Introduction

One of the several proposed models to offset the low count of chromatic dispersion (CD) is the Fokas-Lenells equation (FLE) that introduces nonlinear dispersive effect [1,2,3,4,5,6]. In addition, higher–order dispersion terms are introduced to provide additional dispersive effects that would provide the perfect necessary balance between CD and self–phase modulation (SPM) effect [7,8,9,10,11,12]. The current model that is considered in the paper is the FLE with higher–order dispersion effects that stem from inter–modal dispersion (IMD), third–order dispersion (3OD), fourth–order dispersion (4OD), fifth–order dispersion (4OD), fifth–order dispersion (5OD), sixth–order dispersion (6OD) and seventh–ordr dispersion (7OD). These are included in addition to the pre–existing CD [13,14,15,16,17,18]. One of the detrimental effects that is tacitly overlooked is the effect of soliton radiation [19,20,21,22,23,24]. Another effect is the slow–down of the soliton speed with such higher–order dispersive effects [25,26,27,28,29,30].

The model is next considered with polarization–mode dispersion and along both components the effect of white noise effect is introduced to introduce a flavor of stochasticity along the two components of the model [31,32,33,34,35,36]. Thus, the governing model that will be addressed in the current work would be the FLE with PMD and white noise effect along both components. This created the coupled version of FLE with higher–order dispersion terms along both components and the effect of white noise included. In order for the model to be closer to reality, the a few perturbation effects are also taken into consideration that are of Hamiltonian type.

Two integration approaches are implemented to recover the soliton solutions. These are enhanced Kudryashov’s method and the enhanced direct algebraic approach. These two efficient approaches collectively yield a full spectrum of optical solitons for both components as it was successfully applied to several previous works [1,2,3,4]. The effect of white noise stays confined to the phase component of of the solitons along both components. The derivation of the soliton solutions along with their existence criteria are exhibited in the rest of the paper after the introduction to the governing model and a rapid revisitation of the adopted integration algorithms.

Governing model

In this study, we present the formulation of the governing system for the perturbed Fokas-Lenells equation in birefringent fibers while also considering the impact of multiplicative white noise in the Ito sense. This formulation is being introduced for the first time in the existing literature.

$$\begin{aligned}{} & {} iq_{t}+ia_{11}q_{x}+a_{12}q_{xx}+i a_{13}q_{xxx}+a_{14}q_{xxxx}\nonumber \\{} & {} +ia_{15}q_{xxxxx}+a_{16}q_{xxxxxx}+\left( b_1\left| q\right| ^{2}+c_1\left| r\right| ^{2}\right) \nonumber \\{} & {} \times \left( d_1q+ie_1 q_x\right) +q r^*\left( \gamma _1 r+i \zeta _1 r_x\right) +\sigma q\frac{dW(t)}{dt}\\= & {} i\left[ \nu _1 \left( \left| r\right| ^2 q\right) _x+\lambda _{1}\left( \left| q\right| ^{2}q\right) _{x}+\mu _{1}\left( \left| q\right| ^{2}\right) _{x}q+\theta _{1}\left| r\right| ^{2}q_{x}\right] , \nonumber \end{aligned}$$
(1)

and

$$\begin{aligned}{} & {} ir_{t}+ia_{21}r_{x}+a_{22}r_{xx}+i a_{23}r_{xxx}+a_{24}r_{xxxx}\nonumber \\{} & {} +ia_{25}r_{xxxxx}+a_{26}r_{xxxxxx}+\left( b_2\left| r\right| ^{2}+c_2\left| q\right| ^{2}\right) \nonumber \\{} & {} \times \left( d_2r+ie_2 r_x\right) +r q^*\left( \gamma _2 q+i \zeta _2 q_x\right) +\sigma r\frac{dW(t)}{dt}\\= & {} i\left[ \nu _2 \left( \left| q\right| ^2 r\right) _x+\lambda _{2}\left( \left| r\right| ^{2}r\right) _{x}+\mu _{2}\left( \left| r\right| ^{2}\right) _{x}r+\theta _{2}\left| q\right| ^{2}r_{x}\right] , \nonumber \end{aligned}$$
(2)

The functions q(x, t) and r(x, t) represent the wave profile and have complex values. \(q^*(x,t)\) and \(r^*(x, t)\) are functions that are complex conjugates of each other, where i represents the imaginary unit. The terms (1) and (2) in the system correspond to the linear temporal progressions. The variables \(a_{lk}~\left( l=1,2,~k=1,2,3,4,5,6\right) \) represent the coefficients of IMD, CD, 3OD, 4OD, 5OD, 6OD and 7OD, respectively. The variables \(b_{j}\) and \(d_{j}\) represent the self-phase modulation (SPM), whereas the variables \(c_{j}~(j=1,2)\) and \(e_{j}~(j=1,2)\) represent the cross-phase modulation (XPM). The variables \(\gamma _j ~(j=1,2)\) denote the coefficients of cross-phase modulation (XPM) in addition to the four-wave mixing (4WM) components. The variables \(\zeta _j~ (j=1,2)\) are the coefficients of the four-wave mixing (4WM) terms. The symbol \(\sigma \) represents the noise strength, whereas W(t) denotes the normal Wiener process, with dW(t)/dt representing the white noise. The \(\lambda _{j}\) coefficients correspond to the IMD, the \(s_{j}\) terms reflect the self-steepening effects, and the \(\mu _{j}\) and \(\theta _{j}\) coefficients indicate the higher-order nonlinear dispersion components.

Mathematical analysis

In order to solve (1) and (2), the following solution structures are selected:

$$\begin{aligned} q(x,t)= & {} U_1(\xi ) e^{i\phi (x,t)},\end{aligned}$$
(3)
$$\begin{aligned} r(x,t)= & {} U_2(\xi ) e^{i\phi (x,t)}, \end{aligned}$$
(4)

where, the wave variable \(\xi \) is given by

$$\begin{aligned} \xi =k(x-vt), \end{aligned}$$
(5)

where k and v are constants. Here, \(U_l(\xi ), ~l=1,2\) are real valued functions which represent the amplitude components of the soliton solutions and v is the velocity of the soliton while k is the wave width. The phase component \(\phi (x,t)\) is defined as

$$\begin{aligned} \phi (x,t)=-\kappa x+\omega t+\sigma W(t)-\sigma ^2 t+ \theta _0, \end{aligned}$$
(6)

where \(\kappa \) is the frequency of the solitons, while \(\omega \) represents the wave number, \(\sigma \) is the noise strength \((\sigma \ge 0)\) and \(\theta _0\) is the phase constant. Substituting (3) and (4) into (1) and (2), then decomposing into real and imaginary parts. The real parts give:

$$\begin{aligned}{} & {} k^2 \left( 15 a_{16} \kappa ^4-10 a_{15} \kappa ^3-6 a_{14} \kappa ^2+3 a_{13} \kappa +a_{12}\right) U_1''\nonumber \\{} & {} + \left( \gamma _1+c_1 \left( d_1+e_1 \kappa \right) +\kappa \left( \zeta _1-\nu _1\right) \right) U_1 U_2^2\nonumber \\{} & {} + \Big (-a_{16} \kappa ^6+a_{15} \kappa ^5+a_{14} \kappa ^4-a_{13} \kappa ^3-a_{12} \kappa ^2+a_{11} \kappa \nonumber \\{} & {} +\sigma ^2-\omega \Big )U_1+a_{16} k^6 U_1^{(6)}+k^4 \Big (5 \kappa \left( a_{15}-3 a_{16} \kappa \right) \nonumber \\{} & {} +a_{14}\Big ) U_1^{(4)}+ \left( b_1 \left( d_1+e_1 \kappa \right) -\kappa \lambda _1\right) U_1^3=0,\end{aligned}$$
(7)
$$\begin{aligned}{} & {} k^2 \left( 15 a_{26} \kappa ^4-10 a_{25} \kappa ^3-6 a_{24} \kappa ^2+3 a_{23} \kappa +a_{22}\right) U_2''\nonumber \\{} & {} +\left( \gamma _2+c_2 \left( d_2+e_2 \kappa \right) +\kappa \left( \zeta _2-\nu _2\right) \right) U_1{}^2 U_2 \nonumber \\{} & {} + \Big (-a_{26} \kappa ^6+a_{25} \kappa ^5+a_{24} \kappa ^4-a_{23} \kappa ^3-a_{22} \kappa ^2+a_{21} \kappa \nonumber \\{} & {} +\sigma ^2-\omega \Big )U_2+a_{26} k^6 U_2{}^{(6)}+k^4 \Big (5 \kappa \left( a_{25}-3 a_{26} \kappa \right) \nonumber \\{} & {} +a_{24}\Big ) U_2{}^{(4)}+ \left( b_2 \left( d_2+e_2 \kappa \right) -\kappa \lambda _2\right) U_2{}^3=0, \end{aligned}$$
(8)

and the imaginary parts give:

$$\begin{aligned}{} & {} k \Big (-6 a_{16} \kappa ^5+5 a_{15} \kappa ^4+4 a_{14} \kappa ^3-3 a_{13} \kappa ^2-2 a_{12} \kappa \nonumber \\{} & {} +a_{11} +c_1 e_1 U_2{}^2\!-\nu _1 U_2{}^2\!-v\Big )U_1'\!+k^5 \left( a_{15}\!-6 a_{16} \kappa \right) U_1{}^{(5)}\nonumber \\{} & {} +k^3 \left( 2 \kappa \left( 5 \kappa \left( 2 a_{16} \kappa -a_{15}\right) -2 a_{14}\right) +a_{13}\right) U_1{}^{(3)} +k (b_1 e_1\nonumber \\{} & {} -3 \lambda _1-2 \mu _1)U_1{}^2 U_1' +k \left( \zeta _1-2 \left( \theta _1+\nu _1\right) \right) U_1 U_2 U_2'=0, \end{aligned}$$
(9)
$$\begin{aligned}{} & {} k U_2' \Big (-6 a_{26} \kappa ^5+5 a_{25} \kappa ^4+4 a_{24} \kappa ^3-3 a_{23} \kappa ^2-2 a_{22} \kappa \nonumber \\{} & {} +a_{21}+c_2 e_2 U_1{}^2-\nu _2 U_1{}^2-v\Big )+k^5 \left( a_{25}-6 a_{26} \kappa \right) U_2{}^{(5)}\nonumber \\{} & {} +k^3 \left( 2 \kappa \left( 5 \kappa \left( 2 a_{26} \kappa -a_{25}\right) -2 a_{24}\right) +a_{23}\right) U_2{}^{(3)}+k (b_2 e_2\nonumber \\{} & {} -3 \lambda _2-2 \mu _2)U_2{}^2 U_2'+k \left( \zeta _2-2 \left( \theta _2+\nu _2\right) \right) U_1 U_2 U_1'=0, \end{aligned}$$
(10)

where \(U_1=U_1(\xi )\), \(U_2=U_2(\xi )\) and superscripts stand for the derivative with respect to \(\xi \). Using the balancing principle leads to \(U_2=\varrho ~ U_1\) and \(\varrho \) is nonzero constant, then (7) and (8) become

