Quiescent optical solitons for Fokas–Lenells equation with nonlinear chromatic dispersion and a couple of self-phase modulation structures

Abstract

The focus of the current paper is on the retrieval of quiescent optical solitons from Fokas–Lenells equation with nonlinear chromatic dispersion and having quadratic–cubic as well as quadratic–cubic–quartic forms of self-phase modulation structures. Two integration algorithms are implemented to carry out to seek such soliton solutions. They are the enhanced Kudryashov’s approach and the projective Riccati equation approach. In this context, both linear temporal evolution and generalized temporal evolution effects are addressed. A full spectrum of quiescent optical solitons is thus recovered.

1 Introduction

The evolution of quiescent optical solitons has gained momentum and extreme importance ever since its first appearance about two decades ago [1,2,3,4,5]. The study of such solitons with nonlinear chromatic dispersion (CD) revealed quite a few results that have been reported all across the board [6,7,8,9,10]. Several models with nonlinear CD and various forms of self–phase modulation (SPM) structures have been studied [11,12,13,14,15]. They are the nonlinear Schrödinger’s equation, complex Ginzburg–Landau equation, Sasa–Satsuma equation, Radhakrishnan–Kundu–Lakshmanan equation, NLSE with Kudryashov’s form of SPM, concatenation model with and without SPM, dispersive concatenation model and many more [16,17,18,19,20]. It is now time to move further on. This paper addresses the retrieval of quiescent optical solitons for Fokas–Lenells equation with nonlinear CD. The SPM structures are of two types. They are the quadratic–cubic type as well as quadratic–cubic–quartic type. The paper studies the model with linear temporal evolution as well as the generalized temporal evolution.

Two adopted integration schemes have made this retrieval possible. They are the enhanced Kudryashov’s scheme [21,22,23,24,25] and the projective Riccati equation method [26,27,28,29,30]. The recovered solitons form a complete spectrum of single solitons. These are true for both linear temporal evolutions as well as generalized temporal evolution. The recovered solitons are exhibited in the rest of the paper along with the restrictions on the parameters that naturally emerged during the course of derivation of the soliton solutions and these are labeled as parameter constraints. The paper starts off with a recapitulation of the adopted integration algorithms and subsequently they are implemented to the models and the quiescent solitons are yielded.

The main novelty of this paper lies in its exploration of quiescent optical solitons within the framework of the Fokas–Lenells equation, considering nonlinear CD alongside SPM structures with quadratic-cubic and quadratic-cubic-quartic forms. This marks the first instance where such soliton solutions have been sought within this specific equation and with these particular nonlinearities. Additionally, the paper addresses two integration algorithms, namely the enhanced Kudryashov’s approach and the projective Riccati equation approach, to effectively identify these soliton solutions. By employing these algorithms, the study addresses both linear temporal evolution and generalized temporal evolution effects, thereby providing a comprehensive understanding of the dynamics involved.

The significance of the results lies in the broadening of our understanding of optical solitons in a complex nonlinear setting. By uncovering a full spectrum of quiescent optical solitons, this paper contributes to advancing our knowledge of soliton behavior in optical systems with nonlinear CD and various forms of SPM. These findings could have implications for the design and optimization of optical communication systems and nonlinear optical devices.

2 An overview of the integration algorithms

We adopt a governing model that follows the structure of

$$\begin{aligned} F(q,q_{x},q_{t},q_{xt},q_{xx},...)=0. \end{aligned}$$

(1)

Within the context of this equation, $q=q(x,t)$, we are portraying a wave’s profile, with t representing time and x representing a spatial coordinate [31,32,33,34,35].

Using the wave transformation as follows:

$$\begin{aligned} q(x,t)=U(\xi ),~~~\xi =k (x-\upsilon t), \end{aligned}$$

(2)

results in the transformation of Eq. (1) to

$$\begin{aligned} P(U,-k \upsilon U^{\prime },k U^{\prime },k^{2}U^{\prime \prime },...)=0. \end{aligned}$$

(3)

Within this equation, k is assigned the role of wave width, $\xi$ takes on the role of wave variable, and $\upsilon$ assumes the role of wave velocity.

2.1 Enhanced Kudryashov’s procedure

This subsection offers a detailed overview of the fundamental principles underlying the enhanced Kudryashov’s technique (EK) [36,37,38,39].

Step–1:  The explicit solution to the simplified model represented by Eq. (3) is presented as follows: [40,41,42,43,44,45,46,47,48,49]

$$\begin{aligned} U(\xi )=\eta _{0}+\sum _{i=1}^{N}\left\{ \eta _{i}R(\xi )^{i}+\rho _{i} \left( \frac{R^{\prime }(\xi )}{R(\xi )}\right)^{i}\right\} , \end{aligned}$$

(4)

along with the auxiliary equation provided,

$$\begin{aligned} {R^{\prime }(\xi )}^{2}={R(\xi )}^{2}(1-\chi {R(\xi )}^{2}). \end{aligned}$$

(5)

The quantities $\eta _{0}$, $\chi$, $\eta _{i}$, and $\rho _{i}$ (where $i=1,..., N$) will be given, where N is determined by the balancing procedure described in Eq. (3).

Step–2:  Eq. (5) describes the characteristics of soliton waves:

$$\begin{aligned} R(\xi )=\frac{4d~e^{\xi }}{4d^{2}e^{2\xi }+\chi }. \end{aligned}$$

(6)

Given that d remains constant, while Eq. (6) yields the bright and singular solitons corresponding to $\chi = \pm 4{d^2}$, respectively.

Step–3:  Through the substitution of equation (4) into equation (3), along with equation (5), we can determine the required constants for equations (2) and (4). The specified parametric restrictions can then be incorporated into Eq. (4) in conjunction with Eq. (6) to facilitate integration. This process leads to the retrieval of straddled solitons, which can subsequently be classified as appropriate.

2.2 Projective Riccati equation approach

This section will revisit the projective Riccati equation method and enumerate the algorithmic details.

Step 1:  We assume the existence of a formal solution to Equation (3) of the following form:

$$\begin{aligned} U(\xi )=\eta _{0}+\sum _{i=1}^{N} F^{i-1}(\xi )\bigg (\eta _{i} F(\xi )+\rho _{i} G(\xi )\bigg ). \end{aligned}$$

(7)

The condition states that both $\eta _{N}$ and $\rho _{N}$ cannot be simultaneously zero. Moreover, the functions $F(\xi )$ and $G(\xi )$ are governed by the ordinary differential equations (ODEs):

$$\begin{aligned}{} & {} F^{\prime} (\xi )=- F(\xi ) G(\xi ), \nonumber \\{} & {} G^{\prime} (\xi )=1 - G^2(\xi )-\tau F(\xi ), \end{aligned}$$

(8)

with

$$\begin{aligned} G(\xi )^2= 1-2\tau F(\xi )+R (\tau )F(\xi )^2. \end{aligned}$$

(9)

The determination of the nonzero constant $\tau$ and the positive integer N relies on the balancing principle outlined in Equation (3), wherein the real constants $\eta _0$, $\eta _i$, and $\rho _i$ (where $i=0,1,...,N$) are incorporated.

Step–2:  Below are the solutions for Eq. (8):

Case-1:  $R (\tau )=0$.

$$\begin{aligned} F(\xi )= \frac{1}{2\tau }\text {sech}^2\left[ \frac{\xi }{2}\right] ~~~ \text {and} ~~~ G(\xi )= \tanh \left[ \frac{\xi }{2}\right] , \end{aligned}$$

(10)

or

$$\begin{aligned} F(\xi )=-\frac{1}{2\tau }\text {csch}^2\left[ \frac{\xi }{2}\right] ~~~ \text {and} ~~~ G(\xi )= \coth \left[ \frac{\xi }{2}\right] . \end{aligned}$$

(11)

Case-2:  $R (\tau )=\frac{24}{25}\tau ^2$.

$$\begin{aligned} F(\xi )=\frac{1}{\tau }\frac{5~\text {sech}[\xi ]}{5~\text {sech}[\xi ]\pm 1} ~~~ \text {and} ~~~ G(\xi )=\frac{\pm \tanh [\xi ]}{5~\text {sech}[\xi ]\pm 1}. \end{aligned}$$

(12)

Case-3:  $R (\tau )=\frac{5}{9}\tau ^2$.

$$\begin{aligned} F(\xi )=\frac{1}{\tau }\frac{3~\text {sech}[\xi ]}{3~\text {sech}[\xi ]\pm 2} ~~~ \text {and} ~~~ G(\xi )=\frac{2}{2~ \coth [\xi ]\pm 3~ \text {csch}[\xi ]}. \end{aligned}$$

(13)

Case-4:  $R (\tau )=\tau ^2-1$.

$$\begin{aligned} F(\xi )=\frac{4~\text {sech}[\xi ]}{3~\tanh [\xi ]+4\tau ~\text {sech}[\xi ]+5} ~~~ \text {and} ~~~ G(\xi )=\frac{5~\tanh [\xi ]+3}{3~\tanh [\xi ]+4\tau ~\text {sech}[\xi ]+5}, \end{aligned}$$

(14)

or

$$\begin{aligned} F(\xi )=\frac{\text {sech}[\xi ]}{\tau ~\text {sech}[\xi ]+1} ~~~ \text {and} ~~~ G(\xi )=\frac{\tanh [\xi ]}{\tau ~\text {sech}[\xi ]+1}. \end{aligned}$$

(15)

Case-5:  $R (\tau )=\tau ^2+1$.

$$\begin{aligned} F(\xi )=\frac{\text {csch}[\xi ]}{\tau ~\text {csch}[\xi ]+1} ~~~ \text {and} ~~~ G(\xi )=\frac{\coth [\xi ]}{\tau ~\text {csch}[\xi ]+1}. \end{aligned}$$

(16)

Step–3:  Through the amalgamation of Equation (7) with Equations (8) and (9), followed by their substitution into Equation (3), we arrive at a polynomial equation encompassing the variables $F(\xi )$ and $G(\xi )$. The coefficients derived from this polynomial equation furnish the parameter values for Eqs. (2) and (7), as referenced in the literature.

3 Linear temporal evolution

3.1 Quadratic–cubic SPM

The FLE with both nonlinear CD and quadratic–cubic SPM with linear temporal evolution is as follows:

$$\begin{aligned} iq_{t} + a(\left| q \right| ^{n})_{xx} +i \sigma \left| q \right| ^{2} q_{x} + \left( b_1 \left| q \right| +b_2 \left| q \right| ^{2}\right) q = i \left[ \alpha q_{x}+\lambda (\left| q \right| ^{2} q)_{x}+\mu (\left| q \right| ^2)_{x} q \right] , \end{aligned}$$

(17)

where $q=q(x,t)$ is a complex-valued function that represents the optical wave. x indicates the propagation distance along the optical medium in this context, whereas t represents the time variable. The quantity $b_2$ denotes the nonlinear coefficient, which characterizes the nonlinear self-interaction of the optical field caused by the medium’s intensity dependent refractive index. This nonlinearity is caused by the Kerr effect, which occurs when the refractive index varies in response to the intensity of the light. The nonlinear CD parameter is represented by parameter a, while the power-law parameter n imparts nonlinearity to the CD. The nonlinear dispersion coefficient is represented by the parameter $\sigma$, while the quadratic form of SPM is represented by $b_1$. The term $iq_t$ describes the optical wave’s temporal evolution as it travels through the nonlinear medium. The coefficient $\alpha$ relates to intermodal dispersion, whereas the coefficient $\lambda$ relates to the self-steepening perturbation term. At last, $\mu$ incorporates to the shift in self-frequency. The assumed profile describes the quiescent optical soliton:

$$\begin{aligned} q(x,t)=U(\xi ) e^{i (\omega t+\theta )},~~\xi =K x. \end{aligned}$$

(18)

Representing the amplitude, wave vector, frequency, and phase shift of the optical soliton are $U(\xi )$, K, $\omega$, and $\theta$, respectively. Applying the transformation (18), divides Eq. (17) into the following real and imaginary parts:

$$\begin{aligned} a K^2 (n+1) U^{n+1} U''+a K^2 n (n+1) U^n U^{\prime 2}+b_2 U^4+b_1 U^3-\omega U^2=0, \end{aligned}$$

(19)

and

$$\begin{aligned} -K U U' \left( \alpha +(3 \lambda +2 \mu -\sigma ) U^2\right) =0, \end{aligned}$$

(20)

respectively. From the imaginary part, we can deduce:

$$\begin{aligned} \alpha =0,~~3 \lambda +2 \mu -\sigma =0. \end{aligned}$$

(21)

To satisfy the integrability condition, we consider $n=1$. In this case, Eq. (17) can be rewritten as:

$$\begin{aligned} iq_t + a(\left| q \right| q)_{xx} + i \sigma \left| q \right| ^2 q_{x}+ \left( b_1 \left| q \right| +b_2 \left| q \right| ^2\right) q = i \left[ \lambda (\left| q \right| ^{2}q)_{x}+\mu (\left| q \right| ^2)_{x} q \right] , \end{aligned}$$

(22)

and Eq. (19) can be rewritten as:

$$\begin{aligned} 2 a K^2 U U''+2 a K^2 U^{\prime 2}+b_1 U^2+b_2 U^3-\omega U=0. \end{aligned}$$

(23)

Applying the balance principle between $U U''$ and $U^3$, leads to $N=2$. In the following subsections, the integration algorithms will be utilized to Eq. (23).

3.1.1 Enhanced Kudryashov’s method

This method enables us to express the solution of Eq. (23) as follows:

$$\begin{aligned} U(\xi )=\eta _0 + \eta _1 R(\xi )+ \eta _2 R(\xi )^{2}+ \rho _1 \left( \frac{ R^\prime (\xi )}{ R(\xi )}\right) + \rho _2 \left( \frac{ R^\prime (\xi )}{ R(\xi )}\right) ^2. \end{aligned}$$

(24)

By incorporating Eq. (24) alongside Eq. (5) into Eq. (23), we arrive at the ensuing system of algebraic equations:

$$\begin{aligned}{} & {} 40 a \eta _2 K^2 \rho _2 \chi ^2-20 a \eta _2^{2} K^{2} \chi -20 a K^2 \rho _2^2 \chi ^3+3 b_2 \eta _2 \rho _2^2 \chi ^2-3 b_2 \eta _2^2 \rho _2 \chi +b_2 \eta _2^3-b_2 \rho _2^3 \chi ^3=0, \end{aligned}$$

(25)

$$\begin{aligned}{} & {} 24 a \eta _1 K^2 \rho _2 \chi ^2-24 a \eta _1 \eta _2 K^2 \chi +3 b_2 \eta _1 \rho _2^2 \chi ^2-6 b_2 \eta _1 \eta _2 \rho _2 \chi +3 b_2 \eta _1 \eta _2^2=0, \end{aligned}$$

(26)

$$\begin{aligned}{} & {} 12 a \eta _0 K^2 \rho _2 \chi ^2-44 a \eta _2 K^2 \rho _2 \chi -6 a \eta _1^2 K^2 \chi -12 a \eta _0 \eta _2 K^2 \chi \nonumber \\{} & {} \quad +16 a \eta _2^2 K^2+28 a K^2 \rho _2^2 \chi ^2+6 a K^2 \rho _1^2 \chi ^2\nonumber \\{} & {} \quad +3 b_2 \eta _0 \rho _2^2 \chi ^2-6 b_2 \eta _2 \rho _2^2 \chi -3 b_2 \eta _1^2 \rho _2 \chi -6 b_2 \eta _0 \eta _2 \rho _2 \chi -3 b_2 \eta _2 \rho _1^2 \chi \nonumber \\{} & {} \quad +3 b_2 \eta _2^2 \rho _2+3 b_2 \eta _0 \eta _2^2+3 b_2 \eta _1^2 \eta _2+3 b_2 \rho _2^3 \chi ^2\nonumber \\{} & {} \quad +3 b_2 \rho _1^2 \rho _2 \chi ^2-2 b_1 \eta _2 \rho _2 \chi +b_1 \eta _2^2+b_1 \rho _2^2 \chi ^2=0, \end{aligned}$$

(27)

$$\begin{aligned}{} & {} \quad -24 a \eta _2 K^2 \rho _1 \chi +24 a K^2 \rho _1 \rho _2 \chi ^2-6 b_2 \eta _2 \rho _1 \rho _2 \chi +3 b_2 \eta _2^2 \rho _1+3 b_2 \rho _1 \rho _2^2 \chi ^2=0, \end{aligned}$$

(28)

$$\begin{aligned}{} & {} \quad -22 a \eta _1 K^2 \rho _2 \chi -4 a \eta _0 \eta _1 K^2 \chi +18 a \eta _2 \eta _1 K^2-3 b_2 \eta _1 \rho _1^2 \chi \nonumber \\{} & {} \quad -6 b_2 \eta _1 \rho _2^2 \chi -6 b_2 \eta _0 \eta _1 \rho _2 \chi +6 b_2 \eta _2 \eta _1 \rho _2+b_2 \eta _1^3 \nonumber \\{} & {} \quad +6 b_2 \eta _0 \eta _2 \eta _1-2 b_1 \eta _1 \rho _2 \chi +2 b_1 \eta _2 \eta _1=0, \end{aligned}$$

(29)

$$\begin{aligned}{} & {} \quad -12 a \eta _1 K^2 \rho _1 \chi -6 b_2 \eta _1 \rho _1 \rho _2 \chi +6 b_2 \eta _1 \eta _2 \rho _1=0, \end{aligned}$$

