Solitary Wave Solutions for the Stochastic Fractional-Space KdV in the Sense of the M-Truncated Derivative

Abstract

The stochastic fractional-space Korteweg–de Vries equation (SFSKdVE) in the sense of the M-truncated derivative is examined in this article. In the Itô sense, the SFSKdVE is forced by multiplicative white noise. To produce new trigonometric, hyperbolic, rational, and elliptic stochastic fractional solutions, the tanh–coth and Jacobi elliptic function methods are used. The obtained solutions are useful in interpreting certain fascinating physical phenomena because the KdV equation is essential for understanding the behavior of waves in shallow water. To demonstrate how the multiplicative noise and the M-truncated derivative impact the precise solutions of the SFSKdVE, different 3D and 2D graphical representations are plotted.

1. Introduction

In many fields, such as engineering, mathematics, and physics, complicated phenomena are explained by using nonlinear evolution equations (NEEs). NEEs, including the Korteweg–de Vries (KdV) equation, Jaulent–Miodek equation, Whitham–Broer–Kaup equation, Green–Naghdi equation, Boussinesq equation, and Gardeners equation, appear in fluid dynamics in the setting of shallow-water waves. The most important physical problem in NEEs is obtaining their traveling-wave solutions. Numerous successful methods for solving NEEs have been provided, such as improved $tan(\phi /2)$-expansion [1], sine-Gordon expansion [2,3], the generalized Kudryashov method [4], the mapping method [5,6], the Exp-function [7], the Laplace transform [8], Lie symmetry [9], Ricatti equation expansion [10], the Bernoulli sub-equation function [11], ($G^{\prime }/G$)-expansion [12,13], Riemann-Hilbert problems with the identity jump matrix [14,15,16], and variational methods [17].

Recently, the importance of adding random effects in evaluating, portending, simulating, and modeling complex systems has been widely recognized in climatic dynamics, physics, chemistry, geophysics, biology, and other fields [18,19]. In the presence of random effects or noise, NEEs are exemplary mathematical models for representing complex systems. On the other hand, fractional differential equations (FDEs) are widely utilized in plasma physics, mathematical biology, quantum field theory, neural physics, fluid mechanics, optical fibers, solid state physics, and other fields [20,21,22,23]. In addition, the fractional-order derivative describes many physical phenomena, including sound electrostatics, heat, elasticity, fluid dynamics, electrodynamics, gravity, diffusion, quantum mechanics, and so on. Due to the importance of the fractional-order derivative, many definitions have been suggested, such as the new truncated M-fractional derivative, Atangana–Baleanu derivative in the context of Caputo, the Grunwald–Letnikov derivative, the Caputo derivative, the Riemann–Liouville derivative, He’s fractional derivative, the Riesz derivative, the Weyl derivative, and conformable fractional definitions [24,25,26,27,28,29,30,31].

As a result, it is critical to think about FDEs with some random force. In this paper, we take the fractional-stochastic Korteweg–de Vries equation (SFSKdVE) into account as follows:

$$d\phi +\left[6{\phi }_0{\mathbb{D}}_{M,x}^{\alpha ,\beta }\phi {+}_0{\mathbb{D}}_{M,xxx}^{\alpha ,\beta }\phi \right]dt=\sigma \phi d\mathcal{W},$$

(1)

where $\phi (x,t)$ is a real function of two variables x and t, ${}_0{\mathbb{D}}_{M,x}^{\alpha ,\beta }$ is the M-truncated derivative, $\sigma$ is the intensity of noise, $\mathcal{W}\left(t\right)$ is the white noise, and $\phi d\mathcal{W}$ is a multiplicative noise in the Itô sense.

In this paper, we aim to find the analytical fractional-stochastic solutions of SFSKdVE (1). In order to obtain these solutions, we employ two different techniques, namely, the tanh–coth and Jacobi elliptic function methods. It is common knowledge that stochastic solutions are more precise than deterministic ones. Consequently, the obtained solutions would be tremendously helpful to physicists in describing some important physical phenomena due to the significance of the KdV equation in explaining ion acoustic waves in plasma, acoustic waves on a crystal lattice, long internal waves in a density-stratified ocean, and weakly interacting shallow-water waves [32,33]. In addition, we extend some previously reported results, such as those in [34]. Additionally, we explore the impact of noise and the fractional derivative on the analytical solution of the SFSKdVE (1) by presenting several graphical representations via the MATLAB software.