$$\begin{aligned}{} & {} k^2 \left( 15 a_{16} \kappa ^4-10 a_{15} \kappa ^3-6 a_{14} \kappa ^2+3 a_{13} \kappa +a_{12}\right) U_1''\nonumber \\{} & {} +k^4 \left( 5 \kappa \left( a_{15}-3 a_{16} \kappa \right) +a_{14}\right) U_1{}^{(4)}+ \Big (-a_{16} \kappa ^6+a_{15} \kappa ^5\nonumber \\{} & {} +a_{14} \kappa ^4-a_{13} \kappa ^3-a_{12} \kappa ^2+a_{11} \kappa +\sigma ^2-\omega \Big )U_1\nonumber \\{} & {} +a_{16} k^6 U_1{}^{(6)} +\Big (b_1 \left( d_1+e_1 \kappa \right) +\gamma _1 \varrho ^2+c_1 \varrho ^2 \left( d_1+e_1 \kappa \right) \nonumber \\{} & {} +\zeta _1 \kappa \varrho ^2 -\kappa \lambda _1-\kappa \nu _1 \varrho ^2\Big )U_1{}^3 =0,\end{aligned}$$
(11)
$$\begin{aligned}{} & {} k^2 \varrho \left( 15 a_{26} \kappa ^4-10 a_{25} \kappa ^3-6 a_{24} \kappa ^2+3 a_{23} \kappa +a_{22}\right) U_1''\nonumber \\{} & {} +k^4 \varrho \left( 5 \kappa \left( a_{25}-3 a_{26} \kappa \right) +a_{24}\right) U_1{}^{(4)} +\varrho \Big (-a_{26} \kappa ^6\nonumber \\{} & {} +a_{25} \kappa ^5+a_{24} \kappa ^4-a_{23} \kappa ^3-a_{22} \kappa ^2+a_{21} \kappa \nonumber \\{} & {} +\sigma ^2-\omega \Big )U_1+a_{26} k^6 U_2{}^{(6)}+\varrho \Big (b_2 \varrho ^2 \left( d_2+e_2 \kappa \right) +\gamma _2\nonumber \\{} & {} +c_2 \left( d_2+e_2 \kappa \right) +\zeta _2 \kappa -\kappa \lambda _2 \varrho ^2-\kappa \nu _2\Big )U_1{}^3 =0, \end{aligned}$$
(12)

and (9) and (10) become

$$\begin{aligned}{} & {} k \Big (-6 a_{16} \kappa ^5+5 a_{15} \kappa ^4+4 a_{14} \kappa ^3-3 a_{13} \kappa ^2-2 a_{12} \kappa \nonumber \\{} & {} +a_{11} \!-v\Big )U_1'\!+k^5 \left( a_{15}\!-6 a_{16} \kappa \right) U_1{}^{(5)}\!+k^3 (2 \kappa (5 \kappa (2 a_{16} \kappa \nonumber \\{} & {} -a_{15})-2 a_{14})+a_{13}) U_1{}^{(3)}+k \Big (b_1 e_1+c_1 e_1 \varrho ^2+\zeta _1 \varrho ^2\nonumber \\{} & {} -2 \theta _1 \varrho ^2-3 \lambda _1-2 \mu _1-3 \nu _1 \varrho ^2\Big )U_1{}^2 U_1'=0,\end{aligned}$$
(13)
$$\begin{aligned}{} & {} k \varrho \Big (-6 a_{26} \kappa ^5+5 a_{25} \kappa ^4+4 a_{24} \kappa ^3-3 a_{23} \kappa ^2-2 a_{22} \kappa \nonumber \\{} & {} \!+a_{21} \!-\!v\Big )U_1'\!+k^5 \left( a_{25}\!-\!6 a_{26} \kappa \right) U_1{}^{(5)}\!+\!k^3 \varrho (2 \kappa (5 \kappa (2 a_{26} \kappa \nonumber \\{} & {} -a_{25})-2 a_{24})+a_{23}) U_1{}^{(3)}+k \varrho \Big (b_2 e_2 \varrho ^2+c_2 e_2+\zeta _2\nonumber \\{} & {} -2 \theta _2-3 \lambda _2 \varrho ^2-2 \mu _2 \varrho ^2-3 \nu _2\Big )U_1^2 U_1'=0. \end{aligned}$$
(14)

By comparing (11) and (12) and reducing them to a single equation, we are able to derive the following parametric restrictions:

$$\begin{aligned}{} & {} -a_{16} \kappa ^6\!+a_{15} \kappa ^5\!+a_{14} \kappa ^4\!-a_{13} \kappa ^3\!-a_{12} \kappa ^2\!+a_{11} \kappa \!+\sigma ^2\!-\!\omega \nonumber \\= & {} \varrho \left( \!-a_{26} \kappa ^6\!+a_{25} \kappa ^5\!+a_{24} \kappa ^4\!-\!a_{23} \kappa ^3\!-\!a_{22} \kappa ^2\!+a_{21} \kappa +\sigma ^2\!-\!\omega \right) ,\nonumber \\ \end{aligned}$$
(15)
$$\begin{aligned}{} & {} b_1 \left( d_1\!+e_1 \kappa \right) \!+\gamma _1 \varrho ^2\!+c_1 \varrho ^2 \left( d_1\!+e_1 \kappa \right) \!+\zeta _1 \kappa \varrho ^2\!-\!\kappa \lambda _1\!-\!\kappa \nu _1 \varrho ^2\nonumber \\= & {} \varrho \left( b_2 \varrho ^2 \left( d_2\!+e_2 \kappa \right) +\gamma _2\!+c_2 \left( d_2\!+e_2 \kappa \right) \!+\zeta _2 \kappa \!-\kappa \lambda _2 \varrho ^2\!-\kappa \nu _2\right) ,\nonumber \\ \end{aligned}$$
(16)
$$\begin{aligned}{} & {} 15 a_{16} \kappa ^4-10 a_{15} \kappa ^3-6 a_{14} \kappa ^2+3 a_{13} \kappa +a_{12}\nonumber \\= & {} \varrho \left( 15 a_{26} \kappa ^4-10 a_{25} \kappa ^3-6 a_{24} \kappa ^2+3 a_{23} \kappa +a_{22}\right) , \end{aligned}$$
(17)
$$\begin{aligned}{} & {} 5 \kappa \left( a_{15}\!-\!3 a_{16} \kappa \right) \!+a_{14}\!=\!\varrho \left( 5 \kappa \left( a_{25}\!-\!3 a_{26} \kappa \right) \!+\!a_{24}\right) , \end{aligned}$$
(18)
$$\begin{aligned} a_{16}= & {} a_{26}. \end{aligned}$$
(19)

From the imaginary parts (13) and (14), we can equate the coefficients of the linearly independent functions with being zero then we get the speed of the two components as follows:

$$\begin{aligned} v= & {} -6 a_{26} \kappa ^5+5 a_{25} \kappa ^4+4 a_{24} \kappa ^3-3 a_{23} \kappa ^2\nonumber \\{} & {} -2 a_{22} \kappa +a_{21} ,\end{aligned}$$
(20)
$$\begin{aligned} v= & {} -6 a_{16} \kappa ^5+5 a_{15} \kappa ^4+4 a_{14} \kappa ^3-3 a_{13} \kappa ^2\nonumber \\{} & {} -2 a_{12} \kappa +a_{11} , \end{aligned}$$
(21)

and the soliton frequency as follows:

$$\begin{aligned} \kappa = -\frac{a_{15}}{6 a_{16}}=-\frac{a_{25}}{6 a_{26}} , \end{aligned}$$
(22)

where \(a_{l6}\ne 0~(l=1,2)\) together with the conditions,

$$\begin{aligned}{} & {} b_1 e_1+c_1 e_1 \varrho ^2+\zeta _1 \varrho ^2-2 \theta _1 \varrho ^2-3 \lambda _1-2 \mu _1-3 \nu _1 \varrho ^2\nonumber \\= & {} \varrho \left( b_2 e_2 \varrho ^2+c_2 e_2+\zeta _2-2 \theta _2-3 \lambda _2 \varrho ^2-2 \mu _2 \varrho ^2-3 \nu _2\right) \nonumber \\= & {} 0,\end{aligned}$$
(23)
$$\begin{aligned}{} & {} 2 \kappa \left( 5 \kappa \left( 2 a_{16} \kappa -a_{15}\right) -2 a_{14}\right) +a_{13}\nonumber \\= & {} \varrho \left( 2 \kappa \left( 5 \kappa \left( 2 a_{26} \kappa -a_{25}\right) -2 a_{24}\right) +a_{23}\right) =0. \end{aligned}$$
(24)

Equation (11) can be set as

$$\begin{aligned} k^2 U_1{}^{(6)}+\vartheta _3 U_1''+\vartheta _2 U_1{}^3+\vartheta _1 U_1+\vartheta _4 U_1{}^{(4)}=0, \end{aligned}$$
(25)

with

$$\begin{aligned} {{\left\{ \begin{array}{ll} \vartheta _1=\frac{-a_{16} \kappa ^6+a_{15} \kappa ^5+a_{14} \kappa ^4-a_{13} \kappa ^3-a_{12} \kappa ^2+a_{11} \kappa +\sigma ^2-\omega }{a_{16} k^4}\\ \vartheta _2=\frac{b_1 \left( d_1+e_1 \kappa \right) +\gamma _1 \varrho ^2+c_1 \varrho ^2 \left( d_1+e_1 \kappa \right) +\zeta _1 \kappa \varrho ^2-\kappa \lambda _1-\kappa \nu _1 \varrho ^2}{a_{16} k^4}\\ \vartheta _3=\frac{15 a_{16} \kappa ^4-10 a_{15} \kappa ^3-6 a_{14} \kappa ^2+3 a_{13} \kappa +a_{12}}{a_{16} k^2}\\ \vartheta _4=\frac{5 \kappa \left( a_{15}-3 a_{16} \kappa \right) +a_{14}}{a_{16}}. \end{array}\right. }} \end{aligned}$$
(26)

Integration algorithms: an overview

We could include a governing model which has the structure of

$$\begin{aligned} F(\psi ,\psi _{x},\psi _{t},\psi _{xt},\psi _{xx},...)=0. \end{aligned}$$
(27)

The expression \(\psi =\psi (x,t)\) represents a wave profile, where t and x are time and space coordinates, respectively.

The utilization of wave transformation

$$\begin{aligned} \psi (x,t)=U(\eta ),~~~\eta =k (x-\upsilon t), \end{aligned}$$
(28)

causes a reduction of (27) to

$$\begin{aligned} P(U,-k \upsilon U^{\prime },k U^{\prime },k ^{2}U^{\prime \prime },...)=0. \end{aligned}$$
(29)

In the aforementioned mathematical argument, k denotes the wave width, \(\eta \) represents the wave variable, and \(\upsilon \) symbolizes the wave velocity.

Enhanced Kudryashov’s method

This subsection presents a thorough overview of the basic procedures with the enhanced Kudryashov technique (EK) [1, 2].

Step–1:  The explicit solution for the reduced model (29) is provided as follows.

$$\begin{aligned} U(\eta )=\sigma _{0}+\sum _{i=1}^{N}\left\{ \sigma _{i}R(\eta )^{i}+\rho _i \left( \frac{R^{\prime }(\eta )}{R(\eta )}\right) ^{i}\right\} , \end{aligned}$$
(30)

In conjunction with the auxiliary equation,

$$\begin{aligned} {R^{\prime }(\eta )}^{2}={R(\eta )}^{2}(1-\chi {R(\eta )}^{2}), \end{aligned}$$
(31)

The constants \(\sigma _{0}\), \(\chi \), \(\sigma _{i}\), and \(\rho _{i}\) (where \(i=1,..., N\)) will be provided, with N determined by the balancing procedure in (30).

