Analytical Solutions for Generalized Stochastic HSC-KdV Equations with Variable Coefficients Using Hermite Transform and F-Expansion Method

Abstract

This study focuses on analyzing the generalized HSC-KdV equations characterized by variable coefficients and Wick-type stochastic (Wt.S) elements. To derive white noise functional (WNF) solutions, we employ the Hermite transform, the homogeneous balance principle, and the $F_e$ (F-expansion) technique. Leveraging the inherent connection between hypercomplex system (HCS) theory and white noise (WN) analysis, we establish a comprehensive framework for exploring stochastic partial differential equations (PDEs) involving non-Gaussian parameters (N-GP). As a result, exact solutions expressed through Jacobi elliptic functions (JEFs) and trigonometric and hyperbolic forms are obtained for both the variable coefficients and stochastic forms of the generalized HSC-KdV equations. An illustrative example is included to validate the theoretical findings.

1. Introduction

Let Q be a locally compact space. Consider the quasi-nuclear rigging framework as described in [1]:

$$H_{-q}^{\aleph }\supseteq L_2(Q,d\rho \left(x\right))\supseteq H_q^{\aleph },$$

where the central space $L_2(Q,d\rho \left(x\right))$ is defined over a commutative normal hypercomplex system (HCS) $L_1(Q,dm\left(x\right))$, which is based on a measurable structure with support in Q and a multiplicative measure m. Spaces $H_q^{\aleph }$ and $H_{-q}^{\aleph }$ are formulated using the Delsarte characters ${\aleph }_n\in C\left(Q\right)$, serving as a basis for the construction.

The major subject of this paper is the Wt.S generalized HSC-KdV-E with N-GP:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{U}_t+{{\rm Y}}_1\left(t\right){\diamond }_{\aleph }{U}_{xxx}+{{\rm Y}}_2\left(t\right){\diamond }_{\aleph }U{\diamond }_{\aleph }{U}_x+{{\rm Y}}_3\left(t\right){\diamond }_{\aleph }{\left(V{\diamond }_{\aleph }W\right)}_x=0,\hfill \\ V_t+{{\rm Y}}_4\left(t\right){\diamond }_{\aleph }{V}_{xxx}-{{\rm Y}}_2\left(t\right){\diamond }_{\aleph }U{\diamond }_{\aleph }{V}_x=0,\hfill \\ W_t+{{\rm Y}}_4\left(t\right){\diamond }_{\aleph }{W}_{xxx}-{{\rm Y}}_2\left(t\right){\diamond }_{\aleph }U{\diamond }_{\aleph }{W}_x=0.\hfill \end{array}\right.\end{array}$$

(1)

where “${\diamond }_{\aleph }$” is the ℵ-WP (Wick product) on $H_{-q}^{\aleph }$, and ${{\rm Y}}_{1,2,3,4}\in H_{-q}^{\aleph }$ are non-Gaussian 1-valued functions [2,3,4]. Additionally, ℵ-WP “${\diamond }_{\aleph }$” is used to replace the ordinary product in Equation (1), considering bounded measurable or integrable functions ${{\rm Y}}_1$, ${{\rm Y}}_2$, ${{\rm Y}}_3$, and ${{\rm Y}}_4$. The variable coefficients of the generalized HSC-KdV-E can be obtained as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{u}_t+{{\rm Y}}_1\left(t\right)u_{xxx}+{{\rm Y}}_2\left(t\right)u{u}_x+{{\rm Y}}_3\left(t\right){\left(vw\right)}_x=0,\hfill \\ v_t+{{\rm Y}}_4\left(t\right)v_{xxx}-{{\rm Y}}_2\left(t\right)u{v}_x=0,\hfill \\ w_t+{{\rm Y}}_4\left(t\right)w_{xxx}-{{\rm Y}}_2\left(t\right)u{w}_x=0.\hfill \end{array}\right.\end{array}$$

(2)

on ${\mathbb{R}}_{+}$ [2,3,4].

Because of the fact that the generalized nonlinear equations of the HSC-KdV system have great importance in mathematics and physics, it is interesting to study and analyze its solutions [5,6].

In order to shed light on some scientific applications, we provide a few illustrative examples. The generalized HSC-KdV system has been applied in describing the global properties of mathematical structures [5,7,8], in the thermodynamics field [3,9,10], and in soliton applications [11,12,13]. The study of nonlinear equations of evolution representing different physical models has played an important role in a variety of scientific and physics fields [14,15,16].

In polarity symmetric systems, the propagation of nonlinear waves is expressed through the generalized HSC-KdV-E (Hirota–Satsuma coupled KdV equations) (2). Moreover, when the problem is analyzed within a non-Gaussian stochastic (N-GS) framework, it leads to the formulation of a generalized HSC-KdV equation adapted to the N-GS setting to produce the exact stochastic solutions of the generalized HSC-KdV-E, which has been sought for an N-GWN problem. Accordingly, we proceed to examine the stochastic generalized HSC-KdV equation with variable coefficients, as given in Equation (1).

As is well-known, solitons exhibit particle-like behavior and tend to be stable in the presence of alternate collisions. Nonlinear equations with constant and variable coefficients cannot fully describe realistic physical phenomena. Therefore, studying stochastic nonlinear equations is crucial. In [17], the effect of external noise on the movement of the soliton was discussed. The Cauchy problems related to stochastic PDEs have attracted the attention of many researchers [18,19,20,21,22,23]. For the use of WN functional analysis, see [2,3,24,25,26,27,28,29,30]. For N-GWN analysis, see [4,31].

The main aim of the present paper is to reveal the exact STWSs for the generalized HSC-KdV-E (1) and (2), respectively.

The remainder of the present paper is structured as follows: In Section 2, we introduce the framework of non-Gaussian stochastic partial differential equations (SPDEs) and present the construction of the ℵ-Wick product and its algebraic properties in the rigged function space $H_{-q}^{\aleph }$. In Section 3, we apply the Hermite transform to convert the stochastic generalized HSC-KdV equations into deterministic ones, and then use the $F_e$-expansion method to derive exact stochastic traveling wave solutions (STWSs) of the deterministic system. In Section 4, we utilize the inverse Hermite transform to reconstruct exact non-Gaussian white noise (N-GWN) functional solutions of the original Wick-type stochastic system. Finally, Section 5 presents a summary of the main findings and discusses potential applications and future research directions.

2. SPDEs with N-GP

In this part, we present a new WP on the space $H_{-q}^{\aleph }$ with respect to $\rho$, called the ℵ-WP (ℵ-WP). For the Gaussian case, a more realistic and descriptive model can be obtained by considering the objects of a DF to be ${\left(\mathcal{S}\right)}_{-1}$-valued; this is called the Wt.S differential equation [30]. In general, a non-Gaussian Wt.S model can be obtained through the substitution of ${\left(\mathcal{S}\right)}_{-1}$ in $H_{-}{q}^{\aleph }$ and the WP corresponding to the Gaussian measure on the ℵ-WP.

The idea of WP was introduced by Wick in 1950 [32]. Since then, it has served as a technique for restructuring specific divergent quantities in quantum field theory. Lastly, in [33], the WP was considered a stochastic formula. Presently, the WP benefits several applications, for instance, stochastic ODEs and PDEs [30]. In [34], Dobroshin and Minlos used the WP to comprehensively treat some problems in both mathematical physics and probability theory.

Next, we provide some useful concepts for setting a characterization theorem of the space $H_{-q}^{\aleph }$ [4,31,35,36,37].

Definition 1 

([4]). Let $\varpi ={\sum }_{m=0}^{\infty }{\varpi }_m{q}_m^{\aleph }$, $\mathcal{l}={\sum }_{n=0}^{\infty }{\mathcal{l}}_n{q}_n^{\aleph }\in H_{-q}^{\aleph }$ with ${\varpi }_m,{\mathcal{l}}_n$$\in \mathbb{C}$. The ℵ-WP of ϖ and ℓ, denoted by $\varpi {\diamond }_{\aleph }\mathcal{l}$, is formulated as follows:

$$\varpi {\diamond }_{\aleph }\mathcal{l}=\sum _{m,n=0}^{\infty }{\varpi }_m{\mathcal{l}}_n{q}_{m+n}^{\aleph },(when convergent).$$

(3)

To verify that spaces $H_{-q}^{\aleph }$ and $H_q^{\aleph }$ are closed under ℵ-WP, we give the following lemma.

Lemma 1 

([4]). If $\varpi ,\mathcal{l}\in H_{-q}^{\aleph }$ and ${\rm Y},\psi \in H_q^{\aleph }$, we have

(a) 

$\varpi {\diamond }_{\aleph }\mathcal{l}\in H_{-q}^{\aleph }$;

(b) 

${\rm Y}{\diamond }_{\aleph }\psi \in H_q^{\aleph }$.

Here is a stability analysis framework formulated in a negative-order Sobolev space $H_{-q}^{\aleph }$ (or a generalization $H_{-q}^{\aleph }$—interpreted as a Hilbert space of order $-q$ with the usual dual norm). This means that the solutions that start small in $H_{-q}^{\aleph }$ stay small for all time (local- or global-in-time stability) via standard Banach fixed points or perturbation arguments. Definition 1 provides us the following significant algebraic properties of the ℵ-WP.

