Introduction

The study of nonlinear stochastic partial differential equations (NSPDEs) constitutes a vital area of research, with extensive applications across diverse fields such as modern physics, biology, superfluid dynamics, image processing, optical fiber communications, plasma physics, and finance1,2,3. These wide-ranging applications underscore the importance and relevance of NSPDEs in contemporary scientific inquiry.

A central component of stochastic calculus, and one of the most widely studied stochastic processes, is the Wiener process also known as Brownian motion which possesses both martingale and Markov properties4. This process is fundamental for modeling random phenomena due to its continuity and normally distributed increments over any time interval. As a result, the Wiener process plays a critical role in the modeling of dispersive systems5,6.

Moreover, there exists a profound and well-established relationship between partial differential equations and stochastic processes, where the stochastic component often introduces randomness into systems governed by deterministic laws. This interplay is particularly evident in the formulation and analysis of NSPDEs, where randomness can model uncertainty or noise in real-world systems. Nonlinear Schrödinger equations are among the most extensively utilized models in applied sciences, owing to their wide spectrum of applications in fields such as optics, fluid dynamics, plasma physics, and quantum mechanics7,8,9,10,11,12,13,14,15,16,17,18. The exploration of soliton solutions to these equations is of fundamental significance in nonlinear science, as such solutions reveal the physical mechanisms underlying a variety of complex natural phenomena. Over the years, this area has matured into one of the most vibrant and rapidly evolving research domains19,20,21.

In recent developments, numerous novel solitary wave solutions have been derived using innovative analytical and computational techniques applied to nonlinear models22,23. Particular attention has been given to the study of N-soliton solutions, which serve as a foundation for constructing more intricate wave structures such as lump solutions and rogue waves. These investigations span a wide class of models, including both modified Korteweg–de Vries (mKdV)-type integrable systems and reduced integrable nonlinear Schrödinger-type equations, highlighting the deep interconnectedness of nonlinear wave dynamics across various physical contexts.

The study of nonlinear Schrödinger equations (NLSEs) in the context of optical solitons has emerged as a rapidly expanding area within the field of nonlinear photonics24,25. In recent years, considerable attention has been devoted to examining the effects of various types of nonlinearities on soliton behavior. These include parabolic, Kerr, power-law, polynomial, and saturable nonlinearities, each of which introduces distinct dynamical features into the propagation and stability of optical solitons26. These studies are fundamental to the development of advanced optical technologies and to the exploration of wave behavior in intricate nonlinear environments.

Furthermore, the study of stochastic solitons is crucial for understanding the behavior of coherent structures in realistic environments where noise and uncertainty are inherent, such as in optical fibers, fluid dynamics, and plasma physics. Research in this area often involves analytical techniques, numerical simulations, and probabilistic tools to explore how noise affects soliton stability, interaction, and long-time dynamics. In recent years, numerous fields have recognized the effect of including random effects in modeling and analyzing physical processes27.

In this work, the stochastic resonant NLSE incorporating both space–time dispersion and intermodal dispersion, with multiplicative noise treated in the Itô sense, together with a generalized Kudryashov-type nonlinearity28,29:

$$\begin{aligned} & i Q_{t}+\alpha Q_{xx}+\beta Q_{xt}+\gamma \left( \frac{|Q|_{xx}}{|Q|}\right) Q-i \delta Q_{x}+\chi (Q-i \beta Q_{x})\frac{dW(t)}{dt}\nonumber \\ & +(b_{1}|Q|^{2 n}+b_{2}|Q|^{2 n}+|Q|^{2 n}+b_{3}|Q|^{3 n}+b_{4}|Q|^{4 n}+c_{1}|Q|^{- n}+c_{2}|Q|^{- 2n}\nonumber \\ & +c_{3}|Q|^{-3 n}+c_{4}|Q|^{- 4 n})Q =0. \end{aligned}$$
(1)

The wave function Q(x, t), which is complex-valued and satisfies \(i^{2}=-1\) , describes the evolution of the wave. The parameters \(\alpha ,\beta ,\gamma ,\delta\) and \(\chi\) are considered constant throughout. In Eq. (1), the first term accounts for the linear progression over time, whereas the coefficients \(\alpha\) and \(\beta\) are linked to chromatic dispersion (CD) and space–time dispersion (STD), also referred to as self-steepening effects, respectively. Here, \(\delta\) characterizes the intermodal dispersion (IMD), and \(\gamma\) quantifies the strength of the resonant nonlinearity. The coefficients \(b_{s}\) and \(c_{s}\) (for \(s=1,\ldots ,4\) ) are associated with self-phase modulation (SPM) and related nonlinear effects, while n is the power-law nonlinearity parameter. In this setting, W(t) represents a standard Wiener process and \(\chi\) denotes the noise strength. The white noise component is modeled as the formal time derivative \(\frac{dW(t)}{dt}\).

Generalized Kudryashov’s law nonlinearity refers to a mathematical model for the nonlinear refractive index in optical systems, providing a comprehensive framework that allows for multiple terms of arbitrary powers of intensity to be included in the equations describing the propagation of light pulses30,31,32,33.

The SRNLSE has been studied by few authors such as, Eslami et al.34 discussed the optical soliton solutions of the SRNLSE using New Kudryashov’s method and \(\frac{G^{\prime}}{G}\). Trouba et al.35 investigated the optical soliton solutions of SRNLSE by applying the extended auxiliary equation method.

In the current work, the proposed model is studied for the first time using the modified extended mapping technique, through which we derive a rich variety of exact analytical solutions, including dark and singular solitons, as well as singular periodic solutions, hyperbolic solutions, periodic and rational solutions. The influence of stochastic perturbations is further illustrated by comprehensive two- and three-dimensional graphical representations. This dual contribution-diverse explicit solutions combined with visual clarification of stochastic effects-offers deeper theoretical insights and practical guidance on the robustness of soliton transmission in noisy environments.

The paper is organized into five main sections: the next section offers an overview of the proposed approach, followed by the findings in Section “Solitons and other solutions to the proposed model”. Section “Visual graphs for some solutions” presents examples of 2D and 3D graphical simulations, including contour plots. The conclusion is found in Section “Conclusions”.

Overview of the developed approach

This section offers a concise overview of the MEMT. As one of the more contemporary techniques, the MEMT36,37 has garnered significant attention for its effectiveness in solving nonlinear partial differential equations (NPDEs). To illustrate its steps of application, let us consider a general NPDE of the form:

$$\begin{aligned} \mathcal {F}(\mathfrak {q},\mathfrak {q}_z,\mathfrak {q}_t,\mathfrak {q}_{zt},\mathfrak {q}_{zz},\mathfrak {q}_{zzz},\ldots )=0. \end{aligned}$$
(2)

The solution to Eq. (2) is obtained by implementing the following algorithmic steps.

Step-(1): Using the subsequent wave transformation:

$$\begin{aligned} \mathfrak {q}(z,t)=\mathfrak {Q}(\xi ), \, \xi =z-t \omega , \end{aligned}$$
(3)

where \(\omega\) represents the wave speed.

The NPDE in Eq. (2) can be reduced to an ordinary differential equation (ODE) and takes the following form:

$$\begin{aligned} \Re (\mathfrak {Q},\mathfrak {Q}^{\prime},\mathfrak {Q}^{\prime\prime},\mathfrak {Q}^{\prime\prime\prime},\ldots )=0. \end{aligned}$$
(4)

Step-(2): This method implies that the solution to Eq. (4) can be expressed in the following form:

$$\begin{aligned} \mathfrak {Q}(\xi )=\sum _{j=0}^{N}{A_j \hbar ^j(\xi )}+\sum _{j=-1}^{-N}{B_{-j} \hbar ^j(\xi )}+\sum _{j=2}^{N}{C_j \hbar ^{j-2}(\xi )\hbar ^{\prime}(\xi )}+\sum _{j=-1}^{-N}{D_{-j} \hbar ^{j}(\xi )\hbar ^{\prime}(\xi )}, \end{aligned}$$
(5)

here \(A_j,B_{-j},C_j,D_{-j}\) denote real-valued constants that are to be determined, and the function \(\hbar (\xi )\) is required to satisfy the following condition:

$$\begin{aligned} \hbar ^{\prime}(\xi )=\sqrt{\iota _0+\iota _1 \hbar (\xi )+\iota _2 \hbar ^2(\xi )+\iota _3 \hbar ^3(\xi )+\iota _4 \hbar ^4(\xi )+\iota _6 \hbar ^6(\xi )}. \end{aligned}$$
(6)

Here \(\iota _i,\ (i = 0, 1, 2, 3, 4, 6)\) are constants whose values determine distinct forms of the solution.