Step–2:  Equation (31) gives the soliton waves

$$\begin{aligned} R(\eta )=\frac{4c~e^{\eta }}{4c^{2}e^{2\eta }+\chi }, \end{aligned}$$
(32)

where c is constant and (32) generates the bright and dark soliton solutions for \(\chi = \pm 4{c^2}\), respectively.

Step–3:  By inserting the (30) into the (29), together with the (31), we can derive the requisite constants for the (28) and (30). In order to incorporate the identified parametric restrictions, they can be substituted into (30) together with (32). Consequently, straddled solitons are obtained, which can be further classified as bright, dark, or singular solitons.

Enhanced direct algebraic method

This section presents a thorough overview of the basic procedures with the enhanced direct algebraic technique.

Step-1: We assume that the solution of (29) can be expressed in the form

$$\begin{aligned} U(\xi ) =\alpha _0+ \sum \limits _{i = 1}^N {{\alpha _i}{\theta (\xi )^i}}, \end{aligned}$$
(33)

where \(\theta \) satisfies

$$\begin{aligned} {\theta ^{\prime }(\xi )}^2 = \sum \limits _{l = 0}^4{\tau _l} \theta (\xi )^l, \end{aligned}$$
(34)

where \(\tau _l~ (l=0,1,2,3,4)\) are constants provided that \(\tau _4\ne 0\). Equation (34) provides several kinds of solutions of different types as follows:

Set-1: If we set \({\tau _0} = {\tau _1} = {\tau _3} = 0\), we get bright soliton with \(\tau _2>0\), \(\tau _4<0\) and singular soliton with \(\tau _2>0\), \(\tau _4>0\):

$$\begin{aligned} \theta (\xi )= & {} \sqrt{ - \frac{{{\tau _2}}}{{{\tau _4}}}} ~\text {sech} \left[ {\sqrt{{\tau _2}} \,\xi } \right] ,~~~{\tau _2} > 0,~{\tau _4} < 0,\end{aligned}$$
(35)
$$\begin{aligned} \theta (\xi )= & {} \sqrt{ \frac{{{\tau _2}}}{{{\tau _4}}}} ~\text {csch} \left[ {\sqrt{{\tau _2}} \,\xi } \right] , ~~~{\tau _2}> 0,~{\tau _4} > 0, \end{aligned}$$
(36)

Set-2: If we set \({\tau _0} =\frac{{\tau _2}^2}{4\tau _4},~ {\tau _1} = {\tau _3} = 0\), we have dark and singular solitons for \(\tau _2<0\), \(\tau _4>0\):

$$\begin{aligned} \theta (\xi )= & {} \sqrt{ - \frac{{{\tau _2}}}{{{2\tau _4}}}} ~\tanh \left[ {\sqrt{-{\frac{\tau _2}{2}}} \,\xi } \right] ,~~~{\tau _2} < 0,~{\tau _4} > 0,\end{aligned}$$
(37)
$$\begin{aligned} \theta (\xi )= & {} \sqrt{ - \frac{{{\tau _2}}}{{{2\tau _4}}}} ~\coth \left[ {\sqrt{-{\frac{\tau _2}{2}}} \,\xi } \right] , ~~~{\tau _2} < 0,~{\tau _4} > 0, \end{aligned}$$
(38)

Set-3: If we set \({\tau _1} = {\tau _3} = 0\), we get Jacobi elliptic doubly periodic type solution for different choices of \(\tau _0\) as follows:

$$\begin{aligned} \theta (\xi )= & {} \pm \sqrt{-\frac{m^2 \tau _2}{\left( 2 m^2-1\right) \tau _4}} \text {cn}\left( \left. \sqrt{\frac{\tau _2}{2 m^2-1}} \xi \right| m\right) ;\nonumber \\ \tau _0= & {} \frac{m^2 \left( 1-m^2\right) \tau _2^2}{\left( 2 m^2-1\right) ^2 \tau _4},\end{aligned}$$
(39)
$$\begin{aligned} \theta (\xi )= & {} \pm \sqrt{-\frac{m^2 \tau _2}{\left( 2-m^2\right) \tau _4}} \text {dn}\left( \left. \sqrt{\frac{\tau _2}{2-m^2}} \xi \right| m\right) ;\nonumber \\ \tau _0= & {} \frac{\left( 1-m^2\right) \tau _2^2}{\left( 2-m^2\right) ^2 \tau _4},\end{aligned}$$
(40)
$$\begin{aligned} \theta (\xi )= & {} \pm \sqrt{-\frac{m^2 \tau _2}{\left( m^2+1\right) \tau _4}} \text {sn}\left( \left. \sqrt{-\frac{\tau _2}{m^2+1}} \xi \right| m\right) ;\nonumber \\ \tau _0= & {} \frac{m^2 \tau _2^2}{\left( m^2+1\right) ^2 \tau _4}, \end{aligned}$$
(41)

The value “m” is a free parameter that is usually taken to be real within the range of 0 to 1.

Set-4: If we set \({\tau _1} = {\tau _3} = 0\), we get Weierstrass elliptic doubly periodic type solutions:

$$\begin{aligned} \theta (\xi )= & {} \frac{3\wp '\left( \xi ;g_2,g_3\right) }{\sqrt{\tau _4}\left[ 6\wp \left( \xi ;g_2,g_3\right) +\tau _2\right] },~~~{\tau _4} > 0,\end{aligned}$$
(42)
$$\begin{aligned} \theta (\xi )= & {} \frac{\sqrt{\tau _0}\left[ 6\wp \left( \xi ;g_2,g_3\right) +\tau _2\right] }{3\wp '\left( \xi ;g_2,g_3\right) },~~~{\tau _0}>0, \end{aligned}$$
(43)

where \(g_2=\frac{\tau _2^2}{12}+\tau _0 \tau _4\) and \(g_3=\frac{\tau _2}{216} \left( 36 \tau _0 \tau _4-\tau _2^2\right) \) are called invariants of the Weierstrass elliptic function.

Set-5: If we set, \({\tau _0} = {\tau _1} = 0\), we get straddled soliton solutions with \({\tau _2} > 0\) as follows:

$$\begin{aligned} \theta (\xi )= & {} \frac{-{{\tau _2}\mathrm{sech^2}\left[ {\frac{1}{2}\sqrt{{\tau _2}} \,\xi } \right] }}{{\pm 2 \sqrt{{\tau _2}{\tau _4}} \,\tanh \left[ {\frac{1}{2}\sqrt{{\tau _2}} \,\xi } \right] + {\tau _3}}}, ~~~{\tau _4} > 0,\end{aligned}$$
(44)
$$\begin{aligned} \theta (\xi )= & {} \frac{{{\tau _2}\mathrm{csch^2}\left[ {\frac{1}{2}\sqrt{{\tau _2}} \,\xi } \right] }}{{\pm 2 \sqrt{{\tau _2}{\tau _4}} \,\coth \left[ {\frac{1}{2}\sqrt{{\tau _2}} \,\xi } \right] + {\tau _3}}}, ~~~{\tau _4} > 0,\end{aligned}$$
(45)
$$\begin{aligned} \theta (\xi )= & {} \frac{2 \tau _2 \text {sech}\left[ \sqrt{\tau _2}\xi \right] }{\pm \sqrt{\tau _3^2-4 \tau _2 \tau _4}-\tau _3 \text {sech}\left[ \sqrt{\tau _2}\xi \right] },~~~\tau _3^2\!-\!4 \tau _2 \tau _4\!>\!0,\nonumber \\ \end{aligned}$$
(46)
$$\begin{aligned} \theta (\xi )= & {} \frac{2 \tau _2 \text {csch}\left[ \sqrt{\tau _2}\xi \right] }{\pm \sqrt{4 \tau _2 \tau _4-\tau _3^2}-\tau _3 \text {csch}\left[ \sqrt{\tau _2}\xi \right] },~~~\tau _3^2\!-\!4 \tau _2 \tau _4\!<\!0,\nonumber \\\end{aligned}$$
(47)
$$\begin{aligned} \theta (\xi )= & {} -\frac{\tau _2 \tau _3 \text {sech}^2\left[ \frac{ \sqrt{\tau _2}}{2}\xi \right] }{\tau _3^2-\tau _2 \tau _4 \left( 1-\tanh \left[ \frac{ \sqrt{\tau _2}}{2}\xi \right] \right) ^2},~~~\tau _3\ne 0,\end{aligned}$$
(48)
$$\begin{aligned} \theta (\xi )= & {} \frac{\tau _2 \tau _3 \text {csch}^2\left[ \frac{ \sqrt{\tau _2}}{2}\xi \right] }{\tau _3^2-\tau _2 \tau _4 \left( 1-\coth \left[ \frac{ \sqrt{\tau _2}}{2}\xi \right] \right) ^2},~~~\tau _3\ne 0. \end{aligned}$$
(49)

Step-2: Determine the positive integer number N in (29) by balancing the highest order derivatives and the nonlinear terms in (29)

Step-4: Substitute (33) into (29) along with (34). As a result of this substitution, we get a polynomial of \(\phi \). In this polynomial, we gather all terms of the same powers and equate them to zero. We get an over-determined system of algebraic equations which Mathematica can solve to get the unknown parameters in (28) and (34). Consequently, we obtain the exact solutions of (27).

Optical solitons

The enhanced Kudryashov’s method

Balancing \(U_1^{(6)}\) with \(U_1^3\) in (25) gives \(N=3\). In accordance with the enhanced Kudryashov technique, the solution is expressed in the following structure.

$$\begin{aligned} U_1(\xi )= & {} a_0 \!+ a_1 R(\xi )+ a_2 R(\xi )^2\!+a_3 R(\xi )^3\!+ b_1 \left( \frac{ R^\prime (\xi )}{ R(\xi )}\right) \nonumber \\{} & {} + b_2 \left( \frac{ R^\prime (\xi )}{ R(\xi )}\right) ^2++ b_3 \left( \frac{ R^\prime (\xi )}{ R(\xi )}\right) ^3. \end{aligned}$$
(50)

Plugging (50) together with (31) into (25), a system of algebraic equations is obtained.