(30)

$$\begin{aligned}{} & {} \quad -8 a \eta _0 K^2 \rho _2 \chi +8 a \eta _2 K^2 \rho _2+4 a \eta _1^2 K^2+8 a \eta _0 \eta _2 K^2-8 a K^2 \rho _2^2 \chi \nonumber \\{} & {} \quad -4 a K^2 \rho _1^2 \chi -6 b_2 \eta _0 \rho _2^2 \chi -3 b_2 \eta _0^2 \rho _2 \chi \nonumber \\{} & {} \quad -3 b_2 \eta _0 \rho _1^2 \chi +3 b_2 \eta _2 \rho _2^2+3 b_2 \eta _1^2 \rho _2+6 b_2 \eta _0 \eta _2 \rho _2+3 b_2 \eta _2 \rho _1^2+3 b_2 \eta _0 \eta _1^2\nonumber \\{} & {} \quad +3 b_2 \eta _0^2 \eta _2-3 b_2 \rho _2^3 \chi -6 b_2 \rho _1^2 \rho _2 \chi -2 b_1 \eta _0 \rho _2 \chi \nonumber \\{} & {} \quad +2 b_1 \eta _2 \rho _2+b_1 \eta _1^2+2 b_1 \eta _0 \eta _2-2 b_1 \rho _2^2 \chi -b_1 \rho _1^2 \chi -\eta _2 \omega +\rho _2 \chi \omega =0, \end{aligned}$$

(31)

$$\begin{aligned}{} & {} \quad -4 a \eta _0 K^2 \rho _1 \chi +8 a \eta _2 K^2 \rho _1-12 a K^2 \rho _2 \rho _1 \chi -6 b_2 \eta _0 \rho _2 \rho _1 \chi \nonumber \\{} & {} \quad +3 b_2 \eta _1^2 \rho _1+6 b_2 \eta _0 \eta _2 \rho _1+6 b_2 \eta _2 \rho _2 \rho _1-b_2 \rho _1^3 \chi \nonumber \\{} & {} \quad -6 b_2 \rho _2^2 \rho _1 \chi +2 b_1 \eta _2 \rho _1-2 b_1 \rho _2 \rho _1 \chi =0, \end{aligned}$$

(32)

$$\begin{aligned}{} & {} 2 a \eta _1 K^2 \rho _2+2 a \eta _0 \eta _1 K^2+3 b_2 \eta _1 \rho _1^2+3 b_2 \eta _1 \rho _2^2+6 b_2 \eta _0 \eta _1 \rho _2\nonumber \\{} & {} \quad +3 b_2 \eta _0^2 \eta _1+2 b_1 \eta _1 \rho _2+2 b_1 \eta _0 \eta _1-\eta _1 \omega =0, \end{aligned}$$

(33)

$$\begin{aligned}{} & {} 2 a \eta _1 K^2 \rho _1+6 b_2 \eta _0 \eta _1 \rho _1+6 b_2 \eta _1 \rho _1 \rho _2+2 b_1 \eta _1 \rho _1=0, \end{aligned}$$

(34)

$$\begin{aligned}{} & {} 3 b_2 \eta _0^2 \rho _2+3 b_2 \eta _0 \rho _1^2+3 b_2 \eta _0 \rho _2^2+b_2 \eta _0^3+b_2 \rho _2^3+3 b_2 \rho _1^2 \rho _2\nonumber \\{} & {} \quad +2 b_1 \eta _0 \rho _2+b_1 \eta _0^2+b_1 \rho _1^2+b_1 \rho _2^2-\eta _0 \omega -\rho _2 \omega =0, \end{aligned}$$

(35)

$$\begin{aligned}{} & {} 3 b_2 \eta _0^2 \rho _1+6 b_2 \eta _0 \rho _2 \rho _1+b_2 \rho _1^3+3 b_2 \rho _2^2 \rho _1+2 b_1 \eta _0 \rho _1+2 b_1 \rho _2 \rho _1-\rho _1 \omega =0. \end{aligned}$$

(36)

These equations together yield the subsequent outcomes:

Result-1:

$$\begin{aligned} K=\pm \sqrt{\frac{b_1}{32 a}},~\eta _0=\frac{-8 b_2 \rho _2-5 b_1}{8 b_2},~\eta _1=0,~\eta _2=\frac{\chi \left( 8 b_2 \rho _2+5 b_1\right) }{8 b_2},~\rho _1=0,~\omega =-\frac{15 b_1^2}{64 b_2}. \end{aligned}$$

(37)

This leads us to the soliton solution of the governing equation (17):

$$\begin{aligned} q(x,t)=\left\{ -\frac{5 b_1 \left( \chi -4 d^2 e^{ \pm \frac{1}{2} \sqrt{\frac{b_1}{2 a}} x}\right) ^2}{8 b_2 \left( 4 d^2 e^{ \pm \frac{1}{2} \sqrt{\frac{b_1}{2 a}} x}+\chi \right) ^2}\right\} e^{i \left( -\frac{15 b_1^2 }{64 b_2}t+\theta \right) }. \end{aligned}$$

(38)

Choosing $\chi = \pm 4 d^2$ allows us to retrieve dark and singular solitons for $\frac{b_1}{a}>0$:

$$\begin{aligned} q(x,t)=-\frac{5 b_1}{8 b_2}\tanh ^2\left( \frac{1}{4} \sqrt{\frac{b_1}{2 a}} x \right) e^{i \left( -\frac{15 b_1^2 }{64 b_2}t+\theta \right) }, \end{aligned}$$

(39)

and

$$\begin{aligned} q(x,t)=-\frac{5 b_1}{8 b_2}\coth ^2\left( \frac{1}{4} \sqrt{\frac{b_1}{2 a}} x \right) e^{i \left( -\frac{15 b_1^2 }{64 b_2}t+\theta \right) }. \end{aligned}$$

(40)

Fig. 1

2D plot of a quiescent dark optical soliton solution

3.1.2 Projective Riccati equation method

In line with this approach, the solution is structured as follows:

$$\begin{aligned} U(\xi )=\eta _0+\eta _1 F(\xi )+\eta _2 F(\xi )^2+\rho _1 G(\xi )+\rho _2 F(\xi ) G(\xi ). \end{aligned}$$

(41)

Substituting Eq. (41) alongside Eqs. (8) and (9) into Eq. (23), we arrive at:

$$\begin{aligned}{} & {} 40 a \eta _2 K^2 \rho _2 R(\tau )+3 b_2 \eta _2^2 \rho _2+b_2 \rho _2^3 R(\tau )=0, \end{aligned}$$

(42)

$$\begin{aligned}{} & {} -\quad 56 a \eta _2 K^2 \rho _2 \tau +24 a \eta _1 K^2 \rho _2 R(\tau )+24 a \eta _2 K^2 \rho _1 R(\tau )\nonumber \\{} & {} \quad +6 b_2 \eta _1 \eta _2 \rho _2+3 b_2 \eta _2^2 \rho _1-2 b_2 \rho _2^3 \tau +3 b_2 \rho _1 \rho _2^2 R(\tau )=0, \end{aligned}$$

(43)

$$\begin{aligned}{} & {} \quad -30 a \eta _1 K^2 \rho _2 \tau -30 a \eta _2 K^2 \rho _1 \tau +18 a \eta _2 K^2 \rho _2+12 a \eta _0 K^2 \rho _2 R(\tau )\nonumber \\{} & {} \quad +12 a \eta _1 K^2 \rho _1 R(\tau )+3 b_2 \eta _1^2 \rho _2+6 b_2 \eta _0 \eta _2 \rho _2\nonumber \\{} & {} \quad +6 b_2 \eta _1 \eta _2 \rho _1-6 b_2 \rho _1 \rho _2^2 \tau +b_2 \rho _2^3+3 b_2 \rho _1^2 \rho _2 R(\tau )+2 b_1 \eta _2 \rho _2=0, \end{aligned}$$

(44)

$$\begin{aligned}{} & {} \quad -12 a \eta _1 K^2 \rho _1 \tau -12 a \eta _0 K^2 \rho _2 \tau +8 a \eta _2 K^2 \rho _1+8 a \eta _1 K^2 \rho _2+4 a \eta _0 K^2 \rho _1 R(\tau )\nonumber \\{} & {} \quad +3 b_2 \eta _1^2 \rho _1+6 b_2 \eta _0 \eta _2 \rho _1+6 b_2 \eta _0 \eta _1 \rho _2\nonumber \\{} & {} \quad -6 b_2 \rho _2 \rho _1^2 \tau +3 b_2 \rho _2^2 \rho _1+b_2 \rho _1^3 R(\tau )+2 b_1 \eta _2 \rho _1+2 b_1 \eta _1 \rho _2=0, \end{aligned}$$

(45)

$$\begin{aligned}{} & {} \quad -2 a \eta _0 K^2 \rho _1 \tau +2 a \eta _1 K^2 \rho _1+2 a \eta _0 K^2 \rho _2+6 b_2 \eta _0 \eta _1 \rho _1+3 b_2 \eta _0^2 \rho _2\nonumber \\{} & {} \quad -2 b_2 \rho _1^3 \tau +3 b_2 \rho _2 \rho _1^2+2 b_1 \eta _1 \rho _1\nonumber \\{} & {} \quad +2 b_1 \eta _0 \rho _2-\rho _2 \omega =0, \end{aligned}$$

(46)

$$\begin{aligned}{} & {} 3 b_2 \eta _0^2 \rho _1+b_2 \rho _1^3+2 b_1 \eta _0 \rho _1-\rho _1 \omega =0,20 a \eta _2^2 K^2 R(\tau )\nonumber \\{} & {} \quad +20 a K^2 \rho _2^2 R(\tau )^2+b_2 \eta _2^3+3 b_2 \eta _2 \rho _2^2 R(\tau )=0, \end{aligned}$$

(47)

$$\begin{aligned}{} & {} \quad -36 a \eta _2^2 K^2 \tau +24 a \eta _1 \eta _2 K^2 R(\tau )-60 a K^2 \rho _2^2 \tau R(\tau )+24 a K^2 \rho _1 \rho _2 R(\tau )^2-6 b_2 \eta _2 \rho _2^2 \tau \nonumber \\{} & {} \quad +3 b_2 \eta _1 \eta _2^2+3 b_2 \eta _1 \rho _2^2 R(\tau )+6 b_2 \eta _2 \rho _1 \rho _2 R(\tau )=0, \end{aligned}$$

(48)

$$\begin{aligned}{} & {} \quad -42 a \eta _1 \eta _2 K^2 \tau +16 a \eta _2^2 K^2+42 a K^2 \rho _2^2 \tau ^2+6 a \eta _1^2 K^2 R(\tau )+12 a \eta _0 \eta _2 K^2 R(\tau )+6 a K^2 \rho _1^2 R(\tau )^2\nonumber \\{} & {} \quad +22 a K^2 \rho _2^2 R(\tau )-66 a K^2 \rho _1 \rho _2 \tau R(\tau )-6 b_2 \eta _1 \rho _2^2 \tau -12 b_2 \eta _2 \rho _1 \rho _2 \tau \nonumber \\{} & {} \quad +3 b_2 \eta _2 \rho _2^2+3 b_2 \eta _0 \eta _2^2+3 b_2 \eta _1^2 \eta _2\nonumber \\{} & {} \quad +3 b_2 \eta _2 \rho _1^2 R(\tau )+3 b_2 \eta _0 \rho _2^2 R(\tau )+6 b_2 \eta _1 \rho _1 \rho _2 R(\tau )+b_1 \eta _2^2+b_1 \rho _2^2 R(\tau )=0, \end{aligned}$$

(49)

$$\begin{aligned}{} & {} \quad -10 a \eta _1^2 K^2 \tau -20 a \eta _0 \eta _2 K^2 \tau +18 a \eta _2 \eta _1 K^2+40 a K^2 \rho _1 \rho _2 \tau ^2-28 a K^2 \rho _2^2 \tau +4 a \eta _0 \eta _1 K^2 R(\tau )\nonumber \\{} & {} \quad -14 a K^2 \rho _1^2 \tau R(\tau )+22 a K^2 \rho _1 \rho _2 R(\tau )-12 b_2 \eta _1 \rho _1 \rho _2 \tau -6 b_2 \eta _2 \rho _1^2 \tau \nonumber \\{} & {} \quad -6 b_2 \eta _0 \rho _2^2 \tau +3 b_2 \eta _1 \rho _2^2+6 b_2 \eta _2 \rho _1 \rho _2\nonumber \\{} & {} \quad +b_2 \eta _1^3+6 b_2 \eta _0 \eta _2 \eta _1+3 b_2 \eta _1 \rho _1^2 R(\tau )+6 b_2 \eta _0 \rho _1 \rho _2 R(\tau )+2 b_1 \eta _2 \eta _1-2 b_1 \rho _2^2 \tau +2 b_1 \rho _1 \rho _2 R(\tau )=0, \end{aligned}$$

(50)

$$\begin{aligned}{} & {} \quad -6 a \eta _0 \eta _1 K^2 \tau +4 a \eta _1^2 K^2+8 a \eta _0 \eta _2 K^2+6 a K^2 \rho _1^2 \tau ^2-22 a K^2 \rho _1 \rho _2 \tau +4 a K^2 \rho _2^2+4 a K^2 \rho _1^2 R(\tau )\nonumber \\{} & {} \quad -6 b_2 \eta _1 \rho _1^2 \tau -12 b_2 \eta _0 \rho _1 \rho _2 \tau +3 b_2 \eta _2 \rho _1^2+3 b_2 \eta _0 \rho _2^2+6 b_2 \eta _1 \rho _1 \rho _2+3 b_2 \eta _0 \eta _1^2\nonumber \\{} & {} \quad +3 b_2 \eta _0^2 \eta _2+3 b_2 \eta _0 \rho _1^2 R(\tau )\nonumber \\{} & {} \quad +b_1 \eta _1^2+2 b_1 \eta _0 \eta _2-4 b_1 \rho _1 \rho _2 \tau +b_1 \rho _2^2+b_1 \rho _1^2 R(\tau )-\eta _2 \omega =0, \end{aligned}$$

(51)

$$\begin{aligned}{} & {} 2 a \eta _0 \eta _1 K^2-2 a K^2 \rho _1^2 \tau +2 a K^2 \rho _1 \rho _2-6 b_2 \eta _0 \rho _1^2 \tau +3 b_2 \eta _1 \rho _1^2\nonumber \\{} & {} \quad +6 b_2 \eta _0 \rho _1 \rho _2+3 b_2 \eta _0^2 \eta _1+2 b_1 \eta _0 \eta _1-2 b_1 \rho _1^2 \tau \nonumber \\{} & {} \quad +2 b_1 \rho _1 \rho _2-\eta _1 \omega =0, \end{aligned}$$

(52)

$$\begin{aligned}{} & {} 3 b_2 \eta _0 \rho _1^2+b_2 \eta _0^3+b_1 \eta _0^2+b_1 \rho _1^2-\eta _0 \omega =0. \end{aligned}$$

(53)

These equations collectively yield the subsequent outcomes:

Case-1:   $R (\tau )=0$.

$$\begin{aligned} K=\pm \sqrt{\frac{b_1}{8 a}},~\eta _0=-\frac{5 b_1}{8 b_2},~\eta _1=\frac{5 b_1 \tau }{4 b_2},~\eta _2=\rho _1=\rho _2=0,~\omega =-\frac{15 b_1^2}{64 b_2}. \end{aligned}$$

(54)

Thus, the bright and singular soliton solutions of the governing equation (17) for $\frac{b_1}{a}>0$ emerge:

$$\begin{aligned} q(x,t)= & {} \left\{ -\frac{5 b_1}{8 b_2}+\frac{5 b_1~\text {sech}^2\left( \sqrt{\frac{b_1}{32 a}}x\right) }{8 b_2}\right\} e^{i \left( -\frac{15 b_1^2 }{64 b_2}t+\theta \right) }, \end{aligned}$$

(55)

and

$$\begin{aligned} q(x,t)= & {} \left\{ -\frac{5 b_1}{8 b_2}-\frac{5 b_1~\text {csch}^2\left( \sqrt{\frac{b_1}{32 a}}x\right) }{8 b_2}\right\} e^{i \left( -\frac{15 b_1^2 }{64 b_2}t+\theta \right) }. \end{aligned}$$

(56)

Fig. 2

2D plot of a quiescent bright optical soliton solution

Dark solitons are fascinating nonlinear wave phenomena characterized by localized amplitude minima in a background of higher intensity. In this study, we investigate the characteristics of optical dark soliton solutions governed by Eq. (39) and Eq. (122). The impact of various nonlinear effects such as quadratic SPM, cubic SPM, and nonlinear CD on dark solitons is explored through Figs. 1 and 3. In Fig. 1, we present three subfigures (a), (b), and (c), each highlighting a distinct nonlinear effect on the dark soliton profile. The influence of quadratic SPM on the dark soliton profile (q(x, t)) is depicted in Fig. 1(a). Quadratic SPM causes a modification in the shape of the dark soliton. Specifically, we observe a distortion in the soliton profile as it propagates through the medium due to the nonlinear phase shift induced by the quadratic SPM effect. Figure 1(b) illustrates the impact of cubic SPM on the dark soliton solution. Cubic SPM introduces additional nonlinear phase modulation, leading to further distortion and perturbation of the soliton profile compared to the quadratic case. The cubic SPM effect exacerbates the deviation from the ideal dark soliton behavior. The effect of nonlinear CD on the dark soliton profile is demonstrated in Figure 1(c). Nonlinear CD alters the propagation characteristics of the soliton, leading to spectral broadening. This results in a modification of the soliton’s shape and width as it propagates through the medium. Figure 3 provides insight into the behavior of dark solitons under the influence of generalized temporal evolution. This temporal evolution encompasses various nonlinear effects, including higher-order dispersion terms and temporal nonlinearities, which collectively influence the dynamics of dark soliton propagation.

