Introduction

The concepts of the concatenation model and dispersive concatenation model were sequentially conceived a decade ago [1,2,3,4,5]. Subsequently, extensive studies with these two models were carried out during the past couple of years. These range from the retrieval of soliton solutions and the conservation laws, studying the model with power law of self-phase modulation, addressing the Internet bottleneck effect and minimizing the slowdown of soliton transmission by introducing the spatiotemporal dispersion (STD) in addition to the pre-existing chromatic dispersion (CD). The numerical analysis of solitons for the concatenation model by the Laplace–Adomian decomposition was also conducted. Recently, the effect of white noise in soliton transmission for the concatenation model with Kerr law and power law of SPM yielded interesting results [6,7,8,9,10,11,12,13,14,15]. The current paper addresses the effect of white noise for the dispersive concatenation model that is studied in the current paper with power law of SPM. The results of the current paper follow a previously published prequel paper on dispersive optical solitons with the Kerr law of SPM, as recently reported [13].

The dispersive concatenation model is the combination of the Schrödinger–Hirota equation (SHE), Lakshmanan–Porsezian–Daniel (LPD) model and the dispersive nonlinear Schrödinger’s equation (NLSE) that carries a fifth–order dispersive effect. Therefore, the model is truly dispersive as the name bears. The current paper will address this model with power law of SPM as a sequel to the Kerr law [13]. The white noise in Itô sense is included as a source of stochasticity to the model. Two integration approaches will lead to the soliton solutions. These are the enhanced Kudryashov’s approach and the extended auxiliary equation scheme. These two algorithms would lead to the emergence of bright and singular optical solitons only. These approaches fail to recover the dark optical solitons as expected since it is well known, and experimentally proved in the past, that dark solitons are not supported by any model with power law of SPM unless the power law collapses to Kerr law. The details of the concepts and the derivation of the results are exhibited in the rest of the paper after a succinct recapitulation of the integration methodologies.

Governing model

In [1], the concatenation model with higher-order dispersion effects and power law nonlinearity was studied. In this study, we consider the dimensionless expression of this model by incorporating multiplicative white noise effect, which can be expressed as follows:

$$\begin{aligned} \begin{aligned}&iq_t + aq_{xx} + b \left| q \right| ^{2n} q \\&- i \delta _1 \left[ \sigma _1 q_{xxx} + \sigma _2 \left| q \right| ^{2n} q_x \right] + \sigma q ~\frac{dW(t)}{\rm{{dt}}} \\ +&\delta _2 \left[ \sigma _3 q_{xxxx} + \sigma _4 \left| q \right| ^{2n} q_{xx} + \sigma _5 \left| q \right| ^{2n+2} q \right. \\&\left. + \sigma _6 \left| q_x \right| ^2 q + \sigma _7 q_x^2 q^{*} + \sigma _8 q_{xx}^{*} q^2 \right] \\ -&i\delta _3 \left[ \sigma _9 q_{xxxxx} + \sigma _{10} \left| q \right| ^{2n} q_{xxx} + \sigma _{11} \left| q \right| ^{2n+2} q_x \right. \\&\left. +\sigma _{12} q q_x q_{xx}^{*} + \sigma _{13} q^{*} q_x q_{xx} + \sigma _{14} q q_x^{*} q_{xx} + \sigma _{15} q_x^{2} q_x^{*} \right] =0. \end{aligned} \end{aligned}$$
(1)

Here, q(x, t) represents a complex-valued function that describes the wave profile. In this context, x and t represent the spatial and temporal coordinates, respectively. The parameters a and b correspond to CD and SPM, respectively, of the NLSE with power law of SPM. The symbol \(i=\sqrt{-1}\) is the imaginary unit. The coefficients \(\delta _1,~\delta _2\) and \(\delta _3\) are the nonzero parameters associated with SHE, LPD model and the fifth-order NLSE, respectively. The variable \(\sigma\) is used to represent a nonzero constant value and serves as the sign for indicating the intensity of white noise. Furthermore, the standard Wiener process, represented as W(t), can be defined as the integral of the function \(\Lambda (\eta ) = dW(t)/dt\) concerning the Wiener process \(W(\eta )\), where \(\eta\) is a variable that assumes values smaller than t. In the provided context, the symbol \(\eta\) represents a stochastic variable, whereas \(\Lambda (\eta )\) is utilized to denote typical Gaussian white noise, popularly known as multiplicative white noise.

Mathematical analysis of the governing model

For investigating a model with a multiplicative white noise effect, the following wave structure is selected:

$$\begin{aligned} q(x,t)=U(\xi ) e^{i\left( -\kappa x+\omega t+\sigma W(t)-\sigma ^2 t+ \theta _0\right) }. \end{aligned}$$
(2)

The wave variable \(\xi\) is defined as

$$\begin{aligned} \xi =k(x-vt), \end{aligned}$$
(3)

where k and v represent nonzero constants. In Eq. (2), \(U(\xi )\) denotes a real-valued function representing the soliton solutions’ amplitude components. The variable v corresponds to the speed of the soliton, while k represents the wave width. In the provided context, the symbol \(\kappa\) represents the frequency of the solitons. The symbol \(\omega\) is used to describe the wave number. The symbol \(\sigma\) is employed to signify the noise coefficient. Lastly, the symbol \(\theta _0\) represents the phase constant. The following formulas are obtained by substituting Eq. (2) into Eq. (1) and subsequently decomposing them into their real component

$$\begin{aligned}{} & {} -k^2 \left( a+10 \delta _3 \kappa ^3 \sigma _9-6 \delta _2 \kappa ^2 \sigma _3-3 \delta _1 \kappa \sigma _1\right) U'' \nonumber \\{} & {} + \left( a \kappa ^2+\delta _3 \kappa ^5 \sigma _9-\delta _2 \kappa ^4 \sigma _3-\delta _1 \kappa ^3 \sigma _1-\sigma ^2+\omega \right) U\nonumber \\{} & {} \quad + \left( -b-\delta _3 \kappa ^3 \sigma _{10}+\delta _2 \kappa ^2 \sigma _4+\delta _1 \kappa \sigma _2\right) U^{2 n+1}\nonumber \\{} & {} \quad +k^4 \left( 5 \delta _3 \kappa \sigma _9-\delta _2 \sigma _3\right) U^{(4)}+k^2 \left( 3 \delta _3 \kappa \sigma _{10}-\delta _2 \sigma _4\right) U^{2 n} U''\nonumber \\{} & {} \quad +k^2 \left( \delta _3 \kappa \left( \sigma _{12}+\sigma _{13}-\sigma _{14}\right) -\delta _2 \sigma _8\right) U^2 U''\nonumber \\{} & {} \quad +k^2 \left( \delta _3 \kappa \left( -2 \sigma _{12}+2 \sigma _{13}+2 \sigma _{14}+\sigma _{15}\right) -\delta _2 \left( \sigma _6+\sigma _7\right) \right) U U'^2\nonumber \\{} & {} \quad +\left( \delta _3 \kappa \sigma _{11}-\delta _2 \sigma _5\right) U^{2 n+3}+\kappa ^2 \left( \delta _3 \kappa \left( -\sigma _{12}-\sigma _{13}+\sigma _{14}+\sigma _{15}\right) \right. \nonumber \\{} & {} \quad \left. +\delta _2 \left( -\sigma _6+\sigma _7+\sigma _8\right) \right) U^3=0, \end{aligned}$$
(4)