$$\begin{aligned}{} & {} a_3^3 \vartheta _2-3 a_3 \chi ^3 \left( b_3^2 \vartheta _2+6720 k^2\right) =0, \end{aligned}$$
(51)
$$\begin{aligned}{} & {} -3 \left( a_2-b_2 \chi \right) \left( \chi ^3 \left( b_3^2 \vartheta _2+1680 k^2\right) -a_3^2 \vartheta _2\right) =0, \end{aligned}$$
(52)
$$\begin{aligned}{} & {} b_3 \chi \left( \chi ^3 \left( b_3^2 \vartheta _2+20160 k^2\right) -3 a_3^2 \vartheta _2\right) =0, \end{aligned}$$
(53)
$$\begin{aligned}{} & {} a_3 \Big (\!-2 a_2 b_2 \chi \vartheta _2\!+\!a_2^2 \vartheta _2\!+\!\chi ^2 \Big (b_2^2 \vartheta _2\!+\!3 b_3^2 \vartheta _2\!+\!2 b_1 b_3 \vartheta _2\nonumber \\{} & {} +9960 k^2+120 \vartheta _4\Big )\Big )\!+\!a_1 \left( a_3^2 \vartheta _2\!-\!\chi ^3 \left( b_3^2 \vartheta _2\!+\!240 k^2\right) \!\right) \nonumber \\= & {} 0, \end{aligned}$$
(54)
$$\begin{aligned}{} & {} 6 a_3 b_3 \chi \vartheta _2 \left( b_2 \chi -a_2\right) =0, \end{aligned}$$
(55)
$$\begin{aligned}{} & {} a_2^3 \vartheta _2-3 a_2^2 b_2 \chi \vartheta _2-b_2^3 \chi ^3 \vartheta _2+3 a_0 \vartheta _2 \left( a_3^2-b_3^2 \chi ^3\right) \nonumber \\{} & {} +3 a_2 \Big (2 a_1 a_3 \vartheta _2\!+\chi ^2 \Big (b_2^2 \vartheta _2\!+3 b_3^2 \vartheta _2\!+2 b_1 b_3 \vartheta _2\!+2240 k^2\nonumber \\{} & {} +40 \vartheta _4\Big )\Big )-3 b_2 \Big (2 a_1 a_3 \chi \vartheta _2-a_3^2 \vartheta _2+2 \chi ^3 \Big (2 b_3^2 \vartheta _2\nonumber \\{} & {} +b_1 b_3 \vartheta _2+20 \left( 56 k^2+\vartheta _4\right) \Big )\Big )=0, \end{aligned}$$
(56)
$$\begin{aligned}{} & {} b_3 \Big (-6 a_2 b_2 \chi ^2 \vartheta _2+3 a_2^2 \chi \vartheta _2+6 a_1 a_3 \chi \vartheta _2-3 a_3^2 \vartheta _2\nonumber \\{} & {} +3 b_2^2 \chi ^3 \vartheta _2+4 b_3^2 \chi ^3 \vartheta _2+20880 k^2 \chi ^3+360 \chi ^3 \vartheta _4\Big )\nonumber \\{} & {} +3 b_1 \left( \chi ^3 \left( b_3^2 \vartheta _2+240 k^2\right) -a_3^2 \vartheta _2\right) =0, \end{aligned}$$
(57)
$$\begin{aligned}{} & {} -a_3 \Big (-2 a_0 \vartheta _2 \left( a_2\!-b_2 \chi \right) -\!2 a_2 b_2 \vartheta _2\!+b_1^2 \chi \vartheta _2+2 b_2^2 \chi \vartheta _2\nonumber \\{} & {} +3 b_3^2 \chi \vartheta _2+4 b_1 b_3 \chi \vartheta _2+3724 k^2 \chi +4 \chi \vartheta _3+136 \chi \vartheta _4\Big )\nonumber \\{} & {} +a_1 \Big (-2 a_2 b_2 \chi \vartheta _2\!+a_2^2 \vartheta _2\!+\chi ^2 \Big (b_2^2 \vartheta _2\!+3 b_3^2 \vartheta _2\!+2 b_1 b_3 \vartheta _2\nonumber \\{} & {} +280 k^2+8 \vartheta _4\Big )\Big )+a_1^2 a_3 \vartheta _2=0, \end{aligned}$$
(58)
$$\begin{aligned}{} & {} 6 \vartheta _2 \Big (a_2 \left( a_3 \left( b_1+b_3\right) -a_1 b_3 \chi \right) +\chi \Big (a_1 b_2 b_3 \chi \nonumber \\{} & {} -a_3 \left( b_3 \left( a_0+2 b_2\right) +b_1 b_2\right) \Big )\Big )=0, \end{aligned}$$
(59)
$$\begin{aligned}{} & {} a_0 b_2^2 \chi ^2 \vartheta _2+a_0 \vartheta _2 \left( 2 a_1 a_3+b_3 \left( 2 b_1+3 b_3\right) \chi ^2\right) \nonumber \\{} & {} +a_2^2 \vartheta _2 \left( a_0+b_2\right) +b_2^3 \chi ^2 \vartheta _2-a_2 \Big (\chi \Big (2 a_0 b_2 \vartheta _2+b_1^2 \vartheta _2\nonumber \\{} & {} +2 b_2^2 \vartheta _2+3 b_3^2 \vartheta _2+4 b_1 b_3 \vartheta _2+672 k^2+2 \vartheta _3+40 \vartheta _4\Big )\nonumber \\{} & {} -a_1^2 \vartheta _2\Big )+b_2 \Big (-a_1^2 \chi \vartheta _2+2 a_1 a_3 \vartheta _2+\chi ^2 \Big (b_1^2 \vartheta _2+6 b_3^2 \vartheta _2\nonumber \\{} & {} +6 b_1 b_3 \vartheta _2+672 k^2+2 \vartheta _3+40 \vartheta _4\Big )\Big )=0, \end{aligned}$$
(60)
$$\begin{aligned}{} & {} b_3 \Big (\!-4 a_2 b_2 \chi \vartheta _2-a_1^2 \chi \vartheta _2+a_2^2 \vartheta _2+2 a_1 a_3 \vartheta _2+3 b_2^2 \chi ^2 \vartheta _2\nonumber \\{} & {} +2 b_3^2 \chi ^2 \vartheta _2\!+1504 k^2 \chi ^2\!+4 \chi ^2 \vartheta _3\!+88 \chi ^2 \vartheta _4\Big )\!+b_3 b_1^2 \chi ^2 \vartheta _2\nonumber \\{} & {} +b_1 \Big (\!-2 a_2 b_2 \chi \vartheta _2\!+\!a_2^2 \vartheta _2\!+\!2 a_1 a_3 \vartheta _2\!+\!\chi ^2 \Big (b_2^2 \vartheta _2\!+\!3 b_3^2 \vartheta _2\nonumber \\{} & {} +160 k^2+8 \vartheta _4\Big )\Big )+b_3 \Big (2 a_0 \chi \vartheta _2 \left( b_2 \chi -a_2\right) \Big )=0, \end{aligned}$$
(61)
$$\begin{aligned}{} & {} -a_1 \Big (\!-6 a_0 \vartheta _2 \left( a_2\!-b_2 \chi \right) \!-6 a_2 b_2 \vartheta _2\!+3 b_1^2 \chi \vartheta _2\!+6 b_2^2 \chi \vartheta _2\nonumber \\{} & {} +9 b_3^2 \chi \vartheta _2+12 b_1 b_3 \chi \vartheta _2+182 k^2 \chi +2 \chi \vartheta _3+20 \chi \vartheta _4\Big )\nonumber \\{} & {} +a_3 \Big (3 \Big (2 a_0 b_2 \vartheta _2+a_0^2 \vartheta _2+b_1^2 \vartheta _2+b_2^2 \vartheta _2+b_3^2 \vartheta _2\nonumber \\{} & {} +2 b_1 b_3 \vartheta _2+243 k^2+3 \vartheta _3+27 \vartheta _4\Big )+\vartheta _1\Big )+a_1^3 \vartheta _2=0,\nonumber \\ \end{aligned}$$
(62)
$$\begin{aligned}{} & {} 6 \vartheta _2 \Big (a_1 \left( a_2 \left( b_1+b_3\right) -\chi \left( b_3 \left( a_0+2 b_2\right) +b_1 b_2\right) \right) \nonumber \\{} & {} +a_3 \left( b_1+b_3\right) \left( a_0+b_2\right) \Big )=0, \end{aligned}$$
(63)
$$\begin{aligned}{} & {} \!-6 a_0 b_2^2 \chi \vartheta _2\!-\!3 b_2^3 \chi \vartheta _2\!+\!3 a_0 \vartheta _2 \left( \!a_1^2\!-\!\left( b_1^2\!+\!4 b_3 b_1\!+\!3 b_3^2\right) \! \chi \!\right) \nonumber \\{} & {} +a_2 \Big (6 a_0 b_2 \vartheta _2+3 a_0^2 \vartheta _2+3 b_1^2 \vartheta _2+3 b_2^2 \vartheta _2+3 b_3^2 \vartheta _2\nonumber \\{} & {} +6 b_1 b_3 \vartheta _2+64 k^2+\vartheta _1+4 \vartheta _3+16 \vartheta _4\Big )-b_2 \Big (3 a_0^2 \chi \vartheta _2\nonumber \\{} & {} -3 a_1^2 \vartheta _2+6 b_1^2 \chi \vartheta _2+12 b_3^2 \chi \vartheta _2+18 b_1 b_3 \chi \vartheta _2+64 k^2 \chi \nonumber \\{} & {} +\chi \vartheta _1+4 \chi \vartheta _3+16 \chi \vartheta _4\Big )=0, \end{aligned}$$
(64)
$$\begin{aligned}{} & {} -b_1 \Big (-6 a_2 b_2 \vartheta _2-3 a_1^2 \vartheta _2+6 b_2^2 \chi \vartheta _2+9 b_3^2 \chi \vartheta _2+32 k^2 \chi \nonumber \\{} & {} +2 \chi \vartheta _3+8 \chi \vartheta _4\Big )+b_1^3 (-\chi ) \vartheta _2-6 b_3 b_1^2 \chi \vartheta _2\nonumber \\{} & {} -b_3 \Big (-6 a_0 \vartheta _2 \left( a_2\!-2 b_2 \chi \right) \!-6 a_2 b_2 \vartheta _2\!+3 a_0^2 \chi \vartheta _2\nonumber \\{} & {} -3 a_1^2 \vartheta _2+9 b_2^2 \chi \vartheta _2+4 b_3^2 \chi \vartheta _2+96 k^2 \chi +\chi \vartheta _1+6 \chi \vartheta _3\nonumber \\{} & {} +24 \chi \vartheta _4\Big ) -b_1 \left( -6 a_0 \vartheta _2 \left( a_2-b_2 \chi \right) \right) =0, \end{aligned}$$
(65)
$$\begin{aligned}{} & {} a_1 \Big (6 a_0 b_2 \vartheta _2+3 a_0^2 \vartheta _2+3 b_1^2 \vartheta _2+3 b_2^2 \vartheta _2+3 b_3^2 \vartheta _2\nonumber \\{} & {} +6 b_1 b_3 \vartheta _2+k^2+\vartheta _1+\vartheta _3+\vartheta _4\Big )=0, \end{aligned}$$
(66)
$$\begin{aligned}{} & {} \left( b_1+b_3\right) \Big (\vartheta _2 \left( 6 a_0 b_2+3 a_0^2+b_1^2+3 b_2^2+b_3^2+2 b_1 b_3\right) \nonumber \\{} & {} +\vartheta _1\Big ) =0, \end{aligned}$$
(67)
$$\begin{aligned}{} & {} 6 a_1 \left( b_1+b_3\right) \vartheta _2 \left( a_0+b_2\right) =0, \end{aligned}$$
(68)
$$\begin{aligned}{} & {} \left( a_0+b_2\right) \Big (\vartheta _2 \left( 2 a_0 b_2+a_0^2+3 b_1^2+b_2^2+3 b_3^2+6 b_1 b_3\right) \nonumber \\{} & {} +\vartheta _1\Big )=0, \end{aligned}$$
(69)