Lemma 2 

([4]). For each $\varpi ,\mathcal{l},\zeta \in H_{-q}^{\aleph }$, we get

(a) 

$\varpi {\diamond }_{\aleph }\mathcal{l}=\mathcal{l}{\diamond }_{\aleph }\varpi$ (Commutative law);

(b) 

$\varpi {\diamond }_{\aleph }\left(\mathcal{l}{\diamond }_{\aleph }\zeta \right)=\left(\varpi {\diamond }_{\aleph }\mathcal{l}\right){\diamond }_{\aleph }\zeta$ (Associative law);

(c) 

$\varpi {\diamond }_{\aleph }(\mathcal{l}+\zeta )=\varpi {\diamond }_{\aleph }\mathcal{l}+\varpi {\diamond }_{\aleph }\zeta$ (Distributive law).

Accordingly, we conclude that both spaces $H_{-q}^{\aleph }$ and $H_q^{\aleph }$ possess the structure of topological algebras with respect to the ℵ-Wick product (ℵ-WP).

Considering what was stated above, the ℵ-WP meets all the ordinary algebraic rules for multiplication. As a result, computations may be carried out in the same manner as when using standard products.

3. Exact Traveling Wave Solutions of Equation (1)

Since there are some difficulties when including limit operations, to address such cases, we utilize the ℵ-Hermite transform (ℵ-HT) to convert ℵ-Wick products into standard complex products. This transformation also maps convergence in $H_{-q}^{\aleph }$ to boundedness and translates pointwise convergence into a neighborhood around zero in $\mathbb{C}$. Regarding the original HT, the authors have used it in several contexts, and it was initially presented in [23]. We now define the $aleph$-HT and go through its fundamental characteristics.

Definition 2 

([4]). Let $\varpi ={\sum }_{𝚥=0}^{\infty }{\varpi }_{𝚥}{q}_{𝚥}^{\aleph }\in H_{-q}^{\aleph }$ with ${\varpi }_n\in \mathbb{C}$. Then, the ℵ-HT of ϖ, denoted by ${\mathcal{H}}_{\aleph }\varpi$ or $\widehat{\varpi }$, is presented as

$${\mathcal{H}}_{\aleph }\varpi \left(\varsigma \right)=\widehat{\varpi }\left(\varsigma \right)=\sum _{𝚥=0}^{\infty }{\varpi }_{𝚥}{z}^{𝚥}\in \mathbb{C}(whenconvergent).$$

(4)

Next, for $0<\mathcal{M},q<\infty$, we introduce the neighborhoods ${\mathbb{O}}_q\left(\mathcal{M}\right)$ of the origin in $\mathbb{C}$ as follows:

$${\mathbb{O}}_q\left(\mathcal{M}\right)=\left\{z\in \mathbb{C}:\sum _{𝚥=0}^{\infty }{\left|z^{𝚥}\right|}^2{K}^{q𝚥}<{\mathcal{M}}^2\right\}.$$

(5)

where the constant K (specifically $K=K_q$) is associated with the weighted norm in the neighborhood ${\mathbb{O}}_q\left(\mathcal{M}\right)$. This constant ensures the convergence of the Hermite transform expansion in the functional space $H_{-q}^{\aleph }$.

It is then clear to observe that

$$q\le p,\mathcal{N}\le \mathcal{M}\Rightarrow {\mathbb{O}}_p\left(\mathcal{N}\right)\subseteq {\mathbb{O}}_q\left(\mathcal{M}\right).$$

The following is a summary of the discussion above.

Proposition 1 

([4]). If $\varpi \in H_{-q}^{\aleph }$, then ${\mathcal{H}}_{\aleph }\varpi$ converges for all $z\in {\mathbb{O}}_q\left(\mathcal{M}\right)$ for all $q,\mathcal{M}<\infty$.

The ability of the ℵ-HT to transform the ℵ-WP into an ordinary (complex) product is a powerful feature.

Proposition 2 

([4]). If $\varpi ,\mathcal{l}\in H_{-q}^{\aleph },$ we get

$${\mathcal{H}}_{\aleph }\left(\varpi {\diamond }_{\aleph }\mathcal{l}\right)\left(\varsigma \right)={\mathcal{H}}_{\aleph }\varpi \left(\varsigma \right).{\mathcal{H}}_{\aleph }\mathcal{l}\left(\varsigma \right),$$

(6)

for all z such that ${\mathcal{H}}_{\aleph }\varpi$ and ${\mathcal{H}}_{\aleph }\mathcal{l}$ exist.

In the context of ℵ-Brownian motion, it can be observed that

$$W_{\aleph }\left(t\right)={\displaystyle \frac{d}{dt}}{B}_{\aleph }\left(t\right)\mathrm{in}{H}_{-q}^{\aleph }.$$

(7)

Consequently, operating within the general space $H_{-q}^{\aleph }$ of stochastic distributions ensures the existence of solutions for a wide class of non-Gaussian stochastic (N-GS) differential equations, encompassing ordinary, partial, and multidimensional forms. When elements take values in $H_{-q}^{\aleph }$, differentiation may be understood in the conventional strong sense.

Theorem 1 

([4,30]). Suppose $u(\xi ,t,\varsigma )$ is a solution (in the usual strong, pointwise sense) of the equation

$$\widehat{A}(\xi ,t,{\partial }_t,{\nabla }_{\xi },u,\varsigma )=0,$$

(8)

for $(\xi ,t)$ in some bounded open set $D\subset {\mathbb{R}}^n\times {\mathbb{R}}_{+}$, and for all $z\in {\mathbb{O}}_q\left(\mathcal{M}\right)$, for some $q,\mathcal{M}$.

Furthermore, assume that the function $u(\xi ,t,\varsigma )$, along with all its partial derivatives appearing in Equation (8), are uniformly bounded over the domain $(\xi ,t,\varsigma )\in D\times {\mathbb{O}}_q\left(\mathcal{M}\right)$. Suppose also that u is continuous with respect to $(\xi ,t)\in D$ for each fixed $\varsigma \in {\mathbb{O}}_q\left(\mathcal{M}\right)$, and analytic in $\varsigma$ throughout ${\mathbb{O}}_q\left(\mathcal{M}\right)$ for every $(\xi ,t)\in D$. Then, there exists a function $u(\xi ,t)\in H_{-q}^{\aleph }$ such that

$$u(\xi ,t,\varsigma )={\mathcal{H}}_{\aleph }u(\xi ,t,\varsigma )\mathrm{for}\mathrm{all}(\xi ,t,\varsigma )\in D\times {\mathbb{O}}_q\left(\mathcal{M}\right),$$

and the function $u(\xi ,t)$ satisfies the equation in the strong sense in $H_{-q}^{\aleph }$:

$$A^{{\diamond }_{\aleph }}(\xi ,t,{\partial }_t,{\nabla }_{\xi },U,s)=0\mathrm{in}{H}_{-q}^{\aleph }.$$

(9)

4. Exact STWSs of HSC-KdV

In this section, we employ the $F_e$ technique to derive exact solutions and stochastic traveling wave solutions (STWSs) for the generalized HSC-KdV equation presented in Equation (2). Firstly, we use $chi$-HT to convert Equation (1) into deterministic PDEs. The resultant PDEs can be transformed into nonlinear ODEs by creating an appropriate transformation. Therefore, a family of exact solutions for Equation (2) can be obtained using the suggested $F_e$ technique.

With the ℵ-HT of Equation (1), we obtain the deterministic equations as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\widehat{U}}_t(\xi ,t,\varsigma )+{\widehat{{\rm Y}}}_1(t,\varsigma ){\widehat{U}}_{xxx}(\xi ,t,\varsigma )+{\widehat{{\rm Y}}}_2(t,\varsigma )\widehat{U}(\xi ,t,\varsigma ){\widehat{U}}_x(\xi ,t,\varsigma )+{\widehat{{\rm Y}}}_3(t,\varsigma ){\left[\widehat{V}\widehat{W}\right]}_x(\xi ,t,\varsigma )=0,\hfill \\ {\widehat{V}}_t(\xi ,t,\varsigma )+{\widehat{{\rm Y}}}_4(t,\varsigma ){\widehat{V}}_{xxx}(\xi ,t,\varsigma )-{\widehat{{\rm Y}}}_2(t,\varsigma )u(\xi ,t,\varsigma ){\widehat{V}}_x(\xi ,t,\varsigma )=0,\hfill \\ {\widehat{W}}_t(\xi ,t,\varsigma )+{\widehat{{\rm Y}}}_4(t,\varsigma ){\widehat{W}}_{xxx}(\xi ,t,\varsigma )-{\widehat{{\rm Y}}}_2(t,\varsigma )\widehat{U}(\xi ,t,\varsigma ){\widehat{W}}_x(\xi ,t,\varsigma )=0,\hfill \end{array}\right.\end{array}$$

(10)

where $z\in \mathbb{C}$ is a parameter.

In this part, we apply the $F_e$ method to address Equation (10). To construct solitary stochastic traveling wave solutions (STWSs) of Equation (10), we introduce the following variable transformations:

$$\widehat{U}(\xi ,t,\varsigma ):=u(\xi ,t,\varsigma )=u\left(\zeta (\xi ,t,\varsigma )\right),\widehat{V}(\xi ,t,\varsigma ):=v(\xi ,t,\varsigma )=v\left(\zeta (\xi ,t,\varsigma )\right),$$

$$\widehat{W}(\xi ,t,\varsigma ):=w(\xi ,t,\varsigma )=w\left(\zeta (\xi ,t,\varsigma )\right),$$

$${\widehat{{\rm Y}}}_j(t,\varsigma ):={{\rm Y}}_j(t,\varsigma ),\mathrm{for}j=1,2,3,4,$$

where the traveling wave variable is defined by

$$\zeta (\xi ,t,\varsigma ):=kx+\lambda {\int }_0^t\delta (\tau ,\varsigma )d\tau +c,$$

(11)

with constants $k,\lambda ,c\in \mathbb{R}$ satisfying $k\lambda \ne 0$, and $\delta (t,\varsigma )$ being a non-vanishing function to be specified. With these transformations, Equation (10) is reduced to a corresponding system of ordinary differential equations (ODEs).