Bright solitons, characterized by localized amplitude maxima, are another intriguing manifestation of nonlinear wave phenomena. In this study, we analyze the properties of bright soliton solutions described by Eq. (55) and Eq. (139), focusing on the impact of quadratic SPM, cubic SPM, nonlinear CD, and generalized temporal evolution, as depicted in Figures 2 and 4. Similar to Fig. 1, Fig. 2 comprises three subfigures (a), (b), and (c), each examining the effect of a specific nonlinear phenomenon on the bright soliton profile. Quadratic SPM-induced modifications on the bright soliton profile are showcased in Fig. 2(a). As with the dark soliton case, quadratic SPM induces a nonlinear phase shift, altering the shape of the bright soliton during propagation. Figure 2(b) demonstrates the influence of cubic SPM on bright solitons. Analogous to its effect on dark solitons, cubic SPM leads to additional phase modulation and distortion in the bright soliton profile, contributing to its nonlinear dynamics. The effect of nonlinear CD on bright solitons is explored in Fig. 2(c). Nonlinear CD-induced spectral broadening affects the propagation characteristics of bright solitons, influencing their shape and width over propagation distance. Figure 4 provides an overview of bright soliton behavior under the influence of generalized temporal evolution, encompassing various higher-order dispersion terms and temporal nonlinearities. This figure sheds light on the collective impact of these nonlinear effects on the dynamics of bright soliton propagation. The parameters have specific values: $b_{1}=1$, $b_{2}=1$, $a=1$, and $\theta =1$.

Case-2:   $R (\tau )=\frac{24}{25}\tau ^2$.

$$\begin{aligned} K=\pm \sqrt{\frac{b_1}{8 a}},~\eta _0=-\frac{5 b_1}{8 b_2},~\eta _1=\frac{5 b_1 \tau }{4 b_2},~\eta _2=-\frac{6 b_1 \tau ^2}{5 b_2},~\rho _1=0,~\rho _2=\pm \sqrt{\frac{3}{2}}\frac{ b_1 \tau }{b_2},~\omega =-\frac{15 b_1^2}{64 b_2}. \end{aligned}$$

(57)

Therefore, we achieve the bright–dark soliton solution for the governing equation (17) when $\frac{b_1}{a}>0$ as:

$$\begin{aligned} q(x,t)= & {} \Bigg \{ -\frac{5 b_1}{8 b_2}+\frac{25 b_1 \text {sech}\left( \sqrt{\frac{b_1}{8 a}x}\right) }{4 b_2 \left( 5 \text {sech}\left( \sqrt{\frac{b_1}{8 a}x}\right) \pm 1\right) }-\frac{30 b_1 \text {sech}^2\left( \sqrt{\frac{b_1}{8 a}}x\right) }{b_2 \left( 5 \text {sech}\left( \sqrt{\frac{b_1}{8 a}}x\right) \pm 1\right) ^2}\nonumber \\{} & {} \quad +\frac{5 \sqrt{\frac{3}{2}} b_1 \tanh \left( \sqrt{\frac{b_1}{8 a}}x\right) \text {sech}\left( \sqrt{\frac{b_1}{8 a}}x\right) }{b_2 \left( 1\pm 5 \text {sech}\left( \sqrt{\frac{b_1}{8 a}}x\right) \right) \left( 5 \text {sech}\left( \sqrt{\frac{b_1}{8 a}}x\right) \pm 1\right) }\Bigg \} e^{i \left( -\frac{15 b_1^2 }{64 b_2}t+\theta \right) }. \end{aligned}$$

(58)

Case-3:  $R (\tau )=\frac{5}{9}\tau ^2$.

$$\begin{aligned} K=\pm \sqrt{\frac{b_1}{8 a}},~\eta _0=-\frac{5 b_1}{8 b_2},~\eta _1=\frac{5 b_1 \tau }{4 b_2},~\eta _2=-\frac{25 b_1 \tau ^2}{36 b_2},~\rho _1=0,~\rho _2=\pm \frac{5 \sqrt{5} b_1 \tau }{12 b_2},~\omega =-\frac{15 b_1^2}{64 b_2}. \end{aligned}$$

(59)

Thus, we arrive at the bright–singular soliton solution of the governing equation (17) under the condition $\frac{b_1}{a}>0$:

$$\begin{aligned} q(x,t)= & {} \Bigg \{-\frac{5 b_1}{8 b_2}+\frac{15 b_1 \text {sech}\left( \sqrt{\frac{b_1}{8 a}}x\right) }{4 b_2 \left( 3 \text {sech}\left( \sqrt{\frac{b_1}{8 a}}x\right) \pm 2\right) }-\frac{25 b_1 \text {sech}^2\left( \sqrt{\frac{b_1}{8 a}}x\right) }{4 b_2 \left( 3 \text {sech}\left( \sqrt{\frac{b_1}{8 a}}x\right) \pm 2\right) ^2} \nonumber \\{} & {} \quad + \frac{30 \sqrt{5} b_1 \text {sech}\left( \sqrt{\frac{b_1}{8 a}}x\right) }{ 12 b_2 \left( 3 \text {sech}\left( \sqrt{\frac{b_1}{8 a}}x\right) \pm 2\right) \left( 2 \coth \left( \sqrt{\frac{b_1}{8 a}}x\right) \pm 3 \text {csch}\left( \sqrt{\frac{b_1}{8 a}}x\right) \right) }\Bigg \} e^{i \left( -\frac{15 b_1^2 }{64 b_2}t+\theta \right) }. \end{aligned}$$

(60)

Case-4:  $R (\tau )=\tau ^2-1$.

$$\begin{aligned} K= & {} \pm \sqrt{\frac{b_1}{8 a}},~\eta _0=-\frac{5 b_1}{8 b_2},~\eta _1=\frac{5 b_1 \tau }{4 b_2},~\rho _1=0,~\eta _2=-\frac{5 \left( b_1 \tau ^2-b_1\right) }{4 b_2},\nonumber \\{} & {} \rho _2=\frac{5 \sqrt{b_1^2 \tau ^2-b_1^2}}{4 b_2},~\omega =-\frac{15 b_1^2}{64 b_2}. \end{aligned}$$

(61)

This leads us to recover the bright–dark soliton solution of the governing equation (17) as follows:

$$\begin{aligned} q(x,t)= & {} \Bigg \{-\frac{5 b_1}{8 b_2}-\frac{20 \left( b_1 \tau ^2-b_1\right) \text {sech}^2\left( \sqrt{\frac{b_1}{8 a}}x\right) }{b_2 \left( \left( 4 \tau \text {sech}\left( \sqrt{\frac{b_1}{8 a}}x\right) \pm 3 \tanh \left( \sqrt{\frac{b_1}{8 a}}x\right) \right) +5\right) ^2} \nonumber \\{} & {} \quad +\frac{5 b_1 \tau \text {sech}\left( \sqrt{\frac{b_1}{8 a}}x\right) }{b_2 \left( \left( 4 \tau \text {sech}\left( \sqrt{\frac{b_1}{8 a}}x\right) \pm 3 \tanh \left( \sqrt{\frac{b_1}{8 a}}x\right) \right) +5\right) }\nonumber \\{} & {} \quad +\frac{5 \sqrt{b_1^2 \tau ^2-b_1^2} \text {sech}\left( \sqrt{\frac{b_1}{8 a}}x\right) \left( 3\pm 5 \tanh \left( \sqrt{\frac{b_1}{8 a}}x\right) \right) }{b_2 \left( \left( 4 \tau \text {sech}\left( \sqrt{\frac{b_1}{8 a}}x\right) \pm 3 \tanh \left( \sqrt{\frac{b_1}{8 a}}x\right) \right) +5\right) ^2}\Bigg \} e^{i \left( -\frac{15 b_1^2 }{64 b_2}t+\theta \right) }, \end{aligned}$$

(62)

and

$$\begin{aligned} q(x,t)= & {} \Bigg \{ -\frac{5 b_1}{8 b_2}-\frac{5 \left( b_1 \tau ^2-b_1\right) \text {sech}^2\left( \sqrt{\frac{b_1}{8 a}}x\right) }{4 b_2 \left( \tau \text {sech}\left( \sqrt{\frac{b_1}{8 a}}x\right) +1\right) ^2}+\frac{5 b_1 \tau \text {sech}\left( \sqrt{\frac{b_1}{8 a}}x\right) }{4 b_2 \left( \tau \text {sech}\left( \sqrt{\frac{b_1}{8 a}}x\right) +1\right) }\nonumber \\{} & {} \pm \frac{5 \sqrt{b_1^2 \tau ^2-b_1^2} \tanh \left( \sqrt{\frac{b_1}{8 a}}x\right) \text {sech}\left( \sqrt{\frac{b_1}{8 a}}x\right) }{4 b_2 \left( \tau \text {sech}\left( \sqrt{\frac{b_1}{8 a}}x\right) +1\right) ^2}\Bigg \} e^{i \left( -\frac{15 b_1^2 }{64 b_2}t+\theta \right) }. \end{aligned}$$

(63)

Case-5:  $R (\tau )=\tau ^2+1$.

$$\begin{aligned} K= & {} \pm \sqrt{\frac{b_1}{8 a}},~\eta _0=-\frac{5 b_1}{8 b_2},~\eta _1=\frac{5 b_1 \tau }{4 b_2},~\eta _2=-\frac{5 \left( b_1 \tau ^2+b_1\right) }{4 b_2},~\rho _1=0,\nonumber \\{} & {} ~\rho _2=\frac{5 \sqrt{b_1^2 \tau ^2+b_1^2}}{4 b_2},~\omega =-\frac{15 b_1^2}{64 b_2}. \end{aligned}$$

(64)

Thus, the singular–singular soliton solution of the governing equation (17) for $\frac{b_1}{a}>0$ turns out to be:

$$\begin{aligned} q(x,t)= & {} \Bigg \{-\frac{5 b_1}{8 b_2}-\frac{5 \left( b_1 \tau ^2+b_1\right) \text {csch}^2\left( \pm \sqrt{\frac{b_1}{8 a}}x\right) }{4 b_2 \left( \tau \text {csch}\left( \pm \sqrt{\frac{b_1}{8 a}}x\right) +1\right) ^2}+\frac{5 b_1 \tau \text {csch}\left( \pm \sqrt{\frac{b_1}{8 a}}x\right) }{4 b_2 \left( \tau \text {csch}\left( \pm \sqrt{\frac{b_1}{8 a}}x\right) +1\right) }\nonumber \\{} & {} \quad +\frac{5 \sqrt{b_1^2 \tau ^2+b_1^2} \coth \left( \pm \sqrt{\frac{b_1}{8 a}}x\right) \text {csch}\left( \pm \sqrt{\frac{b_1}{8 a}}x\right) }{4 b_2 \left( \tau \text {csch}\left( \pm \sqrt{\frac{b_1}{8 a}}x\right) +1\right) ^2}\Bigg \} e^{i \left( -\frac{15 b_1^2 }{64 b_2}t+\theta \right) }. \end{aligned}$$

(65)

3.2 Quadratic–cubic–quartic form of SPM

The equation describing the perturbed FLE with nonlinear CD and a quadratic-cubic-quartic SPM is outlined as follows:

$$\begin{aligned} iq_t + a(\left| q \right| ^nq)_{xx} +i \sigma \left| q \right| ^2 q_{x} + \left( b_1 \left| q \right| +b_2 \left| q \right| ^2+b_3 \left| q \right| ^3\right) q = i \left[ \alpha q_{x}+\lambda (\left| q \right| ^2q)_{x}+\mu (\left| q \right| ^2)_{x} q \right] . \end{aligned}$$

(66)

When transformation (18) is employed, Eq. (66) can be split into its real and imaginary components:

$$\begin{aligned} a K^2 (n+1) U^{n+1} U''+a K^2 n (n+1) U^n U^{\prime 2}+b_1 U^3+b_2 U^4+b_3 U^5-\omega U^2=0, \end{aligned}$$

(67)

and

$$\begin{aligned} -K U U' \left( \alpha +(3 \lambda +2 \mu -\sigma ) U^2\right) =0. \end{aligned}$$

(68)

From the imaginary part, we can deduce:

$$\begin{aligned} \alpha =0,~~3 \lambda +2 \mu -\sigma =0. \end{aligned}$$

(69)

Choosing $n=1$ to fulfill the integrability condition, we can represent Eq. (66) in the following form:

$$\begin{aligned} iq_t + a(\left| q \right| q)_{xx} +i \sigma \left| q \right| ^2 q_{x} + \left( b_1 \left| q \right| +b_2 \left| q \right| ^2+b_3 \left| q \right| ^3\right) q = i \left[ \lambda (\left| q \right| ^2q)_{x}+\mu (\left| q \right| ^2)_{x} q \right] , \end{aligned}$$

(70)

and Eq. (67) can be rewritten as:

$$\begin{aligned} 2 a K^2 U U''+2 a K^2 U^{\prime 2}+b_1 U^2+b_2 U^3+b_3 U^4-\omega U=0. \end{aligned}$$

(71)

Applying the balance principle between $U U''$ and $U^4$, leads to $N=1$. In the following subsections, the integration algorithms will be implemented into Eq. (71).

3.2.1 Enhanced Kudryashov’s method

The solution of Eq. (71), as dictated by this method, can be structured as follows:

$$\begin{aligned} U(\xi )=\eta _0 + \eta _1 R(\xi )+ \rho _1 \left( \frac{ R^\prime (\xi )}{ R(\xi )}\right) . \end{aligned}$$

(72)

When we substitute Eq. (72) along with Eq. (5) into Eq. (71), we derive:

$$\begin{aligned}{} & {} -6 a \eta _1^2 K^2 \chi +6 a K^2 \rho _1^2 \chi ^2-6 b_3 \eta _1^2 \rho _1^2 \chi +b_3 \eta _1^4+b_3 \rho _1^4 \chi ^2=0, \end{aligned}$$

(73)

$$\begin{aligned}{} & {} \quad -4 a \eta _0 \eta _1 K^2 \chi -3 b_2 \eta _1 \rho _1^2 \chi +b_2 \eta _1^3-12 b_3 \eta _0 \eta _1 \rho _1^2 \chi +4 b_3 \eta _0 \eta _1^3=0, \end{aligned}$$

(74)

$$\begin{aligned}{} & {} \quad -12 a \eta _1 K^2 \rho _1 \chi -4 b_3 \eta _1 \rho _1^3 \chi +4 b_3 \eta _1^3 \rho _1=0, \end{aligned}$$

(75)

$$\begin{aligned}{} & {} 4 a \eta _1^2 K^2-4 a K^2 \rho _1^2 \chi -3 b_2 \eta _0 \rho _1^2 \chi +3 b_2 \eta _0 \eta _1^2+b_1 \eta _1^2\nonumber \\{} & {} \quad -b_1 \rho _1^2 \chi -6 b_3 \eta _0^2 \rho _1^2 \chi +6 b_3 \eta _1^2 \rho _1^2+6 b_3 \eta _0^2 \eta _1^2-2 b_3 \rho _1^4 \chi =0, \end{aligned}$$

(76)

$$\begin{aligned}{} & {} -4 a \eta _0 K^2 \rho _1 \chi +3 b_2 \eta _1^2 \rho _1-b_2 \rho _1^3 \chi -4 b_3 \eta _0 \rho _1^3 \chi +12 b_3 \eta _0 \eta _1^2 \rho _1=0, \end{aligned}$$

(77)

$$\begin{aligned}{} & {} 2 a \eta _1 \eta _0 K^2+3 b_2 \eta _1 \rho _1^2+3 b_2 \eta _1 \eta _0^2+2 b_1 \eta _1 \eta _0+12 b_3 \eta _1 \eta _0 \rho _1^2+4 b_3 \eta _1 \eta _0^3-\eta _1 \omega =0, \end{aligned}$$

(78)

$$\begin{aligned}{} & {} 3 b_2 \eta _0^2 \rho _1+b_2 \rho _1^3+2 b_1 \eta _0 \rho _1+4 b_3 \eta _0^3 \rho _1+4 b_3 \eta _0 \rho _1^3-\rho _1 \omega =0, \end{aligned}$$

(79)

$$\begin{aligned}{} & {} 2 a \eta _1 K^2 \rho _1+6 b_2 \eta _0 \eta _1 \rho _1+2 b_1 \eta _1 \rho _1+4 b_3 \eta _1 \rho _1^3+12 b_3 \eta _0^2 \eta _1 \rho _1=0, \end{aligned}$$

(80)

$$\begin{aligned}{} & {} 3 b_2 \eta _0 \rho _1^2+b_2 \eta _0^3+b_1 \eta _0^2+b_1 \rho _1^2+6 b_3 \eta _0^2 \rho _1^2+b_3 \eta _0^4+b_3 \rho _1^4-\eta _0 \omega =0. \end{aligned}$$

(81)

The together calculation of these equations yields the subsequent outcomes:

Result-1:

$$\begin{aligned} K=\pm \sqrt{\frac{b_1}{20 a}},~\eta _0=\pm \sqrt{\frac{3 b_1}{10 b_3}},~\eta _1=\pm \sqrt{\frac{3 b_1 \chi }{10 b_3}},~\rho _1=0,~\omega =\frac{\left( \frac{3 b_1}{10}\right) ^{3/2}}{\sqrt{b_3}},~b_2=\mp \sqrt{\frac{10 b_1 b_3}{3}}. \end{aligned}$$

(82)

This leads us to the soliton solution of the governing equation (66) as follows:

$$\begin{aligned} q(x,t)=\pm \sqrt{\frac{3 b_1}{10 b_3}}\left\{ 1+ \frac{4 d e^{\pm \sqrt{\frac{b_1}{20 a}}x } \sqrt{\chi }}{4 d^2 e^{\pm 2 \sqrt{\frac{b_1}{20 a}}x }+\chi }\right\} e^{i \left( \frac{\left( \frac{3 b_1}{10}\right) ^{3/2} }{\sqrt{b_3}}t+\theta \right) }. \end{aligned}$$

(83)

Choosing $\chi =4 d^2$, we recover bright soliton solution for $\frac{b_1}{a}>0$ and $\frac{b_1}{b_3}>0$:

$$\begin{aligned} q(x,t)=\pm \sqrt{\frac{3 b_1}{10 b_3}} \left\{ 1+\text {sech}\left( \sqrt{\frac{b_1}{20 a}}x\right) \right\} e^{i \left( \frac{\left( \frac{3 b_1}{10}\right) ^{3/2} }{\sqrt{b_3}}t+\theta \right) }. \end{aligned}$$

(84)

3.2.2 The new projective Riccati equations method

In line with this procedure, the solution is structured as follows:

$$\begin{aligned} U(\xi )=\eta _0+\eta _1 F(\xi )+\rho _1 G(\xi ). \end{aligned}$$

(85)

By incorporating Eq. (85) in conjunction with Eqs. (8) and (9) into Eq. (71), we arrive at:

$$\begin{aligned}{} & {} 12 a \eta _1 K^2 \rho _1 R(\tau )+4 b_3 \eta _1^3 \rho _1+4 b_3 \eta _1 \rho _1^3 R(\tau )=0. \end{aligned}$$

(86)

$$\begin{aligned}{} & {} -12 a \eta _1 K^2 \rho _1 \tau +4 a \eta _0 K^2 \rho _1 R(\tau )+3 b_2 \eta _1^2 \rho _1\nonumber \\{} & {} \quad +b_2 \rho _1^3 R(\tau )-8 b_3 \eta _1 \rho _1^3 \tau +12 b_3 \eta _0 \eta _1^2 \rho _1+4 b_3 \eta _0 \rho _1^3 R(\tau )=0. \end{aligned}$$

(87)

$$\begin{aligned}{} & {} -2 a \eta _0 K^2 \rho _1 \tau +2 a \eta _1 K^2 \rho _1+6 b_2 \eta _0 \eta _1 \rho _1-2 b_2 \rho _1^3 \tau +2 b_1 \eta _1 \rho _1\nonumber \\{} & {} \quad -8 b_3 \eta _0 \rho _1^3 \tau +4 b_3 \eta _1 \rho _1^3+12 b_3 \eta _0^2 \eta _1 \rho _1=0. \end{aligned}$$

(88)

$$\begin{aligned}{} & {} 3 b_2 \eta _0^2 \rho _1+b_2 \rho _1^3+2 b_1 \eta _0 \rho _1+4 b_3 \eta _0^3 \rho _1+4 b_3 \eta _0 \rho _1^3-\rho _1 \omega =0. \end{aligned}$$

(89)

$$\begin{aligned}{} & {} 6 a \eta _1^2 K^2 R(\tau )+6 a K^2 \rho _1^2 R(\tau )^2+b_3 \eta _1^4+6 b_3 \eta _1^2 \rho _1^2 R(\tau )+b_3 \rho _1^4 R(\tau )^2=0. \end{aligned}$$

(90)

$$\begin{aligned}{} & {} -10 a \eta _1^2 K^2 \tau +4 a \eta _0 \eta _1 K^2 R(\tau )-14 a K^2 \rho _1^2 \tau R(\tau )+b_2 \eta _1^3\nonumber \\{} & {} \quad +3 b_2 \eta _1 \rho _1^2 R(\tau )-12 b_3 \eta _1^2 \rho _1^2 \tau +4 b_3 \eta _0 \eta _1^3\nonumber \\{} & {} \quad +12 b_3 \eta _0 \eta _1 \rho _1^2 R(\tau )-4 b_3 \rho _1^4 \tau R(\tau )=0. \end{aligned}$$

(91)

$$\begin{aligned}{} & {} \quad -6 a \eta _0 \eta _1 K^2 \tau +4 a \eta _1^2 K^2+6 a K^2 \rho _1^2 \tau ^2+4 a K^2 \rho _1^2 R(\tau )-6 b_2 \eta _1 \rho _1^2 \tau \nonumber \\{} & {} \quad +3 b_2 \eta _0 \eta _1^2+3 b_2 \eta _0 \rho _1^2 R(\tau )+b_1 \eta _1^2\nonumber \\{} & {} \quad +b_1 \rho _1^2 R(\tau )-24 b_3 \eta _0 \eta _1 \rho _1^2 \tau +6 b_3 \eta _1^2 \rho _1^2+6 b_3 \eta _0^2 \eta _1^2\nonumber \\{} & {} \quad +4 b_3 \rho _1^4 \tau ^2+6 b_3 \eta _0^2 \rho _1^2 R(\tau )+2 b_3 \rho _1^4 R(\tau )=0. \end{aligned}$$

(92)

$$\begin{aligned}{} & {} 2 a \eta _0 \eta _1 K^2-2 a K^2 \rho _1^2 \tau -6 b_2 \eta _0 \rho _1^2 \tau +3 b_2 \eta _1 \rho _1^2+3 b_2 \eta _0^2 \eta _1\nonumber \\{} & {} \quad +2 b_1 \eta _0 \eta _1-2 b_1 \rho _1^2 \tau -12 b_3 \eta _0^2 \rho _1^2 \tau +12 b_3 \eta _0 \eta _1 \rho _1^2\nonumber \\{} & {} \quad +4 b_3 \eta _0^3 \eta _1-4 b_3 \rho _1^4 \tau -\eta _1 \omega =0. \end{aligned}$$

(93)

$$\begin{aligned}{} & {} 3 b_2 \eta _0 \rho _1^2+b_2 \eta _0^3+b_1 \eta _0^2+b_1 \rho _1^2+6 b_3 \eta _0^2 \rho _1^2+b_3 \eta _0^4+b_3 \rho _1^4-\eta _0 \omega =0. \end{aligned}$$

(94)

The together calculation of these equations yields the subsequent outcomes:

Case-1:   $R (\tau )=\frac{24}{25}\tau ^2$.

$$\begin{aligned} K= & {} \pm \sqrt{-\frac{b_1}{10 a}},~\eta _0=-3 \sqrt{\frac{b_1}{10 b_3}},~\eta _1=\tau \sqrt{\frac{72 b_1}{125 b_3}},~\rho _1=0,\nonumber \\{} & {} \quad ~\omega =\frac{-9}{20} \sqrt{\frac{b_1^3}{10 b_3}},~b_2=\pm \frac{7}{6} \sqrt{\frac{5 b_1 b_3}{2}}. \end{aligned}$$

(95)

Hence, we attain the bright solution to the governing equation (66) for $\frac{b_1}{b_3}>0$ and $\frac{b_1}{a}<0$ as follows:

$$\begin{aligned} q(x,t)= \sqrt{\frac{b_1}{b_3}} \left\{ -\frac{3}{\sqrt{10}}+\frac{6 \sqrt{\frac{2}{5}} \text {sech}\left( \sqrt{-\frac{b_1}{10 a}}x\right) }{5 \text {sech}\left( \sqrt{-\frac{b_1}{10 a}}x\right) \pm 1} \right\} e^{i \left( \frac{-9}{20} \sqrt{\frac{b_1^3}{10 b_3}} t+\theta \right) }. \end{aligned}$$

(96)

Case-2:  $R (\tau )=\frac{5}{9}\tau ^2$.

$$\begin{aligned} K=\pm \sqrt{-\frac{b_1}{52 a}},~\eta _0=\pm \sqrt{\frac{15 b_1}{26 b_3}},~\eta _1=\mp \tau \sqrt{\frac{5 b_1}{78 b_3}},~\rho _1=0,~\omega =\pm \frac{3}{13} \sqrt{\frac{15 b_1^3}{26 b_3}},~b_2=\mp \sqrt{\frac{245 b_1 b_3}{78}}. \end{aligned}$$

(97)

Therefore, the bright soliton solution of the governing equation (66) for $\frac{b_1}{b_3}>0$ and $\frac{b_1}{a}<0$ is as follows:

$$\begin{aligned} q(x,t)= \pm \sqrt{\frac{15 b_1}{26 b_3}} \left\{ 1-\frac{\text {sech}\left( \sqrt{-\frac{b_1}{52 a}}x\right) }{3 \text {sech}\left( \sqrt{-\frac{b_1}{52 a}}x\right) \pm 2}\right\} e^{i \left( \pm \frac{3}{13} \sqrt{\frac{15 b_1^3}{26 b_3}} t+\theta \right) }. \end{aligned}$$

(98)

Case-3:  $R (\tau )=\tau ^2+1$.

$$\begin{aligned}{} & {} K=\pm \sqrt{\frac{-6 \sqrt{-b_1^2 b_3^2 \tau ^2 \left( \tau ^2+1\right) ^2}-b_1 b_3 \tau ^4+4 b_1 b_3 \tau ^2+5 b_1 b_3}{4 \left( a b_3 \tau ^4+26 a b_3 \tau ^2+25 a b_3\right) }},\nonumber \\{} & {} \quad \eta _0=\frac{\sqrt{\frac{3}{2}} \left( \sqrt{-b_1^2 b_3^2 \tau ^2 \left( \tau ^2+1\right) ^2}-b_1 b_3 \tau ^2 \left( \tau ^2+1\right) \right) \sqrt{\frac{6 \sqrt{-b_1^2 b_3^2 \tau ^2 \left( \tau ^2+1\right) ^2}+b_1 b_3 \left( \tau ^4-4 \tau ^2-5\right) }{b_3^2 \left( \tau ^2+25\right) }}}{b_1 b_3 \tau \left( \tau ^2+1\right) ^2},\nonumber \\{} & {} \quad \eta _1=\sqrt{\frac{18 \sqrt{-b_1^2 b_3^2 \tau ^2 \left( \tau ^2+1\right) ^2}+3 b_1 b_3 \left( \tau ^4-4 \tau ^2-5\right) }{2 b_3^2 \left( \tau ^2+25\right) }},~\rho _1=0,\nonumber \\{} & {} \quad ~\omega =-\frac{3 \sqrt{\frac{3}{2}} \left( 4 b_1 b_3 \tau ^2 \left( \tau ^2+1\right) -\left( \tau ^2+5\right) \sqrt{-b_1^2 b_3^2 \tau ^2 \left( \tau ^2+1\right) ^2}\right) \sqrt{\frac{6 \sqrt{-b_1^2 b_3^2 \tau ^2 \left( \tau ^2+1\right) ^2}+b_1 b_3 \left( \tau ^4-4 \tau ^2-5\right) }{b_3^2 \left( \tau ^2+25\right) }}}{2 b_3 \tau \left( \tau ^2+1\right) ^2 \left( \tau ^2+25\right) },\nonumber \\{} & {} \quad b=\frac{5 \left( -2 \sqrt{-b_1^2 b_3^2 \left( \tau ^3+\tau \right) ^2}+b_1 b_3 \tau ^4+b_1 b_3 \tau ^2\right) \sqrt{\frac{6 \sqrt{-b_1^2 b_3^2 \tau ^2 \left( \tau ^2+1\right) ^2}+b_1 b_3 \left( \tau ^4-4 \tau ^2-5\right) }{b_3^2 \left( \tau ^2+25\right) }}}{\sqrt{6} b_1 \tau \left( \tau ^2+1\right) ^2}. \end{aligned}$$

(99)

Hence, the singular soliton solution of the governing equation (66) is attained:

$$\begin{aligned} q(x,t)= & {} \Bigg \{\frac{\sqrt{\frac{3}{2}} \left( \sqrt{-b_1^2 b_3^2 \tau ^2 \left( \tau ^2+1\right) ^2}-b_1 b_3 \tau ^2 \left( \tau ^2+1\right) \right) \sqrt{\frac{6 \sqrt{-b_1^2 b_3^2 \tau ^2 \left( \tau ^2+1\right) ^2}+b_1 b_3 \left( \tau ^4-4 \tau ^2-5\right) }{b_3^2 \left( \tau ^2+25\right) }}}{b_1 b_3 \tau \left( \tau ^2+1\right) ^2}\nonumber \\{} & {} \quad +\frac{\sqrt{\frac{18 \sqrt{-b_1^2 b_3^2 \tau ^2 \left( \tau ^2+1\right) ^2}+3 b_1 b_3 \left( \tau ^4-4 \tau ^2-5\right) }{b_3^2 \left( \tau ^2+25\right) }} \text {csch}\left( \pm \sqrt{\frac{-6 \sqrt{-b_1^2 b_3^2 \tau ^2 \left( \tau ^2+1\right) ^2}-b_1 b_3 \tau ^4+4 b_1 b_3 \tau ^2+5 b_1 b_3}{4 \left( a b_3 \tau ^4+26 a b_3 \tau ^2+25 a b_3\right) }}x\right) }{\sqrt{2} \left( \tau \text {csch}\left( \pm \sqrt{\frac{-6 \sqrt{-b_1^2 b_3^2 \tau ^2 \left( \tau ^2+1\right) ^2}-b_1 b_3 \tau ^4+4 b_1 b_3 \tau ^2+5 b_1 b_3}{4 \left( a b_3 \tau ^4+26 a b_3 \tau ^2+25 a b_3\right) }}x\right) +1\right) }\Bigg \}\nonumber \\{} & {} \quad \times e^{i \left( -\frac{3 \sqrt{\frac{3}{2}} \left( 4 b_1 b_3 \tau ^2 \left( \tau ^2+1\right) -\left( \tau ^2+5\right) \sqrt{-b_1^2 b_3^2 \tau ^2 \left( \tau ^2+1\right) ^2}\right) \sqrt{\frac{6 \sqrt{-b_1^2 b_3^2 \tau ^2 \left( \tau ^2+1\right) ^2}+b_1 b_3 \left( \tau ^4-4 \tau ^2-5\right) }{b_3^2 \left( \tau ^2+25\right) }}}{2 b_3 \tau \left( \tau ^2+1\right) ^2 \left( \tau ^2+25\right) }t+\theta \right) }. \end{aligned}$$

(100)

4 Generalized temporal evolution

4.1 Quadratic—cubic nonlinearity

Describing the perturbed FLE with nonlinear CD and quadratic-cubic SPM and featuring generalized temporal evolution, the equation is as follows:

$$\begin{aligned}{} & {} i \left( q^l \right) _t + a(\left| q \right| ^n q^l)_{xx} + \left( b_1 \left| q \right| + b_2 \left| q \right| ^2 \right) q^l + i \sigma \left| q \right| ^2 \left( q^l \right) _x\nonumber \\{} & {} \quad = i \left[ \alpha \left( q^l \right) _x + \lambda \left( \left| q \right| ^2 q^l \right) _x + \mu \left( \left| q \right| ^2 \right) _x q^l \right] , \end{aligned}$$

(101)

where l is the parameter for the generalized temporal evolution. The previous section covers the case when $l = 1$.

Applying the transformation (18), splits Eq. (101) into the following real and imaginary parts:

$$\begin{aligned} a K^2 \left( l^2+l (2 n-1)+(n-1) n\right) U^n U^{\prime 2}+a K^2 (l+n) U^{n+1} U''+b_2 U^4+b_1 U^3-l \omega U^2=0, \end{aligned}$$

(102)

and

$$\begin{aligned} -K U U' \left( U(K x)^2 (\lambda (l+2)-l \sigma +2 \mu )+\alpha l\right) =0. \end{aligned}$$

(103)

From the imaginary part, we can deduce:

$$\begin{aligned} \alpha =0,~~\lambda (l+2)-l \sigma +2 \mu =0. \end{aligned}$$

(104)

Setting $n=1$ to satisfy the integrability condition leads to the re-writing of Eq. (101) as:

$$\begin{aligned} i \left( q^l \right) _t + a(\left| q \right| q^l)_{xx} + \left( b_1 \left| q \right| + b_2 \left| q \right| ^2 \right) q^l + i \sigma \left| q \right| ^2 \left( q^l \right) _x = i \left[ \lambda \left( \left| q \right| ^2 q^l \right) _x + \mu \left( \left| q \right| ^2 \right) _x q^l \right] , \end{aligned}$$

(105)

and Eq. (102) can be rewritten as:

$$\begin{aligned} a K^2 \left( l^2+l\right) U^{\prime 2}+2 a K^2 U U''+b_2 U^3+b_1 U^2-l \omega U=0. \end{aligned}$$

(106)

Applying the balancing principle between $U U''$ and $U^3$, leads to $N=2$. In the following subsections, the integration algorithms will be utilized to Eq. (106).