The simultaneous solution of these equations leads to the subsequent results:

Result-1:

$$\begin{aligned} a_0= & {} -b_2,~a_1=0,~a_2=b_2 \chi ,~a_3= \pm 24\chi \sqrt{-\frac{35 \chi \vartheta _4}{83 \vartheta _2}},\nonumber \\ b_1= & {} 0,~b_3=0,\nonumber \\ k= & {} \sqrt{-\frac{\vartheta _4}{83}},~\vartheta _1=\frac{11025 \vartheta _4}{83},~\vartheta _3=-\frac{1891\vartheta _4}{83} . \end{aligned}$$
(70)

As a result, the solution of the governing system (1)-(2) is attained.

$$\begin{aligned} q(x,t)= & {} \Bigg \{\!-b_2\!+\!b_2 \chi \left( \!\frac{4 c}{4 c^2 e^{\sqrt{-\frac{\vartheta _4}{83}} (x-v t)}\!+\!\chi e^{-\sqrt{-\frac{\vartheta _4}{83}} (x-v t)}}\!\right) ^2\nonumber \\{} & {} +b_2 \left( \frac{\chi -4 c^2 e^{2 \sqrt{-\frac{\vartheta _4}{83}} (x-v t)}}{4 c^2 e^{2 \sqrt{-\frac{\vartheta _4}{83}} (x-v t)}+\chi }\right) ^2\nonumber \\&\pm&24 \chi \sqrt{-\frac{35 \chi \vartheta _4}{83 \vartheta _2}} \left( \frac{4 c}{4 c^2 e^{\sqrt{-\frac{\vartheta _4}{83}} (x-v t)}\!+\!\chi e^{-\sqrt{-\frac{\vartheta _4}{83}} (x-v t)}}\right) ^3\!\Bigg \}\nonumber \\{} & {} \times e^{i \left( \left\{ \frac{a_{15}}{6 a_{16}}\right\} x+\omega t+\sigma W(t)-\sigma ^2 t+\theta _0 \right) }, \end{aligned}$$
(71)
$$\begin{aligned} r(x,t)=\varrho ~q(x,t). \end{aligned}$$
(72)

Selecting \(\chi =\pm 4 c^2 \) to recover bright soliton for \(\vartheta _4<0\) and \(\vartheta _2>0\).

$$\begin{aligned} q(x,t)= & {} \pm 24 \sqrt{-\frac{35 \vartheta _4}{83 \vartheta _2}} \text {sech}^3\left[ \sqrt{-\frac{\vartheta _4}{83}} (x-v t)\right] \nonumber \\{} & {} \times e^{i \left( \left\{ \frac{a_{15}}{6 a_{16}}\right\} x+\omega t+\sigma W(t)-\sigma ^2 t+\theta _0 \right) },\end{aligned}$$
(73)
$$\begin{aligned} r(x,t)= & {} \varrho ~q(x,t), \end{aligned}$$
(74)

and singular soliton solution with \(\vartheta _4<0\) and \(\vartheta _2<0\).

$$\begin{aligned} q(x,t)= & {} \pm 24 \sqrt{\frac{35 \vartheta _4}{83 \vartheta _2}} \text {csch}^3\left[ \sqrt{-\frac{\vartheta _4}{83}} (x-v t)\right] \nonumber \\{} & {} \times e^{i \left( \left\{ \frac{a_{15}}{6 a_{16}}\right\} x+\omega t+\sigma W(t)-\sigma ^2 t+\theta _0 \right) },\end{aligned}$$
(75)
$$\begin{aligned} r(x,t)= & {} \varrho ~q(x,t). \end{aligned}$$
(76)

Result-2:

$$\begin{aligned} a_0= & {} -b_2,~a_1=\pm 288 \sqrt{\frac{5 \chi \vartheta _4}{1411 \vartheta _2}},~a_2=b_2 \chi ,\nonumber \\ a_3= & {} \mp \chi 24 \sqrt{\frac{85 \chi \vartheta _4}{83 \vartheta _2}},~b_1=0,~b_3=0,\nonumber \\ k= & {} \sqrt{\frac{17 \vartheta _4}{581}},~\vartheta _1=-\frac{102825 \vartheta _4}{9877},~\vartheta _3=\frac{13237 \vartheta _4}{1411}. \end{aligned}$$
(77)

As a result, the solution of the governing system (1)-(2) is attained.

$$\begin{aligned} q(x,t)= & {} \Bigg \{b_2 \chi \left( \frac{4 c}{4 c^2 e^{\sqrt{\frac{17 \vartheta _4}{581}} (x-v t)}+\chi e^{-\sqrt{\frac{17 \vartheta _4}{581}} (x-v t)}}\right) {}^2\nonumber \\{} & {} +b_2 \left( \frac{\chi -4 c^2 e^{2 \sqrt{\frac{17 \vartheta _4}{581}} (x-v t)}}{4 c^2 e^{2 \sqrt{\frac{17 \vartheta _4}{581}} (x-v t)}+\chi }\right) ^2-b_2\nonumber \\&\pm&\frac{1152c \sqrt{\frac{5 \chi \vartheta _4}{1411 \vartheta _2}}}{4 c^2 e^{\sqrt{\frac{17 \vartheta _4}{581}} (x-v t)}+\chi e^{-\sqrt{\frac{17 \vartheta _4}{581}} (x-v t)}}\nonumber \\{} & {} \mp 24 \chi \sqrt{\frac{85 \chi \vartheta _4}{83 \vartheta _2}}\! \left( \!\frac{4 c}{4 c^2 e^{\sqrt{\frac{17 \vartheta _4}{581}} (x-v t)}\!+\!\chi e^{-\sqrt{\frac{17 \vartheta _4}{581}} (x-v t)}}\!\right) ^3\!\Bigg \}\nonumber \\\times & {} e^{i \left( \left\{ \frac{a_{15}}{6 a_{16}}\right\} x+\omega t+\sigma W(t)-\sigma ^2 t+\theta _0 \right) },\end{aligned}$$
(78)
$$\begin{aligned} r(x,t)= & {} \varrho ~q(x,t). \end{aligned}$$
(79)

Selecting \(\chi =\pm 4 c^2 \) to recover bright soliton for \(\vartheta _4>0\) and \(\vartheta _2>0\)..

$$\begin{aligned} q(x,t)= & {} \pm \Bigg \{288 \sqrt{\frac{5 \vartheta _4}{1411 \vartheta _2}} \text {sech}\left( \sqrt{\frac{17 \vartheta _4}{581}} (x-v t)\right) \nonumber \\{} & {} -24 \sqrt{\frac{85 \vartheta _4}{83 \vartheta _2}} \text {sech}^3\left( \sqrt{\frac{17 \vartheta _4}{581}} (x-v t)\right) \Bigg \}\nonumber \\\times & {} e^{i \left( \left\{ \frac{a_{15}}{6 a_{16}}\right\} x+\omega t+\sigma W(t)-\sigma ^2 t+\theta _0 \right) },\end{aligned}$$
(80)
$$\begin{aligned} r(x,t)= & {} \varrho ~q(x,t), \end{aligned}$$
(81)

and singular soliton solutions for \(\vartheta _4>0\) and \(\vartheta _2<0\).

$$\begin{aligned} q(x,t)= & {} \pm \Bigg \{288 \sqrt{-\frac{5 \vartheta _4}{1411 \vartheta _2}} \text {csch}\left( \sqrt{\frac{17 \vartheta _4}{581}} (x-v t)\right) \nonumber \\{} & {} +24 \sqrt{-\frac{85 \vartheta _4}{83 \vartheta _2}} \text {csch}^3\left( \sqrt{\frac{17 \vartheta _4}{581}} (x-v t)\right) \Bigg \}\nonumber \\{} & {} \times e^{i \left( \left\{ \frac{a_{15}}{6 a_{16}}\right\} x+\omega t+\sigma W(t)-\sigma ^2 t+\theta _0 \right) },\end{aligned}$$
(82)
$$\begin{aligned} r(x,t)= & {} \varrho ~q(x,t). \end{aligned}$$
(83)

Result-3:

$$\begin{aligned} a_0= & {} -b_2,~a_1=0,~a_2=b_2 \chi ,~a_3=0,\nonumber \\ b_1= & {} \sqrt{\frac{35 \vartheta _4}{83 \vartheta _2}} \pm 36,~b_3=\sqrt{\frac{35 \vartheta _4}{83 \vartheta _2}} \mp 12,\nonumber \\ k= & {} \frac{1}{2} \sqrt{-\frac{\vartheta _4}{83}},~\vartheta _1=-\frac{20160 \vartheta _4}{83} ,~\vartheta _3=-\frac{3784 \vartheta _4}{83}. \end{aligned}$$
(84)

As a result, the solution of the governing system (1)-(2) is attained.

$$\begin{aligned} q(x,t)= & {} \Bigg \{\!-b_2\pm \frac{36 \sqrt{\frac{35 \vartheta _4}{83 \vartheta _2}} \left( \chi -4 c^2 e^{\sqrt{-\frac{\vartheta _4}{83}}}\right) }{4 c^2 e^{\sqrt{-\frac{\vartheta _4}{83}}}+\chi }\nonumber \\{} & {} +b_2 \chi \left( \frac{4 c}{4 c^2 e^{\frac{1}{2} \sqrt{-\frac{\vartheta _4}{83}}}+\chi e^{-\frac{1}{2} \sqrt{-\frac{\vartheta _4}{83}}}}\right) ^2\nonumber \\{} & {} +b_2 \left( \frac{\chi \!-\!4 c^2 e^{\sqrt{-\frac{\vartheta _4}{83}}}}{4 c^2 e^{\sqrt{-\frac{\vartheta _4}{83}}}+\chi }\right) ^2\!\mp \!12 \sqrt{\frac{35 \vartheta _4}{83 \vartheta _2}} \left( \!\frac{\chi \!-\!4 c^2 e^{\sqrt{-\frac{\vartheta _4}{83}}}}{4 c^2 e^{\sqrt{-\frac{\vartheta _4}{83}}}+\chi }\!\right) ^3\!\Bigg \}\nonumber \\{} & {} \times e^{i \left( \left\{ \frac{a_{15}}{6 a_{16}}\right\} x+\omega t+\sigma W(t)-\sigma ^2 t+\theta _0 \right) },\end{aligned}$$
(85)
$$\begin{aligned} r(x,t)= & {} \varrho ~q(x,t). \end{aligned}$$
(86)

Selecting \(\chi =\pm 4 c^2 \) to recover kink-type solutions for \(\vartheta _4<0\) and \(\vartheta _2<0\).