If we set $\sigma =\beta =0$ and $\alpha =1$, then we get the Korteweg–de Vries (KdV) equation, which is one of the most well-known NEEs:

$${\phi }_t+6\phi {\phi }_x+{\phi }_{xxx}=0.$$

(2)

The KdV Equation (2) represents weakly and nonlinearly interacting shallow-water waves, acoustic waves on a crystal lattice, long internal waves in a density-stratified ocean, and ion acoustic waves in a plasma. Many authors have investigated the solutions of the KdV Equation (2) by using various approaches, such as the domain decomposition method [35], Bäcklund transform [36], finite difference method [37], Galerkin method [38], homotopy perturbation method [39], iterative transform method [40], the Hirota direct method [41], extended tanh method [34], etc.

The following is the order of the article: In Section 2, we define the M-truncated derivative and state its properties. In Section 3, we use the wave transformation to get the wave equation for the SFSKdVE (1). In Section 4, the tanh–coth and Jacobi elliptic function methods are used to obtain the exact fractional-stochastic solutions of the SFSKdVE (1). In the acquired solutions of the SFSKdVE, we can observe the effects of white noise and the M-truncated derivative, as described in Section 5. The conclusions of the article are presented thereafter.

2. M-Truncated Derivative

In [31], Sousa et al. recently suggested a new fractional derivative called the truncated M-fractional derivative. From this point, let us define the truncated Mittag–Leffler function (TMLF) as follows.

Definition 1 

([31,42]). For $\beta >0$ and $z\in \mathbb{C}$, the TMLF with one parameter is defined as

$${}_i{E}_{\beta }\left(z\right)=\sum _{k=0}^i\frac{z^k}{\Gamma (\beta k+1)}.$$

Definition 2 

([31,42]). The M-truncated derivative of order $\alpha \in (0,1)$ for the function $\phi :[0,\infty )\to \mathbb{R}$ is defined as

$${}_i{\mathbb{D}}_{M,x}^{\alpha ,\beta }\phi \left(x\right)=\underset{h\to 0}{lim}\frac{\phi \left(x{+}_i{E}_{\beta }\left(h{x}^{-\alpha }\right)\right)-\phi \left(x\right)}{h}.$$

The M-truncated derivative satisfies the following properties [31,42]: If $\Phi$ and $\Psi$ are differentiable functions and a, b, and $\upsilon$ are real constants, then

(1) ${}_i{\mathbb{D}}_{M,x}^{\alpha ,\beta }(a\Phi +b\Psi )=a_i{\mathbb{D}}_{M,x}^{\alpha ,\beta }(\Phi )+b_i{\mathbb{D}}_{M,x}^{\alpha ,\beta }(\Psi );$

(2) ${}_i{\mathbb{D}}_{M,x}^{\alpha ,\beta }\left(x^{\nu }\right)=\frac{\nu }{\Gamma (\beta +1)}{x}^{\nu -\alpha }$;

(3) ${}_i{\mathbb{D}}_{M,x}^{\alpha ,\beta }(\Phi \Psi )={\Phi }_i{\mathbb{D}}_{M,x}^{\alpha ,\beta }\Psi +{\Psi }_i{\mathbb{D}}_{M,x}^{\alpha ,\beta }\Phi ;$

(4) ${}_i{\mathbb{D}}_{M,x}^{\alpha ,\beta }(\Phi )\left(x\right)=\frac{x^{1-\alpha }}{\Gamma (\beta +1)}\frac{d\Phi }{dx}$;

(5) ${}_i{\mathbb{D}}_{M,x}^{\alpha ,\beta }(\Phi \circ \Psi )\left(x\right)={\Phi }^{{}^{\prime }}{(\Psi \left(x\right))}_i{\mathbb{D}}_{M,x}^{\alpha ,\beta }\Psi \left(x\right)$.