and imaginary component

$$\begin{aligned}{} & {} k \left( 2 a \kappa +5 \delta _3 \kappa ^4 \sigma _9-4 \delta _2 \kappa ^3 \sigma _3-3 \delta _1 \kappa ^2 \sigma _1+v\right) U'\nonumber \\{} & {} \quad +\delta _3 k^5 \sigma _9 U^{(5)}+\delta _3 k^3 \sigma _{10} U^{(3)} U^{2 n}\nonumber \\{} & {} \quad +k^3 \left( 2 \kappa \left( 2 \delta _2 \sigma _3-5 \delta _3 \kappa \sigma _9\right) +\delta _1 \sigma _1\right) U^{(3)}\nonumber \\{} & {} \quad +\delta _3 k^3 \sigma _{15} U'^3+\delta _3 k^3 \left( \sigma _{12}+\sigma _{13}+\sigma _{14}\right) U U' U''\nonumber \\{} & {} \quad +k \left( \kappa \left( 2 \delta _2 \sigma _4-3 \delta _3 \kappa \sigma _{10}\right) +\delta _1 \sigma _2\right) \nonumber \\{} & {} \quad U^{2 n} U'+\delta _3 k \sigma _{11} U^{2 n+2} U'\nonumber \\{} & {} \quad +\kappa k U^2 \left( \delta _3 \kappa \left( \sigma _{12}-3 \sigma _{13}+\sigma _{14}+\sigma _{15}\right) \right. \nonumber \\{} & {} \quad \left. +2 \delta _2 \left( \sigma _7-\sigma _8\right) \right) U'=0. \end{aligned}$$
(5)

From the imaginary part, the soliton speed reads as

$$\begin{aligned} v=-2 a \kappa -5 \delta _3 \kappa ^4 \sigma _9+4 \delta _2 \kappa ^3 \sigma _3+3 \delta _1 \kappa ^2 \sigma _1, \end{aligned}$$
(6)

and the soliton frequency as

$$\begin{aligned} \kappa =-\frac{\delta _2 \left( \sigma _8-\sigma _7\right) }{2 \delta _3 \sigma _{13}}, \end{aligned}$$
(7)

with the following parametric restrictions

$$\begin{aligned} {{\left\{ \begin{array}{ll} \sigma _1=-\frac{2 \delta _2^2 \sigma _3 \left( \sigma _7-\sigma _8\right) }{\delta _1 \delta _3 \sigma _{13}}, \\ \sigma _4=-\frac{\delta _1 \delta _3 \sigma _2 \sigma _{13}}{\delta _2^2 \left( \sigma _7-\sigma _8\right) },\\ \sigma _{12}+\sigma _{13}+\sigma _{14}=0, \\ \sigma _9=\sigma _{10}=\sigma _{11}=\sigma _{15}=0. \end{array}\right. }} \end{aligned}$$
(8)

Under the previous conditions, Eq. (1) will have the following form

$$\begin{aligned} \begin{aligned}&iq_t + aq_{xx} + b \left| q \right| ^{2n} q \\&\quad - i \delta _1 \left[ \sigma _1 q_{xxx} + \sigma _2 \left| q \right| ^{2n} q_x \right] \\&\quad +\sigma q ~\frac{dW(t)}{dt} \\&\quad + \delta _2 \left[ \sigma _3 q_{xxxx} + \sigma _4 \left| q \right| ^{2n} q_{xx} \right. \\&\quad + \sigma _5 \left| q \right| ^{2n+2} q \\&\quad \left. + \sigma _6 \left| q_x \right| ^2 q+ \sigma _7 q_x^2 q^{*} + \sigma _8 q_{xx}^{*} q^2 \right] \\&\quad - i\delta _3 \left[ \sigma _{12} q q_x q_{xx}^{*} + \sigma _{13} q^{*} q_x q_{xx} \right. \\&\quad \left. + \sigma _{14} q q_x^{*} q_{xx} \right] = 0. \end{aligned} \end{aligned}$$
(9)

Additionally, Eq. (4) will take on the following form

$$\begin{aligned} k^2 U^{(4)}+ w _8 U^{2 n} U''+ w _7 U^2 U''+ w _6 U''+ w _5 U U'^2+ w _4 U^{2 n+1}+ w _3 U^{2 n+3}+ w _2 U^3+ w _1 U=0, \end{aligned}$$
(10)

with

$$\begin{aligned} {{\left\{ \begin{array}{ll} w _1=\frac{-a \kappa ^2+\delta _2 \kappa ^4 \sigma _3+\delta _1 \kappa ^3 \sigma _1+\sigma ^2-\omega }{\delta _2 k^2 \sigma _3},\\ w _2=-\frac{\kappa ^2 \left( 2 \delta _3 \kappa \sigma _{14}+\delta _2 \left( -\sigma _6+\sigma _7+\sigma _8\right) \right) }{\delta _2 k^2 \sigma _3},\\ w _3=\frac{\sigma _5}{k^2 \sigma _3},\\ w _4=\frac{b-\delta _2 \kappa ^2 \sigma _4-\delta _1 \kappa \sigma _2}{\delta _2 k^2 \sigma _3},\\ w _5=\frac{\delta _2 \left( \sigma _6+\sigma _7\right) -4 \delta _3 \kappa \left( \sigma _{13}+\sigma _{14}\right) }{\delta _2 \sigma _3},\\ w _6=\frac{a-6 \delta _2 \kappa ^2 \sigma _3-3 \delta _1 \kappa \sigma _1}{\delta _2 \sigma _3},\\ w _7=\frac{2 \delta _3 \kappa \sigma _{14}+\delta _2 \sigma _8}{\delta _2 \sigma _3},\\ w _8=\frac{\sigma _4}{\sigma _3}. \end{array}\right. }} \end{aligned}$$
(11)

Provided that \(\sigma _3\ne 0\). Using the transformation

$$\begin{aligned} U=V^{\frac{1}{n}}, \end{aligned}$$

Equation (10) collapses to

$$\begin{aligned}{} & {} k^2 n^3 V^{(4)} V^3-3 k^2 (n-1) n^2 V^2 V''^2\nonumber \\{} & {} \quad -4 k^2 (n-1) n^2 V^{(3)} V^2 V'+6 k^2 n \left( 2 n^2-3 n+1\right) V V'^2 V''\nonumber \\{} & {} \quad +k^2 \left( -6 n^3+11 n^2-6 n+1\right) V'^4+n^4 w _4 V^6\nonumber \\{} & {} \quad +n^4 w _1 V^4+n^3 w _8 V^5 V''+n^3 w _6 V^3 V''\nonumber \\{} & {} \quad +V^{\frac{2}{n}} \left( n^4 w _3 V^6+n^4 w _2 V^4+n^3 w _7 V^3 V''\right. \nonumber \\{} & {} \quad \left. +n^2 \left( w _5-(n-1) w _7\right) V^2 V'^2\right) \nonumber \\{} & {} \quad -(n-1) n^2 w _8 V^4 V'^2-(n-1) n^2 w _6 V^2 V'^2=0. \end{aligned}$$
(12)