$$\begin{aligned} q(x,t)= & {} \pm \Bigg \{12 \sqrt{\frac{35 \vartheta _4}{83 \vartheta _2}} \tanh ^3\left( \frac{1}{2} \sqrt{-\frac{\vartheta _4}{83}} (x-v t)\right) \nonumber \\{} & {} -36 \sqrt{\frac{35 \vartheta _4}{83 \vartheta _2}} \tanh \left( \frac{1}{2} \sqrt{-\frac{\vartheta _4}{83}} (x-v t)\right) \Bigg \}\nonumber \\{} & {} \times e^{i \left( \left\{ \frac{a_{15}}{6 a_{16}}\right\} x+\omega t+\sigma W(t)-\sigma ^2 t+\theta _0 \right) },\end{aligned}$$
(87)
$$\begin{aligned} r(x,t)= & {} \varrho ~q(x,t), \end{aligned}$$
(88)

and singular soliton solution \(\vartheta _4<0\) and \(\vartheta _2<0\)

$$\begin{aligned} q(x,t)= & {} \pm \Bigg \{12 \sqrt{\frac{35 \vartheta _4}{83 \vartheta _2}} \coth ^3\left( \frac{1}{2} \sqrt{-\frac{\vartheta _4}{83}} (x-v t)\right) \nonumber \\{} & {} -36 \sqrt{\frac{35 \vartheta _4}{83 \vartheta _2}} \coth \left( \frac{1}{2} \sqrt{-\frac{\vartheta _4}{83}} (x-v t)\right) \Bigg \}\nonumber \\{} & {} \times e^{i \left( \left\{ \frac{a_{15}}{6 a_{16}}\right\} x+\omega t+\sigma W(t)-\sigma ^2 t+\theta _0 \right) },\end{aligned}$$
(89)
$$\begin{aligned} r(x,t)= & {} \varrho ~q(x,t). \end{aligned}$$
(90)

Result-4:

$$\begin{aligned} a_0= & {} -b_2,~a_1=0,~a_2=b_2 \chi ,\nonumber \\ a_3= & {} \chi \pm 12 \sqrt{-\frac{35 \chi \vartheta _4}{83 \vartheta _2}},b_1=\sqrt{\frac{35 \vartheta _4}{83 \vartheta _2}} \mp 18,\nonumber \\ b_3= & {} \sqrt{\frac{35 \vartheta _4}{83 \vartheta _2}} \pm 12,~k=\sqrt{-\frac{\vartheta _4}{83}},\nonumber \\ \vartheta _1= & {} -\frac{1260 \vartheta _4}{83} ,~\vartheta _3=-\frac{946 \vartheta _4}{83}. \end{aligned}$$
(91)

As a result, the solution of the governing system (1)-(2) is attained.

$$\begin{aligned} q(x,t)= & {} \Bigg \{b_2 \chi \left( \frac{4 c}{4 c^2 e^{\sqrt{-\frac{\vartheta _4}{83}} (x-v t)}+\chi e^{-\sqrt{-\frac{\vartheta _4}{83}} (x-v t)}}\right) {}^2\nonumber \\{} & {} +b_2 \left( \frac{\chi -4 c^2 e^{2 \sqrt{-\frac{\vartheta _4}{83}} (x-v t)}}{4 c^2 e^{2 \sqrt{-\frac{\vartheta _4}{83}} (x-v t)}+\chi }\right) ^2-b_2\nonumber \\&\mp&\frac{18 \sqrt{\frac{35 \vartheta _4}{83 \vartheta _2}} \left( \chi -4 c^2 e^{2 \sqrt{-\frac{\vartheta _4}{83}} (x-v t)}\right) }{4 c^2 e^{2 \sqrt{-\frac{\vartheta _4}{83}} (x-v t)}+\chi }\chi \nonumber \\&\pm&12 \sqrt{-\frac{35 \chi \vartheta _4}{83 \vartheta _2}} \left( \frac{4 c}{4 c^2 e^{\sqrt{-\frac{\vartheta _4}{83}} (x-v t)}+\chi e^{-\sqrt{-\frac{\vartheta _4}{83}} (x-v t)}}\right) ^3\nonumber \\&\pm&12 \sqrt{\frac{35 \vartheta _4}{83 \vartheta _2}} \left( \frac{\chi -4 c^2 e^{2 \sqrt{-\frac{\vartheta _4}{83}} (x-v t)}}{4 c^2 e^{2 \sqrt{-\frac{\vartheta _4}{83}} (x-v t)}+\chi }\right) ^3\Bigg \}\nonumber \\{} & {} \times e^{i \left( \left\{ \frac{a_{15}}{6 a_{16}}\right\} x+\omega t+\sigma W(t)-\sigma ^2 t+\theta _0 \right) },\end{aligned}$$
(92)
$$\begin{aligned} r(x,t)= & {} \varrho ~q(x,t), \end{aligned}$$
(93)

where \(\chi <0\). Selecting \(\chi =- 4 c^2 \) to recover singular soliton solution for \(\vartheta _4<0\) and \(\vartheta _2<0\).

$$\begin{aligned} q(x,t)= & {} \pm \Bigg \{18\sqrt{\frac{35 \vartheta _4}{83 \vartheta _2}} \coth \left( \sqrt{-\frac{\vartheta _4}{83}} (x-v t)\right) \nonumber \\{} & {} - 12 \sqrt{\frac{35 \vartheta _4}{83 \vartheta _2}} \text {csch}^3\left( \sqrt{-\frac{\vartheta _4}{83}} (x-v t)\right) \nonumber \\{} & {} -12 \sqrt{\frac{35 \vartheta _4}{83 \vartheta _2}} \coth ^3\left( \sqrt{-\frac{\vartheta _4}{83}} (x-v t)\right) \Bigg \}\nonumber \\{} & {} \times e^{i \left( \left\{ \frac{a_{15}}{6 a_{16}}\right\} x+\omega t+\sigma W(t)-\sigma ^2 t+\theta _0 \right) },\end{aligned}$$
(94)
$$\begin{aligned} r(x,t)= & {} \varrho ~q(x,t). \end{aligned}$$
(95)

Enhanced direct algebraic approach

In (25), balancing \(U_1^{(6)}(\xi )\) with \(U_1(\xi )^3\) yields \(N=3\). The solution is expressed in the following structure per the proposed technique in this article.

$$\begin{aligned} U_1(\xi )=\alpha _0 + \alpha _1 \theta (\xi )+ \alpha _2 \theta (\xi )^2+ \alpha _3 \theta (\xi )^3. \end{aligned}$$
(96)

Inserting (96) together with (34) into (25), we get a system of algebraic equations as follows:

$$\begin{aligned} \alpha _3^3 \vartheta _2+20160 \alpha _3 k^2 \tau _4^3=0, \end{aligned}$$
(97)
$$\begin{aligned} 41580 \alpha _3 k^2 \tau _3 \tau _4^2+3 \alpha _2 \left( \alpha _3^2 \vartheta _2+1680 k^2 \tau _4^3\right) =0, \end{aligned}$$
(98)
$$\begin{aligned}{} & {} 3 \Big (\alpha _3 \alpha _2^2 \vartheta _2+3240 \alpha _2 k^2 \tau _3 \tau _4^2+30 \alpha _3 \tau _4 \Big (297 k^2 \tau _3^2\nonumber \\{} & {} +4 \tau _4 \left( 83 k^2 \tau _2+\vartheta _4\Big )\Big )+\alpha _1 \left( \alpha _3^2 \vartheta _2+240 k^2 \tau _4^3\right) \right) \nonumber \\= & {} 0, \end{aligned}$$
(99)
$$\begin{aligned}{} & {} \frac{3}{2} \Big (2 \alpha _0 \alpha _3^2 \vartheta _2\!+840 \alpha _1 k^2 \tau _3 \tau _4^2\!+45 \alpha _3 \Big (77 k^2 \tau _3^3\!+336 k^2 \tau _1 \tau _4^2\nonumber \\{} & {} +4 \tau _4 \tau _3 \Big (133 k^2 \tau _2+2 \vartheta _4\Big )\Big )\Big )+\alpha _2^3 \vartheta _2+6 \alpha _2 \Big (\alpha _1 \alpha _3 \vartheta _2\nonumber \\{} & {} +5 \tau _4 \Big (189 k^2 \tau _3^2+4 \tau _4 \Big (56 k^2 \tau _2+\vartheta _4\Big )\Big )\Big )=0, \end{aligned}$$
(100)
$$\begin{aligned}{} & {} \alpha _1^2 \alpha _3 \vartheta _2+2 \alpha _0 \alpha _2 \alpha _3 \vartheta _2+63 \alpha _3 \tau _3^2 \vartheta _4+3150 \alpha _3 k^2 \tau _2 \tau _3^2\nonumber \\{} & {} +315 \alpha _2 k^2 \tau _3^3+4 \alpha _3 \tau _4 \vartheta _3+136 \alpha _3 \tau _2 \tau _4 \vartheta _4+3724 \alpha _3 k^2 \tau _2^2 \tau _4\nonumber \\{} & {} +56 \alpha _2 \tau _3 \tau _4 \vartheta _4+8568 \alpha _3 k^2 \tau _1 \tau _3 \tau _4+2380 \alpha _2 k^2 \tau _2 \tau _3 \tau _4\nonumber \\{} & {} +6048 \alpha _3 k^2 \tau _0 \tau _4^2+1680 \alpha _2 k^2 \tau _1 \tau _4^2+\alpha _1 \Big (\alpha _2^2 \vartheta _2+210 k^2 \tau _3^2 \tau _4\nonumber \\{} & {} +8 \tau _4^2 \left( 35 k^2 \tau _2+\vartheta _4\right) \Big )=0, \end{aligned}$$
(101)
$$\begin{aligned}{} & {} 4 \alpha _1^2 \alpha _2 \vartheta _2+4 \alpha _0 \alpha _2^2 \vartheta _2+14 \alpha _3 \tau _3 \vartheta _3+350 \alpha _3 \tau _2 \tau _3 \vartheta _4\nonumber \\{} & {} +6734 \alpha _3 k^2 \tau _2^2 \tau _3+70 \alpha _2 \tau _3^2 \vartheta _4+8253 \alpha _3 k^2 \tau _1 \tau _3^2\nonumber \\{} & {} +2030 \alpha _2 k^2 \tau _2 \tau _3^2\!+\!8 \alpha _2 \tau _4 \vartheta _3\!+\!420 \alpha _3 \tau _1 \tau _4 \vartheta _4\!+\!160 \alpha _2 \tau _2 \tau _4 \vartheta _4\nonumber \\{} & {} +19740 \alpha _3 k^2 \tau _1 \tau _2 \tau _4+2688 \alpha _2 k^2 \tau _2^2 \tau _4+26208 \alpha _3 k^2 \tau _0 \tau _3 \tau _4\nonumber \\{} & {} +6636 \alpha _2 k^2 \tau _1 \tau _3 \tau _4+5376 \alpha _2 k^2 \tau _0 \tau _4^2+\alpha _1 \Big (8 \alpha _0 \alpha _3 \vartheta _2\nonumber \\{} & {} +105 k^2 \tau _3^3+\!840 k^2 \tau _1 \tau _4^2\!+20 \tau _4 \tau _3 \left( 49 k^2 \tau _2+\!2 \vartheta _4\right) \Big )\!=\!0,\nonumber \\ \end{aligned}$$
(102)
$$\begin{aligned}{} & {} \frac{1}{2} \alpha _1 \Big (12 \alpha _0 \alpha _2 \vartheta _2+364 k^2 \tau _4 \tau _2^2+210 k^2 \tau _3^2 \tau _2\nonumber \\{} & {} +4 \tau _4 \left( 252 k^2 \Big (\tau _1 \tau _3+\tau _0 \tau _4\right) +\vartheta _3\Big )+5 \left( 3 \tau _3^2+8 \tau _2 \tau _4\right) \vartheta _4\Big )\nonumber \\{} & {} +\frac{1}{2} \alpha _3 \Big (9 \Big (2 k^2 \Big (81 \tau _2^3+\Big (647 \tau _1 \tau _3+1172 \tau _0 \tau _4\Big ) \tau _2\nonumber \\{} & {} +495 \Big (\!\tau _4 \tau _1^2+\tau _0 \tau _3^2\Big )\!\Big )\!+\!2 \tau _2 \vartheta _3\!+\!\Big (18 \tau _2^2\!+\!41 \tau _1 \tau _3\!+\!56 \tau _0 \tau _4\Big ) \vartheta _4\Big )\!\Big )\nonumber \\{} & {} +5 \alpha _2 \Big (7 k^2 \left( 19 \tau _3 \tau _2^2+72 \tau _1 \tau _4 \tau _2+27 \tau _3 \left( \tau _1 \tau _3+4 \tau _0 \tau _4\right) \right) \nonumber \\{} & {} +\tau _3 \vartheta _3+\left( 13 \tau _2 \tau _3+18 \tau _1 \tau _4\right) \vartheta _4\Big )\nonumber \\{} & {} +\alpha _1^3 \vartheta _2+\frac{1}{2} \alpha _3\left( 6 \alpha _0^2 \vartheta _2+2 \vartheta _1\right) =0, \end{aligned}$$
(103)
$$\begin{aligned}{} & {} \frac{3}{2} \alpha _1 \Big (7 k^2 \Big (3 \tau _3 \tau _2^2+20 \tau _1 \tau _4 \tau _2+6 \tau _3 \Big (\tau _1 \tau _3+6 \tau _0 \tau _4\Big )\Big )\nonumber \\{} & {} +\tau _3 \vartheta _3+5 \Big (\tau _2 \tau _3+2 \tau _1 \tau _4\Big ) \vartheta _4\Big )+\frac{15}{2} \alpha _3 \Big (7 k^2 \Big (27 \tau _3 \tau _1^2\nonumber \\{} & {} +\Big (19 \tau _2^2+108 \tau _0 \tau _4\Big ) \tau _1+72 \tau _0 \tau _2 \tau _3\Big )+\tau _1 \vartheta _3+\Big (13 \tau _1 \tau _2\nonumber \\{} & {} +18 \tau _0 \tau _3\Big ) \vartheta _4\Big )+3 \alpha _0 \alpha _1^2 \vartheta _2+\alpha _2\Big (3 \alpha _0^2 \vartheta _2+64 k^2 \tau _2^3\nonumber \\{} & {} +678 k^2 \tau _1 \tau _3 \tau _2+675 k^2 \tau _0 \tau _3^2+16 \tau _2^2 \vartheta _4+4 \tau _2 \vartheta _3\nonumber \\{} & {} +42 \tau _1 \tau _3 \vartheta _4+\vartheta _1\Big )+3 \alpha _0 \alpha _1^2 \vartheta _2+\alpha _2\Big (675 k^2 \tau _1^2 \tau _4\nonumber \\{} & {} +1728 k^2 \tau _0 \tau _2 \tau _4+72 \tau _0 \tau _4 \vartheta _4\Big )=0, \end{aligned}$$
(104)
$$\begin{aligned}{} & {} \alpha _1 \Big (3 \alpha _0^2 \vartheta _2+k^2 \tau _2^3+27 k^2 \tau _1 \tau _3 \tau _2+132 k^2 \tau _0 \tau _4 \tau _2\nonumber \\{} & {} +45 k^2 \tau _0 \tau _3^2+45 k^2 \tau _1^2 \tau _4+\tau _2^2 \vartheta _4+\tau _2 \vartheta _3\Big )\nonumber \\{} & {} +\frac{3}{2} \alpha _3 \Big (14 k^2 \left( 72 \tau _4 \tau _0^2+26 \tau _2^2 \tau _0+72 \tau _1 \tau _3 \tau _0+15 \tau _1^2 \tau _2\right) \nonumber \\{} & {} +4 \tau _0 \vartheta _3+5 \left( 3 \tau _1^2+8 \tau _0 \tau _2\right) \vartheta _4\Big )+3 \alpha _2 \Big (7 k^2 \Big (6 \tau _3 \tau _1^2\nonumber \\{} & {} +3 \Big (\tau _2^2+12 \tau _0 \tau _4\Big ) \tau _1+20 \tau _0 \tau _2 \tau _3\Big )+\tau _1 \vartheta _3+5 \Big (\tau _1 \tau _2\nonumber \\{} & {} +2 \tau _0 \tau _3\Big ) \vartheta _4\Big )+\alpha _1\left( \frac{9}{2} \tau _1 \tau _3 \vartheta _4+12 \tau _0 \tau _4 \vartheta _4+\vartheta _1\right) =0,\nonumber \\ \end{aligned}$$
(105)
$$\begin{aligned}{} & {} 4 \alpha _0 \vartheta _1+4 \alpha _0^3 \vartheta _2+2 \alpha _1 \tau _1 \vartheta _3+72 \alpha _3 \tau _0 \tau _1 \vartheta _4+45 \alpha _3 k^2 \tau _1^3\nonumber \\{} & {} +2 \alpha _1 \tau _1 \tau _2 \vartheta _4\!+\!900 \alpha _3 k^2 \tau _0 \tau _1 \tau _2\!+\!2 \alpha _1 k^2 \tau _1 \tau _2^2\!+\!12 \alpha _1 \tau _0 \tau _3 \vartheta _4\nonumber \\{} & {} +1080 \alpha _3 k^2 \tau _0^2 \tau _3+9 \alpha _1 k^2 \tau _1^2 \tau _3+60 \alpha _1 k^2 \tau _0 \tau _2 \tau _3\nonumber \\{} & {} +144 \alpha _1 k^2 \tau _0 \tau _1 \tau _4+2 \alpha _2 \Big (k^2 \Big (15 \tau _2 \tau _1^2+198 \tau _0 \tau _3 \tau _1\nonumber \\{} & {} +32 \tau _0 \Big (2 \tau _2^2\!+\!9 \tau _0 \tau _4\Big )\!\Big )\!+\!4 \tau _0 \vartheta _3\!+\!\left( 3 \tau _1^2\!+\!16 \tau _0 \tau _2\right) \vartheta _4\Big )\!=\!0.\nonumber \\ \end{aligned}$$
(106)

Solving these equations together yields the following results:

Case-1: Choosing \(\tau _0=\tau _1=\tau _3=0\), yields

Result-1:

$$\begin{aligned} \alpha _0= & {} \alpha _1=\alpha _2=0,~\alpha _3=24 k \tau _4 \sqrt{-\frac{35}{\vartheta _2}},\nonumber \\ \vartheta _1= & {} \frac{11025 \vartheta _4^3}{571787 k^4},~\vartheta _3=\frac{1891 \vartheta _4^2}{6889 k^2},~\tau _2=-\frac{\vartheta _4}{83 k^2}. \end{aligned}$$
(107)

As a result, the solutions of system (1)-(2) reach

$$\begin{aligned} q(x,t)= & {} \pm \frac{24 \vartheta _4}{83 k^2}\sqrt{-\frac{35 \vartheta _4}{83 \tau _4 \vartheta _2}} \text {sech}^3\left[ \sqrt{-\frac{\vartheta _4}{83}} (x-v t)\right] \nonumber \\{} & {} \times e^{i \left( \left\{ \frac{a_{15}}{6 a_{16}}\right\} x+\omega t+\sigma W(t)-\sigma ^2 t+\theta _0 \right) }.\end{aligned}$$
(108)
$$\begin{aligned} r(x,t)= & {} \varrho ~q(x,t). \end{aligned}$$
(109)

This solution is a bright soliton with \(\vartheta _4<0\) and \(\vartheta _2<0\).

$$\begin{aligned} q(x,t)= & {} \pm \frac{24 \vartheta _4}{83 k^2}\sqrt{\frac{35 \vartheta _4}{83 \tau _4 \vartheta _2}} \text {csch}^3\left[ \sqrt{-\frac{\vartheta _4}{83}} (x-v t)\right] \nonumber \\{} & {} \times e^{i \left( \left\{ \frac{a_{15}}{6 a_{16}}\right\} x+\omega t+\sigma W(t)-\sigma ^2 t+\theta _0 \right) }.\end{aligned}$$
(110)
$$\begin{aligned} r(x,t)= & {} \varrho ~q(x,t). \end{aligned}$$
(111)

This solution is a singular soliton with \(\vartheta _4<0\) and \(\vartheta _2<0\).

Result-2:

$$\begin{aligned} \alpha _0= & {} 0,~\alpha _1=\frac{\left( 288 \vartheta _4\right) \sqrt{-\frac{5 \tau _4}{7 \vartheta _2}}}{83 k},\nonumber \\ \alpha _2= & {} 0,~\alpha _3=24 k \tau _4 \sqrt{-\frac{35 \tau _4}{\vartheta _2}},\nonumber \\ \vartheta _1= & {} -\frac{1748025 \vartheta _4^3}{196122941 k^4},~\vartheta _3=\frac{1891 \vartheta _4^2}{6889 k^2},~\tau _2=\frac{17 \vartheta _4}{581 k^2}. \end{aligned}$$
(112)

As a result, the solutions of system (1)-(2) reach

$$\begin{aligned} q(x,t)= & {} \mp \frac{288 \vartheta _4}{83 k^2} \sqrt{\frac{85 \vartheta _4}{4067 \vartheta _2}} \text {sech}\left[ \sqrt{\frac{17 \vartheta _4}{581}} (x-v t)\right] \nonumber \\{} & {} \pm \frac{408 \vartheta _4}{581 k}\sqrt{\frac{85 \vartheta _4}{83 k^2 \vartheta _2}} \text {sech}^3\left[ \sqrt{\frac{17 \vartheta _4}{581}} (x-v t)\right] \nonumber \\{} & {} \times e^{i \left( \left\{ \frac{a_{15}}{6 a_{16}}\right\} x+\omega t+\sigma W(t)-\sigma ^2 t+\theta _0 \right) }.\end{aligned}$$
(113)
$$\begin{aligned} r(x,t)= & {} \varrho ~q(x,t). \end{aligned}$$
(114)

This solution is a bright soliton with \(\vartheta _4>0\) and \(\vartheta _2>0\).

$$\begin{aligned} q(x,t)= & {} \pm \frac{288 \vartheta _4}{83 k^2} \sqrt{-\frac{85 \vartheta _4}{4067 \vartheta _2}} \text {csch}\left[ \sqrt{\frac{17 \vartheta _4}{581}} (x-v t)\right] \nonumber \\{} & {} \pm \frac{408 \vartheta _4}{581 k}\sqrt{-\frac{85 \vartheta _4}{83 k^2 \vartheta _2}} \text {csch}^3\left[ \sqrt{\frac{17 \vartheta _4}{581}} (x-v t)\right] \nonumber \\{} & {} \times e^{i \left( \left\{ \frac{a_{15}}{6 a_{16}}\right\} x+\omega t+\sigma W(t)-\sigma ^2 t+\theta _0 \right) }.\end{aligned}$$
(115)
$$\begin{aligned} r(x,t)= & {} \varrho ~q(x,t). \end{aligned}$$
(116)

This solution is a singular soliton with \(\vartheta _4>0\) and \(\vartheta _2<0\).