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\lambda \delta (t,\varsigma )u^{\prime }+k^3{{\rm Y}}_1(t,\varsigma )u^{‴}+k{{\rm Y}}_2(t,\varsigma )u{u}^{\prime }+k{{\rm Y}}_3(t,\varsigma )[v^{\prime }w+v{w}^{\prime }]=0,\hfill \\ \lambda \delta (t,\varsigma )v^{\prime }+k^3{{\rm Y}}_4(t,\varsigma )v^{‴}-k{{\rm Y}}_2(t,\varsigma )u{v}^{\prime }=0,\hfill \\ \lambda \delta (t,\varsigma )w^{\prime }+k^3{{\rm Y}}_4(t,\varsigma )w^{‴}-k{{\rm Y}}_2(t,\varsigma )u{w}^{\prime }=0,\hfill \end{array}\right.\end{array}$$

(12)

where ′ $={\displaystyle \frac{d}{d\varpi }}$.

Because of the $F_e$ method, the solutions of Equation (12), can be presented in the following forms:

$$u\left(\varpi \right)=\sum _{i=0}^N{a}_i{F}^i\left(\varpi \right),v\left(\varpi \right)=\sum _{i=0}^N{b}_i{F}^i\left(\varpi \right),w\left(\varpi \right)=\sum _{i=0}^N{c}_i{F}^i\left(\varpi \right),$$

(13)

Here, constants $a_i$, $b_i$, and $c_i$ are to be determined at a later stage. By applying the homogeneous balance principle between the highest-order nonlinear terms and the leading partial derivatives—namely $u^{‴}$ and $u{u}^{\prime }$; $v^{‴}$, $u{v}^{\prime }$; $v^{\prime }w+v{w}^{\prime }$; and $w^{‴}$ and $u{w}^{\prime }$—in Equation (12), we deduce that $N=2$. Accordingly, Equation (13) can be reformulated as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}u(\xi ,t,\varsigma )=a_0+a_{1}F\left(\varpi \right)+a_2{F}^2\left(\varpi \right),\hfill \\ v(\xi ,t,\varsigma )=b_0+b_{1}F\left(\varpi \right)+b_2{F}^2\left(\varpi \right),\hfill \\ w(\xi ,t,\varsigma )=c_0+c_{1}F\left(\varpi \right)+c_2{F}^2\left(\varpi \right),\hfill \end{array}\right.\end{array}$$

(14)

Here, $a_0,a_1,a_2,b_0,b_1,b_2,c_0,c_1$, and $c_2$ represent constants that will be specified later. By substituting Equation (14), along with the conditions imposed by the $F_e$ method, into Equation (12) and subsequently gathering all terms corresponding to the same powers of $F^i\left(\varpi \right){\left[F^{\prime }\left(\varpi \right)\right]}^j$, where $i=0,\pm 1,\pm 2,\dots$ and $j=0,1$, we obtain the following:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\left[a_1\lambda \delta (t,\varsigma )+k^{3}Q{a}_1{{\rm Y}}_1(t,\varsigma )+k{a}_0{a}_1{{\rm Y}}_2(t,\varsigma )+k{{\rm Y}}_3(t,\varsigma )(c_0{b}_1+b_0{c}_1)\right]F^{\prime }+[2a_2\lambda \delta (t,\varsigma )\hfill \\ +8Q{k}^3{a}_2{{\rm Y}}_1(t,\varsigma )+2k{a}_0{a}_2{{\rm Y}}_2(t,\varsigma )+k{a}_1^2{{\rm Y}}_2(t,\varsigma )+2k{{\rm Y}}_3(t,\varsigma )(c_1{b}_1+c_0{b}_2+c_2{b}_0)]F{F}^{\prime }\hfill \\ +\left[6P{k}^3{a}_1{{\rm Y}}_1(t,\varsigma )+3k{a}_1{a}_2{{\rm Y}}_2(t,\varsigma )+3k{{\rm Y}}_3(t,\varsigma )(b_1{c}_2+b_2{c}_1)\right]F^2{F}^{\prime }\hfill \\ +\left[24P{k}^3{a}_2{{\rm Y}}_1(t,\varsigma )+2k{a}_2^2{{\rm Y}}_2(t,\varsigma )+4k{b}_2{c}_2{{\rm Y}}_3(t,\varsigma )\right]F^3{F}^{\prime }=0,\hfill \\ \\ \left[b_1\lambda \delta (t,\varsigma )+k^{3}Q{b}_1{{\rm Y}}_4(t,\varsigma )-k{a}_0{b}_1{{\rm Y}}_2(t,\varsigma )\right]F^{\prime }+[2b_2\lambda \delta (t,\varsigma )+8Q{k}^3{b}_2{{\rm Y}}_4(t,\varsigma )\hfill \\ -2k{a}_0{b}_2{{\rm Y}}_2(t,\varsigma )-k{a}_1{b}_1{{\rm Y}}_2(t,\varsigma )]F{F}^{\prime }+[6P{k}^3{b}_1{{\rm Y}}_4(t,\varsigma )-2k{a}_1{b}_2{{\rm Y}}_2(t,\varsigma )\hfill \\ -k{a}_2{b}_1{{\rm Y}}_2(t,\varsigma )]F^2{F}^{\prime }+\left[24P{k}^3{b}_2{{\rm Y}}_4(t,\varsigma )-2k{a}_2{b}_2{{\rm Y}}_2(t,\varsigma )\right]F^3{F}^{\prime }=0,\hfill \\ \\ \left[c_1\lambda \delta (t,\varsigma )+k^{3}Q{c}_1{{\rm Y}}_4(t,\varsigma )-k{a}_0{c}_1{{\rm Y}}_2(t,\varsigma )\right]F^{\prime }+[2c_2\lambda \delta (t,\varsigma )+8Q{k}^3{c}_2{{\rm Y}}_4(t,\varsigma )\hfill \\ -2k{a}_0{c}_2{{\rm Y}}_2(t,\varsigma )-k{a}_1{c}_1{{\rm Y}}_2(t,\varsigma )]F{F}^{\prime }+[6P{k}^3{c}_1{{\rm Y}}_4(t,\varsigma )-2k{a}_1{c}_2{{\rm Y}}_2(t,\varsigma )\hfill \\ -c_{1}k{a}_2{{\rm Y}}_2(t,\varsigma )]F^2{F}^{\prime }+\left[24P{k}^3{c}_2{{\rm Y}}_4(t,\varsigma )-2k{a}_2{c}_2{{\rm Y}}_2(t,\varsigma )\right]F^3{F}^{\prime }=0.\hfill \end{array}\right.\end{array}$$

(15)

When the coefficient of $F^i\left(\varpi \right){\left[F^{\prime }\left(\varpi \right)\right]}^j$ vanishes, we obtain an algebraic system of equations that may be represented by

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{a}_1\lambda \delta (t,\varsigma )+k^{3}Q{a}_1{{\rm Y}}_1(t,\varsigma )+k{a}_0{a}_1{{\rm Y}}_2(t,\varsigma )++k{{\rm Y}}_3(t,\varsigma )(c_0{b}_1+b_0{c}_1)=0,\hfill \\ 2a_2\lambda \delta (t,\varsigma )+8Q{k}^3{a}_2{{\rm Y}}_1(t,\varsigma )+2k{a}_0{a}_2{{\rm Y}}_2(t,\varsigma )+k{a}_1^2{{\rm Y}}_2(t,\varsigma )+2k{{\rm Y}}_3(t,\varsigma )\hfill \\ (c_1{b}_1+c_0{b}_2+c_2{b}_0)=0,\hfill \\ 6P{k}^3{a}_1{{\rm Y}}_1(t,\varsigma )+3k{a}_1{a}_2{{\rm Y}}_2(t,\varsigma )+3k{{\rm Y}}_3(t,\varsigma )(b_1{c}_2+b_2{c}_1)=0,\hfill \\ 24P{k}^3{a}_2{{\rm Y}}_1(t,\varsigma )+2k{a}_2^2{{\rm Y}}_2(t,\varsigma )+4k{b}_2{c}_2{{\rm Y}}_3(t,\varsigma )=0,\hfill \\ \\ b_1\lambda \delta (t,\varsigma )+k^{3}Q{b}_1{{\rm Y}}_4(t,\varsigma )-k{a}_0{b}_1{{\rm Y}}_2(t,\varsigma )=0,\hfill \\ 2b_2\lambda \delta (t,\varsigma )+8Q{k}^3{b}_2{{\rm Y}}_4(t,\varsigma )-2k{a}_0{b}_2{{\rm Y}}_2(t,\varsigma )-k{a}_1{b}_1{{\rm Y}}_2(t,\varsigma )=0,\hfill \\ 6P{k}^3{b}_1{{\rm Y}}_4(t,\varsigma )-2k{a}_1{b}_2{{\rm Y}}_2(t,\varsigma )-k{a}_2{b}_1{{\rm Y}}_2(t,\varsigma )=0,\hfill \\ 24P{k}^3{b}_2{{\rm Y}}_4(t,\varsigma )-2k{a}_2{b}_2{{\rm Y}}_2(t,\varsigma )=0,\hfill \\ \\ c_1\lambda \delta (t,\varsigma )+k^{3}Q{c}_1{{\rm Y}}_4(t,\varsigma )-k{a}_0{c}_1{{\rm Y}}_2(t,\varsigma )=0\hfill \\ 2c_2\lambda \delta (t,\varsigma )+8Q{k}^3{c}_2{{\rm Y}}_4(t,\varsigma )-2k{a}_0{c}_2{{\rm Y}}_2(t,\varsigma )-k{a}_1{c}_1{{\rm Y}}_2(t,\varsigma )=0,\hfill \\ 6P{k}^3{c}_1{{\rm Y}}_4(t,\varsigma )-2k{a}_1{c}_2{{\rm Y}}_2(t,\varsigma )-c_{1}k{a}_2{{\rm Y}}_2(t,\varsigma )=0,\hfill \\ 24P{k}^3{c}_2{{\rm Y}}_4(t,\varsigma )-2k{a}_2{c}_2{{\rm Y}}_2(t,\varsigma )=0.\hfill \end{array}\right.\end{array}$$