Step-(3): The balancing principle is applied to Eq. (4) in order to determine the integer N

Step-(4): By substituting the assumed solutions from Eq. (5) and Eq. (6) into Eq. (4), a system of equations involving the unknowns \(A_j,B_{-j},C_j,D_{-j}\) and \(\omega\) is derived, then equalising the coefficients’ terms of \(\hbar ^{\prime j}(\xi )\hbar ^i(\xi )\) (\(j=0,1\); \(i=0,\pm 1,\pm 2,\ldots\)) to zero. To manage the resultant system raised, utilise the Wolfram Mathematica® package. Once that’s established, we able to identify the constants \(A_j,B_{-j},C_j,D_{-j}\) and \(\omega\) .

Step-(5): Several types of exact solutions could be obtained using Eq. (6) by selecting different values for \(\iota _0,\iota _1,\iota _2,\iota _3,\iota _4,\iota _6\), as follows:

Case 1: When \(\iota _0=\iota _1=\iota _3=\iota _6=0\),

$$\begin{aligned} & \hbar (\xi )=\sqrt{-\frac{\iota _2}{\iota _4}} \text {sech}\left( \sqrt{\iota _2} \xi \right) ,\quad \iota _2>0, \iota _4<0.\\ & \hbar (\xi )=\sqrt{-\frac{\iota _2}{\iota _4}} \sec \left( \sqrt{-\iota _2}\xi \right) , \quad \iota _2<0, \iota _4>0.\\ & \hbar (\xi )=\sqrt{-\frac{\iota _2}{\iota _4}} \csc \left( \sqrt{-\iota _2} \xi \right) , \quad \iota _2<0, \iota _4>0. \end{aligned}$$

Case 2: When \(\iota _1=\iota _3=\iota _6=0, \iota _0=\dfrac{\iota _2^2}{4\iota _4}\),

$$\begin{aligned} & \hbar (\xi )=\sqrt{-\frac{\iota _2}{2 \iota _4}} \tanh \left( \sqrt{-\frac{\iota _2}{2}} \xi \right) ,\quad \iota _2<0, \iota _4>0.\\ & \hbar (\xi )=\sqrt{\frac{\iota _2}{2 \iota _4}} \tan \left( \sqrt{\frac{\iota _2}{2}} \xi \right) ,\quad \iota _2>0, \iota _4>0. \end{aligned}$$

Case 3: When \(\iota _3=\iota _4=\iota _6=0\),

$$\begin{aligned} & \hbar (\xi )=\frac{\iota _1 \sinh \left( 2 \sqrt{\iota _2} \xi \right) }{2 \iota _2}-\frac{\iota _1}{2 \iota _2},\quad \iota _2>0,\iota _0=0.\\ & \hbar (\xi )=\frac{\iota _1 \sin \left( \sqrt{-\iota _2} \xi \right) }{2 \iota _2}-\frac{\iota _1}{2 \iota _2},\quad \iota _2<0,\iota _0=0.\\ & \hbar (\xi )=\sqrt{\frac{\iota _0}{\iota _2}} \sinh \left( \sqrt{\iota _2} \xi \right) ,\quad \iota _0>0,\iota _2>0,\iota _1=0.\\ & \hbar (\xi )=\sqrt{-\frac{\iota _0}{\iota _2}} \sin \left( \sqrt{-\iota _2} \xi \right) ,\quad \iota _0>0,\iota _2<0,\iota _1=0.\\ & \hbar (\xi )=\exp \left( \sqrt{\iota _2} \xi \right) -\frac{\iota _1}{2 \iota _2},\quad \iota _2>0,\iota _0=\frac{\iota _1^2}{4 \iota _2}. \end{aligned}$$

Case 4: When \(\iota _0=\iota _1=\iota _6=0\),

$$\begin{aligned} \hbar (\xi )=\frac{4 \iota _3}{\iota _3^2 \xi ^2-4 \iota _4},\quad \iota _3>0, \iota _4>0. \end{aligned}$$

Case 5: When \(\iota _0=\iota _1=\iota _2=\iota _6=0\),

$$\begin{aligned} & \hbar (\xi )=-\frac{\iota _2 \left( \tanh \left( \frac{1}{2} \sqrt{\iota _2} \xi \right) +1\right) }{\iota _3},\quad \iota _3^2=4\iota _2\iota _4,\iota _2>0.\\ & \hbar (\xi )=-\frac{\iota _2 \left( \coth \left( \frac{1}{2} \sqrt{\iota _2} \xi \right) +1\right) }{\iota _3},\quad \iota _3^2=4\iota _2\iota _4,\iota _2>0.\\ & \hbar (\xi )=\frac{\iota _2 \text {sech}^2\left( \frac{1}{2} \sqrt{\iota _2}\xi \right) }{2 \sqrt{\iota _2 \iota _4} \tanh \left( \frac{1}{2} \sqrt{\iota _2} \xi \right) -\iota _3},\quad \iota _3^2\ne 4\iota _2\iota _4,\iota _2>0,\iota _4>0.\\ & \hbar (\xi )=-\frac{\iota 2 \sec ^2\left( \frac{1}{2} \sqrt{-\iota _2}\xi \right) }{2 \sqrt{-\iota _2 \iota _4} \tan \left( \frac{1}{2} \sqrt{-\iota _2} \xi \right) +\iota _3},\quad \iota _3^2\ne 4\iota _2\iota _4,\iota _2<0,\iota _4>0. \end{aligned}$$

Case 6: When \(\iota _2=\iota _4=\iota _6=0\),

$$\begin{aligned} \hbar (\xi )=\wp \left( \frac{1}{2} \sqrt{\iota _3}\xi ;-\frac{4 \iota _1}{\iota _3},-\frac{4 \iota _0}{\iota _3}\right) ,\quad \iota _3>0. \end{aligned}$$

Case 7: When \(\iota _0=\iota _1=\iota _3=0\),

$$\begin{aligned} & \hbar (\xi )=\sqrt{\frac{2 \iota _2 \text {sech}^2\left( \sqrt{\iota _2} \xi \right) }{2 \sqrt{\iota _4^2-4 \iota _2 \iota _6}-\left( \sqrt{\iota _4^2-4 \iota _2 \iota _6}+\iota _4\right) \text {sech}^2\left( \sqrt{\iota _2}\xi \right) }},\quad \iota _2>0.\\ & \hbar (\xi )=\sqrt{\frac{2 \iota _2 \sec ^2\left( \sqrt{-\iota _2}\xi \right) }{2 \sqrt{\iota _4^2-4 \iota _2 \iota _6}-\left( \sqrt{\iota _4^2-4 \iota _2 \iota _6}-\iota _4\right) \sec ^2\left( \sqrt{-\iota _2}\xi \right) }},\quad \iota _2<0. \end{aligned}$$

Case 8: When \(\iota _1=\iota _3=\iota _6=0\),

No.

\({\iota _0}\)

\({\iota _2}\)

\({\iota _4}\)

\({\hbar (\xi )}\)

1

1

\(-(1+m^2)\)

\(m^2\)

\(\text {cd}(\zeta ,m)\) or \(\text {sn}(\zeta ,m)\)

2

\(m^2-1\)

\(-m^2+2\)

\(-1\)

\(\text {dn}(\zeta ,m)\)

3

\(-m^2\)

\(2m^2-1\)

\(-m^2+1\)

\(\text {nc}(\zeta ,m)\)

4

\(-1\)

\(-m^2+2\)

\(m^2-1\)

\(\text {nd}(\zeta ,m)\)

5

\(m^2-2m^3+m^4\)

\(-\dfrac{4}{m}\)

\(-1+6m-m^2\)

\(\dfrac{mdn(\zeta |m)cn(\zeta |m)}{1+msn(\zeta |m)^2}\)

6

\(\dfrac{1}{4}\)

\(\dfrac{1}{2}m^2-1\)

\(\dfrac{m^4}{4}\)

\(\dfrac{\text {sn}(\zeta |m)}{1+\text {dn}(\zeta |m)}\) or \(\dfrac{\text {cn}(\zeta |m)}{\sqrt{1-m^2}+\text {dn}(\zeta |m)}\)

Step-(6): By substituting the determined constants \(A_j,B_{-j},C_j\) and \(D_{-j}\) into Eq. (5) and incorporating the expressions from Eq. (6), multiple solutions to Eq. (2) can be derived.

The flowchart in Fig. 1 shows a brief description of the used scheme.