Case-2: Choosing \(\tau _0=\frac{\tau _2^2}{4 \tau _4},~\tau _1=\tau _3=0\), yields

$$\begin{aligned} \alpha _0= & {} 0,~\alpha _1= \pm \frac{18 \vartheta _4}{83 k}\sqrt{-\frac{35 \tau _4}{\vartheta _2}},~\alpha _2=0,\nonumber \\ \alpha _3= & {} \mp k \tau _4 24 \sqrt{-\frac{35 \tau _4}{\vartheta _2}},\nonumber \\ \vartheta _1= & {} -\frac{1260 \vartheta _4^3}{571787 k^4},~\vartheta _3=\frac{946 \vartheta _4^2}{6889 k^2},~\tau _2=\frac{\vartheta _4}{166 k^2}.\end{aligned}$$
(117)
$$\begin{aligned} r(x,t)= & {} \varrho ~q(x,t). \end{aligned}$$
(118)

As a result, the solutions of system (1)-(2) reach

$$\begin{aligned} q(x,t)= & {} \pm \frac{18 \vartheta _4}{83 k^2}\sqrt{\frac{35 \vartheta _4}{332 \vartheta _2}} \tanh \left[ \sqrt{-\frac{\vartheta _4}{332}} (x-v t)\right] \nonumber \\{} & {} \pm \frac{6 \vartheta _4}{83 k^2}\sqrt{\frac{35 \vartheta _4}{332 \vartheta _2}} \tanh ^3\left[ \sqrt{-\frac{\vartheta _4}{332}} (x-v t)\right] \nonumber \\{} & {} \times e^{i \left( \left\{ \frac{a_{15}}{6 a_{16}}\right\} x+\omega t+\sigma W(t)-\sigma ^2 t+\theta _0 \right) }.\end{aligned}$$
(119)
$$\begin{aligned} r(x,t)= & {} \varrho ~q(x,t). \end{aligned}$$
(120)

This solution is a kink-tyoe solution with \(\vartheta _4<0\) and \(\vartheta _2<0\).

$$\begin{aligned} q(x,t)= & {} \pm \frac{18 \vartheta _4}{83 k^2}\sqrt{\frac{35 \vartheta _4}{332 \vartheta _2}} \coth \left[ \sqrt{-\frac{\vartheta _4}{332}} (x-v t)\right] \nonumber \\{} & {} \pm \frac{6 \vartheta _4}{83 k^2}\sqrt{\frac{35 \vartheta _4}{332 \vartheta _2}} \coth ^3\left[ \sqrt{-\frac{\vartheta _4}{332}} (x-v t)\right] \nonumber \\{} & {} \times e^{i \left( \left\{ \frac{a_{15}}{6 a_{16}}\right\} x+\omega t+\sigma W(t)-\sigma ^2 t+\theta _0 \right) }.\end{aligned}$$
(121)
$$\begin{aligned} r(x,t)= & {} \varrho ~q(x,t). \end{aligned}$$
(122)

This solution is a singular soliton with \(\vartheta _4<0\) and \(\vartheta _2<0\).

Case-3: Choosing \(\tau _0=\tau _1=0\), yields

$$\begin{aligned} \alpha _0= & {} \pm \frac{6 \vartheta _4}{83 k^2}\sqrt{\frac{35 \vartheta _4}{83 \vartheta _2}},~\alpha _1=0,\nonumber \\ \alpha _2= & {} \pm 36\tau _4 \sqrt{\frac{35 \vartheta _4}{83 \vartheta _2}} , \alpha _3=\pm \frac{24 k \sqrt{35 \tau _4} \tau _4}{\sqrt{-\vartheta _2}},\nonumber \\ \vartheta _1= & {} -\frac{1260 \vartheta _4^3}{571787 k^4},~\vartheta _3=\frac{946 \vartheta _4^2}{6889 k^2},\nonumber \\ \tau _2= & {} -\frac{\vartheta _4}{83 k^2},~\tau _3=\pm \frac{2 \sqrt{\tau _4} \sqrt{-\vartheta _4}}{\sqrt{83} k}.\end{aligned}$$
(123)
$$\begin{aligned} r(x,t)= & {} \varrho ~q(x,t). \end{aligned}$$
(124)

As a result, the solutions of system (1)-(2) reach

$$\begin{aligned} q(x,t)= & {} \pm \frac{3 \vartheta _4}{83 k^2} \sqrt{\frac{35 \vartheta _4}{83 \vartheta _2}} \left( \frac{\text {sech}^6\left[ \frac{1}{2} \sqrt{-\frac{\vartheta _4}{83}} (x-v t)\right] }{\left( \tanh \left[ \frac{1}{2} \sqrt{-\frac{\vartheta _4}{83}} (x-v t)\right] +1\right) ^3}\right. \nonumber \\{} & {} \left. -\frac{3 \text {sech}^4\left[ \frac{1}{2} \sqrt{-\frac{\vartheta _4}{83}} (x-v t)\right] }{\left( \tanh \left[ \frac{1}{2} \sqrt{-\frac{\vartheta _4}{83}} (x-v t)\right] +1\right) ^2}+2\right) \nonumber \\{} & {} \times e^{i \left( \left\{ \frac{a_{15}}{6 a_{16}}\right\} x+\omega t+\sigma W(t)-\sigma ^2 t+\theta _0 \right) }. \end{aligned}$$
(125)
$$\begin{aligned} r(x,t)= & {} \varrho ~q(x,t). \end{aligned}$$
(126)

This solution is a straddled soliton solution with \(\vartheta _4<0\) and \(\vartheta _2<0\).

$$\begin{aligned} q(x,t)= & {} \pm \frac{3 \vartheta _4}{83 k^2} \sqrt{\frac{35 \vartheta _4}{83 \vartheta _2}} \left( -\frac{\text {csch}^6\left[ \frac{1}{2} \sqrt{-\frac{\vartheta _4}{83}} (x-v t)\right] }{\left( \coth \left[ \frac{1}{2} \sqrt{-\frac{\vartheta _4}{83}} (x-v t)\right] +1\right) ^3}\right. \nonumber \\{} & {} \left. -\frac{3 \text {csch}^4\left[ \frac{1}{2} \sqrt{-\frac{\vartheta _4}{83}} (x-v t)\right] }{\left( \coth \left[ \frac{1}{2} \sqrt{-\frac{\vartheta _4}{83}} (x-v t)\right] +1\right) ^2}+2\right) \nonumber \\{} & {} \times e^{i \left( \left\{ \frac{a_{15}}{6 a_{16}}\right\} x+\omega t+\sigma W(t)-\sigma ^2 t+\theta _0 \right) }. \end{aligned}$$
(127)
$$\begin{aligned} r(x,t)= & {} \varrho ~q(x,t). \end{aligned}$$
(128)

This solution is a straddled soliton solution with \(\vartheta _4<0\) and \(\vartheta _2<0\).

$$\begin{aligned} q(x,t)= & {} \pm \frac{3 \vartheta _4}{83 k^2} \sqrt{\frac{35 \vartheta _4}{83 \vartheta _2}} \left( -\frac{\text {csch}^6\left[ \frac{1}{2} \sqrt{-\frac{\vartheta _4}{83}} (x-v t)\right] }{\left( \coth \left[ \frac{1}{2} \sqrt{-\frac{\vartheta _4}{83}} (x-v t)\right] +1\right) ^3}\right. \nonumber \\{} & {} \left. -\frac{3 \text {csch}^4\left[ \frac{1}{2} \sqrt{-\frac{\vartheta _4}{83}} (x-v t)\right] }{\left( \coth \left[ \frac{1}{2} \sqrt{-\frac{\vartheta _4}{83}} (x-v t)\right] +1\right) ^2}+2\right) \nonumber \\{} & {} \times e^{i \left( \left\{ \frac{a_{15}}{6 a_{16}}\right\} x+\omega t+\sigma W(t)-\sigma ^2 t+\theta _0 \right) }. \end{aligned}$$
(129)
$$\begin{aligned} r(x,t)= & {} \varrho ~q(x,t). \end{aligned}$$
(130)

This solution is a straddled soliton solution with \(\vartheta _4<0\) and \(\vartheta _2<0\).

$$\begin{aligned} q(x,t)= & {} \pm \left( -32 \left( \frac{\text {sech}^2\left[ \frac{1}{2} \sqrt{-\frac{\vartheta _4}{83}} (x-v t)\right] }{\left( 1-\tanh \left[ \frac{1}{2} \sqrt{-\frac{\vartheta _4}{83}} (x-v t)\right] \right) ^2-4}\right) ^3\right. \nonumber \\{} & {} \left. -24 \left( \frac{\text {sech}^2\left[ \frac{1}{2} \sqrt{-\frac{\vartheta _4}{83}} (x-v t)\right] }{\left( 1-\tanh \left[ \frac{1}{2} \sqrt{-\frac{\vartheta _4}{83}} (x-v t)\right] \right) ^2-4}\right) ^2\!+\!1\!\right) \nonumber \\{} & {} \times \frac{6 \vartheta _4}{83 k^2} \sqrt{\frac{35 \vartheta _4}{83 \vartheta _2}}~e^{i \left( \left\{ \frac{a_{15}}{6 a_{16}}\right\} x+\omega t+\sigma W(t)-\sigma ^2 t+\theta _0 \right) }.\end{aligned}$$
(131)
$$\begin{aligned} r(x,t)= & {} \varrho ~q(x,t). \end{aligned}$$
(132)
$$\begin{aligned} q(x,t)= & {} \pm \left( -32 \left( \frac{\text {csch}^2\left[ \frac{1}{2} \sqrt{-\frac{\vartheta _4}{83}} (x-v t)\right] }{\left( 1-\coth \left[ \frac{1}{2} \sqrt{-\frac{\vartheta _4}{83}} (x-v t)\right] \right) ^2-4}\right) ^3\right. \nonumber \\{} & {} \left. -24 \left( \frac{\text {csch}^2\left[ \frac{1}{2} \sqrt{-\frac{\vartheta _4}{83}} (x-v t)\right] }{\left( 1-\coth \left[ \frac{1}{2} \sqrt{-\frac{\vartheta _4}{83}} (x-v t)\right] \right) ^2-4}\right) ^2\!+\!1\!\right) \nonumber \\{} & {} \times \frac{6 \vartheta _4}{83 k^2} \sqrt{\frac{35 \vartheta _4}{83 \vartheta _2}}~e^{i \left( \left\{ \frac{a_{15}}{6 a_{16}}\right\} x+\omega t+\sigma W(t)-\sigma ^2 t+\theta _0 \right) }.\end{aligned}$$
(133)
$$\begin{aligned} r(x,t)= & {} \varrho ~q(x,t). \end{aligned}$$
(134)

Conclusions

This work was about the retrieval of optical solitons for highly dispersive version of FLE with PMD and perturbation terms in presence of white noise. Two integration approaches have made this retrieval possible. A full spectrum of optical solitons have been thus recovered and they are listed along with their respective existence criteria. The observation is that the effect of white noise stays in the phase component of such solitoins and is never a part of the amplitude component of the solitons. The results of the paper thus carry a lot of promise. The model will be further considered in additional optoelectronic devices such as dispersion–flattened fibers and Bragg gratings, optical couplers and also in magneto–optic waveguides. The results of such research activities will be disseminated across the board with time after they are connected and compared wit the pre–existing ones [37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57].