4.1.1 The enhanced Kudryashov method

Thus, this method offers a solution to Eq. (106) in the following form:

$$\begin{aligned} U(\xi )=\eta _0 + \eta _1 R(\xi )+ \eta _2 R(\xi )^{2}+ \rho _1 \left( \frac{ R^\prime (\xi )}{ R(\xi )}\right) + \rho _2 \left( \frac{ R^\prime (\xi )}{ R(\xi )}\right) ^2. \end{aligned}$$

(107)

By incorporating Eq. (107) in conjunction with Eq. (5) into Eq. (106), we arrive at:

$$\begin{aligned}{} & {} 24 a \eta _2 K^2 \rho _2 \chi ^2-12 a \eta _2^2 K^2 \chi +8 a \eta _2 K^2 l^2 \rho _2 \chi ^2-4 a \eta _2^2 K^2 l^2 \chi -4 a K^2 l^2 \rho _2^2 \chi ^3+8 a \eta _2 K^2 l \rho _2 \chi ^2\nonumber \\{} & {} \quad -4 a \eta _2^2 K^2 l \chi -4 a K^2 l \rho _2^2 \chi ^3-12 a K^2 \rho _2^2 \chi ^3+3 b_2 \eta _2 \rho _2^2 \chi ^2-3 b_2 \eta _2^2 \rho _2 \chi +b_2 \eta _2^3-b_2 \rho _2^3 \chi ^3=0. \end{aligned}$$

(108)

$$\begin{aligned}{} & {} 16 a \eta _1 K^2 \rho _2 \chi ^2-16 a \eta _1 \eta _2 K^2 \chi +4 a \eta _1 K^2 l^2 \rho _2 \chi ^2-4 a \eta _1 \eta _2 K^2 l^2 \chi +4 a \eta _1 K^2 l \rho _2 \chi ^2\nonumber \\{} & {} \quad -4 a \eta _1 \eta _2 K^2 l \chi +3 b_2 \eta _1 \rho _2^2 \chi ^2-6 b_2 \eta _1 \eta _2 \rho _2 \chi +3 b_2 \eta _1 \eta _2^2=0. \end{aligned}$$

(109)

$$\begin{aligned}{} & {} 12 a \eta _0 K^2 \rho _2 \chi ^2-28 a \eta _2 K^2 \rho _2 \chi -4 a \eta _1^2 K^2 \chi -12 a \eta _0 \eta _2 K^2 \chi \nonumber \\{} & {} \quad +8 a \eta _2^2 K^2-8 a \eta _2 K^2 l^2 \rho _2 \chi -a \eta _1^2 K^2 l^2 \chi \nonumber \\{} & {} \quad +4 a \eta _2^2 K^2 l^2+4 a K^2 l^2 \rho _2^2 \chi ^2+a K^2 l^2 \rho _1^2 \chi ^2-8 a \eta _2 K^2 l \rho _2 \chi \nonumber \\{} & {} \quad -a \eta _1^2 K^2 l \chi +4 a \eta _2^2 K^2 l+4 a K^2 l \rho _2^2 \chi ^2\nonumber \\{} & {} \quad +a K^2 l \rho _1^2 \chi ^2+20 a K^2 \rho _2^2 \chi ^2+4 a K^2 \rho _1^2 \chi ^2+3 b_2 \eta _0 \rho _2^2 \chi ^2\nonumber \\{} & {} \quad -6 b_2 \eta _2 \rho _2^2 \chi -3 b_2 \eta _1^2 \rho _2 \chi -2 b_1 \eta _2 \rho _2 \chi \nonumber \\{} & {} \quad -6 b_2 \eta _0 \eta _2 \rho _2 \chi -3 b_2 \eta _2 \rho _1^2 \chi +3 b_2 \eta _2^2 \rho _2+b_1 \eta _2^2\nonumber \\{} & {} \quad +3 b_2 \eta _0 \eta _2^2+3 b_2 \eta _1^2 \eta _2+3 b_2 \rho _2^3 \chi ^2+b_1 \rho _2^2 \chi ^2+3 b_2 \rho _1^2 \rho _2 \chi ^2=0. \end{aligned}$$

(110)

$$\begin{aligned}{} & {} -16 a \eta _2 K^2 \rho _1 \chi -4 a \eta _2 K^2 l^2 \rho _1 \chi +4 a K^2 l^2 \rho _1 \rho _2 \chi ^2-4 a \eta _2 K^2 l \rho _1 \chi +4 a K^2 l \rho _1 \rho _2 \chi ^2\nonumber \\{} & {} \quad +16 a K^2 \rho _1 \rho _2 \chi ^2-6 b_2 \eta _2 \rho _1 \rho _2 \chi +3 b_2 \eta _2^2 \rho _1+3 b_2 \rho _1 \rho _2^2 \chi ^2=0. \end{aligned}$$

(111)

$$\begin{aligned}{} & {} -14 a \eta _1 K^2 \rho _2 \chi -4 a \eta _0 \eta _1 K^2 \chi +10 a \eta _2 \eta _1 K^2-4 a \eta _1 K^2 l^2 \rho _2 \chi \nonumber \\{} & {} \quad +4 a \eta _2 \eta _1 K^2 l^2-4 a \eta _1 K^2 l \rho _2 \chi +4 a \eta _2 \eta _1 K^2 l\nonumber \\{} & {} \quad -3 b_2 \eta _1 \rho _1^2 \chi -6 b_2 \eta _1 \rho _2^2 \chi -2 b_1 \eta _1 \rho _2 \chi -6 b_2 \eta _0 \eta _1 \rho _2 \chi \nonumber \\{} & {} \quad +6 b_2 \eta _2 \eta _1 \rho _2+b_2 \eta _1^3+2 b_1 \eta _2 \eta _1+6 b_2 \eta _0 \eta _2 \eta _1=0. \end{aligned}$$

(112)

$$\begin{aligned}{} & {} -8 a \eta _1 K^2 \rho _1 \chi -2 a \eta _1 K^2 l^2 \rho _1 \chi -2 a \eta _1 K^2 l \rho _1 \chi -6 b_2 \eta _1 \rho _1 \rho _2 \chi +6 b_2 \eta _1 \eta _2 \rho _1=0. \end{aligned}$$

(113)

$$\begin{aligned}{} & {} -8 a \eta _0 K^2 \rho _2 \chi +8 a \eta _2 K^2 \rho _2+2 a \eta _1^2 K^2+8 a \eta _0 \eta _2 K^2\nonumber \\{} & {} \quad +a \eta _1^2 K^2 l^2+a \eta _1^2 K^2 l-8 a K^2 \rho _2^2 \chi -4 a K^2 \rho _1^2 \chi \nonumber \\{} & {} \quad -6 b_2 \eta _0 \rho _2^2 \chi -3 b_2 \eta _0^2 \rho _2 \chi -2 b_1 \eta _0 \rho _2 \chi -3 b_2 \eta _0 \rho _1^2 \chi +3 b_2 \eta _2 \rho _2^2+3 b_2 \eta _1^2 \rho _2+2 b_1 \eta _2 \rho _2+6 b_2 \eta _0 \eta _2 \rho _2\nonumber \\{} & {} \quad +3 b_2 \eta _2 \rho _1^2+b_1 \eta _1^2+3 b_2 \eta _0 \eta _1^2+3 b_2 \eta _0^2 \eta _2+2 b_1 \eta _0 \eta _2-3 b_2 \rho _2^3 \chi -2 b_1 \rho _2^2 \chi -6 b_2 \rho _1^2 \rho _2 \chi \nonumber \\{} & {} \quad -b_1 \rho _1^2 \chi -\eta _2 l \omega +l \rho _2 \chi \omega =0. \end{aligned}$$

(114)

$$\begin{aligned}{} & {} -4 a \eta _0 K^2 \rho _1 \chi +8 a \eta _2 K^2 \rho _1-12 a K^2 \rho _2 \rho _1 \chi -6 b_2 \eta _0 \rho _2 \rho _1 \chi +3 b_2 \eta _1^2 \rho _1+2 b_1 \eta _2 \rho _1\nonumber \\{} & {} \quad +6 b_2 \eta _0 \eta _2 \rho _1+6 b_2 \eta _2 \rho _2 \rho _1+b_2 \rho _1^3 (-\chi )-6 b_2 \rho _2^2 \rho _1 \chi -2 b_1 \rho _2 \rho _1 \chi =0. \end{aligned}$$

(115)

$$\begin{aligned}{} & {} 2 a \eta _1 K^2 \rho _2+2 a \eta _0 \eta _1 K^2+3 b_2 \eta _1 \rho _1^2+3 b_2 \eta _1 \rho _2^2+2 b_1 \eta _1 \rho _2\nonumber \\{} & {} \quad +6 b_2 \eta _0 \eta _1 \rho _2+3 b_2 \eta _0^2 \eta _1+2 b_1 \eta _0 \eta _1-\eta _1 l \omega =0. \end{aligned}$$

(116)

$$\begin{aligned}{} & {} 2 a \eta _1 K^2 \rho _1+2 b_1 \eta _1 \rho _1+6 b_2 \eta _0 \eta _1 \rho _1+6 b_2 \eta _1 \rho _1 \rho _2=0. \end{aligned}$$

(117)

$$\begin{aligned}{} & {} 3 b_2 \eta _0^2 \rho _2+3 b_2 \eta _0 \rho _1^2+3 b_2 \eta _0 \rho _2^2+2 b_1 \eta _0 \rho _2+b_2 \eta _0^3+b_1 \eta _0^2+b_2 \rho _2^3+b_1 \rho _1^2\nonumber \\{} & {} \quad +b_1 \rho _2^2+3 b_2 \rho _1^2 \rho _2-\eta _0 l \omega -l \rho _2 \omega =0. \end{aligned}$$

(118)

$$\begin{aligned}{} & {} 3 b_2 \eta _0^2 \rho _1+2 b_1 \eta _0 \rho _1+6 b_2 \eta _0 \rho _2 \rho _1\nonumber \\{} & {} \quad +b_2 \rho _1^3+3 b_2 \rho _2^2 \rho _1+2 b_1 \rho _2 \rho _1-l \rho _1 \omega =0. \end{aligned}$$

(119)

These yield the subsequent outcomes as:

Result–1:

$$\begin{aligned}{} & {} K=\pm \sqrt{\frac{b_1}{8 a l^2+8 a l+16 a}},~\eta _0=\frac{-2 b_2 l^2 \rho _2-b_1 l^2-2 b_2 l \rho _2-b_1 l-4 b_2 \rho _2-3 b_1}{2 b_2 \left( l^2+l+2\right) },~\eta _1=0,\nonumber \\{} & {} \quad \eta _2=\frac{\chi \left( 2 b_2 l^2 \rho _2+b_1 l^2+2 b_2 l \rho _2+b_1 l+4 b_2 \rho _2+3 b_1\right) }{2 b_2 \left( l^2+l+2\right) },~\rho _1=0,\nonumber \\{} & {} \quad ~\omega =-\frac{b_1^2 \left( l^4+2 l^3+5 l^2+4 l+3\right) }{4 b_2 l \left( l^2+l+2\right) ^2}. \end{aligned}$$

(120)

Consequently, the soliton solution of the governing equation (101) is attained:

$$\begin{aligned} q(x,t)=\left\{ -\frac{b_1 \left( l^2+l+3\right) \left( \chi -4 d^2 e^{2 \sqrt{\frac{b_1}{8 a l^2+8 a l+16 a}}x}\right) {}^2}{2 b_2 \left( l^2+l+2\right) \left( 4 d^2 e^{2 \sqrt{\frac{b_1}{8 a l^2+8 a l+16 a}}x}+\chi \right) {}^2}\right\} e^{i \left( -\frac{b_1^2 \left( l^4+2 l^3+5 l^2+4 l+3\right) }{4 b_2 l \left( l^2+l+2\right) ^2}t+\theta \right) }. \end{aligned}$$

(121)

Selecting $\chi =\pm 4 d^2$ recovers dark and singular soliton solutions for $\frac{b_1}{8 a l^2+8 a l+16 a}>0$:

$$\begin{aligned} q(x,t)= & {} -\frac{b_1 \left( l^2+l+3\right) }{2 b_2 \left( l^2+l+2\right) }\tanh ^2\left( \sqrt{\frac{b_1}{8 a l^2+8 a l+16 a}} x \right) e^{i \left( -\frac{b_1^2 \left( l^4+2 l^3+5 l^2+4 l+3\right) }{4 b_2 l \left( l^2+l+2\right) ^2}t+\theta \right) }, \end{aligned}$$

(122)

$$\begin{aligned} q(x,t)= & {} -\frac{b_1 \left( l^2+l+3\right) }{2 b_2 \left( l^2+l+2\right) }\coth ^2\left( \sqrt{\frac{b_1}{8 a l^2+8 a l+16 a}} x \right) e^{i \left( -\frac{b_1^2 \left( l^4+2 l^3+5 l^2+4 l+3\right) }{4 b_2 l \left( l^2+l+2\right) ^2}t+\theta \right) }. \end{aligned}$$

(123)

Fig. 3

Effect of the generalized temporal evolution

4.1.2 The new projective Riccati equations method

Under this approach, the solution is structured as follows:

$$\begin{aligned} U(\xi )=\eta _0+\eta _1 F(\xi )+\eta _2 F(\xi )^2+\rho _1 G(\xi )+\rho _2 F(\xi ) G(\xi ). \end{aligned}$$

(124)

When Eq. (124) is substituted alongside Eqs. (8) and (9) into Eq. (106), we obtain:

$$\begin{aligned}{} & {} 8 a \eta _2 K^2 l^2 \rho _2 R(\tau )+8 a \eta _2 K^2 l \rho _2 R(\tau )+24 a \eta _2 K^2 \rho _2 R(\tau )+3 b_2 \eta _2^2 \rho _2+b_2 \rho _2^3 R(\tau )=0. \end{aligned}$$

(125)

$$\begin{aligned}{} & {} -32 a \eta _2 K^2 \rho _2 \tau -12 a \eta _2 K^2 l^2 \rho _2 \tau +4 a \eta _1 K^2 l^2 \rho _2 R(\tau )+4 a \eta _2 K^2 l^2 \rho _1 R(\tau )-12 a \eta _2 K^2 l \rho _2 \tau \nonumber \\{} & {} \quad +4 a \eta _1 K^2 l \rho _2 R(\tau )+4 a \eta _2 K^2 l \rho _1 R(\tau )\nonumber \\{} & {} \quad +16 a \eta _1 K^2 \rho _2 R(\tau )+16 a \eta _2 K^2 \rho _1 R(\tau )+6 b_2 \eta _1 \eta _2 \rho _2+3 b_2 \eta _2^2 \rho _1\nonumber \\{} & {} \quad -2 b_2 \rho _2^3 \tau +3 b_2 \rho _1 \rho _2^2 R(\tau )=0. \end{aligned}$$

(126)

$$\begin{aligned}{} & {} -18 a \eta _1 K^2 \rho _2 \tau -22 a \eta _2 K^2 \rho _1 \tau +10 a \eta _2 K^2 \rho _2-6 a \eta _1 K^2 l^2 \rho _2 \tau -4 a \eta _2 K^2 l^2 \rho _1 \tau +4 a \eta _2 K^2 l^2 \rho _2\nonumber \\{} & {} \quad +2 a \eta _1 K^2 l^2 \rho _1 R(\tau )-6 a \eta _1 K^2 l \rho _2 \tau -4 a \eta _2 K^2 l \rho _1 \tau \nonumber \\{} & {} \quad +4 a \eta _2 K^2 l \rho _2+2 a \eta _1 K^2 l \rho _1 R(\tau )+12 a \eta _0 K^2 \rho _2 R(\tau )\nonumber \\{} & {} \quad +8 a \eta _1 K^2 \rho _1 R(\tau )+3 b_2 \eta _1^2 \rho _2+2 b_1 \eta _2 \rho _2+6 b_2 \eta _0 \eta _2 \rho _2\nonumber \\{} & {} \quad +6 b_2 \eta _1 \eta _2 \rho _1-6 b_2 \rho _1 \rho _2^2 \tau +b_2 \rho _2^3+3 b_2 \rho _1^2 \rho _2 R(\tau )=0. \end{aligned}$$

(127)

$$\begin{aligned}{} & {} -8 a \eta _1 K^2 \rho _1 \tau -12 a \eta _0 K^2 \rho _2 \tau +8 a \eta _2 K^2 \rho _1+4 a \eta _1 K^2 \rho _2\nonumber \\{} & {} \quad -2 a \eta _1 K^2 l^2 \rho _1 \tau +2 a \eta _1 K^2 l^2 \rho _2-2 a \eta _1 K^2 l \rho _1 \tau \nonumber \\{} & {} \quad +2 a \eta _1 K^2 l \rho _2+4 a \eta _0 K^2 \rho _1 R(\tau )+3 b_2 \eta _1^2 \rho _1+2 b_1 \eta _2 \rho _1\nonumber \\{} & {} \quad +6 b_2 \eta _0 \eta _2 \rho _1+2 b_1 \eta _1 \rho _2+6 b_2 \eta _0 \eta _1 \rho _2-6 b_2 \rho _2 \rho _1^2 \tau \nonumber \\{} & {} \quad +3 b_2 \rho _2^2 \rho _1+b_2 \rho _1^3 R(\tau )=0. \end{aligned}$$

(128)

$$\begin{aligned}{} & {} -2 a \eta _0 K^2 \rho _1 \tau +2 a \eta _1 K^2 \rho _1+2 a \eta _0 K^2 \rho _2+2 b_1 \eta _1 \rho _1+6 b_2 \eta _0 \eta _1 \rho _1+3 b_2 \eta _0^2 \rho _2\nonumber \\{} & {} \quad +2 b_1 \eta _0 \rho _2-2 b_2 \rho _1^3 \tau +3 b_2 \rho _2 \rho _1^2-l \rho _2 \omega =0. \end{aligned}$$