3. Traveling-Wave Equation for SFSKdVE

The wave equation for SFSKdVE (1) is obtained by assuming the following wave transformation:

$$\phi (x,t)=\psi \left(\mu \right)e^{[\sigma \mathcal{W}\left(t\right)-\frac{1}{2}{\sigma }^{2}t]},\mu =c(\frac{\Gamma (\beta +1)}{\alpha }{x}^{\alpha }-\lambda t),$$

(3)

where the function $\psi$ is deterministic, and $c,\lambda$ are undefined constants. We note that

$$d\phi =[-c\lambda {\psi }^{\prime }dt+\sigma \psi d\mathcal{W}]e^{[\sigma \mathcal{W}\left(t\right)-\frac{1}{2}{\sigma }^{2}t]},$$

(4)

and

$${}_0{\mathbb{D}}_{M,x}^{\alpha ,\beta }\phi =c{\psi }^{\prime }{e}^{[\sigma \mathcal{W}\left(t\right)-\frac{1}{2}{\sigma }^{2}t]},{}_0{\mathbb{D}}_{M,x}^{3\alpha ,\beta }\phi =c^3\psi ‴e^{[\sigma \mathcal{W}\left(t\right)-\frac{1}{2}{\sigma }^{2}t]}.$$

(5)

By inserting Equation (3) into Equation (1) and utilizing (4)–(5), we obtain

$$c^3\psi ‴-c\lambda {\psi }^{\prime }+6c\psi {\psi }^{\prime }{e}^{[\sigma \mathcal{W}\left(t\right)-\frac{1}{2}{\sigma }^{2}t]}=0.$$

(6)

We take the expectation on both sides:

$$c^2\psi ‴-\lambda {\psi }^{\prime }+6\psi {\psi }^{\prime }{e}^{-\frac{1}{2}{\sigma }^{2}t}\mathbb{E}{e}^{\left[\sigma \mathcal{W}\right(t\left)\right]}=0.$$

(7)

Since $\mathcal{W}\left(t\right)$ is a Gaussian process, $\mathbb{E}\left(e^{\sigma \mathcal{W}\left(t\right)}\right)=e^{\frac{1}{2}{\sigma }^{2}t}$. Hence, Equation (7) becomes

$$c^2\psi ‴-\lambda {\psi }^{\prime }+6\psi {\psi }^{\prime }=0.$$

(8)

By integrating Equation (8) once, we get

$$c^2\psi ″-\lambda \psi +3{\psi }^2=0,$$

(9)

where we set the constant of integration equal to zero.

4. Exact Solutions of the SFSKdVE

Here, we utilize two different methods—the tanh–coth method and Jacobi elliptic function (JEF) method—in order to acquire the fractional-stochastic solutions for the SFSKdVE (1).

4.1. Tanh–Coth Method

Here, we apply the tanh–coth method (for more details, see [43]). Let the solution $\psi$ of Equation (9) have the form

$$\psi \left(\mu \right)=\sum _{k=0}^M{\ell }_k{\Phi }^k,$$

(10)

where $\Phi =tanh\mu$ (or $\Phi =coth\mu$), and ${\ell }_k$ is an undefined constant for $k=0,1\dots ,M$ such that ${\ell }_M\ne 0.$

To determine the parameter M, we balance ${\psi }^2$ with $\psi$ in Equation (9) to get

$$2M=M+2.$$

Hence,

$$M=2.$$

(11)

We rewrite Equation (10) by using Equation (11):

$$\psi \left(\mu \right)={\ell }_0+{\ell }_1\Phi +{\ell }_2{\Phi }^2.$$

(12)

By substituting Equation (12) into Equation (9), we get

$$\begin{array}{ccc}& & (6{\ell }_2{c}^2+3{\ell }_2^2){\Phi }^4-(2{\ell }_1{c}^2-6{\ell }_1{\ell }_2){\Phi }^3\hfill \\ & & -(8{\ell }_2{c}^2+\lambda {\ell }_2-3{\ell }_1^2-6{\ell }_0{\ell }_2){\Phi }^2\hfill \\ & & -(2{\ell }_1{c}^2+\lambda {\ell }_1-6{\ell }_0{\ell }_1)\Phi \hfill \end{array}$$

$$+(2{\ell }_2{c}^2-\lambda {\ell }_0+3{\ell }_0^2)=0.$$

By equating each coefficient of ${\Phi }^k(k=4,3,2,1,0)$ to zero, we obtain

$$6{\ell }_2{c}^2+3{\ell }_2^2=0,$$

$$-2{\ell }_1{c}^2+6{\ell }_1{\ell }_2=0,$$

$$-8{\ell }_2{c}^2-\lambda {\ell }_2+3{\ell }_1^2+6{\ell }_0{\ell }_2=0,$$

$$-2{\ell }_1{c}^2-\lambda {\ell }_1+6{\ell }_0{\ell }_1=0,$$

and

$$2{\ell }_2{c}^2-\lambda {\ell }_0+3{\ell }_0^2=0.$$

The following two sets are obtained by solving these equations:

First set:

$${\ell }_0=\frac{-\lambda }{6},{\ell }_1=0,{\ell }_2=\frac{\lambda }{2},c=\sqrt{\frac{-\lambda }{2}}.$$

(13)

Second set:

$${\ell }_0=\frac{\lambda }{2},{\ell }_1=0,{\ell }_2=\frac{-\lambda }{2},c=\sqrt{\frac{\lambda }{2}}.$$

(14)

First set: There are two cases depending on $\lambda :$

First case: If $\lambda <0:$ The solution of Equation (9) in this case is

$$\psi \left(\mu \right)=\frac{-\lambda }{6}+\frac{\lambda }{2}{tanh}^2\left(\mu \right),$$

or

$$\psi \left(\mu \right)=\frac{-\lambda }{6}+\frac{\lambda }{2}{coth}^2\left(\mu \right).$$

Thus, the exact fractional-stochastic solution of the SFSKdVE (1) is

$${\phi }_{1,1}(x,t)=[\frac{-\lambda }{6}+\frac{\lambda }{2}{tanh}^2(\frac{\Gamma (\beta +1)\sqrt{-\lambda }}{2\alpha }{x}^{\alpha }-\frac{\lambda \sqrt{-\lambda }}{2}t)]e^{[\sigma \mathcal{W}\left(t\right)-\frac{1}{2}{\sigma }^{2}t]},$$

(15)

or

$${\phi }_{1,2}(x,t)=[\frac{-\lambda }{6}+\frac{\lambda }{2}{coth}^2(\frac{\Gamma (\beta +1)\sqrt{-\lambda }}{2\alpha }{x}^{\alpha }-\frac{\lambda \sqrt{-\lambda }}{2}t)]e^{[\sigma \mathcal{W}\left(t\right)-\frac{1}{2}{\sigma }^{2}t]}.$$

(16)

Second case: If $\lambda >0:$ The solution of Equation (9) is

$$\psi \left(\mu \right)=\frac{-\lambda }{6}+\frac{\lambda }{2}{tan}^2\left(\mu \right),$$

or

$$\psi \left(\mu \right)=\frac{-\lambda }{6}+\frac{\lambda }{2}{cot}^2\left(\mu \right).$$

Thus, the exact fractional-stochastic solution of the SFSKdVE (1) is

$${\phi }_{1,3}(x,t)=[\frac{-\lambda }{6}+\frac{\lambda }{2}{tan}^2(\frac{\Gamma (\beta +1)\sqrt{\lambda }}{2\alpha }{x}^{\alpha }-\frac{\lambda \sqrt{\lambda }}{2}t)]e^{[\sigma \mathcal{W}\left(t\right)-\frac{1}{2}{\sigma }^{2}t]},$$

(17)

or

$${\phi }_{1,4}(x,t)=[\frac{-\lambda }{6}+\frac{\lambda }{2}{cot}^2(\frac{\Gamma (\beta +1)\sqrt{\lambda }}{2\alpha }{x}^{\alpha }-\frac{\lambda \sqrt{\lambda }}{2}t)]e^{[\sigma \mathcal{W}\left(t\right)-\frac{1}{2}{\sigma }^{2}t]}.$$

(18)

Second set: There are two cases depending on $\lambda :$

First case: If $\lambda <0:$ The solution of Equation (9) is

$$\psi \left(\mu \right)=\frac{\lambda }{2}-\frac{\lambda }{2}{tanh}^2\left(\mu \right)=\frac{\lambda }{2}{\mathrm{sech}}^2\left(\mu \right),$$

or

$$\psi \left(\mu \right)=\frac{\lambda }{2}-\frac{\lambda }{2}{coth}^2\left(\mu \right)=\frac{\lambda }{2}{\mathrm{csch}}^2\left(\mu \right).$$