In terms of integrability, it is observed that we possess

$$\begin{aligned} w _2= w _3= w _5= w _7=0 \end{aligned}$$
(13)

This assumption leads \(\sigma _5=0\). In this case, Eq. (12) reads

$$\begin{aligned}{} & {} k^2 n^3 V^{(4)} V^3-3 k^2 (n-1) n^2 V^2 V''^2\nonumber \\{} & {} \quad -4 k^2 (n-1) n^2 V^{(3)} V^2 V'+6 k^2 n \left( 2 n^2-3 n+1\right) V V'^2 V''\nonumber \\{} & {} \quad +k^2 \left( -6 n^3+11 n^2-6 n+1\right) V'^4\nonumber \\{} & {} \quad +n^4 w _4 V^6+n^4 w _1 V^4+n^3 w _8 V^5 V''+n^3 w _6 V^3 V''\nonumber \\- & {} (n-1) n^2 w _8 V^4 V'^2-(n-1) n^2 w _6 V^2 V'^2=0. \end{aligned}$$
(14)

Balancing \(V^3 V^{\prime \prime \prime \prime }\) with \(V^5 V^{\prime \prime }\) or \(V^6\) in Eq. (14) gives \(N=1\) or \(N=2\).

An outline of the integration algorithms

We could include a governing model which has the structure of

$$\begin{aligned} F(u,u_{x},u_{t},u_{xt},u_{xx},...)=0, \end{aligned}$$
(15)

where \(u=u(x,t)\) represents a wave profile, where t and x describe the time and space variables, respectively.

The use of the wave transformation

$$\begin{aligned} u(x,t)=U(\xi ),~~~\xi =k (x-\upsilon t), \end{aligned}$$
(16)

causes a reduction of Eq. (15) to

$$\begin{aligned} P(U,-k \upsilon U^{\prime },k U^{\prime },k ^{2}U^{\prime \prime },...)=0. \end{aligned}$$
(17)

In that expression, k represents the wave width, \(\xi\) represents the wave variable, and \(\upsilon\) represents the wave velocity.

The enhanced Kudryashov’s method

This subsection presents a thorough overview of the basic procedures with the enhanced Kudryashov technique.

Step 1:  The explicit solution for the reduced model Eq. (17) is provided as follows

$$\begin{aligned} U(\xi )=\sigma _{0}+\sum _{i=1}^{N}\left\{ \sigma _{i}R(\xi )^{i}+\rho _i \left( \frac{R^{\prime }(\xi )}{R(\xi )}\right) ^{i}\right\} , \end{aligned}$$
(18)

along with the auxiliary equation

$$\begin{aligned} {R^{\prime }(\xi )}^{2}={R(\xi )}^{2}(1-\chi {R(\xi )}^{2}). \end{aligned}$$
(19)

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 Eq. (17).

Step 2:  Eq. (19) gives the soliton waves

$$\begin{aligned} R(\xi )=\frac{4c}{4c^{2}e^{\xi }+\chi e^{-\xi }}, \end{aligned}$$
(20)

where c is nonzero constant.

Step 3:  By inserting Eq. (18) into Eq. (17), together with Eq. (19), we can derive the requisite constants for Eq. (16) and (18). In order to incorporate the identified parametric restrictions, they can be substituted into Eq. (18) together with Eq. (20). Consequently, straddled solitons are obtained, which can be further classified as bright, dark, or singular solitons.

The extended auxiliary equation scheme

This subsection presents a thorough overview of the basic procedures with the extended auxiliary equation technique.

Step 2: We assume that the solution of Eq. (17) can be expressed in the form

$$\begin{aligned} U(\xi ) =\alpha _0+ \sum \limits _{i = 1}^N {\left\{ {\alpha _i}{\theta (\xi )^i}+{\beta _i}{\theta (\xi )^{-i}}\right\} }, \end{aligned}$$
(21)

where \(\theta (\xi )\) satisfies

$$\begin{aligned} {\theta ^{\prime }(\xi )}^2 = \sum \limits _{l = 0}^4{\tau _l} \theta (\xi )^l. \end{aligned}$$
(22)

This equation gives various kinds of solutions as follows

Case 1 \({\tau _0} = {\tau _1} = {\tau _3} = 0.\) Bright and singular soliton solutions are obtained:

$$\begin{aligned} \theta (\xi ) = \sqrt{ - \frac{{{\tau _2}}}{{{\tau _4}}}} ~\text {sech} \left[ {\sqrt{{\tau _2}} \,\xi } \right] ,~~~{\tau _2} > 0,~{\tau _4} < 0, \end{aligned}$$
(23)

and

$$\begin{aligned} \theta (\xi ) = \sqrt{ \frac{{{\tau _2}}}{{{\tau _4}}}} ~\text {csch} \left[ {\sqrt{{\tau _2}} \,\xi } \right] , ~~~{\tau _2}> 0,~{\tau _4} > 0. \end{aligned}$$
(24)

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

Dark and singular soliton solutions are obtained:

$$\begin{aligned} \theta (\xi ) = \sqrt{ - \frac{{{\tau _2}}}{{{2\tau _4}}}} ~\tanh \left[ {\sqrt{-{\frac{\tau _2}{2}}} \,\xi } \right] ,~~~{\tau _2} < 0,~{\tau _4} > 0, \end{aligned}$$
(25)

and

$$\begin{aligned} \theta (\xi ) = \sqrt{ - \frac{{{\tau _2}}}{{{2\tau _4}}}} ~\coth \left[ {\sqrt{-{\frac{\tau _2}{2}}} \,\xi } \right] , ~~~{\tau _2} < 0,~{\tau _4} > 0. \end{aligned}$$
(26)

Case 3 \(\tau _1=\tau _3=0.\) A Weierstrass elliptic doubly periodic type solution is obtained:

$$\begin{aligned} \theta (\xi ) =\frac{3\wp '\left( \xi ;g_2,g_3\right) }{\sqrt{\tau _4}\left[ 6\wp \left( \xi ;g_2,g_3\right) +\tau _2\right] }, \end{aligned}$$
(27)

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

Case 4 \({\tau _0} = {\tau _1} = 0,~{\tau _2,~\tau _4} > 0, ~{\tau _3} \ne \pm 2 \sqrt{{\tau _2}{\tau _4}}\) Straddled soliton solutions are obtained:

$$\begin{aligned} \theta (\xi ) = \frac{-{{\tau _2}{{{\mathop \textrm{sech}\nolimits } }^2}\left[ {\frac{1}{2}\sqrt{{\tau _2}} \,\xi } \right] }}{{\pm 2 \sqrt{{\tau _2}{\tau _4}} \,\tanh \left[ {\frac{1}{2}\sqrt{{\tau _2}} \,\xi } \right] + {\tau _3}}}, \end{aligned}$$
(28)

and

$$\begin{aligned} \theta (\xi ) = \frac{{{\tau _2}{{{\mathop \textrm{csch}\nolimits } }^2}\left[ {\frac{1}{2}\sqrt{{\tau _2}} \,\xi } \right] }}{{\pm 2 \sqrt{{\tau _2}{\tau _4}} \,\coth \left[ {\frac{1}{2}\sqrt{{\tau _2}} \,\xi } \right] + {\tau _3}}}. \end{aligned}$$
(29)

Step 3: Determine the positive integer number N in Eq. (8) by balancing the highest order derivatives and the nonlinear terms in Eq. (17).