(129)

$$\begin{aligned}{} & {} 3 b_2 \eta _0^2 \rho _1+2 b_1 \eta _0 \rho _1+b_2 \rho _1^3-l \rho _1 \omega =0. \end{aligned}$$

(130)

$$\begin{aligned}{} & {} 4 a \eta _2^2 K^2 l^2 R(\tau )+4 a K^2 l^2 \rho _2^2 R(\tau )^2+4 a \eta _2^2 K^2 l R(\tau )+4 a K^2 l \rho _2^2 R(\tau )^2+12 a \eta _2^2 K^2 R(\tau )\nonumber \\{} & {} \quad +12 a K^2 \rho _2^2 R(\tau )^2+b_2 \eta _2^3+3 b_2 \eta _2 \rho _2^2 R(\tau )=0. \end{aligned}$$

(131)

$$\begin{aligned}{} & {} -20 a \eta _2^2 K^2 \tau -8 a \eta _2^2 K^2 l^2 \tau +4 a \eta _1 \eta _2 K^2 l^2 R(\tau )-12 a K^2 l^2 \rho _2^2 \tau R(\tau )\nonumber \\{} & {} \quad +4 a K^2 l^2 \rho _1 \rho _2 R(\tau )^2-8 a \eta _2^2 K^2 l \tau \nonumber \\{} & {} \quad +4 a \eta _1 \eta _2 K^2 l R(\tau )-12 a K^2 l \rho _2^2 \tau R(\tau )+4 a K^2 l \rho _1 \rho _2 R(\tau )^2+16 a \eta _1 \eta _2 K^2 R(\tau )-36 a K^2 \rho _2^2 \tau R(\tau )\nonumber \\{} & {} \quad +16 a K^2 \rho _1 \rho _2 R(\tau )^2-6 b_2 \eta _2 \rho _2^2 \tau +3 b_2 \eta _1 \eta _2^2+3 b_2 \eta _1 \rho _2^2 R(\tau )+6 b_2 \eta _2 \rho _1 \rho _2 R(\tau )=0. \end{aligned}$$

(132)

$$\begin{aligned}{} & {} -26 a \eta _1 \eta _2 K^2 \tau +8 a \eta _2^2 K^2-8 a \eta _1 \eta _2 K^2 l^2 \tau +4 a \eta _2^2 K^2 l^2\nonumber \\{} & {} \quad +9 a K^2 l^2 \rho _2^2 \tau ^2+a \eta _1^2 K^2 l^2 R(\tau )+a K^2 l^2 \rho _1^2 R(\tau )^2\nonumber \\{} & {} \quad +4 a K^2 l^2 \rho _2^2 R(\tau )-10 a K^2 l^2 \rho _1 \rho _2 \tau R(\tau )-8 a \eta _1 \eta _2 K^2 l \tau +4 a \eta _2^2 K^2 l+9 a K^2 l \rho _2^2 \tau ^2+a \eta _1^2 K^2 l R(\tau )\nonumber \\{} & {} \quad +a K^2 l \rho _1^2 R(\tau )^2+4 a K^2 l \rho _2^2 R(\tau )-10 a K^2 l \rho _1 \rho _2 \tau R(\tau )\nonumber \\{} & {} \quad +24 a K^2 \rho _2^2 \tau ^2+4 a \eta _1^2 K^2 R(\tau )+12 a \eta _0 \eta _2 K^2 R(\tau )\nonumber \\{} & {} \quad +4 a K^2 \rho _1^2 R(\tau )^2+14 a K^2 \rho _2^2 R(\tau )-46 a K^2 \rho _1 \rho _2 \tau R(\tau )\nonumber \\{} & {} \quad -6 b_2 \eta _1 \rho _2^2 \tau -12 b_2 \eta _2 \rho _1 \rho _2 \tau +3 b_2 \eta _2 \rho _2^2+b_1 \eta _2^2+3 b_2 \eta _0 \eta _2^2\nonumber \\{} & {} \quad +3 b_2 \eta _1^2 \eta _2+3 b_2 \eta _2 \rho _1^2 R(\tau )+3 b_2 \eta _0 \rho _2^2 R(\tau )+6 b_2 \eta _1 \rho _1 \rho _2 R(\tau )+b_1 \rho _2^2 R(\tau )=0. \end{aligned}$$

(133)

$$\begin{aligned}{} & {} -6 a \eta _1^2 K^2 \tau -20 a \eta _0 \eta _2 K^2 \tau +10 a \eta _2 \eta _1 K^2-2 a \eta _1^2 K^2 l^2 \tau \nonumber \\{} & {} \quad +4 a \eta _2 \eta _1 K^2 l^2+6 a K^2 l^2 \rho _1 \rho _2 \tau ^2-6 a K^2 l^2 \rho _2^2 \tau \nonumber \\{} & {} \quad -2 a K^2 l^2 \rho _1^2 \tau R(\tau )+2 a K^2 l^2 \rho _1 \rho _2 R(\tau )-2 a \eta _1^2 K^2 l \tau +4 a \eta _2 \eta _1 K^2 l+6 a K^2 l \rho _1 \rho _2 \tau ^2-6 a K^2 l \rho _2^2 \tau \nonumber \\{} & {} \quad -2 a K^2 l \rho _1^2 \tau R(\tau )+2 a K^2 l \rho _1 \rho _2 R(\tau )+28 a K^2 \rho _1 \rho _2 \tau ^2\nonumber \\{} & {} \quad -16 a K^2 \rho _2^2 \tau +4 a \eta _0 \eta _1 K^2 R(\tau )-10 a K^2 \rho _1^2 \tau R(\tau )\nonumber \\{} & {} \quad +18 a K^2 \rho _1 \rho _2 R(\tau )-12 b_2 \eta _1 \rho _1 \rho _2 \tau -6 b_2 \eta _2 \rho _1^2 \tau -6 b_2 \eta _0 \rho _2^2 \tau +3 b_2 \eta _1 \rho _2^2+6 b_2 \eta _2 \rho _1 \rho _2\nonumber \\{} & {} \quad +b_2 \eta _1^3+2 b_1 \eta _2 \eta _1+6 b_2 \eta _0 \eta _2 \eta _1-2 b_1 \rho _2^2 \tau +3 b_2 \eta _1 \rho _1^2 R(\tau )\nonumber \\{} & {} \quad +6 b_2 \eta _0 \rho _1 \rho _2 R(\tau )+2 b_1 \rho _1 \rho _2 R(\tau )=0. \end{aligned}$$

(134)

$$\begin{aligned}{} & {} -6 a \eta _0 \eta _1 K^2 \tau +2 a \eta _1^2 K^2+8 a \eta _0 \eta _2 K^2+a \eta _1^2 K^2 l^2+a K^2 l^2 \rho _1^2 \tau ^2-2 a K^2 l^2 \rho _1 \rho _2 \tau \nonumber \\{} & {} \quad +a K^2 l^2 \rho _2^2+a \eta _1^2 K^2 l\nonumber \\{} & {} \quad +a K^2 l \rho _1^2 \tau ^2-2 a K^2 l \rho _1 \rho _2 \tau +a K^2 l \rho _2^2+4 a K^2 \rho _1^2 \tau ^2-18 a K^2 \rho _1 \rho _2 \tau +2 a K^2 \rho _2^2+4 a K^2 \rho _1^2 R(\tau )\nonumber \\{} & {} \quad -6 b_2 \eta _1 \rho _1^2 \tau -12 b_2 \eta _0 \rho _1 \rho _2 \tau +3 b_2 \eta _2 \rho _1^2+3 b_2 \eta _0 \rho _2^2+6 b_2 \eta _1 \rho _1 \rho _2+b_1 \eta _1^2+3 b_2 \eta _0 \eta _1^2\nonumber \\{} & {} \quad +3 b_2 \eta _0^2 \eta _2+2 b_1 \eta _0 \eta _2-4 b_1 \rho _1 \rho _2 \tau +b_1 \rho _2^2+3 b_2 \eta _0 \rho _1^2 R(\tau )+b_1 \rho _1^2 R(\tau )-\eta _2 l \omega =0. \end{aligned}$$

(135)

$$\begin{aligned}{} & {} 2 a \eta _0 \eta _1 K^2-2 a K^2 \rho _1^2 \tau +2 a K^2 \rho _1 \rho _2-6 b_2 \eta _0 \rho _1^2 \tau +3 b_2 \eta _1 \rho _1^2+6 b_2 \eta _0 \rho _1 \rho _2+3 b_2 \eta _0^2 \eta _1\nonumber \\{} & {} \quad +2 b_1 \eta _0 \eta _1-2 b_1 \rho _1^2 \tau +2 b_1 \rho _1 \rho _2-\eta _1 l \omega =0. \end{aligned}$$

(136)

$$\begin{aligned}{} & {} 3 b_2 \eta _0 \rho _1^2+b_2 \eta _0^3+b_1 \eta _0^2+b_1 \rho _1^2-\eta _0 l \omega =0. \end{aligned}$$

(137)

The solutions derived from these equations leave us with:

Case–1:   $R (\tau )=0$.

$$\begin{aligned}{} & {} K=\pm \sqrt{\frac{b_1}{2 a l^2+2 a l+4 a}},~\eta _0=-\frac{b_1 \left( l^2+l+3\right) }{2 b_2 \left( l^2+l+2\right) }\nonumber \\{} & {} \quad ~\eta _1=\frac{b_1 \left( l^2+l+3\right) \tau }{b_2 \left( l^2+l+2\right) },~\eta _2=\rho _1=\rho _2=0,\nonumber \\{} & {} \quad \omega =-\frac{b_1^2 \left( l^4+2 l^3+5 l^2+4 l+3\right) }{4 b_2 l \left( l^2+l+2\right) ^2}. \end{aligned}$$

(138)

The bright and singular soliton solutions of the governing equation (101) for $\frac{b_1}{8 a l^2+8 a l+16 a}>0$ therefore emerge as:

$$\begin{aligned} q(x,t)= \left\{ -\frac{b_1 \left( l^2+l+3\right) }{2 b_2 \left( l^2+l+2\right) }+\frac{b_1 \left( l^2+l+3\right) \text {sech}^2\left( \sqrt{\frac{b_1}{8 a l^2+8 a l+16 a}}x\right) }{2 b_2 \left( l^2+l+2\right) }\right\} e^{i \left( -\frac{b_1^2 \left( l^4+2 l^3+5 l^2+4 l+3\right) }{4 b_2 l \left( l^2+l+2\right) ^2}t+\theta \right) }, \end{aligned}$$

(139)

and

$$\begin{aligned} q(x,t)= \left\{ -\frac{b_1 \left( l^2+l+3\right) }{2 b_2 \left( l^2+l+2\right) }-\frac{b_1 \left( l^2+l+3\right) \text {csch}^2\left( \sqrt{\frac{b_1}{8 a l^2+8 a l+16 a}}x\right) }{2 b_2 \left( l^2+l+2\right) }\right\} e^{i \left( -\frac{b_1^2 \left( l^4+2 l^3+5 l^2+4 l+3\right) }{4 b_2 l \left( l^2+l+2\right) ^2}t+\theta \right) }. \end{aligned}$$

(140)

Fig. 4

Effect of the generalized temporal evolution

Case–2:   $R (\tau )=\frac{24}{25}\tau ^2$.

$$\begin{aligned}{} & {} K=\pm \sqrt{\frac{b_1}{2 a l^2+2 a l+4 a}},~\eta _0=-\frac{b_1 \left( l^2+l+3\right) }{2 b_2 \left( l^2+l+2\right) },\nonumber \\{} & {} \quad \eta _1=\frac{b_1 \left( l^2+l+3\right) \tau }{b_2 \left( l^2+l+2\right) },~\eta _2=-\frac{24 b_1 \left( l^2+l+3\right) \tau ^2}{25 b_2 \left( l^2+l+2\right) },\nonumber \\{} & {} \quad \rho _1=0,~\rho _2=-\frac{2 \sqrt{6} \sqrt{b_1^2 \left( -\left( l^2+l+3\right) ^2\right) \tau ^2}}{5 \sqrt{b_2^2 \left( -\left( l^2+l+2\right) ^2\right) }},~\omega =-\frac{b_1^2 \left( l^2+l+1\right) \left( l^2+l+3\right) }{4 b_2 l \left( l^2+l+2\right) ^2}. \end{aligned}$$

(141)

The bright–dark soliton solution of the governing equation (101) for $\frac{b_1}{2 a l^2+2 a l+4 a}>0$ thus reads:

$$\begin{aligned} q(x,t)= & {} \Bigg \{ -\frac{b_1 \left( l^2+l+3\right) }{2 b_2 \left( l^2+l+2\right) }+\frac{5 b_1 \left( l^2+l+3\right) \text {sech}\left( \sqrt{\frac{b_1}{2 a l^2+2 a l+4 a}}x\right) }{b_2 \left( l^2+l+2\right) \left( 5 \text {sech}\left( \sqrt{\frac{b_1}{2 a l^2+2 a l+4 a}}x\right) \pm 1\right) }\nonumber \\{} & {} \quad \mp \frac{2 \sqrt{6} \sqrt{b_1^2 \left( -\left( l^2+l+3\right) ^2\right) \tau ^2} \tanh \left( \sqrt{\frac{b_1}{2 a l^2+2 a l+4 a}}x\right) \text {sech}\left( \sqrt{\frac{b_1}{2 a l^2+2 a l+4 a}}x\right) }{\tau \sqrt{b_2^2 \left( -\left( l^2+l+2\right) ^2\right) } \left( 1\pm 5 \text {sech}\left( \sqrt{\frac{b_1}{2 a l^2+2 a l+4 a}}x\right) \right) \left( 5 \text {sech}\left( \sqrt{\frac{b_1}{2 a l^2+2 a l+4 a}}x\right) \pm 1\right) }\nonumber \\{} & {} \quad -\frac{24 b_1 \left( l^2+l+3\right) \text {sech}^2\left( \sqrt{\frac{b_1}{2 a l^2+2 a l+4 a}}x\right) }{b_2 \left( l^2+l+2\right) \left( 5 \text {sech}\left( \sqrt{\frac{b_1}{2 a l^2+2 a l+4 a}}x\right) \pm 1\right) {}^2}\Bigg \} e^{i \left( -\frac{b_1^2 \left( l^2+l+1\right) \left( l^2+l+3\right) }{4 b_2 l \left( l^2+l+2\right) ^2}t+\theta \right) }. \end{aligned}$$

(142)

Case–3:  $R (\tau )=\frac{5}{9}\tau ^2$.

$$\begin{aligned}{} & {} K=\pm \sqrt{\frac{b_1}{2 a l^2+2 a l+4 a}},~\eta _0=-\frac{b_1 \left( l^2+l+3\right) }{2 b_2 \left( l^2+l+2\right) },\nonumber \\{} & {} \quad \eta _1=\frac{b_1 \left( l^2+l+3\right) \tau }{b_2 \left( l^2+l+2\right) },~\eta _2=-\frac{5 b_1 \left( l^2+l+3\right) \tau ^2}{9 b_2 \left( l^2+l+2\right) },\nonumber \\{} & {} \quad \rho _1=0,~\rho _2=\frac{\sqrt{5} \sqrt{b_1^2 \left( -\left( l^2+l+3\right) ^2\right) \tau ^2}}{3 \sqrt{b_2^2 \left( -\left( l^2+l+2\right) ^2\right) }},~\omega =-\frac{b_1^2 \left( l^2+l+1\right) \left( l^2+l+3\right) }{4 b_2 l \left( l^2+l+2\right) ^2}. \end{aligned}$$

(143)

The bright–singular soliton solution of the governing equation (101) for $\frac{b_1}{2 a l^2+2 a l+4 a}>0$ thus comes out to be:

$$\begin{aligned} q(x,t)= & {} \Bigg \{-\frac{b_1 \left( l^2+l+3\right) }{2 b_2 \left( l^2+l+2\right) }+\frac{3 b_1 \left( l^2+l+3\right) \text {sech}\left( \sqrt{\frac{b_1}{2 a l^2+2 a l+4 a}}x\right) }{b_2 \left( l^2+l+2\right) \left( 3 \text {sech}\left( \sqrt{\frac{b_1}{2 a l^2+2 a l+4 a}}x\right) \pm 2\right) }\nonumber \\{} & {} \quad -\frac{5 b_1 \left( l^2+l+3\right) \text {sech}^2\left( \sqrt{\frac{b_1}{2 a l^2+2 a l+4 a}}x\right) }{b_2 \left( l^2+l+2\right) \left( 3 \text {sech}\left( \sqrt{\frac{b_1}{2 a l^2+2 a l+4 a}}x\right) \pm 2\right) {}^2}\nonumber \\{} & {} \quad +\frac{2 \sqrt{5} \sqrt{b_1^2 \left( -\left( l^2+l+3\right) ^2\right) \tau ^2}}{\tau \sqrt{b_2^2 \left( -\left( l^2+l+2\right) ^2\right) }\left( 3 \text {sech}\left( \sqrt{\frac{b_1}{2 a l^2+2 a l+4 a}}x\right) \pm 2\right) }\times \nonumber \\{} & {} \quad \frac{ \text {sech}\left( \sqrt{\frac{b_1}{2 a l^2+2 a l+4 a}}x\right) }{ \left( 2 \coth \left( \sqrt{\frac{b_1}{2 a l^2+2 a l+4 a}}x\right) \pm 3 \text {csch}\left( \sqrt{\frac{b_1}{2 a l^2+2 a l+4 a}}x\right) \right) }\Bigg \} e^{i \left( -\frac{b_1^2 \left( l^2+l+1\right) \left( l^2+l+3\right) }{4 b_2 l \left( l^2+l+2\right) ^2}t+\theta \right) }. \end{aligned}$$