Fig. 1
figure 1

Flowchart of the proposed method.

Full size image

The MEMT offers several advantages over traditional methods for solving nonlinear differential equations. Unlike the Inverse Scattering Transform, which is restricted to integrable equations and relies on complex spectral analysis, MEMT provides a more direct and broadly applicable approach. In contrast to Hirota’s Bilinear Method, which requires cumbersome bilinear transformations and perturbative expansions, MEMT simplifies the process of deriving exact solutions. Similarly, compared to Lie Symmetry Analysis, which involves extensive group-theoretic computations and often yields only symmetry-invariant solutions, MEMT can systematically generate a wider variety of solutions, including solitons, periodic waves, and elliptic functions. This efficiency and versatility make MEMT a powerful and practical tool for nonlinear wave analysis.

Solitons and other solutions to the proposed model

The solutions of Eq. (1) may be obtained by assuming the following wave transformation:

$$\begin{aligned} Q(x,t)=\phi (\xi ) e ^{i \left( -k x+ \omega t+\chi W(t)-\chi ^{2}t\right) }, \, \xi =x-c t, \end{aligned}$$
(7)

The parameters \(k,\omega\) and c are real constants denoting the wave number, frequency, and velocity of the soliton, respectively. The pulse profile is characterized by the real-valued function \(\phi =\phi (\xi )\).

Using the transformation described by Eq. (7) after that, Eq. (1) will be transformed into the nonlinear ordinary differential equations (NODEs), whose real and imaginary parts are separated to produce the following equations:

$$\begin{aligned} & (\gamma -c \beta +\alpha )\phi ^{\prime \prime }+\left( (\beta \kappa -1)(\omega -\chi ^{2})-\delta k- \alpha k^{2}\right) \phi +b_{1}\phi ^{n+1}+b_{2}\phi ^{2n+1}+3\phi ^{3n+1}\nonumber \\ & \quad +b_{4}\phi ^{4n+1}+c_{1}\phi ^{1-n}+c_{2}\phi ^{1-2n}+c_{3}\phi ^{1-3n}+c_{4}\phi ^{1-4n}=0, \end{aligned}$$
(8)

and the imaginary parts are:

$$\begin{aligned} & \left( (\beta k -1) c -2 \alpha k -\delta + \beta (\omega -\chi ^{2})\right) \phi ^{\prime }(\tau )=0, \end{aligned}$$
(9)

From Eq. (9),

$$\begin{aligned} & c=\frac{2 \alpha k+\delta -\beta (\omega -\chi ^{2})}{\beta k-1}, \, \beta k\ne 1. \end{aligned}$$
(10)

By balancing \(\phi ^{\prime \prime }\) and \(\phi ^{4n+1}\) in Eq. (8) we derive the balance \(N=\frac{1}{ n}\). Since N is not an integer, we proceed by taking:

$$\begin{aligned} & \phi (\xi )=\left[ \varphi (\xi )\right] ^{\frac{1}{n}}, \, \varphi (\xi )>0. \end{aligned}$$
(11)

Inserting (11)into Eq. (8) yields :

$$\begin{aligned} & 2 n \pi _{0} \varphi \varphi ^{\prime \prime }+(1-2 n) (\varphi ^{\prime })^{2}+4 n^{2}c_{4}+4 n^{2}c_{2}\varphi +4 n^{2}\pi _{1}\varphi ^{2}+4 n^{2}b_{2}\varphi ^{3}+4 n^{2}b_{4}\varphi ^{4}\nonumber \\ & \quad +4 n^{2}c_{3}\varphi ^{\frac{1}{2}}+4 n^{2}c_{1}\varphi ^{\frac{3}{2}}+4 n^{2}b_{1}\varphi ^{\frac{5}{2}}+4 n^{2}b_{3}\varphi ^{\frac{7}{2}}=0, \end{aligned}$$
(12)

Where

$$\begin{aligned} & \pi _{0}=\gamma -c \beta + \alpha ,\, \pi _{1}=(\beta k-1)(\omega -\chi ^{2})-\delta k-\alpha k^{2}. \end{aligned}$$
(13)

To ensure integrability, the following condition must be imposed38:

$$\begin{aligned} & b_{1}=b_{3}=c_{1}=c_{3}=0. \end{aligned}$$
(14)

Therefore, Eq. (1) is rewritten as:

$$\begin{aligned} & i Q_{t}+ \alpha Q_{xx}+\beta Q_{xt}+\left( \frac{|Q|_{xx}}{|Q|}\right) Q-i\delta Q_{x}+\chi (Q-i \beta Q_{x})\frac{dW(t)}{dt}\nonumber \\ & \quad +(b_{2}|Q|^{2 n}+(b_{4}|Q|^{4 n}+(c_{2}|Q|^{-2 n}+(c_{4}|Q|^{-4 n})Q=0. \end{aligned}$$
(15)

Consequently, Eq. (12) becomes:

$$\begin{aligned} & 2 n \pi _{0} \varphi \varphi ^{\prime \prime }+(1-2 n) \pi _{0} (\varphi ^{\prime })^{2}+4 n^{2}c_{4}+4 n^{2}c_{2}\varphi +4 n^{2}\pi _{1}\varphi ^{2}+4 n^{2}b_{2}\varphi ^{3}+4 n^{2}b_{4}\varphi ^{4} =0. \end{aligned}$$
(16)

A balance between the terms \(\varphi ^{\prime \prime }\) and \(\varphi ^{4}\) yields \(N=1\). Accordingly, the solution of Eq. (16) can be formulated from Eq. (5) as follows:

$$\begin{aligned} \mathfrak {Q}(\xi )=A_0+A_1 \hbar (\xi )+\frac{B_1}{\hbar (\xi )}+D_1\left( \frac{ \hbar ^{\prime}(\xi )}{\hbar (\xi )}\right) , \end{aligned}$$
(17)

Where \(A_1\), \(B_1\), and \(C_1\) are all unknown constants, they may be estimated by applying the restriction \(A_{1}^{2}+B_{1}^{2}+C_{1}^{2}\ne 0\).

Case-(1): If we set \(\iota _0=\iota _1=\iota _3=\iota _6=0\), then we have:

Result 1.1: \(B_1=A_1=0 ,\) \(D_1=-A_0,\) \(b_4=\frac{\pi _0 (10 n-1)}{4 A_0^2 n^2},\) \(\iota _2=1-\frac{2 c_4 n^2}{\sqrt{\pi _0 A_0^2 c_4 n^2 (2 n-1)}},\) \(\pi _1=\frac{4 A_0^3 b_2 n^2 (1-2 n)+2 (5 n-1) \sqrt{\pi _0 A_0^2 c_4 n^2 (2 n-1)}+\pi _0 A_0^2 \left( -38 n^2+23 n-2\right) }{2 A_0^2 n^2 (2 n-1)} .\)

According to the preceding set of solutions, we can discover explicit solutions to the suggested equation as follows:

(1.1.1) If \(\pi _0 c_4 \ne 0\) , \(A_0\ne 0\) and \(\frac{2 c_4 n}{A_0 \sqrt{\pi _0 c_4 (2 n-1)}}<1\), then we get dark soliton solutions are formulated as:

$$\begin{aligned} & Q_{1.1.1}(x,t)=\left( A_0 \left( \sqrt{1-\frac{2 c_4 n}{A_0 \sqrt{\pi _0 c_4 (2 n-1)}}} \tanh \left( \sqrt{1-\frac{2 c_4 n}{A_0 \sqrt{\pi _0 c_4 (2 n-1)}}} (x-c t)\right) +1\right) \right) ^{\frac{1}{n}} \nonumber \\ & \quad \times e^{i \left( -kx+\omega t +\chi W(t)-\chi ^{2}t\right) }, \end{aligned}$$
(18)

(1.1.2) If \(\pi _0 c_4 \ne 0\) , \(A_0\ne 0\) and \(\frac{2 c_4 n}{A_0 \sqrt{\pi _0 c_4 (2 n-1)}}>1\), then we get singular periodic solutions is formulated as:

$$\begin{aligned} & Q_{1.1.2}(x,t)=\left( A_0 \left( 1-\sqrt{\frac{2 c_4 n}{A_0 \sqrt{\pi _0 c_4 (2 n-1)}}+1} \tan \left( \sqrt{\frac{2 c_4 n}{A_0 \sqrt{\pi _0 c_4 (2 n-1)}}-1} (x-c t)\right) \right) \right) ^{\frac{1}{n}}\nonumber \\ & \quad \times e^{i \left( -kx+\omega t +\chi W(t)-\chi ^{2}t\right) }, \end{aligned}$$
(19)