Step 4: Substitute (21) into (17) along with (22). As a result of this substitution, we get a polynomial of \(\theta (\xi )\). In this polynomial we gather all terms of same powers and equating them to be zero, we get an over-determined system of algebraic equations which can be solved together to get the unknown parameters \(k,v,\alpha _0,\alpha _i\) and \(\beta _i ~(i=1,2,...)\). Consequently, we obtain the exact solutions of (15).

Application to the governing model

This section uses the two preceding techniques discussed in this work. Using these methods, we predict the emergence of bright and dark soliton solutions. This section will be divided into two subsections. At first, we will study the governing model in the case when the balance constant \(N=1\). Subsequently, we will investigate the governing model under the issue where the balance constant \(N=2\).

Case 1 \(N=1\)

The enhanced Kudryashov method

In accordance with the enhanced Kudryashov technique, the solution is expressed in the following structure

$$\begin{aligned} V(\xi )=a_0 +a_1 R(\xi )+ b_1 \left( \frac{ R^\prime (\xi )}{ R(\xi )}\right) . \end{aligned}$$
(30)

Plugging Eq. (30) together with Eq. (19) into Eq. (14), we get a system of algebraic equations. Solving these equations together yields the following result:

$$\begin{aligned}{} & {} a_0=0,~a_1=\pm \sqrt{-\frac{ w _6 \left( 6 n^3+11 n^2+6 n+1\right) \chi }{ w _8 \left( 4 n^3+2 n^2+n+1\right) - w _4 n^2 \left( 6 n^2+5 n+1\right) }},\nonumber \\{} & {} b_1=0,~k=n \sqrt{\frac{ w _8 w _6 (n+1)}{ w _4 n^2 \left( 6 n^2+5 n+1\right) - w _8 \left( 4 n^3+2 n^2+n+1\right) }},\nonumber \\{} & {} w _1=\frac{ w _6 (2 n+1) \left( w _4 (3 n+1)-2 w _8\right) }{ w _8 \left( 4 n^3+2 n^2+n+1\right) - w _4 n^2 \left( 6 n^2+5 n+1\right) }. \end{aligned}$$
(31)

As a consequence, we obtain the exact solutions of Eq. (1) as follows

$$\begin{aligned} q(x,t)= & {} \left\{ \frac{ \pm 4 c \sqrt{-\frac{ w _6 \left( 6 n^3+11 n^2+6 n+1\right) \chi }{ w _8 \left( 4 n^3+2 n^2+n+1\right) - w _4 n^2 \left( 6 n^2+5 n+1\right) }}}{4 c^2 e^{n \sqrt{\frac{ w _8 w _6 (n+1)}{ w _4 n^2 \left( 6 n^2+5 n+1\right) - w _8 \left( 4 n^3+2 n^2+n+1\right) }} (x-v t)}+\chi e^{-n \sqrt{\frac{ w _8 w _6 (n+1)}{ w _4 n^2 \left( 6 n^2+5 n+1\right) - w _8 \left( 4 n^3+2 n^2+n+1\right) }} (x-v t)}}\right\} ^{\frac{1}{n}}\nonumber \\\times & {} e^{i \left( \left\{ \frac{\delta _2 \left( \sigma _8-\sigma _7\right) }{2 \delta _3 \sigma _{13}}\right\} x+\omega t+\sigma W(t)-\sigma ^2 t+ \theta _0\right) }. \end{aligned}$$
(32)

Set \(\chi =\pm 4 c^2\) in solution (32). Consequently, for \(w _4 n^2 \left( 6 n^2+5 n+1\right) - w _8 \left( 4 n^3+2 n^2+n+1\right)\) and \({ w _8 w _6 (n+1)}>0\), we have bright soliton with \({ w _6}>0\) and singular soliton with \({ w _6}<0\)

$$q(x,t) = {\text{ }}\left\{ { \pm \sqrt { - \frac{{w_{6} \left( {6n^{3} + 11n^{2} + 6n + 1} \right)}}{{w_{8} \left( {4n^{3} + 2n^{2} + n + 1} \right) - w_{4} n^{2} \left( {6n^{2} + 5n + 1} \right)}}} \times {\text{ sech}}\left[ {n\sqrt {\frac{{w_{8} w_{6} (n + 1)}}{{w_{4} n^{2} \left( {6n^{2} + 5n + 1} \right) - w_{8} \left( {4n^{3} + 2n^{2} + n + 1} \right)}}} (x - vt)} \right]{\text{ }}^{{\frac{1}{n}}} } \right\} \times {\text{ }}e^{i} \left( {\left\{ {\frac{{\delta _{2} \left( {\sigma _{8} - \sigma _{7} } \right)}}{{2\delta _{3} \sigma _{{13}} }}} \right\}x + \omega t + \sigma W(t) - \sigma ^{2} t + \theta _{0} } \right)$$
(33)

and

$$q(x,t) = {\text{ }}\left\{ { \pm \sqrt {\frac{{w_{6} \left( {6n^{3} + 11n^{2} + 6n + 1} \right)}}{{w_{8} \left( {4n^{3} + 2n^{2} + n + 1} \right) - w_{4} n^{2} \left( {6n^{2} + 5n + 1} \right)}}} \times {\text{csch}}\left[ {n\sqrt {\frac{{w_{8} w_{6} (n + 1)}}{{w_{4} n^{2} \left( {6n^{2} + 5n + 1} \right) - w_{8} \left( {4n^{3} + 2n^{2} + n + 1} \right)}}} (x - vt)} \right]} \right\}^{{\frac{1}{n}}} e^{{i\left( {\left\{ {\frac{{\delta _{2} \left( {\sigma _{8} - \sigma _{7} } \right)}}{{2\delta _{3} \sigma _{{13}} }}} \right\}x + \omega t + \sigma W(t) - \sigma ^{2} t + \theta _{0} } \right)}} .{\text{ }}$$
(34)

The extended auxiliary equation method

In accordance with the extended auxiliary equation technique, the solution is expressed in the following structure

$$\begin{aligned} V(\xi )=\alpha _0 + \alpha _1 \theta (\xi )+ \frac{ \beta _1}{ \theta (\xi )}. \end{aligned}$$
(35)

Plugging Eq. (35) together with Eq. (22) into Eq. (14), we get a system of algebraic equations. Solving these equations together yields the following results:

Case 1 \(\tau _0=\tau _1=\tau _3=0.\)

$$\begin{aligned}{} & {} \alpha _0=\beta _1=0,~\tau _2=\frac{n^2 \left( w _4 k^2 \left( 6 n^2+5 n+1\right) - w _8 w _6 (n+1)\right) }{ w _8 k^2 \left( 4 n^3+2 n^2+n+1\right) },~\tau _4=-\frac{\alpha _1^2 w _8 n^2}{k^2 \left( 6 n^2+5 n+1\right) },\nonumber \\{} & {} w _1=-\frac{(2 n+1) \left( w _4 k^2 (3 n+1)+2 w _8 w _6 n^2\right) \left( w _4 k^2 (n (6 n+5)+1)- w _8 w _6 (n+1)\right) }{ w _8^2 k^2 \left( 4 n^3+2 n^2+n+1\right) ^2}. \end{aligned}$$
(36)

For this case, the solution of (1) reads

$$q(x,t) = \left\{ { \pm \sqrt {\frac{{(2n + 1)(3n + 1)\left( {w_{4} k^{2} (n(6n + 5) + 1) - w_{8} w_{6} (n + 1)} \right)}}{{w_{8}^{2} \left( {4n^{3} + 2n^{2} + n + 1} \right)}} \times {\text{sech}}\left[ {n\sqrt {\frac{{w_{4} k^{2} \left( {6n^{2} + 5n + 1} \right) - w_{8} w_{6} (n + 1)}}{{w_{8} \left( {4n^{3} + 2n^{2} + n + 1} \right)}}} (x - vt)} \right]} } \right\}^{{\frac{1}{n}}} e^{{i\left( {\left\{ {\frac{{\delta _{2} \left( {\sigma _{8} - \sigma _{7} } \right)}}{{2\delta _{3} \sigma _{{13}} }}} \right\}x + \omega t + \sigma W(t) - \sigma ^{2} t + \theta _{0} } \right)}} ,$$
(37)

and

$$q(x,t) = {\text{ }}\left\{ { \pm \sqrt { - \frac{{(2n + 1)(3n + 1)\left( {w_{4} k^{2} (n(6n + 5) + 1) - w_{8} w_{6} (n + 1)} \right)}}{{w_{8}^{2} \left( {4n^{3} + 2n^{2} + n + 1} \right)}}} \times {\text{csch}}\left[ {n\sqrt {\frac{{w_{4} k^{2} \left( {6n^{2} + 5n + 1} \right) - w_{8} w_{6} (n + 1)}}{{w_{8} \left( {4n^{3} + 2n^{2} + n + 1} \right)}}} (x - vt)} \right]} \right\}^{{\frac{1}{n}}} e^{{i\left( {\left\{ {\frac{{\delta _{2} \left( {\sigma _{8} - \sigma _{7} } \right)}}{{2\delta _{3} \sigma _{{13}} }}} \right\}x + \omega t + \sigma W(t) - \sigma ^{2} t + \theta _{0} } \right)}} .{\text{ }}$$
(38)

These solitons with \(\tau _2>0\) are bright for \(\tau _4<0\) and singular for \(\tau _4>0\).

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

$$\begin{aligned}{} & {} \alpha _0=0,~\alpha _1=\frac{\sqrt{-k^2 w _8^3 \left( 6 n^2+5 n+1\right) \left( 4 n^3+2 n^2+n+1\right) ^2 \tau _4}}{ w _8^2 n \left( 4 n^3+2 n^2+n+1\right) },\nonumber \\{} & {} \beta _1=\frac{n \sqrt{6 n^2+5 n+1} \left( w _4 k^2 \left( 6 n^2+5 n+1\right) - w _8 w _6 (n+1)\right) }{4 \sqrt{ w _8^3 \left( -k^2\right) \left( 4 n^3+2 n^2+n+1\right) ^2 \tau _4}},\nonumber \\{} & {} \tau _2=-\frac{n^2 \left( w _4 k^2 \left( 6 n^2+5 n+1\right) - w _8 w _6 (n+1)\right) }{2 w _8 k^2 \left( 4 n^3+2 n^2+n+1\right) },\nonumber \\{} & {} w _1=-\frac{(2 n+1) \left( w _4 k^2 (3 n+1)+2 w _8 w _6 n^2\right) \left( w _4 k^2 (n (6 n+5)+1)- w _8 w _6 (n+1)\right) }{ w _8^2 k^2 \left( 4 n^3+2 n^2+n+1\right) ^2}. \end{aligned}$$
(39)

For this case, the solution of (1) reads

$$q(x,t) = \left\{ { \pm \sqrt {\frac{{(2n + 1)(3n + 1)\left( {w_{8} w_{6} (n + 1) - w_{4} k^{2} (n(6n + 5) + 1)} \right)}}{{w_{8}^{2} \left( {4n^{3} + 2n^{2} + n + 1} \right)}}} \times {\text{csch}}\left[ {n\sqrt {\frac{{w_{4} k^{2} (n(6n + 5) + 1) - w_{8} w_{6} (n + 1)}}{{w_{8} \left( {4n^{3} + 2n^{2} + n + 1} \right)}}} (x - vt)} \right]} \right\}^{{\frac{1}{n}}} e^{{i\left( {\left\{ {\frac{{\delta _{2} \left( {\sigma _{8} - \sigma _{7} } \right)}}{{2\delta _{3} \sigma _{{13}} }}} \right\}x + \omega t + \sigma W(t) - \sigma ^{2} t + \theta _{0} } \right)}} .$$
(40)

The obtained soliton is singular with \(\tau _2<0\) and \(\tau _4>0\).

Case 3 \(\tau _1=0,~\tau _3=0\)

$$\begin{aligned}{} & {} \alpha _0=0,~\tau _0=0,~\beta _1=0,~\tau _2=\frac{n^2 \left( w _4 k^2 \left( 6 n^2+5 n+1\right) - w _8 w _6 (n+1)\right) }{ w _8 k^2 \left( 4 n^3+2 n^2+n+1\right) },\nonumber \\{} & {} \quad ~\tau _4=-\frac{\alpha _1^2 w _8 n^2}{k^2 \left( 6 n^2+5 n+1\right) },\nonumber \\{} & {} w _1=-\frac{(2 n+1) \left( w _4 k^2 (3 n+1)+2 w _8 w _6 n^2\right) \left( w _4 k^2 (n (6 n+5)+1)- w _8 w _6 (n+1)\right) }{ w _8^2 k^2 \left( 4 n^3+2 n^2+n+1\right) ^2}. \end{aligned}$$
(41)

For this case, the solution of (1) reads

$$q(x,t) = {\text{ }}\left\{ {\frac{{3\wp ^{\prime}\left( {\xi ;g_{2} ,g_{3} } \right)}}{{\sqrt { - \frac{{w_{8} n^{2} }}{{k^{2} \left( {6n^{2} + 5n + 1} \right)}}} \left[ {6\wp \left( {\xi ;g_{2} ,g_{3} } \right) + \frac{{n^{2} \left( {w_{4} k^{2} \left( {6n^{2} + 5n + 1} \right) - w_{8} w_{6} (n + 1)} \right)}}{{w_{8} k^{2} \left( {4n^{3} + 2n^{2} + n + 1} \right)}}} \right]}}} \right\}^{{\frac{1}{n}}} \times e^{{i\left( {\left\{ {\frac{{\delta _{2} \left( {\sigma _{8} - \sigma _{7} } \right)}}{{2\delta _{3} \sigma _{{13}} }}} \right\}x + \omega t + \sigma W(t) - \sigma ^{2} t + \theta _{0} } \right)}} ,{\text{ }}$$
(42)

where

$$\begin{aligned} g_2=\frac{\tau _2^2}{12},~g_3=-\frac{\tau _2^3}{216}. \end{aligned}$$
(43)