(16)

By solving the system in (16), we get

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{a}_0,b_0,\mathrm{and}{c}_0\mathrm{are}\mathrm{arbitrary}\mathrm{constants},\hfill \\ a_1,b_1,\mathrm{and}{c}_1\mathrm{are}\mathrm{free}\mathrm{parameters},\hfill \\ a_2={\displaystyle \frac{12P{k}^2{{\rm Y}}_4(t,\varsigma )}{{{\rm Y}}_2(t,\varsigma )}},\hfill \\ b_2={\displaystyle \frac{3k^{2}Q{{\rm Y}}_4(t,\varsigma )}{2a_1{b}_1}},\hfill \\ c_2=-{\displaystyle \frac{3c_1{k}^{2}P{{\rm Y}}_4(t,\varsigma )}{a_1}},\hfill \\ \delta ={\displaystyle \frac{k(a_0{{\rm Y}}_2(t,\varsigma )-k^{2}Q{{\rm Y}}_4(t,\varsigma ))}{\lambda }}.\hfill \end{array}\right.\end{array}$$

(17)

Substituting coefficients (17) into (14) leads to the general solutions of Equation (1) as follows:

$$\begin{array}{c}\hfill u(\xi ,t,\varsigma )=a_0+a_{1}F\left(\varpi \right)+{\displaystyle \frac{12P{k}^2{{\rm Y}}_4(t,\varsigma )}{{{\rm Y}}_2(t,\varsigma )}}{F}^2\left(\varpi \right),\end{array}$$

(18)

$$\begin{array}{c}\hfill v(\xi ,t,\varsigma )=b_0+b_{1}F\left(\varpi \right)+{\displaystyle \frac{3k^{2}Q{{\rm Y}}_4(t,\varsigma )}{2a_1{b}_1}}{F}^2\left(\varpi \right),\end{array}$$

(19)

$$\begin{array}{c}\hfill w(\xi ,t,\varsigma )=c_0+c_{1}F\left(\varpi \right)-{\displaystyle \frac{3c_{1}P{k}^2{{\rm Y}}_4(t,\varsigma )}{a_1}}{F}^2\left(\varpi \right),\end{array}$$

(20)

with

$$\begin{array}{c}\hfill \varpi (\xi ,t,\varsigma )=k\left[\xi +{\int }_0^t(a_0{{\rm Y}}_2(\tau ,\varsigma )-k^{2}Q{{\rm Y}}_4(\tau ,\varsigma ))d\tau \right]+c.\end{array}$$

From Appendix A, for a few specific instances, we provide the following solutions:

Case 1. 

Set $P={\displaystyle \frac{1}{4}}$, $Q={\displaystyle \frac{m^2-2}{2}},$ and $R={\displaystyle \frac{m^2}{4}}$. Then, $F\left(\varpi \right)\to ns\left(\varpi \right)\pm ds\left(\varpi \right)$. Hence, we have

$$\begin{array}{c}\hfill u(\xi ,t,\varsigma )=a_0+a_1\left[ns\left({\varpi }_1\right)\pm ds\left({\varpi }_1\right)\right]+{\displaystyle \frac{3k^2{{\rm Y}}_4(t,\varsigma )}{{{\rm Y}}_2(t,\varsigma )}}{\left[ns\left({\varpi }_1\right)\pm ds\left({\varpi }_1\right)\right]}^2,\end{array}$$

(21)

$$\begin{array}{c}\hfill v(\xi ,t,\varsigma )=b_0+b_1\left[ns\left({\varpi }_1\right)\pm ds\left({\varpi }_1\right)\right]+{\displaystyle \frac{3k^2(m^2-2){{\rm Y}}_4(t,\varsigma )}{4a_1{b}_1}}{\left[ns\left({\varpi }_1\right)\pm ds\left({\varpi }_1\right)\right]}^2,\end{array}$$

(22)

$$\begin{array}{c}\hfill w(\xi ,t,\varsigma )=c_0+c_1\left[ns\left({\varpi }_1\right)\pm ds\left({\varpi }_1\right)\right]-{\displaystyle \frac{3c_1{k}^2{{\rm Y}}_4(t,\varsigma )}{4a_1}}{\left[ns\left({\varpi }_1\right)\pm ds\left({\varpi }_1\right)\right]}^2,\end{array}$$

(23)

with

$$\begin{array}{c}\hfill {\varpi }_1(\xi ,t,\varsigma )=k\left[\xi +{\int }_0^t{\displaystyle \frac{[2a_0{{\rm Y}}_2(\tau ,\varsigma )-k^2(m^2-2){{\rm Y}}_4(\tau ,\varsigma )]}{2}}d\tau \right]+c.\end{array}$$

I. In the limiting case as $m\to 0$, the expressions are $ns\left(\varpi \right)\to csc\left(\varpi \right)$ and $ds\left(\varpi \right)\to csc\left(\varpi \right)$. In considering the addition case, $ns\left(\varpi \right)\pm ds\left(\varpi \right)$ tends toward $2csc\left(\varpi \right)$. Consequently, Equations (21)–(23) simplify accordingly:

$$\begin{array}{c}\hfill u(\xi ,t,\varsigma )=a_0+2\left[a_1+{\displaystyle \frac{6k^2{{\rm Y}}_4(t,\varsigma )}{{{\rm Y}}_2(t,\varsigma )}}csc\left({\varpi }_2\right)\right]csc\left({\varpi }_2\right),\end{array}$$

(24)

$$\begin{array}{c}\hfill v(\xi ,t,\varsigma )=b_0+2\left[b_1-{\displaystyle \frac{3k^2{{\rm Y}}_4(t,\varsigma )}{a_1{b}_1}}csc\left({\varpi }_2\right)\right]csc\left({\varpi }_2\right),\end{array}$$

(25)

$$\begin{array}{c}\hfill w(\xi ,t,\varsigma )=c_0+2c_1\left[1-{\displaystyle \frac{3k^2{{\rm Y}}_4(t,\varsigma )}{2a_1}}csc\left({\varpi }_2\right)\right]csc\left({\varpi }_2\right),\end{array}$$

(26)

with

$$\begin{array}{c}\hfill {\varpi }_2(\xi ,t,\varsigma )=k\left[\xi +{\int }_0^t\left(a_0{{\rm Y}}_2(\tau ,\varsigma )+k^2{{\rm Y}}_4(\tau ,\varsigma )\right)d\tau \right]+c.\end{array}$$

II. In the limiting scenario where $m\to 1$, we observe that $ns\left(\varpi \right)\pm ds\left(\varpi \right)$ approaches $coth\left(\varpi \right)\pm csch\left(\varpi \right)$. Accordingly, Equations (21)–(23) are transformed into the following forms:

$$\begin{array}{c}\hfill u(\xi ,t,\varsigma )=a_0+a_1\left[coth\left({\varpi }_3\right)\pm csch\left({\varpi }_3\right)\right]+{\displaystyle \frac{3k^2{{\rm Y}}_4(t,\varsigma )}{{{\rm Y}}_2(t,\varsigma )}}{\left[coth\left({\varpi }_3\right)\pm csch\left({\varpi }_3\right)\right]}^2,\end{array}$$

(27)

$$\begin{array}{c}\hfill v(\xi ,t,\varsigma )=b_0+b_1\left[coth\left({\varpi }_3\right)\pm csch\left({\varpi }_3\right)\right]-{\displaystyle \frac{3k^2{{\rm Y}}_4(t,\varsigma )}{4a_1{b}_1}}{\left[coth\left({\varpi }_3\right)\pm csch\left({\varpi }_3\right)\right]}^2,\end{array}$$

(28)

$$\begin{array}{c}\hfill w(\xi ,t,\varsigma )=c_0+c_1\left[coth\left({\varpi }_3\right)\pm csch\left({\varpi }_3\right)\right]-{\displaystyle \frac{3c_1{k}^2{{\rm Y}}_4(t,\varsigma )}{4a_1}}{\left[coth\left({\varpi }_3\right)\pm csch\left({\varpi }_3\right)\right]}^2,\end{array}$$

(29)

with

$$\begin{array}{c}\hfill {\varpi }_3(\xi ,t,\varsigma )=k\left[\xi +{\displaystyle \frac{1}{2}}{\int }_0^t\left(2a_0{{\rm Y}}_2(\tau ,\varsigma )+k^2{{\rm Y}}_4(\tau ,\varsigma )\right)d\tau \right]+c.\end{array}$$

Case 2. 