(1.1.3) If \(\pi _0 c_4 \ne 0\) , \(A_0\ne 0\) and \(\frac{2 c_4 n}{A_0 \sqrt{\pi _0 c_4 (2 n-1)}}>1\), then we get singular soliton is formulated as:

$$\begin{aligned} & Q_{1.1.3}(x,t)=\left( A_0 \left( \sqrt{\frac{2 c_4 n}{A_0 \sqrt{\pi _0 c_4 (2 n-1)}}-1} \cot \left( \sqrt{\frac{2 c_4 n}{A_0 \sqrt{\pi _0 c_4 (2 n-1)}}-1} (x-c t)\right) +1\right) \right) ^{\frac{1}{n}}\nonumber \\ & \quad \times e^{i \left( -kx+\omega t +\chi W(t)-\chi ^{2}t\right) }, \end{aligned}$$
(20)

Result 1.2: \(A_1=-\frac{A_0^2 \sqrt{\pi _0 \iota _4} \sqrt{-n^2+\frac{3 n}{2}-\frac{1}{2}}}{\sqrt{n^2 \left( A_0 c_2 (2 n-1)+2 c_4 (n-1)\right) }},\,D_1=B_1=0,\,b_2=-\frac{2 (n+1) \left( A_0 c_2 (2 n-1)+2 c_4 (n-1)\right) }{A_0^3 \left( 2 n^2-3 n+1\right) },\)

\(b_4=\frac{(2 n+1) \left( A_0 c_2 (2 n-1)+2 c_4 (n-1)\right) }{2 A_0^4 \left( 2 n^2-3 n+1\right) },\,\pi _1=\frac{5 A_0 c_2 (2 n-1)+8 c_4 (n-1)}{2 A_0^2 \left( 2 n^2-3 n+1\right) },\,\iota _2=\frac{2 n^2 \left( A_0 c_2 (2 n-1)+4 c_4 (n-1)\right) }{\pi _0 A_0^2 \left( 2 n^2-3 n+1\right) }.\)

According to the preceding set of solutions, we can discover explicit solutions to the suggested equation as follows:

(1.2.1) If \(A_0\ne 0\) and \(\pi _0\ne 0\), then we get bright soliton is formulated as:

$$\begin{aligned} & Q_{1.2.1}(x,t)=A_0 \left( 1-\text {sech}\left( \sqrt{\frac{2 n^2 \left( A_0 c_2 (2 n-1)+4 c_4 (n-1)\right) }{\pi _0 A_0^2 \left( 2 n^2-3 n+1\right) }} (x-c t)\right) \right) ^{\frac{1}{n}}\nonumber \\ & \quad \times e^{i \left( -kx+\omega t +\chi W(t)-\chi ^{2}t\right) }, \end{aligned}$$
(21)

(1.2.2) If \(A_0\ne 0\) and \(\pi _0<0\), then we get singular periodic solutions is formulated as:

$$\begin{aligned} & Q_{1.2.2}(x,t)=A_0 \left( 1-\sec \left( \sqrt{-\frac{2 n^2 \left( A_0 c_2 (2 n-1)+4 c_4 (n-1)\right) }{\pi _0 A_0^2 \left( 2 n^2-3 n+1\right) }} (x-c t)\right) \right) ^{\frac{1}{n}}\nonumber \\ & \quad \times e^{i \left( -kx+\omega t +\chi W(t)-\chi ^{2}t\right) }, \end{aligned}$$
(22)

or

$$\begin{aligned} & Q_{1.2.3}(x,t)=\left( 1-\csc \left( \sqrt{-\frac{2 n^2 \left( A_0 c_2 (2 n-1)+4 c_4 (n-1)\right) }{\pi _0 A_0^2 \left( 2 n^2-3 n+1\right) }} (x-c t)\right) \right) ^{\frac{1}{ n}}\nonumber \\ & \quad \times e^{i \left( -kx+\omega t +\chi W(t)-\chi ^{2}t\right) }, \end{aligned}$$
(23)

Case-(2): If \(\iota _1=\iota _3=\iota _6=0\) and \(\iota _0=\frac{\iota _2^2}{4 \iota _4}\), then we have

Result 2.1: \(A_1=\frac{2 A_0 (n-1) \sqrt{\pi _0 \left( -c_4\right) \iota _4}}{\sqrt{c_2^2 \left( -n^2\right) (2 n-1)}}\), \(D_1=B_1=0,\) \(b_2=\frac{c_2^2 \left( 2 n^2+n-1\right) }{4 A_0 c_4 (n-1)^2},\) \(b_4=\frac{c_2^2 \left( 1-4 n^2\right) }{16 A_0^2 c_4 (n-1)^2},\) \(\pi _1=\frac{c_2 \left( A_0 c_2 (1-2 n)+2 c_4 (n-1)\right) }{4 A_0 c_4 (n-1)^2},\) \(\iota _2=-\frac{c_2 n^2 \left( A_0 c_2 (2 n-1)+4 c_4 (n-1)\right) }{2 \pi _0 A_0 c_4 (n-1)^2}.\)

From the recovered solution set, the following solutions can be identified:

(2.1.1) If \(c _2\ne 0\) , \(\pi _0 A_0 c_4\ne 0\) , \(n\ne \frac{1}{2}\) and \(n-1\ne 0\), then we get dark soliton is formulated as:

$$\begin{aligned} & Q_{2.1.1}(x,t)= \left( \frac{\sqrt{\frac{A_0 c_2 \left( A_0 c_2 (1-2 n)-4 c_4 (n-1)\right) }{1-2 n}} \tanh \left( \frac{n \sqrt{\frac{c_2 \left( A_0 c_2 (2 n-1)+4 c_4 (n-1)\right) }{\pi _0 A_0 c_4}} (x-c t)}{2 (n-1)}\right) }{c_2}+A_0\right) ^{\frac{1}{ n}} \nonumber \\ & \quad \times e^{i \left( -kx+\omega t +\chi W(t)-\chi ^{2}t\right) }, \end{aligned}$$
(24)

(2.1.2) If \(c _2\ne 0\) , \(\pi _0 A_0 c_4\ne 0\), \(\pi _0<0\), \(n\ne \frac{1}{2}\) and \(n-1\ne 0\), then we get singular periodic solution is formulated as:

$$\begin{aligned} & Q_{2.1.2}(x,t)=\left( \frac{\sqrt{\frac{A_0 c_2 \left( A_0 c_2 (1-2 n)-4 c_4 (n-1)\right) }{1-2 n}} \tan \left( \frac{n \sqrt{-\frac{c_2 \left( A_0 c_2 (2 n-1)+4 c_4 (n-1)\right) }{\pi _0 A_0 c_4}} (x-c t)}{2 (n-1)}\right) }{c_2}+A_0\right) ^{\frac{1}{ n}}\nonumber \\ & \quad \times e^{i \left( -kx+\omega t +\chi W(t)-\chi ^{2}t\right) }. \end{aligned}$$
(25)

Result 2.2: \(A_1=D_1=0\), \(B_1=\frac{\sqrt{\frac{n^2 \left( A_0 c_2+4 c_4\right) ^2 \left( A_0 c_2 (2 n-1)+4 c_4 (n-1)\right) ^2}{\pi _0 c_4 (n-1)^2}}}{2 \sqrt{\iota _4 (2 n-1) \left( A_0 c_2+4 c_4\right) ^2}},\) \(b_2=\frac{c_2^2 \left( 2 n^2+n-1\right) }{4 A_0 c_4 (n-1)^2},\) \(b_4=\frac{c_2^2 \left( 1-4 n^2\right) }{16 A_0^2 c_4 (n-1)^2},\) \(\pi _1=\frac{c_2 \left( A_0 c_2 (1-2 n)+2 c_4 (n-1)\right) }{4 A_0 c_4 (n-1)^2},\) \(\iota _2=-\frac{c_2 n^2 \left( A_0 c_2 (2 n-1)+4 c_4 (n-1)\right) }{2 \pi _0 A_0 c_4 (n-1)^2}.\)

From the recovered solution set, the following solutions can be identified:

(2.2.1) If \(c _2\ne 0\) , \(\pi _0 c_4\ne 0\), \(n\ne \frac{1}{2}\) and \(n-1\ne 0\), then we get singular soliton is formulated as:

$$\begin{aligned} & Q_{2.2.1}(x,t)= \left( \sqrt{\frac{A_0 c_2 (2 n-1)+4 c_4 (n-1)}{c_2 (2 n-1)}} \coth \left( \frac{n \sqrt{\frac{c_2 \left( A_0 c_2 (2 n-1)+4 c_4 (n-1)\right) }{\pi _0 c_4}} (x-c t)}{2 (n-1)}\right) +a_0\right) ^{\frac{1}{ n}}\nonumber \\ & \quad \times e^{i \left( -kx+\omega t +\chi W(t)-\chi ^{2}t\right) }, \end{aligned}$$
(26)

(2.2.2) If \(c _2\ne 0\) , \(\pi _0 c_4\ne 0\), \(\pi _0<0\), \(n\ne \frac{1}{2}\) and \(n-1\ne 0\), then we get singular periodic solution is formulated as:

$$\begin{aligned} & Q_{2.2.2}(x,t)=\left( \sqrt{\frac{A_0 c_2 (2 n-1)+4 c_4 (n-1)}{c_2 (1-2 n)}} \cot \left( \frac{n \sqrt{\frac{c_2 \left( A_0 c_2 (1-2 n)+4 c_4 (1-n)\right) }{\pi _0 c_4}} (x-c t)}{2 (n-1)}\right) +A_0\right) ^{\frac{1}{n}}\nonumber \\ & \quad \times e^{i \left( -kx+\omega t +\chi W(t)-\chi ^{2}t\right) }. \end{aligned}$$
(27)

Result 2.3: \(A_1=\frac{2 \pi _0 A_0 c_4 \iota _4 (n-1)^2 \mathbb {N}}{c_2 n^2 \ell \left( A_0 c_2 (2 n-1)+4 c_4 (n-1)\right) },\) \(D_1=0,\)\(B_1=-\frac{\mathbb {N}}{8 \ell },\) \(b_2=\frac{c_2^2 \left( 2 n^2+n-1\right) }{4 A_0 c_4 (n-1)^2},\) \(b_4=\frac{c_2^2 \left( 1-4 n^2\right) }{16 A_0^2 c_4 (n-1)^2},\) \(\pi _1=\frac{c_2 \left( A_0 c_2 (1-2 n)+2 c_4 (n-1)\right) }{4 A_0 c_4 (n-1)^2},\) \(\iota _2=-\frac{c_2 n^2 \left( A_0 c_2 (2 n-1)+4 c_4 (n-1)\right) }{8 \pi _0 A_0 c_4 (n-1)^2}.\)

From the recovered solution set, the following solutions can be identified:

(2.3.1) If \(n \ell \ne 0\) , \(\mathbb {N}\ne 0\), \(\pi _0 A_0 c_4 \ne 0\), \(n\ne \frac{1}{2}\) and \(n\ne 1\), then we get singular soliton is formulated as:

$$\begin{aligned} & Q_{2.3.1}(x,t)= \left( \frac{(1-n) \mathbb {N} \sqrt{\frac{\pi _0 A_0 c_4 \iota _4}{c_2 \left( A_0 c_2 (2 n-1)+4 c_4 (n-1)\right) }} \text {csch}\left( \frac{n \sqrt{\frac{c_2 \left( A_0 c_2 (2 n-1)+4 c_4 (n-1)\right) }{\pi _0 A_0 c_4}} (x-c t)}{2 (n-1)}\right) }{n \ell }-A_0\right) ^{\frac{1}{ n}}\nonumber \\ & \quad \times e^{i \left( -kx+\omega t +\chi W(t)-\chi ^{2}t\right) }, \end{aligned}$$
(28)

(2.3.2) If \(n \ell \ne 0\) , \(\mathbb {N}\ne 0\), \(\pi _0 A_0 c_4 \ne 0\), \(n\ne \frac{1}{2}\) and \(n\ne 1\), then we get singular periodic solution is formulated as:

$$\begin{aligned} & Q_{2.3.2}(x,t)=\left( \frac{(n-1) \mathbb {N} \sqrt{-\frac{\pi _0 A_0 c_4 \iota _4}{c_2 \left( A_0 c_2 (2 n-1)+4 c_4 (n-1)\right) }} \csc \left( \frac{n \sqrt{-\frac{c_2 \left( A_0 c_2 (2 n-1)+4 c_4 (n-1)\right) }{\pi _0 A_0 c_4}} (x-c t)}{2 (n-1)}\right) }{n \ell }+A_0\right) ^{\frac{1}{ n}}\nonumber \\ & \quad \times e^{i \left( -kx+\omega t +\chi W(t)-\chi ^{2}t\right) }. \end{aligned}$$
(29)

Where \(\mathbb {N}=\sqrt{\frac{n^2 \left( A_0 c_2 (2 n-1)+4 c_4 (n-1)\right) ^2 \left( -6 A_0 c_4 c_2 \left( 2 n^2-3 n+1\right) +A_0^2 c_2^2 (1-2 n)^2-16 c_4^2 (n-1)^2\right) }{c_4 (n-1)}}\) and \(\ell =\sqrt{\pi _0 \iota _4 \left( 2 n^2-3 n+1\right) \left( -6 A_0 c_4 c_2 \left( 2 n^2-3 n+1\right) +A_0^2 c_2^2 (1-2 n)^2-16 c_4^2 (n-1)^2\right) }\).

Case-(3): If   \(\iota _0\) = \(\iota _3\) = \(\iota _4\) = \(\iota _6\) = 0, then we obtain \(A_1=B_1=0,\) \(D_1=-A_0\) \(b_4=\frac{-4 A_0^3 b_2 \left( 16 n^3+4 n^2-4 n-1\right) +2 \pi _1 A_0^2 \left( -16 n^3-4 n^2+4 n+1\right) -c_4 (2 n+1) (4 n+1)^2+\mho }{8 A_0^4 (2 n-1) (4 n+1)^2},\) \(\pi _0=\frac{4 A_0^2 n^2 (2 n-1) (2 n+1) \left( 2 A_0 b_2+\pi _1\right) ^2 \left( 3 c_4 (2 n+1) (4 n+1)^2+\mho \right) }{\left( c_4 (2 n+1) (4 n+1)^2+\mho \right) \left( 2 A_0^3 b_2 \left( 16 n^3+4 n^2-4 n-1\right) +\pi _1 A_0^2 \left( 16 n^3+4 n^2-4 n-1\right) +32 c_4 n^3+32 c_4 n^2+10 c_4 n+c_4+\mho \right) },\) \(\iota _2= \frac{2 A_0^3 b_2 \left( 16 n^3+4 n^2-4 n-1\right) +\pi _1 A_0^2 \left( 16 n^3+4 n^2-4 n-1\right) +32 c_4 n^3+32 c_4 n^2+10 c_4 n+c_4+\mho }{A_0^2 (4 n+1) \left( 4 n^2-1\right) \left( 2 A_0 b_2+\pi _1\right) }\).

According to the preceding set of solutions, we can discover explicit solutions to the suggested equations as follows:

(3.1) If \(A _0\ne 0\) and \((4 n+1) (4 n^2-1) (2 A_0 b_2+\pi _1)\ne 0\), then we get hyperbolic solution is formulated as:

$$\begin{aligned} & Q_{3.1.1}(x,t)=(\frac{1}{\sinh (\frac{2}{A_0}\sqrt{\frac{2 A_0^3 b_2(16 n^3+4 n^2-4 n-1)+\pi _1 A_0^2 (16 n^3+4 n^2-4 n-1)+c_4 (2 n+1) (4 n+1)^2+\mho }{ (4 n+1) (4 n^2-1) (2 A_0 b_2+\pi _1)}}(x-c t))-1}\nonumber \\ & \quad \times (A_0 (\sinh (\frac{2}{A_0} \sqrt{\frac{2 A_0^3 b_2 (16 n^3+4 n^2-4 n-1)+\pi _1 A_0^2 (16 n^3+4 n^2-4 n-1)+c_4 (2 n+1) (4 n+1)^2+\mho }{(4 n+1) (4 n^2-1) (2 A_0 b_2+\pi _1)}}\nonumber \\ & \quad \times (x-c t))\nonumber \\ & \quad -\frac{2}{A_0} \sqrt{\frac{2 A_0^3 b_2 (16 n^3+4 n^2-4 n-1)+\pi _1 A_0^2 (16 n^3+4 n^2-4 n-1)+c_4 (2 n+1) (4 n+1)^2+\mho }{ (4 n+1) (4 n^2-1) (2 A_0 b_2+\pi _1)}} \nonumber \\ & \quad \times \cosh (\frac{2}{A_0} \sqrt{\frac{2 A_0^3 b_2(16 n^3+4 n^2-4 n-1)+\pi _1 A_0^2 (16 n^3+4 n^2-4 n-1)+c_4 (2 n+1) (4 n+1)^2+\mho }{ (4 n+1) (4 n^2-1) (2 A_0 b_2+\pi _1)}}\nonumber \\ & \quad \times (x-c t))-1)))^{\frac{1}{ n}}e^{i \left( -kx+\omega t +\chi W(t)-\chi ^{2}t\right) }, \end{aligned}$$
(30)