The Weierstrass elliptic doubly periodic type solution (42) with its restricted invariants (43) can be converted to a singular soliton solution

$$q(x,t) = \left\{ { \pm \sqrt {\frac{{(2n + 1)(3n + 1)\left( {w_{8} w_{6} (n + 1) - w_{4} k^{2} (n(6n + 5) + 1)} \right)}}{{w_{8}^{2} \left( {4n^{3} + 2n^{2} + n + 1} \right)}}} \times {\text{csch}}\left[ {n\sqrt {\frac{{w_{4} k^{2} (n(6n + 5) + 1) - w_{8} w_{6} (n + 1)}}{{w_{8} \left( {4n^{3} + 2n^{2} + n + 1} \right)}}} (x - vt)} \right]} \right\}^{{\frac{1}{n}}} \times e^{{i\left( {\left\{ {\frac{{\delta _{2} \left( {\sigma _{8} - \sigma _{7} } \right)}}{{2\delta _{3} \sigma _{{13}} }}} \right\}x + \omega t + \sigma W(t) - \sigma ^{2} t + \theta _{0} } \right)}} .{\text{ }}$$
(44)

Case 4 \(\tau _0=0,~\tau _1=0\)

$$\begin{aligned}{} & {} \alpha _0=\beta _1=0,~\tau _4=-\frac{\alpha _1^2 w _8 n^2}{k^2 \left( 6 n^2+5 n+1\right) },\nonumber \\{} & {} \quad \tau _2=\frac{n^2 \left( w _4 k^2 (n (6 n+5)+1)- w _8 w _6 (n+1)\right) }{ w _8 k^2 \left( 4 n^3+2 n^2+n+1\right) },~\tau _3=0,\nonumber \\{} & {} \quad w _1=-\frac{(2 n+1) \left( w _4 k^2 (3 n+1)+2 w _8 w _6 n^2\right) \left( w _4 k^2 (n (6 n+5)+1)- w _8 w _6 (n+1)\right) }{ w _8^2 k^2 \left( 4 n^3+2 n^2+n+1\right) ^2}. \end{aligned}$$
(45)

For this case, the obtained solitons are bright for \(\tau _2>0\) and \(\tau _4<0\) and singular for \(\tau _2>0\) and \(\tau _4>0\)

$$q(x,t) = \left\{ { \pm \sqrt {\frac{{(2n + 1)(3n + 1)\left( {w_{4} k^{2} (n(6n + 5) + 1) - w_{8} w_{6} (n + 1)} \right)}}{{w_{8}^{2} \left( {4n^{3} + 2n^{2} + n + 1} \right)}}} \times {\text{sech}}\left[ {n\sqrt {\frac{{w_{4} k^{2} \left( {6n^{2} + 5n + 1} \right) - w_{8} w_{6} (n + 1)}}{{w_{8} \left( {4n^{3} + 2n^{2} + n + 1} \right)}}} (x - vt)} \right]} \right\}^{{\frac{1}{n}}} \times {\text{ }}e^{{i\left( {\left\{ {\frac{{\delta _{2} \left( {\sigma _{8} - \sigma _{7} } \right)}}{{2\delta _{3} \sigma _{{13}} }}} \right\}x + \omega t + \sigma W(t) - \sigma ^{2} t + \theta _{0} } \right)}} ,$$
(46)

and

$$q(x,t) = {\text{ }}\left\{ { \pm \sqrt { - \frac{{(2n + 1)(3n + 1)\left( {w_{4} k^{2} (n(6n + 5) + 1) - w_{8} w_{6} (n + 1)} \right)}}{{w_{8}^{2} \left( {4n^{3} + 2n^{2} + n + 1} \right)}}} \times {\text{csch}}\left[ {n\sqrt {\frac{{w_{4} k^{2} \left( {6n^{2} + 5n + 1} \right) - w_{8} w_{6} (n + 1)}}{{w_{8} \left( {4n^{3} + 2n^{2} + n + 1} \right)}}} (x - vt)} \right]} \right\}^{{\frac{1}{n}}} e^{{i\left( {\left\{ {\frac{{\delta _{2} \left( {\sigma _{8} - \sigma _{7} } \right)}}{{2\delta _{3} \sigma _{{13}} }}} \right\}x + \omega t + \sigma W(t) - \sigma ^{2} t + \theta _{0} } \right)}} .$$
(47)

Case 2\(N=2\)

The enhanced Kudryashov method

In accordance with the enhanced Kudryashov technique, the solution is expressed in the following structure

$$\begin{aligned} V(\xi )=a_0 + a_1 R(\xi )+ b_1 \left( \frac{ R^\prime (\xi )}{ R(\xi )}\right) + a_2 R(\xi )^2+ b_2 \left( \frac{ R^\prime (\xi )}{ R(\xi )}\right) ^2. \end{aligned}$$
(48)

Insert (48) together with (19) into Eq. (14) to get a system of algebraic equations. Solving these equations together yields the following result:

$$\begin{aligned}{} & {} a_0=-b_2,~a_2=b_2 \chi \pm \sqrt{\frac{(n+1) (n+2) (3 n+2) w _6 \chi ^2}{n^2 (n (n+2)+2) w _4}}\nonumber \\,{} & {} \quad k=\frac{1}{2} \sqrt{-\frac{n^2 w _6}{n (n+2)+2}},\nonumber \\{} & {} \quad w _1=-\frac{4 (n+1)^2 w _6}{n^2 \left( n^2+2 n+2\right) },~ w _8=0. \end{aligned}$$
(49)

As a consequence, the solution of Eq. (1) reaches

$$q(x,t) = \left\{ {\chi \left( {b_{2} \pm \sqrt {\frac{{(n + 1)(n + 2)(3n + 2)w_{6} }}{{n^{2} (n(n + 2) + 2)w_{4} }}} } \right)\quad \left( {\frac{{4c}}{{4c^{2} e^{{\frac{1}{2}\sqrt { - \frac{{n^{2} w_{6} }}{{n(n + 2) + 2}}} (x - vt)}} + \chi e^{{ - \frac{1}{2}\sqrt { - \frac{{n^{2} w_{6} }}{{n(n + 2) + 2}}} (x - vt)}} }}} \right)^{2} \quad + b_{2} \left( {\frac{{\chi - 4c^{2} e^{{\sqrt { - \frac{{n^{2} w_{6} }}{{n(n + 2) + 2}}} (x - vt)}} }}{{\chi + 4c^{2} e^{{\sqrt { - \frac{{n^{2} w_{6} }}{{n(n + 2) + 2}}} (x - vt}} )}}} \right)^{2} - b_{2} } \right\}^{{\frac{1}{n}}} \quad ~e^{{i\left( {\left\{ {\frac{{\delta _{2} \left( {\sigma _{8} - \sigma _{7} } \right)}}{{2\delta _{3} \sigma _{{13}} }}} \right\}x + \omega t + \sigma W(t) - \sigma ^{2} t + \theta _{0} } \right)}} .{\text{ }}$$
(50)