Set $P=1,Q=-(1+m^2),$ and $R=m^2$. Then, $F\left(\varpi \right)\to ns\left(\varpi \right)$. Hence, we get

$$\begin{array}{c}\hfill u(\xi ,t,\varsigma )=a_0+a_{1}ns\left({\varpi }_4\right)+{\displaystyle \frac{12k^2{{\rm Y}}_4(t,\varsigma )}{{{\rm Y}}_2(t,\varsigma )}}n{s}^2\left({\varpi }_4\right),\end{array}$$

(30)

$$\begin{array}{c}\hfill v(\xi ,t,\varsigma )=b_0+b_{1}ns\left({\varpi }_4\right)-{\displaystyle \frac{3k^2(1+m^2){{\rm Y}}_4(t,\varsigma )}{2a_1{b}_1}}n{s}^2\left({\varpi }_4\right),\end{array}$$

(31)

$$\begin{array}{c}\hfill w(\xi ,t,\varsigma )=c_0+c_{1}ns\left({\varpi }_4\right)-{\displaystyle \frac{3c_1{k}^2{{\rm Y}}_4(t,\varsigma )}{a_1}}n{s}^2\left({\varpi }_4\right),\end{array}$$

(32)

with

$$\begin{array}{c}\hfill {\varpi }_4(\xi ,t,\varsigma )=k\left[\xi +{\int }_0^t(a_0{{\rm Y}}_2(\tau ,\varsigma )+k^2(1+m^2){{\rm Y}}_4(\tau ,\varsigma ))d\tau \right]+c.\end{array}$$

I. When taking the limit $m\to 0$, the function $ns\left(\varpi \right)$ converges to $csc\left(\varpi \right)$. As a result, Equations (30)–(32) simplify accordingly:

$$\begin{array}{c}\hfill u(\xi ,t,\varsigma )=a_0+a_{1}csc\left({\varpi }_2\right)+{\displaystyle \frac{12k^2{{\rm Y}}_4(t,\varsigma )}{{{\rm Y}}_2(t,\varsigma )}}{csc}^2\left({\varpi }_2\right),\end{array}$$

(33)

$$\begin{array}{c}\hfill v(\xi ,t,\varsigma )=b_0+b_{1}csc\left({\varpi }_2\right)-{\displaystyle \frac{3k^2{{\rm Y}}_4(t,\varsigma )}{2a_1{b}_1}}{csc}^2\left({\varpi }_2\right),\end{array}$$

(34)

$$\begin{array}{c}\hfill w(\xi ,t,\varsigma )=c_0+c_{1}csc\left({\varpi }_2\right)-{\displaystyle \frac{3c_1{k}^2{{\rm Y}}_4(t,\varsigma )}{a_1}}{csc}^2\left({\varpi }_2\right),\end{array}$$

(35)

II. In the limiting case as $m\to 1$, the function $ns\left(\varpi \right)$ tends to $coth\left(\varpi \right)$. Consequently, Equations (30)–(32) reduce to the following forms:

$$\begin{array}{c}\hfill u(\xi ,t,\varsigma )=a_0+a_{1}coth\left({\varpi }_5\right)+{\displaystyle \frac{12k^2{{\rm Y}}_4(t,\varsigma )}{{{\rm Y}}_2(t,\varsigma )}}{coth}^2\left({\varpi }_5\right),\end{array}$$

(36)

$$\begin{array}{c}\hfill v(\xi ,t,\varsigma )=b_0+b_{1}coth\left({\varpi }_5\right)-{\displaystyle \frac{6k^2{{\rm Y}}_4(t,\varsigma )}{2a_1{b}_1}}{coth}^2\left({\varpi }_5\right),\end{array}$$

(37)

$$\begin{array}{c}\hfill w(\xi ,t,\varsigma )=c_0+c_{1}coth\left({\varpi }_5\right)-{\displaystyle \frac{3c_1{k}^2{{\rm Y}}_4(t,\varsigma )}{a_1}}{coth}^2\left({\varpi }_5\right),\end{array}$$

(38)

with

$$\begin{array}{c}\hfill {\varpi }_5(\xi ,t,\varsigma )=k\left[\xi +{\int }_0^t(a_0{{\rm Y}}_2(\tau ,\varsigma )+2k^2{{\rm Y}}_4(\tau ,\varsigma ))d\tau \right]+c.\end{array}$$

Case 3. 

Set $P=1,Q=2-m^2,$ and $R=1-m^2$. Then, $F\left(\varpi \right)\to cs\left(\varpi \right)$. Hence, we obtain

$$\begin{array}{c}\hfill u(\xi ,t,\varsigma )=a_0+a_{1}cs\left({\varpi }_6\right)+{\displaystyle \frac{12k^2{{\rm Y}}_4(t,\varsigma )}{{{\rm Y}}_2(t,\varsigma )}}c{s}^2\left({\varpi }_6\right),\end{array}$$

(39)

$$\begin{array}{c}\hfill v(\xi ,t,\varsigma )=b_0+b_{1}cs\left({\varpi }_6\right)+{\displaystyle \frac{3k^2(2-m^2){{\rm Y}}_4(t,\varsigma )}{2a_1{b}_1}}c{s}^2\left({\varpi }_6\right),\end{array}$$

(40)

$$\begin{array}{c}\hfill w(\xi ,t,\varsigma )=c_0+c_{1}cs\left({\varpi }_6\right)-{\displaystyle \frac{3c_1{k}^2{{\rm Y}}_4(t,\varsigma )}{a_1}}c{s}^2\left({\varpi }_6\right),\end{array}$$

(41)

with

$$\begin{array}{c}\hfill {\varpi }_6(\xi ,t,\varsigma )=k\left[\xi +{\int }_0^t(a_0{{\rm Y}}_2(\tau ,\varsigma )-k^2(2-m^2){{\rm Y}}_4(\tau ,\varsigma ))d\tau \right]+c.\end{array}$$

I. In the limit as $m\to 0$, the function $cs\left(\varpi \right)$ approaches $cot\left(\varpi \right)$. Accordingly, Equations (39)–(41) are simplified as follows:

$$\begin{array}{c}\hfill u(\xi ,t,\varsigma )=a_0+a_{1}cot\left({\varpi }_7\right)+{\displaystyle \frac{12k^2{{\rm Y}}_4(t,\varsigma )}{{{\rm Y}}_2(t,\varsigma )}}{cot}^2\left({\varpi }_7\right),\end{array}$$

(42)

$$\begin{array}{c}\hfill v(\xi ,t,\varsigma )=b_0+b_{1}cot\left({\varpi }_7\right)+{\displaystyle \frac{3k^2{{\rm Y}}_4(t,\varsigma )}{a_1{b}_1}}{cot}^2\left({\varpi }_7\right),\end{array}$$

(43)

$$\begin{array}{c}\hfill w(\xi ,t,\varsigma )=c_0+c_{1}cot\left({\varpi }_7\right)-{\displaystyle \frac{3c_1{k}^2{{\rm Y}}_4(t,\varsigma )}{a_1}}{cot}^2\left({\varpi }_7\right),\end{array}$$

(44)

with

$$\begin{array}{c}\hfill {\varpi }_7(\xi ,t,\varsigma )=k\left[\xi +{\int }_0^t(a_0{{\rm Y}}_2(\tau ,\varsigma )-2k^2{{\rm Y}}_4(\tau ,\varsigma ))d\tau \right]+c.\end{array}$$

II. As $m\to 1$, the function $cs\left(\varpi \right)$ converges to $csch\left(\varpi \right)$. Consequently, Equations (39)–(41) take the following simplified forms:

$$\begin{array}{c}\hfill u(\xi ,t,\varsigma )=a_0+a_{1}csch\left({\varpi }_8\right)+{\displaystyle \frac{12k^2{{\rm Y}}_4(t,\varsigma )}{{{\rm Y}}_2(t,\varsigma )}}csc{h}^2\left({\varpi }_8\right),\end{array}$$

(45)

$$\begin{array}{c}\hfill v(\xi ,t,\varsigma )=b_0+b_{1}csch\left({\varpi }_8\right)+{\displaystyle \frac{3k^2{{\rm Y}}_4(t,\varsigma )}{2a_1{b}_1}}csc{h}^2\left({\varpi }_8\right),\end{array}$$

(46)

$$\begin{array}{c}\hfill w(\xi ,t,\varsigma )=c_0+c_{1}csch\left({\varpi }_8\right)-{\displaystyle \frac{3c_1{k}^2{{\rm Y}}_4(t,\varsigma )}{a_1}}csc{h}^2\left({\varpi }_8\right),\end{array}$$

(47)

with

$$\begin{array}{c}\hfill {\varpi }_8(\xi ,t,\varsigma )=k\left[\xi +{\int }_0^t(a_0{{\rm Y}}_2(\tau ,\varsigma )-k^2{{\rm Y}}_4(\tau ,\varsigma ))d\tau \right]+c.\end{array}$$

Some different solutions of Equation (1) occur as a result of changing the coefficient values for $P,Q$, and R (see Appendix A, Appendix B and Appendix C) [4,16]. The previously mentioned instances are only meant to demonstrate the broad applications of our method.

5. N-GWN Functional Solutions of Equation (1)

In this section, we utilize the outcomes from Section 2 and Section 3 and apply the ℵ-Hermite transform to derive exact non-Gaussian white noise (N-GWN) functional solutions for the Wick-type stochastic (Wt.S) generalized HSC-KdV equations given in Equation (1).

Due to the analytical properties of exponential, trigonometric, and hyperbolic functions, there exists a bounded open domain $D\subset \mathbb{R}\times {\mathbb{R}}_{+}$, along with parameters $q<\infty$ and $\mathcal{M}>0$, such that the functions $u(\xi ,t,\varsigma )$, $v(\xi ,t,\varsigma )$, and $w(\xi ,t,\varsigma )$, which solve Equation (10), along with all their partial derivatives involved in Equation (10), are uniformly bounded for all $(\xi ,t,\varsigma )\in D\times {\mathbb{O}}_q\left(\mathcal{M}\right)$.