(3.2) If \(A _0\ne 0\) and \((4 n+1) (4 n^2-1) (2 A_0 b_2+\pi _1)\ne 0\), then we get trigonometric solution is formulated as:

$$\begin{aligned} & Q_{3.1.2}(x,t)=(\frac{1}{\sin (\frac{2}{A_0} \sqrt{\frac{2 A_0^3 b_2(16 n^3+4 n^2-4 n-1)+\pi _1 A_0^2 (16 n^3+4 n^2-4 n-1)+c_4 (2 n+1) (4 n+1)^2+\mho }{ (4 n+1) (4 n^2-1) (2 A_0 b_2+\pi _1)}}(x-c t))-1}\nonumber \\ & \quad \times (A_0 (\sin ( \frac{2}{A_0} \sqrt{\frac{2 A_0^3 b_2 (16 n^3+4 n^2-4 n-1)+\pi _1 A_0^2 (16 n^3+4 n^2-4 n-1)+c_4 (2 n+1) (4 n+1)^2-\mho }{ (4 n+1) (4 n^2-1) (2 A_0 b_2+\pi _1)}}\nonumber \\ & \quad \times (x-c t))\nonumber \\ & \quad -\frac{2}{A_0} \sqrt{\frac{2 A_0^3 b_2 (16 n^3+4 n^2-4 n-1)+\pi _1 A_0^2 (16 n^3+4 n^2-4 n-1)+c_4 (2 n+1) (4 n+1)^2-\mho }{ (4 n+1) (4 n^2-1) (2 A_0 b_2+\pi _1)}} \nonumber \\ & \quad \times \cos (\frac{2}{A_0} \sqrt{\frac{2 A_0^3 b_2(16 n^3+4 n^2-4 n-1)+\pi _1 A_0^2 (16 n^3+4 n^2-4 n-1)+c_4 (2 n+1) (4 n+1)^2-\mho }{ (4 n+1) (4 n^2-1) (2 A_0 b_2+\pi _1)}}\nonumber \\ & \quad \times (x-c t))-1)))^{\frac{1}{n}}e^{i \left( -kx+\omega t +\chi W(t)-\chi ^{2}t\right) }. \end{aligned}$$
(31)

Where \(\mho =\sqrt{c_4 (2 n+1)^2 (4 n+1)^3 \left( 8 A_0^3 b_2 (2 n-1)+4 \pi _1 A_0^2 (2 n-1)+c_4 (4 n+1)\right) }\).

Case-(4): If \(\iota _0\) = \(\iota _1\) = \(\iota _2\) = \(\iota _6\) = 0, then we have \(D_1\) = \(B_1\) = 0, \(c_2=-\frac{A_0^2 (n-1) \left( 8 A_0 b_4 (n+1)+b_2 (6 n+3)\right) }{(n+1) (2 n+1)},\) \(c_4=\frac{A_0^3 (2 n-1) \left( 3 A_0 b_4 (n+1)+b_2 (2 n+1)\right) }{(n+1) (2 n+1)},\) \(\pi _1=-\frac{6 A_0^2 b_4}{2 n+1}-\frac{3 A_0 b_2}{n+1},\) \(\iota _3=-\frac{4 A_1 n^2 \left( 4 A_0 b_4 (n+1)+b_2 (2 n+1)\right) }{\pi _0 (n+1) (2 n+1)},\) \(\iota _4=-\frac{4 A_1^2 b_4 n^2}{2 \pi _0 n+\pi _0}.\)

According to the preceding set of solutions, we can discover explicit solutions to the suggested equations as follows:

(4.1) If \(b_4 (n+1)\ne 0\) and \(\pi _0 (2 n+1)\ne 0\), then we get a rational solution is formulated as:

$$\begin{aligned} & Q_{4.1.1}(x,t)=\left( A_0-\frac{4 A_0 b_4 (n+1)+b_2 (2 n+1)}{\frac{n^2 \left( 4 A_0 b_4 (n+1)+b_2 (2 n+1)\right) ^2 (x-c t)^2}{\pi _0 (2 n+1)}+b_4 (n+1)}\right) ^{\frac{1}{ n}}e^{i \left( -kx+\omega t +\chi W(t)-\chi ^{2}t\right) }. \end{aligned}$$
(32)

Case-(5.1): If \(\iota _0\) = \(\iota _1\) = \(\iota _6\) =0, \(\iota _3^2=4 \iota _2 \iota _4,\) then we have \(A_1=-\frac{\sqrt{\pi _0 \iota _4 (-(2 n+1))}}{2 \sqrt{b_4} n},\) \(D_1\) = \(B_1\) = 0, \(c_2=-\frac{A_0 (n-1) \left( 6 A_0 b_2 (2 n+1) n^2+16 A_0^2 b_4 (n+1) n^2+\pi _0 \iota _2 \left( 2 n^2+3 n+1\right) \right) }{2 n^2 \left( 2 n^2+3 n+1\right) },\) \(c_4=\frac{A_0^2 (2 n-1) \left( 4 A_0 b_2 (2 n+1) n^2+12 A_0^2 b_4 (n+1) n^2+\pi _0 \iota _2 \left( 2 n^2+3 n+1\right) \right) }{4 n^2 \left( 2 n^2+3 n+1\right) },\) \(\pi _1=-\frac{6 A_0^2 b_4}{2 n+1}-\frac{3 A_0 b_2}{n+1}-\frac{\pi _0 \iota _2}{4 n^2},\) \(\iota _3=\frac{2 \sqrt{\iota _4} n \sqrt{-(2 n+1)} \left( 4 A_0 b_4 (n+1)+b_2 (2 n+1)\right) }{\sqrt{\pi _0 b_4} \left( 2 n^2+3 n+1\right) }\).

According to the preceding set of solutions, we can discover explicit solutions to the suggested equation as follows:

(5.1) If \(n \ne 0\), \(\pi _{0} \ne 0\) and \(\iota _2>0\), then we get dark soliton solution are formulated as:

$$\begin{aligned} & Q_{5.1.1}(x,t)= \left( \frac{\pi _0 \iota _2 \left( 2 n^2+3 n+1\right) \left( \tanh \left( \frac{1}{2} \sqrt{\iota _2} (x-c t)\right) +1\right) }{4 n^2 \left( 4 A_0 b_4 (n+1)+b_2 (2 n+1)\right) }+A_0\right) ^{\frac{1}{n}}\nonumber \\ & \quad \times e^{i \left( -kx+\omega t +\chi W(t)-\chi ^{2}t\right) }, \end{aligned}$$
(33)

(5.2) If \(n \ne 0\), \(\pi _{0} \ne 0\) and \(\iota _2>0\), then we get singular solution are formulated as:

$$\begin{aligned} & Q_{5.2.1}(x,t)= \left( \frac{\pi _0 \iota _2 \left( 2 n^2+3 n+1\right) \left( \coth \left( \frac{1}{2} \sqrt{\iota _2} (x-c t)\right) +1\right) }{4 n^2 \left( 4 A_0 b_4 (n+1)+b_2 (2 n+1)\right) }+A_0\right) ^{\frac{1}{ n}}\nonumber \\ & \quad \times e^{i \left( -kx+\omega t +\chi W(t)-\chi ^{2}t\right) }. \end{aligned}$$
(34)

Case-(5.2): If \(\iota _0\) = \(\iota _1\) = \(\iota _6\) = 0,    then we have \(A_1=-\frac{\sqrt{\pi _0 \iota _4 (-(2 n+1))}}{2 \sqrt{b_4} n},\) \(D_1\) = \(B_1\) = 0, \(c_2=-\frac{A_0 (n-1) \left( 6 A_0 b_2 (2 n+1) n^2+16 A_0^2 b_4 (n+1) n^2+\pi _0 \iota _2 \left( 2 n^2+3 n+1\right) \right) }{2 n^2 \left( 2 n^2+3 n+1\right) },\) \(c_4=\frac{A_0^2 (2 n-1) \left( 4 A_0 b_2 (2 n+1) n^2+12 A_0^2 b_4 (n+1) n^2+\pi _0 \iota _2 \left( 2 n^2+3 n+1\right) \right) }{4 n^2 \left( 2 n^2+3 n+1\right) },\) \(\pi _1=-\frac{6 A_0^2 b_4}{2 n+1}-\frac{3 A_0 b_2}{n+1}-\frac{\pi _0 \iota _2}{4 n^2},\) \(\iota _3=\frac{2 \sqrt{\iota _4} n \sqrt{-(2 n+1)} \left( 4 A_0 b_4 (n+1)+b_2 (2 n+1)\right) }{\sqrt{\pi _0 b_4} \left( 2 n^2+3 n+1\right) }\).