Thus, the exact fractional-stochastic solution of the SFSKdVE (1) is

$${\phi }_{2,1}(x,t)=\frac{\lambda }{2}{\mathrm{sech}}^2(\frac{\Gamma (\beta +1)\sqrt{-\lambda }}{2\alpha }{x}^{\alpha }-\frac{\lambda \sqrt{-\lambda }}{2}t)e^{[\sigma \mathcal{W}\left(t\right)-\frac{1}{2}{\sigma }^{2}t]},$$

(19)

or

$${\phi }_{2,2}(x,t)=\frac{\lambda }{2}{\mathrm{csch}}^2(\frac{\Gamma (\beta +1)\sqrt{-\lambda }}{2\alpha }{x}^{\alpha }-\frac{\lambda \sqrt{-\lambda }}{2}t)e^{[\sigma \mathcal{W}\left(t\right)-\frac{1}{2}{\sigma }^{2}t]}.$$

(20)

Second case:$\lambda >0:$ The solution of Equation (9) is

$$\psi \left(\mu \right)=\frac{\lambda }{2}-\frac{\lambda }{2}{tan}^2\left(\mu \right),$$

or

$$\psi \left(\mu \right)=\frac{\lambda }{2}-\frac{\lambda }{2}{cot}^2\left(\mu \right).$$

Thus, the exact fractional-stochastic solution of the SFSKdVE (1) is

$${\phi }_{2,3}(x,t)=[\frac{\lambda }{2}-\frac{\lambda }{2}{tan}^2(\frac{\Gamma (\beta +1)\sqrt{\lambda }}{2\alpha }{x}^{\alpha }-\frac{\lambda \sqrt{\lambda }}{2}t)]e^{[\sigma \mathcal{W}\left(t\right)-\frac{1}{2}{\sigma }^{2}t]},$$

(21)

or

$${\phi }_{2,4}(x,t)=[\frac{\lambda }{2}-\frac{\lambda }{2}{cot}^2(\frac{\Gamma (\beta +1)\sqrt{\lambda }}{2\alpha }{x}^{\alpha }-\frac{\lambda \sqrt{\lambda }}{2}t)]e^{[\sigma \mathcal{W}\left(t\right)-\frac{1}{2}{\sigma }^{2}t]}.$$

(22)

Remark 1. 

If we set $\alpha =1$ and $\sigma =\beta =0$, then we have a solution similar to that stated in [34]:

$$\phi (x,t)=[\frac{-\lambda }{6}+\frac{\lambda }{2}{tanh}^2(\frac{\sqrt{-\lambda }}{2}x-\frac{\lambda \sqrt{-\lambda }}{2}t)]\mathit{for}\lambda <0,$$

and

$$\phi (x,t)=\frac{\lambda }{2}{\mathrm{sech}}^2(\frac{\sqrt{\lambda }}{2}x-\frac{\lambda \sqrt{\lambda }}{2}t)\mathit{for}\lambda >0.$$

4.2. JEF Method

Here, we utilize the JEF method (for more details, see [44]). Considering the solutions to Equation (9), the method takes the following form (with $M=2$):

$$\psi \left(\mu \right)={\hslash }_0+{\hslash }_1\chi \left(\mu \right)+{\hslash }_2{\chi }^2\left(\mu \right),$$

(23)

where ${\hslash }_0,{\hslash }_1$, and ${\hslash }_2$ are undefined constants and $\chi \left(\mu \right)=sn(\mu ,m)$ is a Jacobi elliptic sine function for $0<m<1$. By differentiating Equation (23) twice, we get

$$\psi ″\left(\mu \right)=-2{\hslash }_2-{\hslash }_1(m^2+1)\chi +2{\hslash }_1{m}^2{\chi }^3+2{\hslash }_2{m}^2{\chi }^4.$$

(24)

By plugging Equations (23) and (24) into Equation (9), we have

$$\begin{gathered} (2m^2{c}^2{\hslash }_2+3{\hslash }_2^2){\chi }^4+(2c^2{m}^2{\hslash }_1+6{\hslash }_1{\hslash }_2){\chi }^3+(6{\hslash }_0{\hslash }_2-{\hslash }_2\lambda +3{\hslash }_1^2){\chi }^2 \\ -[(m^2+1)c^2{\hslash }_1+{\hslash }_1\lambda -3{\hslash }_0{\hslash }_1]\chi -(2{\hslash }_2{c}^2+{\hslash }_0\lambda -3{\hslash }_0^2)=0. \end{gathered}$$