Moreover, these functions are continuous with respect to $(\xi ,t)\in D$ for each fixed $\varsigma \in {\mathbb{O}}_q\left(\mathcal{M}\right)$, and analytic in $\varsigma$ for every $(\xi ,t)\in D$.

According to Theorem 1, there exist functions $u(\xi ,t),V(\xi ,t),W(\xi ,t)\in H_{-q}^{\aleph }$ satisfying

$$U(\xi ,t)={\mathcal{H}}_{\aleph }^{-1}u(\xi ,t,\varsigma ),V(\xi ,t)={\mathcal{H}}_{\aleph }^{-1}v(\xi ,t,\varsigma ),W(\xi ,t)={\mathcal{H}}_{\aleph }^{-1}w(\xi ,t,\varsigma ),$$

and these functions are strong solutions for Equation (1) in the space $H_{-q}^{\aleph }$.

Hence, by applying the inverse ℵ-Hermite transform to the results derived in Section 3, we obtain exact N-GWN functional solutions of Equation (1) as follows.

• N-GWN Functional Solutions of JEF Type:

  $$\begin{array}{c}\hfill U_1(\xi ,t)=a_0+a_1\left[n{s}^{{\diamond }_{\aleph }}\left({\varpi }_1\right)\pm d{s}^{{\diamond }_{\aleph }}\left({\varpi }_1\right)\right]+{\displaystyle \frac{3k^2{{\rm Y}}_4\left(t\right)}{{{\rm Y}}_2\left(t\right)}}{\diamond }_{\aleph }{\left[n{s}^{{\diamond }_{\aleph }}\left({\varpi }_1\right)\pm d{s}^{{\diamond }_{\aleph }}\left({\varpi }_1\right)\right]}^{{\diamond }_{\aleph }2},\end{array}$$

  (48)

  $$\begin{array}{c}\hfill V_1(\xi ,t)=b_0+b_1\left[n{s}^{{\diamond }_{\aleph }}\left({\varpi }_1\right)\pm d{s}^{{\diamond }_{\aleph }}\left({\varpi }_1\right)\right]+{\displaystyle \frac{3k^2(m^2-2){{\rm Y}}_4\left(t\right)}{4a_1{b}_1}}{\diamond }_{\aleph }{\left[n{s}^{{\diamond }_{\aleph }}\left({\varpi }_1\right)\pm d{s}^{{\diamond }_{\aleph }}\left({\varpi }_1\right)\right]}^{{\diamond }_{\aleph }2},\end{array}$$

  (49)

  $$\begin{array}{c}\hfill W_1(\xi ,t)=c_0+c_1\left[n{s}^{{\diamond }_{\aleph }}\left({\varpi }_1\right)\pm d{s}^{{\diamond }_{\aleph }}\left({\varpi }_1\right)\right]-{\displaystyle \frac{3c_1{k}^2{{\rm Y}}_4\left(t\right)}{4a_1}}{\diamond }_{\aleph }{\left[n{s}^{{\diamond }_{\aleph }}\left({\varpi }_1\right)\pm d{s}^{{\diamond }_{\aleph }}\left({\varpi }_1\right)\right]}^{{\diamond }_{\aleph }2},\end{array}$$

  (50)

  $$\begin{array}{c}\hfill U_2(\xi ,t)=a_0+a_{1}n{s}^{{\diamond }_{\aleph }}\left({\varpi }_2\right)+{\displaystyle \frac{12k^2{{\rm Y}}_4\left(t\right)}{{{\rm Y}}_2\left(t\right)}}{\diamond }_{\aleph }n{s}^{{\diamond }_{\aleph }2}\left({\varpi }_2\right),\end{array}$$

  (51)

  $$\begin{array}{c}\hfill V_2(\xi ,t)=b_0+b_{1}n{s}^{{\diamond }_{\aleph }}\left({\varpi }_2\right)-{\displaystyle \frac{3k^2(1+m^2){{\rm Y}}_4\left(t\right)}{2a_1{b}_1}}{\diamond }_{\aleph }n{s}^{{\diamond }_{\aleph }2}\left({\varpi }_2\right),\end{array}$$

  (52)

  $$\begin{array}{c}\hfill W_2(\xi ,t)=c_0+c_{1}n{s}^{{\diamond }_{\aleph }}\left({\varpi }_2\right)-{\displaystyle \frac{3c_1{k}^2{{\rm Y}}_4\left(t\right)}{a_1}}{\diamond }_{\aleph }n{s}^{{\diamond }_{\aleph }2}\left({\varpi }_2\right),\end{array}$$

  (53)

  $$\begin{array}{c}\hfill U_3(\xi ,t)=a_0+a_{1}c{s}^{{\diamond }_{\aleph }}\left({\varpi }_3\right)+{\displaystyle \frac{12k^2{{\rm Y}}_4\left(t\right)}{{{\rm Y}}_2\left(t\right)}}{\diamond }_{\aleph }c{s}^{{\diamond }_{\aleph }2}\left({\varpi }_3\right),\end{array}$$

  (54)

  $$\begin{array}{c}\hfill V_3(\xi ,t)=b_0+b_{1}c{s}^{{\diamond }_{\aleph }}\left({\varpi }_3\right)+{\displaystyle \frac{3k^2(2-m^2){{\rm Y}}_4\left(t\right)}{2a_1{b}_1}}{\diamond }_{\aleph }c{s}^{{\diamond }_{\aleph }2}\left({\varpi }_3\right),\end{array}$$

  (55)

  $$\begin{array}{c}\hfill W_3(\xi ,t)=c_0+c_{1}c{s}^{{\diamond }_{\aleph }}\left({\varpi }_3\right)-{\displaystyle \frac{3c_1{k}^2{{\rm Y}}_4\left(t\right)}{a_1}}{\diamond }_{\aleph }c{s}^{{\diamond }_{\aleph }2}\left({\varpi }_3\right),\end{array}$$

  (56)

  with

  $$\begin{array}{c}\hfill {\varpi }_1(\xi ,t)=k\left[\xi +{\displaystyle \frac{1}{2}}{\int }_0^t\left(2a_0{{\rm Y}}_2\left(\tau \right)-k^2(m^2-2){{\rm Y}}_4\left(\tau \right)\right)d\tau \right]+c,\end{array}$$

  $$\begin{array}{c}\hfill {\varpi }_2(\xi ,t)=k\left[\xi +{\int }_0^t\left(a_0{{\rm Y}}_2\left(\tau \right)+k^2(1+m^2){{\rm Y}}_4\left(\tau \right)\right)d\tau \right]+c,\end{array}$$

  $$\begin{array}{c}\hfill {\varpi }_3(\xi ,t)=k\left[\xi +{\int }_0^t\left(a_0{{\rm Y}}_2\left(\tau \right)-k^2(2-m^2){{\rm Y}}_4\left(\tau \right)\right)d\tau \right]+c.\end{array}$$

• N-GWN Functional Solutions of Trigonometric Type:

  $$\begin{array}{c}\hfill U_4(\xi ,t)=a_0+2\left[a_1+{\displaystyle \frac{6k^2{{\rm Y}}_4\left(t\right)}{{{\rm Y}}_2\left(t\right)}}{\diamond }_{\aleph }{csc}^{{\diamond }_{\aleph }}\left({\varpi }_4\right)\right]{csc}^{{\diamond }_{\aleph }}\left({\varpi }_4\right),\end{array}$$

  (57)

  $$\begin{array}{c}\hfill V_4(\xi ,t)=b_0+2\left[b_1-{\displaystyle \frac{3k^2{{\rm Y}}_4\left(t\right)}{a_1{b}_1}}{\diamond }_{\aleph }{csc}^{{\diamond }_{\aleph }}\left({\varpi }_4\right)\right]{csc}^{{\diamond }_{\aleph }}\left({\varpi }_4\right),\end{array}$$

  (58)

  $$\begin{array}{c}\hfill W_4(\xi ,t)=c_0+2c_1\left[1-{\displaystyle \frac{3k^2{{\rm Y}}_4\left(t\right)}{2a_1}}{\diamond }_{\aleph }{csc}^{{\diamond }_{\aleph }}\left({\varpi }_4\right)\right]{csc}^{{\diamond }_{\aleph }}\left({\varpi }_4\right),\end{array}$$

  (59)

  $$\begin{array}{c}\hfill U_5(\xi ,t)=a_0+a_1{csc}^{{\diamond }_{\aleph }}\left({\varpi }_4\right)+{\displaystyle \frac{12k^2{{\rm Y}}_4\left(t\right)}{{{\rm Y}}_2\left(t\right)}}{\diamond }_{\aleph }{csc}^{{\diamond }_{\aleph }2}\left({\varpi }_4\right),\end{array}$$

  (60)

  $$\begin{array}{c}\hfill V_5(\xi ,t)=b_0+b_1{csc}^{{\diamond }_{\aleph }}\left({\varpi }_4\right)-{\displaystyle \frac{3k^2{{\rm Y}}_4\left(t\right)}{2a_1{b}_1}}{\diamond }_{\aleph }{csc}^{{\diamond }_{\aleph }2}\left({\varpi }_4\right),\end{array}$$

  (61)

  $$\begin{array}{c}\hfill W_5(\xi ,t)=c_0+c_1{csc}^{{\diamond }_{\aleph }}\left({\varpi }_4\right)-{\displaystyle \frac{3c_1{k}^2{{\rm Y}}_4\left(t\right)}{a_1}}{\diamond }_{\aleph }{csc}^{{\diamond }_{\aleph }2}\left({\varpi }_4\right),\end{array}$$