According to the preceding set of solutions, we can discover explicit solutions to the suggested equation as follows:

(5.3) If \(n \sqrt{b_4} \ne 0\), \(\sqrt{b_4 \pi _0}\), \(\iota _4>0\) and \(\iota _2<0\), then we get a hyperbolic solution are formulated as:

$$\begin{aligned} & Q_{5.3.1}(x,t)=\left( A_0-\frac{\iota _2 \sqrt{\pi _0 \iota _4 (-(2 n+1))} \text {sech}^2\left( \frac{1}{2} \sqrt{\iota _2} (x-c t)\right) }{2 \sqrt{b_4} n \left( 2 \sqrt{\iota _2 \iota _4} \tanh \left( \frac{1}{2} \sqrt{\iota _2} (x-c t)\right) -\frac{2 \sqrt{\iota _4} \sqrt{-2 n-1} n \left( 4 A_0 b_4 (n+1)+b_2 (2 n+1)\right) }{\sqrt{\pi _0 b_4} \left( 2 n^2+3 n+1\right) }\right) }\right) ^{\frac{1}{n}} \nonumber \\ & \quad \times e^{i \left( -kx+\omega t +\chi W(t)-\chi ^{2}t\right) }. \end{aligned}$$
(35)

(5.4) If \(n \sqrt{b_4} \ne 0\), \(\sqrt{b_4 \pi _0}\), \(\iota _4>0\) and \(\iota _2<0\), then we get singular periodic solution are formulated as:

$$\begin{aligned} & Q_{5.4.1}(x,t)=\left( A_0-\frac{\iota _2 \sqrt{\pi _0 \iota _4 (-(2 n+1))} \text {sec}^2\left( \frac{1}{2} \sqrt{-\iota _2} (x-c t)\right) }{2 \sqrt{b_4} n \left( \frac{2 \sqrt{\iota _4} n \sqrt{-2 n-1} \left( 4 A_0 b_4 (n+1)+b_2 (2 n+1)\right) }{\sqrt{\pi _0 b_4} \left( 2 n^2+3 n+1\right) }+2 \sqrt{-\iota _2 \iota _4} \tan \left( \frac{1}{2} \sqrt{-\iota _2} (x-c t)\right) \right) }\right) ^{\frac{1}{ n}} \nonumber \\ & \quad \times e^{i \left( -kx+\omega t +\chi W(t)-\chi ^{2}t\right) }. \end{aligned}$$
(36)

Case-(6): If \(\iota _2=\iota _4=\iota _6=0\), then we have \(A_1=D_1=0,\) \(B_1=\frac{3 \pi _0 A_0 \iota _1-\sqrt{3} \sqrt{\pi _0 A_0^2 \left( 3 \pi _0 \iota _1^2-32 \pi _1 \iota _0 n^2\right) }}{8 \pi _1 n^2},\) \(b_4=\frac{(2 n+1) \left( A_0 \left( 16 \pi _1 \iota _0 n^2-3 \pi _0 \iota _1^2\right) -\sqrt{3} \iota _1 \sqrt{\pi _0 A_0^2 \left( 3 \pi _0 \iota _1^2-32 \pi _1 \iota _0 n^2\right) }\right) }{96 A_0^3 \iota _0 n^2},\) \(b_2=-\frac{(n+1) \left( A_0 \left( 32 \pi _1 \iota _0 n^2-3 \pi _0 \iota _1^2\right) -\sqrt{3} \iota _1 \sqrt{\pi _0 A_0^2 \left( 3 \pi _0 \iota _1^2-32 \pi _1 \iota _0 n^2\right) }\right) }{48 A_0^2 \iota _0 n^2},\) \(c_2=\frac{(n-1) \left( \sqrt{3} \left( 2 \pi _1 \iota _1 n^2-3 \pi _0 \iota _0 \iota _3\right) \sqrt{\pi _0 A_0^2 \left( 3 \pi _0 \iota _1^2-32 \pi _1 \iota _0 n^2\right) }+A_0 \left( \iota _0 \left( 9 \pi _0^2 \iota _1 \iota _3+64 \pi _1^2 n^4\right) +6 \pi _0 \pi _1 \iota _1^2 n^2\right) \right) }{96 \pi _1 \iota _0 n^4},\) \(c_4=-\frac{A_0 (2 n-1) \left( \sqrt{3} \left( \pi _1 \iota _1 n^2-3 \pi _0 \iota _0 \iota _3\right) \sqrt{\pi _0 A_0^2 \left( 3 \pi _0 \iota _1^2-32 \pi _1 \iota _0 n^2\right) }+A_0 \left( \iota _0 \left( 9 \pi _0^2 \iota _1 \iota _3+16 \pi _1^2 n^4\right) +3 \pi _0 \pi _1 \iota _1^2 n^2\right) \right) }{96 \pi _1 \iota _0 n^4}\).

According to the preceding set of solutions, we can discover explicit solutions to the suggested equation as follows:

(6.1) If \(\iota _3>0\) and \(n^{2}\pi _1\ne 0\) , then we get Weierstrass elliptic solutions are formulated as:

$$\begin{aligned} & Q_{6.1.1}(x,t)= \left( A_0 \left( \frac{\sqrt{3} \sqrt{\pi _0 \left( 3 \pi _0 \iota _1^2-32 \pi _1 \iota _0 n^2\right) }-3 \pi _0 \iota _1}{4 \pi _1 \sqrt{\iota _3} n^2 \wp (c t-x)}+1\right) \right) ^{\frac{1}{n}}e^{i \left( -kx+\omega t +\chi W(t)-\chi ^{2}t\right) }. \end{aligned}$$
(37)

Case-(7): If \(\iota _1=\iota _3=0\), then we have \(D_1=\iota _6=B_1=0,\) \(b_4=-\frac{\pi _0 \iota _4 (2 n+1)}{4 A_1^2 n^2},\) \(b_2=\frac{\pi _0 A_0 \iota _4 (n+1)}{A_1^2 n^2},\) \(c_2=\frac{2 A_0 (n-1) \left( \pi _0 A_0^2 \iota _4+\pi _1 A_1^2 n^2\right) }{A_1^2 n^2},\) \(c_4=-\frac{(2 n-1) \left( 5 \pi _0 A_0^4 \iota _4-\pi _0 A_1^4 \iota _0+4 \pi _1 A_1^2 A_0^2 n^2\right) }{4 A_1^2 n^2},\) \(\iota _2=-\frac{6 A_0^2 \iota _4}{A_1^2}-\frac{4 \pi _1 n^2}{\pi _0},\) \(\iota _2=-\frac{6 A_0^2 \iota _4}{A_1^2}-\frac{4 \pi _1 n^2}{\pi _0}\).

According to the preceding set of solutions, we can discover explicit solutions to the suggested equation as follows:

(7.1) If \(A_1\ne 0\) and \(\pi _0\ne 0\) , then we get singular soliton are formulated as:

$$\begin{aligned} & Q_{7.1.1}(x,t)= \left( A_1 \sqrt{-\frac{6 A_0^2}{A_1^2}-\frac{4 \pi _1 n^2}{\pi _0 \iota _4}} \text {csch}\left( \sqrt{-\frac{6 A_0^2 \iota _4}{A_1^2}-\frac{4 \pi _1 n^2}{\pi _0}} (x-c t)\right) +A_0\right) ^{\frac{1}{n}} \nonumber \\ & \quad \times e^{i \left( -kx+\omega t +\chi W(t)-\chi ^{2}t\right) }. \end{aligned}$$
(38)

(7.2) If \(A_1\ne 0\) and \(\pi _0\ne 0\) , then we get singular periodic solution are formulated as:

$$\begin{aligned} & Q_{7.2.1}(x,t)= \left( A_1 \sqrt{-\frac{6 A_0^2}{A_1^2}-\frac{4 \pi _1 n^2}{\pi _0 \iota _4}} \sec \left( \sqrt{\frac{6 A_0^2 \iota _4}{A_1^2}+\frac{4 \pi _1 n^2}{\pi _0}} (x-c t)\right) +A_0\right) ^{\frac{1}{ n}}\nonumber \\ & \quad \times e^{i \left( -kx+\omega t +\chi W(t)-\chi ^{2}t\right) }. \end{aligned}$$
(39)

Case-(8): If \(\iota _1=\iota _3=\iota _6=0\), then we have \(B_1=D_1=0,\) \(b_4=-\frac{\pi _0 \iota _4 (2 n+1)}{4 A_1^2 n^2},\) \(b_2=\frac{\pi _0 A_0 \iota _4 (n+1)}{A_1^2 n^2},\) \(c_2=\frac{2 A_0 (n-1) \left( \pi _0 A_0^2 \iota _4+\pi _1 A_1^2 n^2\right) }{A_1^2 n^2},\) \(c_4=-\frac{(2 n-1) \left( 5 \pi _0 A_0^4 \iota _4-\pi _0 A_1^4 \iota _0+4 \pi _1 A_1^2 A_0^2 n^2\right) }{4 A_1^2 n^2},\) \(\iota _2=-\frac{6 A_0^2 \iota _4}{A_1^2}-\frac{4 \pi _1 n^2}{\pi _0}\).

