2\) are arbitrary real-valued constants. Thus, equations (5) and (6) gets shaped up as:

$$\begin{aligned}{} & {} \frac{i}{\left( q^n \right) _t} + a \left( \frac{\left| q \right| ^n}{ q^n } \right) _{xx} + \frac{b_1 \left| q \right| + b_2 \left| q \right| ^2 }{ q^n} = \alpha \left| q_x \right| ^2 \left( q^n \right) ^{*}\nonumber \\{} & {} \quad + \frac{\beta \left( q^n \right) ^{*}}{4\left| q \right| ^2} \left[ 2\left| q \right| ^2 \left( \left| q \right| ^2 \right) _{xx} - \left\{ \left( \left| q \right| ^2 \right) _{x}\right\} ^2\right] + \frac{\gamma }{ q^n}, \end{aligned}$$
(61)

and

$$\begin{aligned}{} & {} b_1 \phi (x) + b_2 \phi ^2(x) - \gamma + n \lambda - \phi ^{2n}(x) \left[ \alpha \left\{ \phi ^{\prime }(x) \right\} ^2\right. \nonumber \\{} & {} \quad \left. + \beta \phi (x) \phi ^{\prime \prime }(x) \right] = 0. \end{aligned}$$
(62)

This equation with a single Lie point symmetry, namely \(\partial /\partial x\), that it supports leads to the implicit solution in terms of Appell hypergeometric function as:

$$\begin{aligned}&x =&\pm \frac{\phi ^{1+n}}{n+1} \sqrt{- \frac{\alpha -n \beta }{\gamma - n \lambda }} F_1\left( n+1;\frac{1}{2},\frac{1}{2};n+2;\right. \nonumber \\{} & {} \quad \left. -\frac{A_1}{A_2-\sqrt{A_3 \left( A_4+A_3 b_1^2\right) }},-\frac{A_1}{A_2+\sqrt{A_3 \left( A_4+A_3 b_1^2\right) }}\right) , \end{aligned}$$
(63)

where

$$\begin{aligned} A_1= & {} \left\{ 2 \alpha -(2n-1) \beta \right\} (\alpha -n \beta ) b_2 \phi ,\end{aligned}$$
(64)
$$\begin{aligned} A_2= & {} (\alpha -n \beta ) \left\{ \alpha - (n-1) \beta \right\} b_1,\end{aligned}$$
(65)
$$\begin{aligned} A_3= & {} (\alpha -n \beta ) \left\{ \alpha - (n-1) \beta \right\} , \end{aligned}$$
(66)

and

$$\begin{aligned} A_4 = \left\{ 2\alpha -(2n-1) \beta \right\} ^2 (\gamma -n \lambda ) b_2. \end{aligned}$$
(67)

The convergence criteria given by (27) gives here

$$\begin{aligned} \max \left( \left| \frac{A_1}{A_2-\sqrt{A_3 \left( A_4+A_3 b_1^2\right) }} \right| , \left| \frac{A_1}{A_2+\sqrt{A_3 \left( A_4+A_3 b_1^2\right) }} \right| \right) < 1, \end{aligned}$$
(68)

and the criteria (15) must also remain valid here for the solutions to exist.

Parabolic non-local law

For such a form of nonlinear refractive index change, the functional F takes the form:

$$\begin{aligned} F(\left| q \right| ^2)= & {} b_1 \left| q \right| ^2 +b_2 \left| q \right| ^{4} +b_3 \left( \left| q \right| ^{2} \right) _{xx}, \end{aligned}$$
(69)

for \(b_j\) where \(j = 1, 2, 3\) to be arbitrary real-valued constants. This structure of the functional leads to the relations (5) and (6) to be structured as:

$$\begin{aligned}{} & {} \frac{i}{\left( q^n \right) _t} + a \left( \frac{\left| q \right| ^n}{ q^n } \right) _{xx} + \frac{ b_1 \left| q \right| ^2 +b_2 \left| q \right| ^{4} +b_3 \left( \left| q \right| ^{2} \right) _{xx}}{ q^n}\nonumber \\{} & {} \quad = \alpha \left| q_x \right| ^2 \left( q^n \right) ^{*} + \frac{\beta \left( q^n \right) ^{*}}{4\left| q \right| ^2} \left[ 2\left| q \right| ^2 \left( \left| q \right| ^2 \right) _{xx} - \left\{ \left( \left| q \right| ^2 \right) _{x}\right\} ^2\right] \nonumber \\ {}{} & {} \quad + \frac{\gamma }{ q^n}, \end{aligned}$$
(70)