  (62)

  $$\begin{array}{c}\hfill U_6(\xi ,t)=a_0+a_1{cot}^{{\diamond }_{\aleph }}\left({\varpi }_5\right)+{\displaystyle \frac{12k^2{{\rm Y}}_4\left(t\right)}{{{\rm Y}}_2\left(t\right)}}{\diamond }_{\aleph }{cot}^{{\diamond }_{\aleph }2}\left({\varpi }_5\right),\end{array}$$

  (63)

  $$\begin{array}{c}\hfill V_6(\xi ,t)=b_0+b_1{cot}^{{\diamond }_{\aleph }}\left({\varpi }_5\right)+{\displaystyle \frac{3k^2{{\rm Y}}_4\left(t\right)}{a_1{b}_1}}{\diamond }_{\aleph }{cot}^{{\diamond }_{\aleph }2}\left({\varpi }_5\right),\end{array}$$

  (64)

  $$\begin{array}{c}\hfill W_6(\xi ,t)=c_0+c_1{cot}^{{\diamond }_{\aleph }}\left({\varpi }_5\right)-{\displaystyle \frac{3c_1{k}^2{{\rm Y}}_4\left(t\right)}{a_1}}{\diamond }_{\aleph }{cot}^{{\diamond }_{\aleph }2}\left({\varpi }_5\right),\end{array}$$

  (65)

  with

  $$\begin{array}{c}\hfill {\varpi }_4(\xi ,t)=k\left[\xi +{\int }_0^t\left(a_0{{\rm Y}}_2\left(\tau \right)+k^2{{\rm Y}}_4\left(\tau \right)\right)d\tau \right]+c,\end{array}$$

  $$\begin{array}{c}\hfill {\varpi }_5(\xi ,t)=k\left[\xi +{\int }_0^t\left(a_0{{\rm Y}}_2\left(\tau \right)-2k^2{{\rm Y}}_4\left(\tau \right)\right)d\tau \right]+c.\end{array}$$

• N-GWN Functional Solutions of Hyperbolic Type:

  $$\begin{array}{c}\hfill U_7(\xi ,t)=a_0+a_1\left[{coth}^{{\diamond }_{\aleph }}\left({\varpi }_6\right)\pm csc{h}^{{\diamond }_{\aleph }}\left({\varpi }_6\right)\right]+{\displaystyle \frac{3k^2{{\rm Y}}_4\left(t\right)}{{{\rm Y}}_2\left(t\right)}}{\diamond }_{\aleph }{\left[{coth}^{{\diamond }_{\aleph }}\left({\varpi }_6\right)\pm csc{h}^{{\diamond }_{\aleph }}\left({\varpi }_6\right)\right]}^{{\diamond }_{\aleph }2},\end{array}$$

  (66)

  $$\begin{array}{c}\hfill V_7(\xi ,t)=b_0+b_1\left[{coth}^{{\diamond }_{\aleph }}\left({\varpi }_6\right)\pm csc{h}^{{\diamond }_{\aleph }}\left({\varpi }_6\right)\right]-{\displaystyle \frac{3k^2{{\rm Y}}_4\left(t\right)}{4a_1{b}_1}}{\diamond }_{\aleph }{\left[{coth}^{{\diamond }_{\aleph }}\left({\varpi }_6\right)\pm csc{h}^{{\diamond }_{\aleph }}\left({\varpi }_6\right)\right]}^{{\diamond }_{\aleph }2},\end{array}$$

  (67)

  $$\begin{array}{c}\hfill W_7(\xi ,t)=c_0+c_1\left[{coth}^{{\diamond }_{\aleph }}\left({\varpi }_6\right)\pm csc{h}^{{\diamond }_{\aleph }}\left({\varpi }_6\right)\right]-{\displaystyle \frac{3c_1{k}^2{{\rm Y}}_4\left(t\right)}{4a_1}}{\diamond }_{\aleph }{\left[{coth}^{{\diamond }_{\aleph }}\left({\varpi }_6\right)\pm csc{h}^{{\diamond }_{\aleph }}\left({\varpi }_6\right)\right]}^{{\diamond }_{\aleph }2}\end{array}$$

  (68)

  and

  $$\begin{array}{c}\hfill U_8(\xi ,t)=a_0+a_1{coth}^{{\diamond }_{\aleph }}\left({\varpi }_7\right)+{\displaystyle \frac{12k^2{{\rm Y}}_4\left(t\right)}{{{\rm Y}}_2\left(t\right)}}{\diamond }_{\aleph }{coth}^{{\diamond }_{\aleph }2}\left({\varpi }_7\right),\end{array}$$

  (69)

  $$\begin{array}{c}\hfill V_8(\xi ,t)=b_0+b_1{coth}^{{\diamond }_{\aleph }}\left({\varpi }_7\right)-{\displaystyle \frac{6k^2{{\rm Y}}_4\left(t\right)}{2a_1{b}_1}}{\diamond }_{\aleph }{coth}^{{\diamond }_{\aleph }2}\left({\varpi }_7\right),\end{array}$$

  (70)

  $$\begin{array}{c}\hfill W_8(\xi ,t)=c_0+c_1{coth}^{{\diamond }_{\aleph }}\left({\varpi }_7\right)-{\displaystyle \frac{3c_1{k}^2{{\rm Y}}_4\left(t\right)}{a_1}}{\diamond }_{\aleph }{coth}^{{\diamond }_{\aleph }2}\left({\varpi }_7\right),\end{array}$$

  (71)

  $$\begin{array}{c}\hfill U_9(\xi ,t)=a_0+a_{1}csc{h}^{{\diamond }_{\aleph }}\left({\varpi }_8\right)+{\displaystyle \frac{12k^2{{\rm Y}}_4\left(t\right)}{{{\rm Y}}_2\left(t\right)}}{\diamond }_{\aleph }csc{h}^{{\diamond }_{\aleph }2}\left({\varpi }_8\right),\end{array}$$

  (72)

  $$\begin{array}{c}\hfill V_9(\xi ,t)=b_0+b_{1}csc{h}^{{\diamond }_{\aleph }}\left({\varpi }_8\right)+{\displaystyle \frac{3k^2{{\rm Y}}_4\left(t\right)}{2a_1{b}_1}}{\diamond }_{\aleph }csc{h}^{{\diamond }_{\aleph }2}\left({\varpi }_8\right),\end{array}$$

  (73)

  $$\begin{array}{c}\hfill W_9(\xi ,t)=c_0+c_{1}csc{h}^{{\diamond }_{\aleph }}\left({\varpi }_8\right)-{\displaystyle \frac{3c_1{k}^2{{\rm Y}}_4\left(t\right)}{a_1}}{\diamond }_{\aleph }csc{h}^{{\diamond }_{\aleph }2}\left({\varpi }_8\right),\end{array}$$

  (74)

  with

  $$\begin{array}{c}\hfill {\varpi }_6(\xi ,t)=k\left[\xi +{\displaystyle \frac{1}{2}}{\int }_0^t\left(2a_0{{\rm Y}}_2\left(\tau \right)+k^2{{\rm Y}}_4\left(\tau \right)\right)d\tau \right]+c,\end{array}$$

  $$\begin{array}{c}\hfill {\varpi }_7(\xi ,t)=k\left[\xi +{\int }_0^t\left(a_0{{\rm Y}}_2\left(\tau \right)+2k^2{{\rm Y}}_4\left(\tau \right)\right)d\tau \right]+c,\end{array}$$

  $$\begin{array}{c}\hfill {\varpi }_8(\xi ,t)=k\left[\xi +{\int }_0^t\left(a_0{{\rm Y}}_2\left(\tau \right)-k^2{{\rm Y}}_4\left(\tau \right)\right)d\tau \right]+c.\end{array}$$

It is evident that varying the forms of ${{\rm Y}}_1$, ${{\rm Y}}_2$, ${{\rm Y}}_3$, and ${{\rm Y}}_4$ leads to distinct solutions of Equation (1), as derived from the formulas presented in the previous section. To illustrate this, we provide the following example.

Example 1. 

Suppose that ${{\rm Y}}_1\left(t\right)={\alpha }_1{{\rm Y}}_4\left(t\right)$, ${{\rm Y}}_2\left(t\right)={\alpha }_2{{\rm Y}}_4\left(t\right)$, ${{\rm Y}}_3\left(t\right)={\alpha }_3{{\rm Y}}_4\left(t\right)$, and ${{\rm Y}}_4\left(t\right)=\Gamma \left(t\right)+{\alpha }_4{W}_{\aleph }\left(t\right)$, where ${\alpha }_1,{\alpha }_2,{\alpha }_3\ne 0$ and ${\alpha }_4$ are arbitrary constants, $\Gamma \left(t\right)$ is integrable or a bounded measurable function on ${\mathbb{R}}_{+}$, and $W_{\aleph }\left(t\right)={\dot{B}}_{\aleph }\left(t\right)$ is the one-parameter ℵ-WN, and $B_{\aleph }\left(t\right)$ is the one-parameter ℵ-Brownian motion. We have the ℵ-Hermite transform ${\widehat{W}}_{\aleph }(t,\varsigma )={\sum }_{n=0}^n{\aleph }_n\left(t\right)z^n$. Since ${exp}^{\diamond }\left(B_t\right)=exp\left[B_t-{\displaystyle \frac{t^2}{2}}\right],$ we have ${cot}^{\diamond }\left(B_t\right)=cot\left[B_{\aleph }\left(t\right)-{\displaystyle \frac{t^2}{2}}\right]$, ${csc}^{\diamond }\left(B_t\right)=csc\left[B_{\aleph }\left(t\right)-{\displaystyle \frac{t^2}{2}}\right]$, ${coth}^{\diamond }\left(B_t\right)=coth\left[B_{\aleph }\left(t\right)-{\displaystyle \frac{t^2}{2}}\right]$, and $csc{h}^{\diamond }\left(B_t\right)=csch\left[B_{\aleph }\left(t\right)-{\displaystyle \frac{t^2}{2}}\right]$. We have a non-Wick version of the STWSs of Equation (1):

$$\begin{array}{c}\hfill U_{10}(\xi ,t)=a_0+2\left[a_1+{\displaystyle \frac{6k^2}{{\alpha }_2}}csc\left({\Pi }_1\right)\right]csc\left({\Pi }_1\right),\end{array}$$