Set \(\chi =\pm 4 c^2\) to recover bright and singular solitons for \(w _4<0\) and \(w _6<0\)

$$q(x,t) = {\text{ }}\left\{ { \pm \sqrt {\frac{{(n + 1)(n + 2)(3n + 2)w_{6} }}{{n^{2} (n(n + 2) + 2)w_{4} }}} {\text{sech}}^{2} \left[ {\frac{1}{2}\sqrt { - \frac{{n^{2} w_{6} }}{{n(n + 2) + 2}}} (x - vt)} \right]} \right\}^{{\frac{1}{n}}} \times e^{{i\left( {\left\{ {\frac{{\delta _{2} \left( {\sigma _{8} - \sigma _{7} } \right)}}{{2\delta _{3} \sigma _{{13}} }}} \right\}x + \omega t + \sigma W(t) - \sigma ^{2} t + \theta _{0} } \right)}} ,$$
(51)

and

$$q(x,t) = {\text{ }}\left\{ { \mp \sqrt {\frac{{(n + 1)(n + 2)(3n + 2)w_{6} }}{{n^{2} (n(n + 2) + 2)w_{4} }}} {\text{csch}}^{2} \left[ {\frac{1}{2}\sqrt { - \frac{{n^{2} w_{6} }}{{n(n + 2) + 2}}} (x - vt)} \right]} \right\}^{{\frac{1}{n}}} \times e^{{i\left( {\left\{ {\frac{{\delta _{2} \left( {\sigma _{8} - \sigma _{7} } \right)}}{{2\delta _{3} \sigma _{{13}} }}} \right\}x + \omega t + \sigma W(t) - \sigma ^{2} t + \theta _{0} } \right)}} .{\text{ }}$$
(52)

The extended auxiliary equation method

In accordance with the extended auxiliary equation technique, the solution is expressed in the following structure

$$\begin{aligned} V(\xi )= & {} \alpha _0 + \alpha _1 \theta (\xi )+ \frac{ \beta _1}{ \theta (\xi )} \nonumber \\{} & {} + \alpha _2 \theta (\xi )^2+ \frac{ \beta _2}{ \theta (\xi )^2}. \end{aligned}$$
(53)

Plugging Eq. (53) together with Eq. (22) into Eq. (14), we get a system of algebraic equations. Solving these equations together yields the following results:

Case 1 \(\tau _0=\tau _1=\tau _3=0.\)

$$\begin{aligned} \alpha _0= & {} \alpha _1=\beta _1=\beta _2=0,\nonumber \\ \alpha _2= & {} \pm \frac{2 k\sqrt{3 n^3+11 n^2+12 n+4} \tau _4}{\sqrt{- w _4} n^2},\nonumber \\ \tau _2= & {} -\frac{ w _6 n^2}{4 k^2 \left( n^2+2 n+2\right) },\nonumber \\ w _8= & {} 0, w _1=\frac{ w _6^2 (n+1)^2}{k^2 \left( n^2+2 n+2\right) ^2}. \end{aligned}$$
(54)

For this case, the solution of (1) reads

$$q(x,t) = {\text{ }}\left\{ { \pm \frac{{w_{6} \sqrt {(n + 1)(n + 2)(3n + 2)} }}{{2\sqrt { - w_{4} } k(n(n + 2) + 2)}}{\text{sech}}^{2} \left[ {\frac{n}{2}\sqrt { - \frac{{w_{6} }}{{n^{2} + 2n + 2}}} (x - vt)} \right]} \right\}^{{\frac{1}{n}}} \times e^{{i\left( {\left\{ {\frac{{\delta _{2} \left( {\sigma _{8} - \sigma _{7} } \right)}}{{2\delta _{3} \sigma _{{13}} }}} \right\}x + \omega t + \sigma W(t) - \sigma ^{2} t + \theta _{0} } \right)}} ,$$
(55)

and

$$q(x,t) = {\text{ }}\left\{ { \mp \frac{{w_{6} \sqrt {(n + 1)(n + 2)(3n + 2)} }}{{2\sqrt { - w_{4} } k(n(n + 2) + 2)}}{\text{csch}}^{2} \left[ {\frac{n}{2}\sqrt { - \frac{{w_{6} }}{{n^{2} + 2n + 2}}} (x - vt)} \right]} \right\}^{{\frac{1}{n}}} \times e^{{i\left( {\left\{ {\frac{{\delta _{2} \left( {\sigma _{8} - \sigma _{7} } \right)}}{{2\delta _{3} \sigma _{{13}} }}} \right\}x + \omega t + \sigma W(t) - \sigma ^{2} t + \theta _{0} } \right)}} .{\text{ }}$$
(56)

These solutions are bright and singular solitons with \(w _4<0\) and \(w _6<0\).

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

$$\begin{aligned}{} & {} \alpha _1=\beta _1=\beta _2=0,~\alpha _0=\pm \frac{ w _6}{2} \sqrt{-\frac{(n+1) (n+2) (3 n+2)}{ w _4 k^2 (n (n+2)+2)^2}},\nonumber \\{} & {} \quad \tau _2=\frac{ w _6 n^2}{2 k^2 (n (n+2)+2)},\nonumber \\{} & {} \quad \tau _4=\mp \frac{\alpha _2 \sqrt{- w _4} n^2}{2 k \sqrt{3 n^3+11 n^2+12 n+4}},\nonumber \\{} & {} \quad w _8=0,~ w _1=\frac{ w _6^2 (n+1)^2}{k^2 \left( n^2+2 n+2\right) ^2}. \end{aligned}$$
(57)

For this case, the solution of (1) reads

$$q(x,t) = {\text{ }}\left\{ { \pm \frac{{w_{6} \sqrt {(n + 1)(n + 2)(3n + 2)} }}{{2\sqrt { - w_{4} } k(n(n + 2) + 2)}}{\text{sech}}^{2} \left[ {\frac{n}{2}\sqrt { - \frac{{w_{6} }}{{n^{2} + 2n + 2}}} (x - vt)} \right]} \right\}^{{\frac{1}{n}}} \times e^{{i\left( {\left\{ {\frac{{\delta _{2} \left( {\sigma _{8} - \sigma _{7} } \right)}}{{2\delta _{3} \sigma _{{13}} }}} \right\}x + \omega t + \sigma W(t) - \sigma ^{2} t + \theta _{0} } \right)}} ,{\text{ }}$$
(58)

and

$$q(x,t) = {\text{ }}\left\{ { \mp \frac{{w_{6} \sqrt {(n + 1)(n + 2)(3n + 2)} }}{{2\sqrt { - w_{4} } k(n(n + 2) + 2)}}{\text{csch}}^{2} \left[ {\frac{n}{2}\sqrt { - \frac{{w_{6} }}{{n^{2} + 2n + 2}}} (x - vt)} \right]} \right\}^{{\frac{1}{n}}} \times e^{{i\left( {\left\{ {\frac{{\delta _{2} \left( {\sigma _{8} - \sigma _{7} } \right)}}{{2\delta _{3} \sigma _{{13}} }}} \right\}x + \omega t + \sigma W(t) - \sigma ^{2} t + \theta _{0} } \right)}} .{\text{ }}$$
(59)