(144)

Case–4:  $R (\tau )=\tau ^2-1$.

$$\begin{aligned}{} & {} K=\pm \sqrt{\frac{b_1}{2 a l^2+2 a l+4 a}},~\eta _0=-\frac{b_1 \left( l^2+l+3\right) }{2 b_2 \left( l^2+l+2\right) },~\eta _1=\frac{b_1 \left( l^2+l+3\right) \tau }{b_2 \left( l^2+l+2\right) },~\rho _1=0,\nonumber \\{} & {} \quad \eta _2=-\frac{b_1 \left( l^2+l+3\right) \left( \tau ^2-1\right) }{b_2 \left( l^2+l+2\right) },~\rho _2=\frac{b_1 \left( l^2+l+3\right) \sqrt{\tau ^2-1}}{b_2 \left( l^2+l+2\right) },\nonumber \\{} & {} \quad \omega =-\frac{b_1^2 \left( l^4+2 l^3+5 l^2+4 l+3\right) }{4 b_2 l \left( l^2+l+2\right) ^2}. \end{aligned}$$

(145)

The bright–dark soliton solution of the governing equation (101) for $\frac{b_1}{2 a l^2+2 a l+4 a}$ falls out as:

$$\begin{aligned} q(x,t)= & {} \Bigg \{-\frac{b_1 \left( l^2+l+3\right) }{2 b_2 \left( l^2+l+2\right) }-\frac{16 b_1 \left( l^2+l+3\right) \left( \tau ^2-1\right) \text {sech}^2\left( \sqrt{\frac{b_1}{2 a l^2+2 a l+4 a}}x\right) }{b_2 \left( l^2+l+2\right) \left( \left( 4 \tau \text {sech}\left( \sqrt{\frac{b_1}{2 a l^2+2 a l+4 a}}x\right) \pm 3 \tanh \left( \sqrt{\frac{b_1}{2 a l^2+2 a l+4 a}}x\right) \right) +5\right) {}^2} \nonumber \\{} & {} \quad +\frac{4 b_1 \left( l^2+l+3\right) \tau \text {sech}\left( \sqrt{\frac{b_1}{2 a l^2+2 a l+4 a}}x\right) }{b_2 \left( l^2+l+2\right) \left( \left( 4 \tau \text {sech}\left( \sqrt{\frac{b_1}{2 a l^2+2 a l+4 a}}x\right) \pm 3 \tanh \left( \sqrt{\frac{b_1}{2 a l^2+2 a l+4 a}}x\right) \right) +5\right) }\nonumber \\{} & {} \quad +\frac{4 b_1 \left( l^2+l+3\right) \sqrt{\tau ^2-1} \text {sech}\left( \sqrt{\frac{b_1}{2 a l^2+2 a l+4 a}}x\right) \left( 3\pm 5 \tanh \left( \sqrt{\frac{b_1}{2 a l^2+2 a l+4 a}}x\right) \right) }{b_2 \left( l^2+l+2\right) \left( \left( 4 \tau \text {sech}\left( \sqrt{\frac{b_1}{2 a l^2+2 a l+4 a}}x\right) \pm 3 \tanh \left( \sqrt{\frac{b_1}{2 a l^2+2 a l+4 a}}x\right) \right) +5\right) {}^2}\Bigg \}\nonumber \\{} & {} \quad \times e^{i \left( -\frac{b_1^2 \left( l^4+2 l^3+5 l^2+4 l+3\right) }{4 b_2 l \left( l^2+l+2\right) ^2}t+\theta \right) }, \end{aligned}$$

(146)

and

$$\begin{aligned} q(x,t)= & {} \Bigg \{-\frac{b_1 \left( l^2+l+3\right) }{2 b_2 \left( l^2+l+2\right) }-\frac{b_1 \left( l^2+l+3\right) \left( \tau ^2-1\right) \text {sech}^2\left( x \sqrt{\frac{b_1}{2 a l^2+2 a l+4 a}}\right) }{b_2 \left( l^2+l+2\right) \left( \tau \text {sech}\left( x \sqrt{\frac{b_1}{2 a l^2+2 a l+4 a}}\right) +1\right) {}^2}\nonumber \\{} & {} \quad +\frac{b_1 \left( l^2+l+3\right) \tau \text {sech}\left( x \sqrt{\frac{b_1}{2 a l^2+2 a l+4 a}}\right) }{b_2 \left( l^2+l+2\right) \left( \tau \text {sech}\left( x \sqrt{\frac{b_1}{2 a l^2+2 a l+4 a}}\right) +1\right) }\nonumber \\{} & {} \quad \pm \frac{b_1 \left( l^2+l+3\right) \sqrt{\tau ^2-1} \tanh \left( x \sqrt{\frac{b_1}{2 a l^2+2 a l+4 a}}\right) \text {sech}\left( x \sqrt{\frac{b_1}{2 a l^2+2 a l+4 a}}\right) }{b_2 \left( l^2+l+2\right) \left( \tau \text {sech}\left( x \sqrt{\frac{b_1}{2 a l^2+2 a l+4 a}}\right) +1\right) {}^2}\Bigg \} e^{i \left( -\frac{b_1^2 \left( l^4+2 l^3+5 l^2+4 l+3\right) }{4 b_2 l \left( l^2+l+2\right) ^2}t+\theta \right) }. \end{aligned}$$

(147)

Case-5:  $R (\tau )=\tau ^2+1$.

$$\begin{aligned}{} & {} K=\pm \sqrt{\frac{b_1}{2 a l^2+2 a l+4 a}},~\eta _0=-\frac{b_1 \left( l^2+l+3\right) }{2 b_2 \left( l^2+l+2\right) },~\eta _1=\frac{b_1 \left( l^2+l+3\right) \tau }{b_2 \left( l^2+l+2\right) },\nonumber \\{} & {} \quad ~\eta _2=-\frac{b_1 \left( l^2+l+3\right) \left( \tau ^2+1\right) }{b_2 \left( l^2+l+2\right) },~\rho _1=0,~\rho _2=\frac{b_1 \left( l^2+l+3\right) \sqrt{\tau ^2+1}}{b_2 \left( l^2+l+2\right) },\nonumber \\{} & {} \quad \omega =-\frac{b_1^2 \left( l^4+2 l^3+5 l^2+4 l+3\right) }{4 b_2 l \left( l^2+l+2\right) ^2}. \end{aligned}$$

(148)

Hence, the singular–singular soliton solution of the governing equation (101) for $\frac{b_1}{2 a l^2+2 a l+4 a}>0$ reads:

$$\begin{aligned} q(x,t)= & {} \Bigg \{-\frac{b_1 \left( l^2+l+3\right) }{2 b_2 \left( l^2+l+2\right) }-\frac{b_1 \left( l^2+l+3\right) \left( \tau ^2+1\right) \text {csch}^2\left( x \sqrt{\frac{b_1}{2 a l^2+2 a l+4 a}}\right) }{b_2 \left( l^2+l+2\right) \left( 1\pm \tau \text {csch}\left( x \sqrt{\frac{b_1}{2 a l^2+2 a l+4 a}}\right) \right) {}^2}\nonumber \\{} & {} \quad +\frac{b_1 \left( l^2+l+3\right) \sqrt{\tau ^2+1} \coth \left( x \sqrt{\frac{b_1}{2 a l^2+2 a l+4 a}}\right) \text {csch}\left( x \sqrt{\frac{b_1}{2 a l^2+2 a l+4 a}}\right) }{b_2 \left( l^2+l+2\right) \left( 1\pm \tau \text {csch}\left( x \sqrt{\frac{b_1}{2 a l^2+2 a l+4 a}}\right) \right) {}^2}\nonumber \\{} & {} \quad \pm \frac{b_1 \left( l^2+l+3\right) \tau \text {csch}\left( x \sqrt{\frac{b_1}{2 a l^2+2 a l+4 a}}\right) }{b_2 \left( l^2+l+2\right) \left( 1\pm \tau \text {csch}\left( x \sqrt{\frac{b_1}{2 a l^2+2 a l+4 a}}\right) \right) }\Bigg \} e^{i \left( -\frac{b_1^2 \left( l^4+2 l^3+5 l^2+4 l+3\right) }{4 b_2 l \left( l^2+l+2\right) ^2}t+\theta \right) }. \end{aligned}$$

(149)

4.2 Quadratic–cubic–quartic nonlinearity

The perturbed FLE with nonlinear CD and quadratic–cubic–quartic SPM and having generalized temporal evolution is

$$\begin{aligned}{} & {} i \left( q^l \right) _t + a(\left| q \right| ^n q^l)_{xx} + \left( b_1 \left| q \right| + b_2 \left| q \right| ^2 + b_3 \left| q \right| ^3 \right) q^l + i \sigma \left| q \right| ^2 \left( q^l \right) _x\nonumber \\{} & {} \quad \quad = i \left[ \alpha \left( q^l \right) _x + \lambda \left( \left| q \right| ^2 q^l \right) _x + \mu \left( \left| q \right| ^2 \right) _x q^l \right] . \end{aligned}$$

(150)

Applying the transformation (18), decomposes Eq. (150) into the following real and imaginary parts:

$$\begin{aligned}{} & {} a K^2 \left( l^2+l (2 n-1)+(n-1) n\right) U^n U^{\prime 2}+a K^2 (l+n) U^{n+1} U''\nonumber \\{} & {} \quad +b_3 U^5+b_2 U^4+b_1 U^3-l \omega U^2=0, \end{aligned}$$

(151)

and

$$\begin{aligned} -K U U' \left( U^2 (\lambda (l+2)-l \sigma +2 \mu )+\alpha l\right) =0. \end{aligned}$$

(152)

From the imaginary part, we can derive:

$$\begin{aligned} \alpha =0,~~\lambda (l+2)-l \sigma +2 \mu =0. \end{aligned}$$

(153)

To satisfy the integrability condition, we consider $n=1$. In this case, Eq. (150) turns out to be:

$$\begin{aligned}{} & {} i \left( q^l \right) _t + a(\left| q \right| q^l)_{xx} + \left( b_1 \left| q \right| + b_2 \left| q \right| ^2 + b_3 \left| q \right| ^3 \right) q^l + i \sigma \left| q \right| ^2 \left( q^l \right) _x\nonumber \\{} & {} \quad = i \left[ \lambda \left( \left| q \right| ^2 q^l \right) _x + \mu \left( \left| q \right| ^2 \right) _x q^l \right] , \end{aligned}$$

(154)

and Eq. (151) can be rewritten as:

$$\begin{aligned} a K^2 \left( l^2+l\right) U^{\prime 2}+2 a K^2 U U''+b_3 U^4+b_2 U^3+b_1 U^2-l \omega U=0. \end{aligned}$$

(155)

Applying the balancing principle between $U U''$ and $U^4$, leads to $N=1$. In the following subsections, the integration algorithms will be utilized to Eq. (155).

4.2.1 Enhanced Kudryashov’s method

Thus, this method offers a solution to Eq. (155) in the following form:

$$\begin{aligned} U(\xi )=\eta _0 + \eta _1 R(\xi )+ \rho _1 \left( \frac{ R^{\prime} (\xi )}{ R(\xi )}\right) . \end{aligned}$$

(156)

By incorporating Eq. (156) in conjunction with Eq. (5) into Eq. (155), we obtain:

$$\begin{aligned}{} & {} -4 a \eta _1^2 K^2 \chi -a \eta _1^2 K^2 l^2 \chi +a K^2 l^2 \rho _1^2 \chi ^2-a \eta _1^2 K^2 l \chi +a K^2 l \rho _1^2 \chi ^2+4 a K^2 \rho _1^2 \chi ^2\nonumber \\{} & {} \quad -6 b_3 \eta _1^2 \rho _1^2 \chi +b_3 \eta _1^4+b_3 \rho _1^4 \chi ^2=0. \end{aligned}$$

(157)

$$\begin{aligned}{} & {} -4 a \eta _0 \eta _1 K^2 \chi -3 b_2 \eta _1 \rho _1^2 \chi -12 b_3 \eta _0 \eta _1 \rho _1^2 \chi +b_2 \eta _1^3+4 b_3 \eta _0 \eta _1^3=0. \end{aligned}$$

(158)

$$\begin{aligned}{} & {} -8 a \eta _1 K^2 \rho _1 \chi -2 a \eta _1 K^2 l^2 \rho _1 \chi -2 a \eta _1 K^2 l \rho _1 \chi -4 b_3 \eta _1 \rho _1^3 \chi +4 b_3 \eta _1^3 \rho _1=0. \end{aligned}$$

(159)

$$\begin{aligned}{} & {} \quad 2 a \eta _1^2 K^2+a \eta _1^2 K^2 l^2+a \eta _1^2 K^2 l-4 a K^2 \rho _1^2 \chi -6 b_3 \eta _0^2 \rho _1^2 \chi -3 b_2 \eta _0 \rho _1^2 \chi +6 b_3 \eta _1^2 \rho _1^2\nonumber \\{} & {} \quad +6 b_3 \eta _0^2 \eta _1^2+b_1 \eta _1^2+3 b_2 \eta _0 \eta _1^2-2 b_3 \rho _1^4 \chi -b_1 \rho _1^2 \chi =0. \end{aligned}$$

(160)

$$\begin{aligned}{} & {} -4 a \eta _0 K^2 \rho _1 \chi -4 b_3 \eta _0 \rho _1^3 \chi +3 b_2 \eta _1^2 \rho _1+12 b_3 \eta _0 \eta _1^2 \rho _1+b_2 \rho _1^3 (-\chi )=0. \end{aligned}$$

(161)

$$\begin{aligned}{} & {} 2 a \eta _1 \eta _0 K^2+12 b_3 \eta _1 \eta _0 \rho _1^2+3 b_2 \eta _1 \rho _1^2+4 b_3 \eta _1 \eta _0^3+3 b_2 \eta _1 \eta _0^2+2 b_1 \eta _1 \eta _0-\eta _1 l \omega =0. \end{aligned}$$

(162)

$$\begin{aligned}{} & {} 4 b_3 \eta _0^3 \rho _1+3 b_2 \eta _0^2 \rho _1+4 b_3 \eta _0 \rho _1^3+2 b_1 \eta _0 \rho _1+b_2 \rho _1^3-l \rho _1 \omega =0. \end{aligned}$$

(163)

$$\begin{aligned}{} & {} 2 a \eta _1 K^2 \rho _1+4 b_3 \eta _1 \rho _1^3+12 b_3 \eta _0^2 \eta _1 \rho _1+2 b_1 \eta _1 \rho _1+6 b_2 \eta _0 \eta _1 \rho _1=0. \end{aligned}$$

(164)

$$\begin{aligned}{} & {} 6 b_3 \eta _0^2 \rho _1^2+3 b_2 \eta _0 \rho _1^2+b_3 \eta _0^4+b_2 \eta _0^3+b_1 \eta _0^2+b_3 \rho _1^4+b_1 \rho _1^2-\eta _0 l \omega =0. \end{aligned}$$

(165)

All of these together yield the subsequent outcomes:

Result-1:

$$\begin{aligned}{} & {} K=\pm \sqrt{\frac{b_1}{5 a l^2+5 a l+10 a}},~\eta _0=\pm \sqrt{\frac{b_1 \left( l^2+l+4\right) }{5 b_3 l^2+5 b_3 l+10 b_3}},~\eta _1=\pm \sqrt{\frac{b_1 \left( l^2+l+4\right) \chi }{5 \left( b_3 l^2+b_3 l+2 b_3\right) }},\nonumber \\{} & {} \quad \rho _1=0,~\omega =\frac{2 b_1^{3/2} b_3 \left( l^2+l+1\right) \sqrt{l^2+l+4}}{5 \sqrt{5} l \left( b_3 \left( l^2+l+2\right) \right) {}^{3/2}},~b_2=\mp \frac{4 \sqrt{b_1} b_3 \left( l^2+l+3\right) }{\sqrt{5} \sqrt{l^2+l+4} \sqrt{b_3 \left( l^2+l+2\right) }}. \end{aligned}$$

(166)

The resulting soliton solution of the governing equation (150) comes out to be:

$$\begin{aligned} q(x,t)=\pm \sqrt{\frac{b_1 \left( l^2+l+4\right) }{5 b_3 l^2+5 b_3 l+10 b_3}}\left\{ \frac{4 d \sqrt{\chi } e^{ \pm \sqrt{\frac{b_1}{5 a l^2+5 a l+10 a}}x}}{4 d^2 e^{ \pm 2 \sqrt{\frac{b_1}{5 a l^2+5 a l+10 a}}x}+\chi }+1\right\} e^{i \left( \frac{2 b_1^{3/2} b_3 \left( l^2+l+1\right) \sqrt{l^2+l+4}}{5 \sqrt{5} l \left( b_3 \left( l^2+l+2\right) \right) {}^{3/2}}t+\theta \right) }. \end{aligned}$$

(167)