(75)

$$\begin{array}{c}\hfill V_{10}(\xi ,t)=b_0+2\left[b_1-{\displaystyle \frac{3k^2\{\Gamma \left(t\right)+{\alpha }_4{W}_{\aleph }\left(t\right)\}}{a_1{b}_1}}csc\left({\Pi }_1\right)\right]csc\left({\Pi }_1\right),\end{array}$$

(76)

$$\begin{array}{c}\hfill W_{10}(\xi ,t)=c_0+2c_1\left[1-{\displaystyle \frac{3k^2\{\Gamma \left(t\right)+{\alpha }_4{W}_{\aleph }\left(t\right)\}}{2a_1}}csc\left({\Pi }_1\right)\right]csc\left({\Pi }_1\right),\end{array}$$

(77)

$$\begin{array}{c}\hfill U_{11}(\xi ,t)=a_0+a_{1}csc\left({\Pi }_1\right)+{\displaystyle \frac{12k^2}{{\alpha }_2}}{csc}^2\left({\Pi }_1\right),\end{array}$$

(78)

$$\begin{array}{c}\hfill V_{11}(\xi ,t)=b_0+b_{1}csc\left({\Pi }_1\right)-{\displaystyle \frac{3k^2\{\Gamma \left(t\right)+{\alpha }_4{W}_{\aleph }\left(t\right)\}}{2a_1{b}_1}}{csc}^2\left({\Pi }_1\right),\end{array}$$

(79)

$$\begin{array}{c}\hfill W_{11}(\xi ,t)=c_0+c_{1}csc\left({\Pi }_1\right)-{\displaystyle \frac{3c_1{k}^2\{\Gamma \left(t\right)+{\alpha }_4{W}_{\aleph }\left(t\right)\}}{a_1}}{csc}^2\left({\Pi }_1\right),\end{array}$$

(80)

$$\begin{array}{c}\hfill U_{12}(\xi ,t)=a_0+a_{1}cot\left({\Pi }_2\right)+{\displaystyle \frac{12k^2}{{\alpha }_2}}{cot}^2\left({\Pi }_2\right),\end{array}$$

(81)

$$\begin{array}{c}\hfill V_{12}(\xi ,t)=b_0+b_{1}cot\left({\Pi }_2\right)+{\displaystyle \frac{3k^2\{\Gamma \left(t\right)+{\alpha }_4{W}_{\aleph }\left(t\right)\}}{a_1{b}_1}}{cot}^2\left({\Pi }_2\right),\end{array}$$

(82)

$$\begin{array}{c}\hfill W_{12}(\xi ,t)=c_0+c_{1}cot\left({\Pi }_2\right)-{\displaystyle \frac{3c_1{k}^2\{\Gamma \left(t\right)+{\alpha }_4{W}_{\aleph }\left(t\right)\}}{a_1}}{cot}^2\left({\Pi }_2\right),\end{array}$$

(83)

$$\begin{array}{c}\hfill U_{13}(\xi ,t)=a_0+a_1\left[coth\left({\Pi }_3\right)\pm csch\left({\Pi }_3\right)\right]+{\displaystyle \frac{3k^2}{{\alpha }_2}}{\left[coth\left({\Pi }_3\right)\pm csch\left({\Pi }_3\right)\right]}^2,\end{array}$$

(84)

$$\begin{array}{c}\hfill V_{13}(\xi ,t)=b_0+b_1\left[coth\left({\Pi }_3\right)\pm csch\left({\Pi }_3\right)\right]-{\displaystyle \frac{3k^2\{\Gamma \left(t\right)+{\alpha }_4{W}_{\aleph }\left(t\right)\}}{4a_1{b}_1}}{\left[coth\left({\Pi }_3\right)\pm csch\left({\Pi }_3\right)\right]}^2,\end{array}$$

(85)

$$\begin{array}{c}\hfill W_{13}(\xi ,t)=c_0+c_1\left[coth\left({\Pi }_3\right)\pm csch\left({\Pi }_3\right)\right]-{\displaystyle \frac{3c_1{k}^2\{\Gamma \left(t\right)+{\alpha }_4{W}_{\aleph }\left(t\right)\}}{4a_1}}{\left[coth\left({\Pi }_3\right)\pm csch\left({\Pi }_3\right)\right]}^2,\end{array}$$

(86)

$$\begin{array}{c}\hfill U_{14}(\xi ,t)=a_0+a_{1}coth\left({\Pi }_4\right)+{\displaystyle \frac{12k^2}{{\alpha }_2}}{coth}^2\left({\Pi }_4\right),\end{array}$$

(87)

$$\begin{array}{c}\hfill V_{14}(\xi ,t)=b_0+b_{1}coth\left({\Pi }_4\right)-{\displaystyle \frac{6k^2\{\Gamma \left(t\right)+{\alpha }_4{W}_{\aleph }\left(t\right)\}}{2a_1{b}_1}}{coth}^2\left({\Pi }_4\right),\end{array}$$

(88)

$$\begin{array}{c}\hfill W_{14}(\xi ,t)=c_0+c_{1}coth\left({\Pi }_4\right)-{\displaystyle \frac{3c_1{k}^2\{\Gamma \left(t\right)+{\alpha }_4{W}_{\aleph }\left(t\right)\}}{a_1}}{coth}^2\left({\Pi }_4\right),\end{array}$$

(89)

$$\begin{array}{c}\hfill U_{15}(\xi ,t)=a_0+a_{1}csch\left({\Pi }_5\right)+{\displaystyle \frac{12k^2}{{\alpha }_2}}csc{h}^2\left({\Pi }_5\right),\end{array}$$

(90)

$$\begin{array}{c}\hfill V_{15}(\xi ,t)=b_0+b_{1}csch\left({\Pi }_5\right)+{\displaystyle \frac{3k^2\{\Gamma \left(t\right)+{\alpha }_4{W}_{\aleph }\left(t\right)\}}{2a_1{b}_1}}csc{h}^2\left({\Pi }_5\right),\end{array}$$

(91)

$$\begin{array}{c}\hfill W_{15}(\xi ,t)=c_0+c_{1}csch\left({\Pi }_5\right)-{\displaystyle \frac{3c_1{k}^2\{\Gamma \left(t\right)+{\alpha }_4{W}_{\aleph }\left(t\right)\}}{a_1}}csc{h}^2\left({\Pi }_5\right),\end{array}$$

(92)

with

$$\begin{array}{c}\hfill {\Pi }_1(\xi ,t)=k\left[\xi +(a_0{\alpha }_2+k^2){\int }_0^t\left(\Gamma \left(\tau \right)d\tau +{\alpha }_4[B_{\aleph }\left(t\right)-{\displaystyle \frac{t^2}{2}}]\right)\right]+c,\end{array}$$

$$\begin{array}{c}\hfill {\Pi }_2(\xi ,t)=k\left[\xi +(a_0{\alpha }_2-2k^2){\int }_0^t\left(\Gamma \left(\tau \right)d\tau +{\alpha }_4[B_{\aleph }\left(t\right)-{\displaystyle \frac{t^2}{2}}]\right)\right]+c.\end{array}$$

$$\begin{array}{c}\hfill {\Pi }_3(\xi ,t)=k\left[\xi +{\displaystyle \frac{2a_0{\alpha }_2+k^2}{2}}{\int }_0^t\left(\Gamma \left(\tau \right)d\tau +{\alpha }_4[B_{\aleph }\left(t\right)-{\displaystyle \frac{t^2}{2}}]\right)\right]+c,\end{array}$$

$$\begin{array}{c}\hfill {\Pi }_4(\xi ,t)=k\left[\xi +(a_0{\alpha }_2+2k^2){\int }_0^t\left(\Gamma \left(\tau \right)d\tau +{\alpha }_4[B_{\aleph }\left(t\right)-{\displaystyle \frac{t^2}{2}}]\right)\right]+c,\end{array}$$

$$\begin{array}{c}\hfill {\Pi }_5(\xi ,t)=k\left[\xi +(a_0{\alpha }_2-k^2){\int }_0^t\left(\Gamma \left(\tau \right)d\tau +{\alpha }_4[B_{\aleph }\left(t\right)-{\displaystyle \frac{t^2}{2}}]\right)\right]+c.\end{array}$$

6. Summary and Discussion

Generalized HSC-KdV-E can explain nonlinear wave movement in polarity-symmetric systems, with N-GS generalized HSC-KdV-E obtained in an N-GWN environment. We developed a non-Gaussian Wick calculus using HCS theory, introducing ℵ-WP and ℵ-HT on generalized functions, and by proving characterization theorems. We established a framework for studying SPDEs with N-GP. Finally, a variety of families of precise STWSs were produced using the framework and the $F_e$ approach for the generalized HSC-KdV-E (2) and N-GWN functional solutions of the Wt.S generalized HSC-KdV Equation (1). There are more non-linear PDEs in mathematical physics that the planner may be used with. Depending on the settings, the $F_e$ technique can produce a variety of precise answers.