Case 3 \(\tau _1=\tau _3=0.\)

$$\begin{aligned}{} & {} \alpha _0=0,~\alpha _1=\beta _1=\beta _2=0,~\tau _0=0,\nonumber \\{} & {} \quad ~\tau _2=-\frac{ w _6 n^2}{4 k^2 \left( n^2+2 n+2\right) },\nonumber \\{} & {} \quad \tau _4=\pm \frac{\alpha _2 \sqrt{ w _4} n^2}{2 \sqrt{-k^2 \left( 3 n^3+11 n^2+12 n+4\right) }},\nonumber \\{} & {} \quad w _8=0,~ w _1=\frac{ w _6^2 (n+1)^2}{k^2 \left( n^2+2 n+2\right) ^2}. \end{aligned}$$
(60)

For this case, the solution of (1) reads

$$q(x,t) = {\text{ }}\left\{ { \mp \frac{{92\sqrt { - k^{2} \left( {3n^{3} + 11n^{2} + 12n + 4} \right)} \left( {\wp ^{\prime}\left( {\xi ;g_{2} ,g_{3} } \right)} \right)^{2} }}{{n^{2} \sqrt {w_{4} } \left[ {6\wp \left( {\xi ;g_{2} ,g_{3} } \right) - \frac{{w_{6} n^{2} }}{{4k^{2} \left( {n^{2} + 2n + 2} \right)}}} \right]^{2} }}} \right\}^{{\frac{1}{n}}} e^{{i\left( {\left\{ {\frac{{\delta _{2} \left( {\sigma _{8} - \sigma _{7} } \right)}}{{2\delta _{3} \sigma _{{13}} }}} \right\}x + \omega t + \sigma W(t) - \sigma ^{2} t + \theta _{0} } \right)}} ,$$
(61)

where

$$\begin{aligned} g_2=\frac{\tau _2^2}{12},~g_3=-\frac{\tau _2^3}{216}. \end{aligned}$$
(62)

The Weierstrass elliptic doubly periodic type solution (61) with its restricted invariants (62) can be converted to a singular soliton solution

$$q(x,t) = {\text{ }}\left\{ { \mp \frac{{23w_{6} \sqrt { - k^{2} (n + 1)(n + 2)(3n + 2)} }}{{9\sqrt {w_{4} } k^{2} (n(n + 2) + 2)}}{\text{csch}}^{2} \left[ {\frac{n}{2}\sqrt { - \frac{{w_{6} }}{{n^{2} + 2n + 2}}} (x - vt)} \right]} \right\}^{{\frac{1}{n}}} \times ^{{i\left( {\left\{ {\frac{{\delta _{2} \left( {\sigma _{8} - \sigma _{7} } \right)}}{{2\delta _{3} \sigma _{{13}} }}} \right\}x + \omega t + \sigma W(t) - \sigma ^{2} t + \theta _{0} } \right)}} .{\text{ }}$$
(63)

Case 4 \(\tau _0=\tau _1=0.\)

$$\begin{aligned}{} & {} \alpha _0=0,~\alpha _1=\beta _1=\beta _2=0,~\tau _3=0,\nonumber \\{} & {} \quad \tau _2=-\frac{ w _6 n^2}{4 k^2 \left( n^2+2 n+2\right) },\nonumber \\{} & {} \quad \tau _4=\pm \frac{\alpha _2 \sqrt{ w _4} n^2}{2 \sqrt{-k^2 \left( 3 n^3+11 n^2+12 n+4\right) }},\nonumber \\{} & {} \quad w _8=0,~ w _1=\frac{ w _6^2 (n+1)^2}{k^2 \left( n^2+2 n+2\right) ^2}. \end{aligned}$$
(64)

For this case, the solution of (1) reads

$$\begin{aligned} q(x,t)= & {} \Bigg \{\pm \frac{ w _6 \sqrt{(n+1) (n+2) (3 n+2)}}{2 \sqrt{- w _4} k (n (n+2)+2)} \text {sech}^2\nonumber \\{} & {} \left[ \frac{n}{2} \sqrt{-\frac{ w _6}{n^2+2 n+2}} (x-v t)\right] \Bigg \}^{\frac{1}{n}}\nonumber \\{} & {} \times e^{i \left( \left\{ \frac{\delta _2 \left( \sigma _8-\sigma _7\right) }{2 \delta _3 \sigma _{13}}\right\} x+\omega t+\sigma W(t)-\sigma ^2 t+ \theta _0\right) }, \end{aligned}$$
(65)

and

$$\begin{aligned} q(x,t)= & {} \Bigg \{\mp \frac{ w _6 \sqrt{(n+1) (n+2) (3 n+2)}}{2 \sqrt{- w _4} k (n (n+2)+2)} \text {csch}^2\nonumber \\{} & {} \left[ \frac{n}{2} \sqrt{-\frac{ w _6}{n^2+2 n+2}} (x-v t)\right] \Bigg \}^{\frac{1}{n}}\nonumber \\{} & {} \times e^{i \left( \left\{ \frac{\delta _2 \left( \sigma _8-\sigma _7\right) }{2 \delta _3 \sigma _{13}}\right\} x+\omega t+\sigma W(t)-\sigma ^2 t+ \theta _0\right) }. \end{aligned}$$
(66)

These solutions are bright and singular solitons with \(w _4<0\) and \(w _6<0\).

The enhanced Kudryashov’s approach and the extended auxiliary equation technique were unsuccessful in retrieving dark solitons within the constraints of the governing model.

Conclusions

The paper studied the dispersive concatenation model with power law of SPM in presence of white noise. The bright and singular soliton solutions to the model were retrieved using a couple of integration algorithms. They are the enhanced Kudryashov’s approach and the extended auxiliary equation scheme. These methodologies failed to furnish the dark soliton solutions to the model. This is because of the fact that the model with power law of SPM can recover dark solitons only if the power law collapses to Kerr law, a fact that has been experimentally proven earlier. There is another observation that has been made in the work. The effect of white noise is confined to the phase component of the bright and singular soliton solutions and does not affect the amplitude components of such solitons.

The results of the paper are thus indeed promising. The studies with the current model can be extended further. For example, this research can be carried out with additional optoelectronic devices such as optical couplers, optical metamaterials, PCF, gap solitons and many others. Later, the studies can be extended to fibers with differential group delay as well as dispersion–flattened fibers. The numerical scheme such as variational iteration method and/or Laplace–Adomian decomposition schemes will be implemented to achieve a numerical picture to the studies. The bifurcation analysis and additional analytical aspects are yet to be covered. These results will be disseminated after they are available and aligned with the pre–existing ones [16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58]