and

$$\begin{aligned}{} & {} \gamma - \lambda n - b_1 \phi ^2(x) - b_2 \phi ^4(x) - b_3 \left[ \phi (x) \phi ^{\prime \prime } (x) + \left\{ \phi ^{\prime }(x) \right\} ^2 \right] \nonumber \\{} & {} \quad + \phi ^{2n} (x) \left[ \alpha \left\{ \phi ^{\prime }(x) \right\} ^2 + \beta \phi (x) \phi ^{\prime \prime }(x) \right] = 0. \end{aligned}$$
(71)

For integrability choosing

$$\begin{aligned} \alpha - \beta = 0, \end{aligned}$$
(72)

the ODE for \(\phi (x)\) simplifies to

$$\begin{aligned}{} & {} \gamma -\lambda n - b_2 \phi ^4(x) - b_1 \phi ^2(x) - \left\{ 2b_3 - \beta \phi ^{2n}(x) \right\} \nonumber \\ {}{} & {} \quad \left[ \phi (x) \phi ^{\prime \prime }(x) + \left\{ \phi ^{\prime }(x) \right\} ^2 \right] = 0. \end{aligned}$$
(73)

Equation (73) has a Lie point symmetry given by \(\partial /\partial x\) which leads to its implicit solution in quadratures containing Gauss’ hypergeometric functions.

$$\begin{aligned} x=\pm 2 \int \sqrt{\frac{3 b_3}{6 (\gamma -n \lambda ) \, _2F_1\left( 1,\frac{1}{n};\frac{n+1}{n};\frac{\beta \phi ^{2 n}}{2 b_3}\right) -3b_1\phi ^2 \, _2F_1\left( 1,\frac{2}{n};\frac{n+2}{n};\frac{\beta \phi ^{2 n}}{2 b_3}\right) - 2b_2 \phi ^4 \, _2F_1\left( 1,\frac{3}{n};\frac{n+3}{n};\frac{\beta \phi ^{2 n}}{2 b_3}\right) }} \, d\phi . \end{aligned}$$
(74)

The existence of the implicit solution is guaranteed by (13) that gives that give its bounds to be

$$\begin{aligned} - \left| \frac{2b_3}{\beta } \right| ^{\frac{1}{2n}}< \phi (x) < \left| \frac{2b_3}{\beta } \right| ^{\frac{1}{2n}}. \end{aligned}$$
(75)

Saturating law

The functional F for the saturating law of SPM is

$$\begin{aligned} F(\left| q \right| ^2)= & {} \frac{ b_1\left| q \right| ^2}{b_2 + b_3 \left| q \right| ^2}, \end{aligned}$$
(76)

for real-valued constants \(b_j\) where \(j = 1, 2, 3\). This makes the complex CGL equation and the ODE for \(\phi (x)\) transform from (5) and (6) respectively to

$$\begin{aligned}{} & {} \frac{i}{\left( q^n \right) _t} + a \left( \frac{\left| q \right| ^n}{ q^n } \right) _{xx} + \left( \frac{ b_1\left| q \right| ^2}{b_2 + b_3 \left| q \right| ^2} \right) q \nonumber \\{} & {} \quad = \alpha \left| q_x \right| ^2 \left( q^n \right) ^{*} + \frac{\beta \left( q^n \right) ^{*}}{4\left| q \right| ^2} \left[ 2\left| q \right| ^2 \left( \left| q \right| ^2 \right) _{xx} - \left\{ \left( \left| q \right| ^2 \right) _{x}\right\} ^2\right] \nonumber \\ {}{} & {} \quad + \frac{\gamma }{ q^n}, \end{aligned}$$
(77)

and

$$\begin{aligned} \gamma - \lambda n - \frac{b_1 \phi ^2}{b_2 + b_3 \phi ^2} + \phi ^{2n} (x) \left[ \alpha \left\{ \phi ^{\prime }(x) \right\} ^2 + \beta \phi (x) \phi ^{\prime \prime }(x) \right] = 0. \end{aligned}$$
(78)