Selecting $\chi =4 d^2$, one recovers bright soliton solution for $\frac{b_1 \left( l^2+l+4\right) }{5 b_3 l^2+5 b_3 l+10 b_3}>0$ and $\frac{b_1}{5 a l^2+5 a l+10 a}>0$:

$$\begin{aligned} q(x,t)=\pm \sqrt{\frac{b_1 \left( l^2+l+4\right) }{5 b_3 l^2+5 b_3 l+10 b_3}} \left\{ \text {sech}\left( \sqrt{\frac{b_1}{5 a l^2+5 a l+10 a}}x\right) +1\right\} e^{i \left( \frac{2 b_1^{3/2} b_3 \left( l^2+l+1\right) \sqrt{l^2+l+4}}{5 \sqrt{5} l \left( b_3 \left( l^2+l+2\right) \right) {}^{3/2}}t+\theta \right) }. \end{aligned}$$

(168)

4.2.2 Projective Riccati equation method

In line with this algorithm, the solution takes the following structure

$$\begin{aligned} U(\xi )=\eta _0+\eta _1 F(\xi )+\rho _1 G(\xi ). \end{aligned}$$

(169)

When Eq. (169) is substituted alongside Eqs. (8) and (9) into Eq. (155), we obtain:

$$\begin{aligned}{} & {} 2 a \eta _1 K^2 l^2 \rho _1 R(\tau )+2 a \eta _1 K^2 l \rho _1 R(\tau )+8 a \eta _1 K^2 \rho _1 R(\tau )+4 b_3 \eta _1^3 \rho _1+4 b_3 \eta _1 \rho _1^3 R(\tau )=0. \end{aligned}$$

(170)

$$\begin{aligned}{} & {} -8 a \eta _1 K^2 \rho _1 \tau -2 a \eta _1 K^2 l^2 \rho _1 \tau -2 a \eta _1 K^2 l \rho _1 \tau +4 a \eta _0 K^2 \rho _1 R(\tau )-8 b_3 \eta _1 \rho _1^3 \tau \nonumber \\{} & {} \quad +3 b_2 \eta _1^2 \rho _1+12 b_3 \eta _0 \eta _1^2 \rho _1+4 b_3 \eta _0 \rho _1^3 R(\tau )+b_2 \rho _1^3 R(\tau )=0. \end{aligned}$$

(171)

$$\begin{aligned}{} & {} -2 a \eta _0 K^2 \rho _1 \tau +2 a \eta _1 K^2 \rho _1-8 b_3 \eta _0 \rho _1^3 \tau +4 b_3 \eta _1 \rho _1^3\nonumber \\{} & {} \quad +12 b_3 \eta _0^2 \eta _1 \rho _1+2 b_1 \eta _1 \rho _1+6 b_2 \eta _0 \eta _1 \rho _1-2 b_2 \rho _1^3 \tau =0. \end{aligned}$$

(172)

$$\begin{aligned}{} & {} 4 b_3 \eta _0^3 \rho _1+3 b_2 \eta _0^2 \rho _1+4 b_3 \eta _0 \rho _1^3+2 b_1 \eta _0 \rho _1+b_2 \rho _1^3-l \rho _1 \omega =0. \end{aligned}$$

(173)

$$\begin{aligned}{} & {} a \eta _1^2 K^2 l^2 R(\tau )+a K^2 l^2 \rho _1^2 R(\tau )^2+a \eta _1^2 K^2 l R(\tau )+a K^2 l \rho _1^2 R(\tau )^2+4 a \eta _1^2 K^2 R(\tau )\nonumber \\{} & {} \quad +4 a K^2 \rho _1^2 R(\tau )^2+b_3 \eta _1^4+6 b_3 \eta _1^2 \rho _1^2 R(\tau )+b_3 \rho _1^4 R(\tau )^2=0. \end{aligned}$$

(174)

$$\begin{aligned}{} & {} -6 a \eta _1^2 K^2 \tau -2 a \eta _1^2 K^2 l^2 \tau -2 a K^2 l^2 \rho _1^2 \tau R(\tau )-2 a \eta _1^2 K^2 l \tau -2 a K^2 l \rho _1^2 \tau R(\tau )+4 a \eta _0 \eta _1 K^2 R(\tau )\nonumber \\{} & {} \quad -10 a K^2 \rho _1^2 \tau R(\tau )-12 b_3 \eta _1^2 \rho _1^2 \tau +b_2 \eta _1^3+4 b_3 \eta _0 \eta _1^3\nonumber \\{} & {} \quad +3 b_2 \eta _1 \rho _1^2 R(\tau )+12 b_3 \eta _0 \eta _1 \rho _1^2 R(\tau )-4 b_3 \rho _1^4 \tau R(\tau )=0. \end{aligned}$$

(175)

$$\begin{aligned}{} & {} -6 a \eta _0 \eta _1 K^2 \tau +2 a \eta _1^2 K^2+a \eta _1^2 K^2 l^2+a K^2 l^2 \rho _1^2 \tau ^2\nonumber \\{} & {} \quad +a \eta _1^2 K^2 l+a K^2 l \rho _1^2 \tau ^2+4 a K^2 \rho _1^2 \tau ^2+4 a K^2 \rho _1^2 R(\tau )\nonumber \\{} & {} \quad -6 b_2 \eta _1 \rho _1^2 \tau -24 b_3 \eta _0 \eta _1 \rho _1^2 \tau +6 b_3 \eta _1^2 \rho _1^2+6 b_3 \eta _0^2 \eta _1^2+b_1 \eta _1^2+3 b_2 \eta _0 \eta _1^2\nonumber \\{} & {} \quad +4 b_3 \rho _1^4 \tau ^2+6 b_3 \eta _0^2 \rho _1^2 R(\tau )+3 b_2 \eta _0 \rho _1^2 R(\tau )\nonumber \\{} & {} \quad +2 b_3 \rho _1^4 R(\tau )+b_1 \rho _1^2 R(\tau )=0. \end{aligned}$$

(176)

$$\begin{aligned}{} & {} \quad 2 a \eta _0 \eta _1 K^2-2 a K^2 \rho _1^2 \tau -12 b_3 \eta _0^2 \rho _1^2 \tau -6 b_2 \eta _0 \rho _1^2 \tau +3 b_2 \eta _1 \rho _1^2+12 b_3 \eta _0 \eta _1 \rho _1^2\nonumber \\{} & {} \quad +4 b_3 \eta _0^3 \eta _1+3 b_2 \eta _0^2 \eta _1+2 b_1 \eta _0 \eta _1-4 b_3 \rho _1^4 \tau -2 b_1 \rho _1^2 \tau -\eta _1 l \omega =0. \end{aligned}$$

(177)

$$\begin{aligned}{} & {} 6 b_3 \eta _0^2 \rho _1^2+3 b_2 \eta _0 \rho _1^2+b_3 \eta _0^4+b_2 \eta _0^3+b_1 \eta _0^2+b_3 \rho _1^4+b_1 \rho _1^2-\eta _0 l \omega =0. \end{aligned}$$

(178)

These calculations collectively yield the subsequent outcomes:

Case-1:   $R (\tau )=\frac{24}{25}\tau ^2$.

$$\begin{aligned}{} & {} K=\pm \sqrt{-\frac{2 b_1}{5 a \left( l^2+l+2\right) }},~\eta _0=-\sqrt{\frac{3 b_1 \left( l^2+l+4\right) }{5 b_3 \left( l^2+l+2\right) }},~\eta _1=\frac{1}{5} (4 \tau ) \sqrt{\frac{3 b_1 \left( l^2+l+4\right) }{5 b_3 \left( l^2+l+2\right) }},~\rho _1=0,\nonumber \\{} & {} \quad \omega =-\frac{\sqrt{\frac{3}{5}} b_1^{3/2} b_3 \left( l^2+l+1\right) \sqrt{l^2+l+4}}{5 l \left( b_3 \left( l^2+l+2\right) \right) {}^{3/2}},~b_2=\pm \frac{7 \sqrt{b_1} b_3 \left( l^2+l+3\right) }{\sqrt{15} \sqrt{l^2+l+4} \sqrt{b_3 \left( l^2+l+2\right) }}. \end{aligned}$$

(179)

The bright soliton solution of the governing equation (150) for $\frac{b_1 \left( l^2+l+4\right) }{b_3 \left( l^2+l+2\right) }>0$ and $\frac{ b_1}{ a \left( l^2+l+2\right) }<0$ is achieved as

$$\begin{aligned} q(x,t)= -\sqrt{\frac{3}{5}} \sqrt{\frac{b_1 \left( l^2+l+4\right) }{b_3 \left( l^2+l+2\right) }} \left\{ 1-\frac{4 \text {sech}\left( \sqrt{-\frac{2 b_1}{5 a \left( l^2+l+2\right) }}x\right) }{5 \text {sech}\left( \sqrt{-\frac{2 b_1}{5 a \left( l^2+l+2\right) }}x\right) \pm 1} \right\} e^{i \left( -\frac{\sqrt{\frac{3}{5}} b_1^{3/2} b_3 \left( l^2+l+1\right) \sqrt{l^2+l+4}}{5 l \left( b_3 \left( l^2+l+2\right) \right) {}^{3/2}} t+\theta \right) }. \end{aligned}$$

(180)

Case-2:  $R (\tau )=\frac{5}{9}\tau ^2$.

$$\begin{aligned}{} & {} K=\pm \sqrt{\frac{b_1}{13 \left( -a l^2-a l-2 a\right) }},~\eta _0=\pm \sqrt{\frac{5 b_1 \left( l^2+l+4\right) }{13 b_3 \left( l^2+l+2\right) }},\nonumber \\{} & {} \quad \eta _1=\mp \frac{\tau }{3}\sqrt{\frac{5 b_1 \left( l^2+l+4\right) }{13 b_3 \left( l^2+l+2\right) }},\rho _1=0,\nonumber \\{} & {} \quad \omega =\sqrt{\frac{5 b_1^3 \left( l^2+l+4\right) }{13 b_3 \left( l^2+l+2\right) ^3}} \pm \frac{4 \left( l^2+l+1\right) }{13 l},\nonumber \\{} & {} \quad ~b_2=\mp 14 \left( l^2+l+3\right) \sqrt{\frac{b_1 b_3}{65 \left( l^2+l+2\right) \left( l^2+l+4\right) }}. \end{aligned}$$

(181)

Thus, the bright soliton solution of the governing equation (150) for $\frac{b_1 \left( l^2+l+4\right) }{b_3 \left( l^2+l+2\right) }>0$ and $\frac{b_1}{ \left( a l^2+a l+2 a\right) }<0$ is

$$\begin{aligned} q(x,t)=\pm \sqrt{\frac{5}{13}} \sqrt{\frac{b_1 \left( l^2+l+4\right) }{b_3 \left( l^2+l+2\right) }} \left\{ 1-\frac{\text {sech}\left( \sqrt{\frac{b_1}{13 \left( -a l^2-a l-2 a\right) }}x\right) }{3 \text {sech}\left( \sqrt{\frac{b_1}{13 \left( -a l^2-a l-2 a\right) }}x\right) \pm 2}\right\} e^{i \left( \pm \frac{4 \left( l^2+l+1\right) }{13 l}\sqrt{\frac{5 b_1^3 \left( l^2+l+4\right) }{13 b_3 \left( l^2+l+2\right) ^3}} t+\theta \right) }. \end{aligned}$$

(182)

Case–3:  $R (\tau )=\tau ^2+1$.

$$\begin{aligned}{} & {} K=\pm \sqrt{\frac{-6 \sqrt{-b_1^2 b_3^2 \tau ^2 \left( \tau ^2+1\right) ^2}-b_1 b_3 \tau ^4+4 b_1 b_3 \tau ^2+5 b_1 b_3}{4 \left( a b_3 \tau ^4+26 a b_3 \tau ^2+25 a b_3\right) }},\nonumber \\{} & {} \quad \eta _0=\frac{\sqrt{\frac{3}{2}} \left( \sqrt{-b_1^2 b_3^2 \tau ^2 \left( \tau ^2+1\right) ^2}-b_1 b_3 \tau ^2 \left( \tau ^2+1\right) \right) \sqrt{\frac{6 \sqrt{-b_1^2 b_3^2 \tau ^2 \left( \tau ^2+1\right) ^2}+b_1 b_3 \left( \tau ^4-4 \tau ^2-5\right) }{b_3^2 \left( \tau ^2+25\right) }}}{b_1 b_3 \tau \left( \tau ^2+1\right) ^2},\nonumber \\{} & {} \quad \eta _1=\sqrt{\frac{18 \sqrt{-b_1^2 b_3^2 \tau ^2 \left( \tau ^2+1\right) ^2}+3 b_1 b_3 \left( \tau ^4-4 \tau ^2-5\right) }{2 b_3^2 \left( \tau ^2+25\right) }},~\rho _1=0,\nonumber \\{} & {} \quad ~\omega =-\frac{3 \sqrt{\frac{3}{2}} \left( 4 b_1 b_3 \tau ^2 \left( \tau ^2+1\right) -\left( \tau ^2+5\right) \sqrt{-b_1^2 b_3^2 \tau ^2 \left( \tau ^2+1\right) ^2}\right) \sqrt{\frac{6 \sqrt{-b_1^2 b_3^2 \tau ^2 \left( \tau ^2+1\right) ^2}+b_1 b_3 \left( \tau ^4-4 \tau ^2-5\right) }{b_3^2 \left( \tau ^2+25\right) }}}{2 b_3 \tau \left( \tau ^2+1\right) ^2 \left( \tau ^2+25\right) },\nonumber \\{} & {} \quad b=\frac{5 \left( -2 \sqrt{-b_1^2 b_3^2 \left( \tau ^3+\tau \right) ^2}+b_1 b_3 \tau ^4+b_1 b_3 \tau ^2\right) \sqrt{\frac{6 \sqrt{-b_1^2 b_3^2 \tau ^2 \left( \tau ^2+1\right) ^2}+b_1 b_3 \left( \tau ^4-4 \tau ^2-5\right) }{b_3^2 \left( \tau ^2+25\right) }}}{\sqrt{6} b_1 \tau \left( \tau ^2+1\right) ^2}. \end{aligned}$$

(183)

Hence, the singular soliton solution of the governing equation (66) is written as:

$$\begin{aligned} q(x,t)= & {} \Bigg \{\frac{\sqrt{\frac{3}{2}} \left( \sqrt{-b_1^2 b_3^2 \tau ^2 \left( \tau ^2+1\right) ^2}-b_1 b_3 \tau ^2 \left( \tau ^2+1\right) \right) \sqrt{\frac{6 \sqrt{-b_1^2 b_3^2 \tau ^2 \left( \tau ^2+1\right) ^2}+b_1 b_3 \left( \tau ^4-4 \tau ^2-5\right) }{b_3^2 \left( \tau ^2+25\right) }}}{b_1 b_3 \tau \left( \tau ^2+1\right) ^2}\nonumber \\{} & {} \quad +\frac{\sqrt{\frac{18 \sqrt{-b_1^2 b_3^2 \tau ^2 \left( \tau ^2+1\right) ^2}+3 b_1 b_3 \left( \tau ^4-4 \tau ^2-5\right) }{b_3^2 \left( \tau ^2+25\right) }} \text {csch}\left( \pm \sqrt{\frac{-6 \sqrt{-b_1^2 b_3^2 \tau ^2 \left( \tau ^2+1\right) ^2}-b_1 b_3 \tau ^4+4 b_1 b_3 \tau ^2+5 b_1 b_3}{4 \left( a b_3 \tau ^4+26 a b_3 \tau ^2+25 a b_3\right) }}x\right) }{\sqrt{2} \left( \tau \text {csch}\left( \pm \sqrt{\frac{-6 \sqrt{-b_1^2 b_3^2 \tau ^2 \left( \tau ^2+1\right) ^2}-b_1 b_3 \tau ^4+4 b_1 b_3 \tau ^2+5 b_1 b_3}{4 \left( a b_3 \tau ^4+26 a b_3 \tau ^2+25 a b_3\right) }}x\right) +1\right) }\Bigg \}\nonumber \\{} & {} \quad \times e^{i \left( -\frac{3 \sqrt{\frac{3}{2}} \left( 4 b_1 b_3 \tau ^2 \left( \tau ^2+1\right) -\left( \tau ^2+5\right) \sqrt{-b_1^2 b_3^2 \tau ^2 \left( \tau ^2+1\right) ^2}\right) \sqrt{\frac{6 \sqrt{-b_1^2 b_3^2 \tau ^2 \left( \tau ^2+1\right) ^2}+b_1 b_3 \left( \tau ^4-4 \tau ^2-5\right) }{b_3^2 \left( \tau ^2+25\right) }}}{2 b_3 \tau \left( \tau ^2+1\right) ^2 \left( \tau ^2+25\right) }t+\theta \right) }. \end{aligned}$$

(184)

5 Conclusions

The current paper recovered the quiescent optical solitons for the Fokas–Lenells equation, which includes nonlinear CD and two different forms of SPM. The projective Riccati equation approach and the enhanced Kudryashov’s scheme have made this retrieval possible. A full spectrum of optical solitons has thus been located. The temporal evolutions are considered to be both linear and generalized. The results from the second phase collapse to the ones in the initial phase after setting the generalized temporal evolution parameter to unity. The results of this paper lay a strong foundation for further development with the model. Later, this model will be addressed with the generalized form of SPM structures, yielding a generalized version of the results reported here. The results of such research activities will be disseminated once they are available, and the results are connected and aligned with the pre-existing ones [50,51,52,53,54].