By balancing the coefficient of ${\chi }^n(n=4,3,2,1,0)$ to zero, we have

$$6m^2{c}^2{\hslash }_2+3{\hslash }_2^2=0,$$

$$2c^2{m}^2{\hslash }_1+6{\hslash }_1{\hslash }_2=0,$$

$$-4c^2{\hslash }_2(m^2+1)+6{\hslash }_0{\hslash }_2-{\hslash }_2\lambda +3{\hslash }_1^2=0,$$

$$(m^2+1)c^2{\hslash }_1+{\hslash }_1\lambda -3{\hslash }_0{\hslash }_1=0,$$

and

$$2{\hslash }_2{c}^2-{\hslash }_0\lambda +3{\hslash }_0^2=0.$$

The following is the solution of these equations:

$${\hslash }_0=\frac{\lambda +4c^2(m^2+1)}{6},{\hslash }_1=0,{\hslash }_2=-2c^2{m}^2,c^2=\frac{\lambda }{4\sqrt{m^4-m^2+1}}.$$

Thus, by using (23), Equation (9) has the solution

$$\psi \left(\mu \right)=\frac{\lambda +4c^2(m^2+1)}{6}-2c^2{m}^{2}s{n}^2(\mu ,m).$$

Hence, the solutions of the SFSKdVE (1) are

$$\phi (x,t)=[\frac{\lambda +4c^2(m^2+1)}{6}-2c^2{m}^{2}s{n}^2(\frac{\Gamma (\beta +1)\sqrt{\lambda }}{2m\alpha }{x}^{\alpha }-\frac{\lambda \sqrt{\lambda }}{2m}t,m)]e^{[\sigma \mathcal{W}\left(t\right)-\frac{1}{2}{\sigma }^{2}t]}.$$

(25)

If $m\to 1$, then Equation (25) becomes

$$\begin{array}{ccc}\hfill \phi (x,t)& =& [\frac{\lambda }{2}-\frac{\lambda }{2}{tanh}^2(\frac{\Gamma (\beta +1)\sqrt{\lambda }}{2\alpha }{x}^{\alpha }-\frac{\lambda \sqrt{\lambda }}{2}t)]e^{[\sigma \mathcal{W}\left(t\right)-\frac{1}{2}{\sigma }^{2}t]}\hfill \\ & =& \frac{\lambda }{2}{\mathrm{sech}}^2(\frac{\Gamma (\beta +1)\sqrt{\lambda }}{2\alpha }{x}^{\alpha }-\frac{\lambda \sqrt{\lambda }}{2}t)]e^{[\sigma \mathcal{W}\left(t\right)-\frac{1}{2}{\sigma }^{2}t]}.\hfill \end{array}$$

(26)

In a similar manner, we can change $sn$ in (23) with $cn,$ where $cn(\mu ,m)$ is a Jacobi elliptic cosine function, in order to obtain the following solutions of Equation (9):

$$\phi (x,t)=[\frac{\lambda -4c^2(2m^2-1)}{6}+2c^2{m}^{2}c{n}^2((\frac{c\Gamma (\beta +1)}{\alpha }{x}^{\alpha }-c\lambda t),m)]e^{[\sigma \mathcal{W}\left(t\right)-\frac{1}{2}{\sigma }^{2}t]},$$

where

$$c^2=\frac{\lambda }{4\sqrt{m^4-m^2+1}}.$$

Therefore, the solutions of the the SFSKdVE (1) are:

$$\phi (x,t)=[\frac{\lambda -4c^2(2m^2-1)}{6}+2c^2{m}^{2}c{n}^2((\frac{c\Gamma (\beta +1)}{\alpha }{x}^{\alpha }-c\lambda t),m)]e^{[\sigma \mathcal{W}\left(t\right)-\frac{1}{2}{\sigma }^{2}t]}.$$

(27)

If $m\to 1$, then the solution (27) becomes

$$\phi (x,t)=[\frac{\lambda }{2}{\mathrm{sech}}^2(\frac{\Gamma (\beta +1)\sqrt{\lambda }}{2\alpha }{x}^{\alpha }-\frac{\lambda \sqrt{\lambda }}{2}t)]e^{[\sigma \mathcal{W}\left(t\right)-\frac{1}{2}{\sigma }^{2}t]}.$$

(28)

Remark 2. 