Equation (78) permits a single Lie point symmetry \(\partial /\partial x\) which leads to its implicit solution in quadratures that contains Gauss’ hypergeometric functions:

$$\begin{aligned} x=\pm \int \phi ^n \sqrt{\frac{(\alpha - n\beta ) b_3}{\left\{ 1 - \, _2F_1\left( 1,-n+\frac{\alpha }{\beta };1-n+\frac{\alpha }{\beta };-\frac{\phi ^2 b_3}{b_2}\right) \right\} b_1 - (\gamma -n \lambda ) b_3}} \, d\phi . \end{aligned}$$
(79)

From the condition (13) applied to (79), the solutions are bounded as in

$$\begin{aligned} - \left| \frac{b_2}{b_3} \right| ^{\frac{1}{2}}< \phi (x) < \left| \frac{b_2}{b_3} \right| ^{\frac{1}{2}}. \end{aligned}$$
(80)

Exponential law

The nonlinear functional F for the exponential law of SPM is

$$\begin{aligned} F(\left| q \right| ^2)= & {} \frac{1}{b} \left( 1 - e^{-b \left| q \right| ^2} \right) , \end{aligned}$$
(81)

for real-valued parameter parameter b where (41) holds. Thus, the CGLE takes the form:

$$\begin{aligned}{} & {} \frac{i}{\left( q^n \right) _t} + a \left( \frac{\left| q \right| ^n}{ q^n } \right) _{xx} + \frac{ \left( 1 - e^{-b \left| q \right| ^2} \right) }{bq^n}\nonumber \\{} & {} \quad = \alpha \left| q_x \right| ^2 \left( q^n \right) ^{*} + \frac{\beta \left( q^n \right) ^{*}}{4\left| q \right| ^2} \left[ 2\left| q \right| ^2 \left( \left| q \right| ^2 \right) _{xx} - \left\{ \left( \left| q \right| ^2 \right) _{x}\right\} ^2\right] \nonumber \\ {}{} & {} \quad + \frac{\gamma }{ q^n}, \end{aligned}$$
(82)

while the ODE in \(\phi (x)\) is given as

$$\begin{aligned} b (\gamma - \lambda n) + e^{- b\phi ^2(x)} -1 + b \phi ^{2n} (x) \left[ \alpha \left\{ \phi ^{\prime }(x) \right\} ^2 + \beta \phi (x) \phi ^{\prime \prime }(x) \right] = 0. \end{aligned}$$
(83)

Equation (83), the ODE for \(\phi (x)\), admits a single Lie point symmetry, namely \(\partial /\partial x\). This leads to its implicit integral in terms of quadratures involving exponential integral function as

$$\begin{aligned} x=\pm \int \phi ^n \sqrt{\frac{b \beta (\alpha -n \beta )}{\beta (1-b \gamma +b n \lambda )+(\alpha -n \beta ) E_{1+n-\frac{\alpha }{\beta }}\left( b \phi ^2\right) }} \, d\phi , \end{aligned}$$
(84)

where the exponential integral function is defined as \(E_m(z)\) is defined as

$$\begin{aligned} E_m(z)=\int _1^{\infty } \frac{e^{t (-z)}}{t^m} \, dt. \end{aligned}$$
(85)

Conclusions

The current paper is on the retrieval of quiescent optical solitons for the CGLE with generalized temporal evolution having nonlinear CD and a variety of structures for the SPM. The results are recovered by the aid of Lie symmetry. The solutions are in terms of a few special functions, or quadratures, while some of the results involve quadratures with special functions in it. Two additional forms of SPM are considered in the paper. They are saturable law and exponential law. This complete package of results are important and serves as a stepping stone for further future development. Later, the model will be taken up with the inclusion of perturbation terms that are from all across the board such as Hamiltonian as well as non-Hamiltonian type as well as non-local type. Moreoevrer, additional structures of SPM will be later considered and several additional models from optoelectronics will be addressed. The results of such research activities will be disseminated after they are all lined up with the latest reported results [15,16,17,18,19,20,21,22,23,24,25].