According to the preceding set of solutions, we can discover explicit solutions to the suggested equation as follows:

(8.1) Uncovers different solution types for the given system. Jacobi’s elliptic function solutions are observed when \(m=0\).

$$\begin{aligned} & Q_{8.1.1}(x,t)= \left( A_1 \cos (x-c t)+A_0\right) ^{\frac{1}{ n}} e^{i \left( -kx+\omega t +\chi W(t)-\chi ^{2}t\right) }, \end{aligned}$$
(40)
$$\begin{aligned} & Q_{8.1.2}(x,t)= \left( A_0-A_1 \sin (x-c t)\right) ^{\frac{1}{ n}} e^{i \left( -kx+\omega t +\chi W(t)-\chi ^{2}t\right) }, \end{aligned}$$
(41)
$$\begin{aligned} & Q_{8.1.3}(x,t)= \left( A_0-A_1 \csc (x-c t)\right) ^{\frac{1}{ n}} e^{i \left( -kx+\omega t +\chi W(t)-\chi ^{2}t\right) }, \end{aligned}$$
(42)
$$\begin{aligned} & Q_{8.1.4}(x,t)= \left( A_1 \sec (x-c t)+A_0\right) ^{\frac{1}{ n}} e^{i \left( -kx+\omega t +\chi W(t)-\chi ^{2}t\right) }, \end{aligned}$$
(43)
$$\begin{aligned} & Q_{8.1.5}(x,t)= \left( A_0-A_1 \tan (x-c t)\right) ^{\frac{1}{ n}} e^{i \left( -kx+\omega t +\chi W(t)-\chi ^{2}t\right) }. \end{aligned}$$
(44)

(8.1) Uncovers different solution types for the given system. Jacobi’s elliptic function solutions are observed when \(m=1\).

$$\begin{aligned} & Q_{8.2.1}(x,t)= \left( A_0-A_1 \tanh (c t-x)\right) ^{\frac{1}{ n}} e^{i \left( -kx+\omega t +\chi W(t)-\chi ^{2}t\right) }, \end{aligned}$$
(45)
$$\begin{aligned} & Q_{8.2.2}(x,t)= \left( A_0-A_1 \coth (c t-x)\right) ^{\frac{1}{ n}} e^{i \left( -kx+\omega t +\chi W(t)-\chi ^{2}t\right) }, \end{aligned}$$
(46)
$$\begin{aligned} & Q_{8.2.3}(x,t)= \left( A_1 \text {sech}(c t-x)+A_0\right) ^{\frac{1}{ n}} e^{i \left( -kx+\omega t +\chi W(t)-\chi ^{2}t\right) }, \end{aligned}$$
(47)
$$\begin{aligned} & Q_{8.2.4}(x,t)= \left( A_1 \cosh (c t-x)+A_0\right) ^{\frac{1}{ n}} e^{i \left( -kx+\omega t +\chi W(t)-\chi ^{2}t\right) }. \end{aligned}$$
(48)

Visual graphs for some solutions

This section examines the influence of noise on the recovered solutions by presenting several stochastic solutions in two and three dimensions, as depicted in Figs. 2, 3, 4, 5, 6, 7, 8 and 9. This is achieved by assigning appropriate values to the relevant parameters.

Fig. 2
figure 2

2D and 3D Modulus of bright soliton for Eq. (21) when \(A_0=1;c=1;\) \(n=2;c_2=0.2;c_4=0.6;\)\(\pi _0=3;k=2;\)\(\omega =1.4;\iota _4=-0.6\).

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Fig. 3
figure 3

2D bright soliton for Eq. (21) when \(\chi =0,0.2,0.6,1\) and (\(A_0=1;c=1;n=2;c_2=0.2;\) \(c_4=0.6;\pi _0=3;\) \(k=2;\omega =1.4;\iota _4=-0.6\)).

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Fig. 4
figure 4

3D Re-component of bright soliton for Eq. (21) concerning to x-axis when \(\chi =0,0.2,0.6,1\) and (\(A_0=1;c=1;n=2;\) \(c_2=0.2;c_4=0.6;\pi _0=3;\) \(k=2;\omega =1.4;\iota _4=-0.6\)).

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Fig. 5
figure 5

3D Im-component of bright soliton for Eq. (21) concerning to x-axis when \(\chi =0,0.2,0.6,1\) and (\(A_0=1;c=1;n=2;\) \(c_2=0.2;c_4=0.6;\pi _0=3;\) \(k=2;\omega =1.4;\iota _4=-0.6\)).

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Fig. 6
figure 6

2D and 3D Modulus of dark soliton for Eq. (33) when \(A_0=1;c=1;n=2;\) \(c_2=0.2;c_4=0.6;\) \(b_2=0.8;b_4=-0.2;\pi _0=3;k=2;\) \(\omega =2;\iota _2=0.2;\iota _4=0.2\).

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Fig. 7
figure 7

2D dark soliton for Eq. (33) when \(\chi =0,0.2,0.6,1\) and (\(A_0=1;c=1;n=2;\) \(c_2=0.2;c_4=0.6;\) \(b_2=0.8;b_4=-0.2;\pi _0=3;k=2;\) \(\omega =2;\iota _2=0.2;\iota _4=0.2\)).

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Fig. 8
figure 8

3D Re-component of dark soliton for Eq. (33) concerning to x-axis when \(\chi =0,0.2,0.6,1\) and (\(A_0=1;c=1;n=2;\) \(c_2=0.2;c_4=0.6;\) \(b_2=0.8;b_4=-0.2;\pi _0=3;\) \(k=2;\omega =2;\iota _2=0.2;\iota _4=0.2\)).

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Fig. 9
figure 9

3D Im-component of dark soliton for Eq. (33) concerning to x-axis when \(\chi =0,0.2,0.6,1\) and (\(A_0=1;c=1;n=2;\) \(c_2=0.2;c_4=0.6;\) \(b_2=0.8;b_4=-0.2;\pi _0=3;k=2;\) \(\omega =2;\iota _2=0.2;\iota _4=0.2\)).

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Conclusions

In this study, we investigated optical soliton solutions of the stochastic NLSE, which incorporates spatiotemporal dispersion, intermodal dispersion, nonlinearity, and multiplicative white noise, analyzed under the framework of the generalized Kudryashov law using the MEMT. Our analysis provided a comprehensive framework for understanding how higher-order nonlinearities (cubic, quintic, septic) and stochastic perturbations influence soliton propagation. We derived bright, dark, and singular soliton solutions, as well as periodic, singular periodic, hyperbolic solution, rational solution, Jacobi’s elliptic function solutions, and Weierstrass elliptic wave solutions under stochastic noise, demonstrating how septic nonlinearity modifies soliton profiles compared to cubic-quintic cases. This technique proved effective in handling the complex interplay of nonlinearities and stochasticity, offering diverse exact solutions compared to other methods.

Despite these contributions, the study has limitations. It is restricted to one-dimensional propagation, assumes multiplicative white noise only, and does not include numerical stability analysis. Future research should therefore address higher-dimensional extensions, explore colored or non-Gaussian noise models, and validate analytical results with detailed simulations. Extending the framework to coupled or multi-wave systems may also uncover new interaction dynamics relevant to practical communication and photonic technologies. Future research may extend this framework by considering stochastic fractional differential equations, which provide a richer mathematical structure for modeling memory effects and anomalous diffusion under random perturbations.