If we set $\alpha =1$ and $\sigma =\beta =0$ in Equation (26) or in Equation (28), we have the same solution as that stated in [34]:

$$\phi (x,t)=\frac{\lambda }{2}{\mathrm{sech}}^2(\frac{\sqrt{\lambda }}{2}x-\frac{\lambda \sqrt{\lambda }}{2}t)].$$

5. The Influence of Noise and the Fractional Derivative

The impacts of white noise and the fractional derivative on the exact solutions of the SFSKdVE (1) are discussed here. According to the relevant studies [45,46,47], nowadays, the destabilizing and stabilizing effects brought on by noisy terms in deterministic systems are well recognized. There is no longer any doubt that these effects are crucial for comprehending the long-term behavior of genuine systems. To illustrate the behavior of these solutions, we offer a variety of graphical representations. We simulated some figures for some of the solutions that were achieved, such as Equations (25) and (26) for different $\sigma$ (noise strength). Let us first fix the parameters $\lambda =1$ and $m=0.5$. In addition, let $x\in [0,5]$ and $t\in [0,2].$

First, the effects of noise:

In Figure 1, the surface has some irregularities and is not flat when there is no noise (i.e., $\sigma =0$).

Figure 1. A 3D-plot of the solution $\phi (x,t)$ in Equation (25), which is a singular periodic soliton, and Equation (26), which is a singular and bright soliton.

However, we can see in Figure 2 and Figure 3 that, after minor transit behaviors, the periodic surface becomes more planar when the noise strength exceeds zero.

Figure 2. A 3D-plot of the solution $\phi (x,t)$ in Equation (25) for $\sigma =1,2$.

Figure 3. A 3D-plot of the solution $\phi (x,t)$ in Equation (26) for $\sigma =1,2$.

Second, the effects of the M-truncated derivative: In Figure 4 and Figure 5, we can observe that, if $\sigma =0$ and $m=0.5$, the surface moves to the left and shrinks when $\alpha$ is decreasing:

Figure 4. For Equation (25), (a–c) indicate the 3D profiles with $\sigma =0$, (d) denotes the 2D plot for different values of $\alpha$ at $t=1$, and the curves of the solution move to the left. (a) $\alpha =1,\beta =0,$ (b)$\alpha =0.7,\beta =0.9,$ and (c) $\alpha =0.5,\beta =0.9$.

Figure 5. For Equation (26), (a–c) indicate the 3D profiles with $\sigma =0$, (d) denotes the 2D plot for different values of $\alpha$ at $t=1$, and the curves of the solution move to the left. (a) $\alpha =1,\beta =0,$ (b)$\alpha =0.7,\beta =0.9,$ and (c) $\alpha =0.5,\beta =0.9$.

From the above figures, we deduce that multiplicative white noise stabilizes the solutions, whereas the fractional space—in the sense of the M-truncated derivative—has an effect on the surface and makes it move to the left.

6. Conclusions

We investigated the fractional-stochastic KdV equation (Equation (1)) in the sense of the M-truncated derivative. The given stochastic term in Equation (1) is multiplicative white noise in the Itô sense. We were able to obtain the exact solutions by using the tanh–coth and JEF methods. These solutions are critical in the characterization of a variety of fascinating and complicated physical phenomena due to the significance of the KdV equation in explaining acoustic waves on a crystal lattice, ion acoustic waves in plasma, lengthy internal waves in a density-stratified ocean, and weakly interacting shallow-water waves. In addition, we generalized some earlier studies, such as [34]. The impacts of multiplicative white noise and the fractional derivative on the analytical solution of the SFSKdVE (1) were finally demonstrated by using a MATLAB package. We concluded that multiplicative white noise stabilizes the solutions, whereas the fractional space—in the sense of the M-truncated derivative—moves the surface to the right as the fractional order increases. In upcoming work, we can investigate the KdV equation by using either additive noise or multiplicative colored noise.