New stochastic solitary solutions for the modified Korteweg-de Vries equation with stochastic term/random variable coefficients

Wael W. Mohammed; Farah M. Al-Askar; Wael W. Mohammed; Farah M. Al-Askar

doi:10.3934/math.2024995

AIMS Mathematics

2024, Volume 9, Issue 8: 20467-20481. doi: 10.3934/math.2024995

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New stochastic solitary solutions for the modified Korteweg-de Vries equation with stochastic term/random variable coefficients

Wael W. Mohammed ^{1,2
,
,},
Farah M. Al-Askar ³

1.
Department of Mathematics, College of Science, University of Ha'il, Ha'il 2440, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
3.
Department of Mathematical Science, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia

Received: 11 May 2024 Revised: 31 May 2024 Accepted: 11 June 2024 Published: 24 June 2024
MSC : 60H10, 60H15, 35Q51, 35A20, 83C15

In this research, we are examining the stochastic modified Korteweg-de Vries (SMKdV) equation forced in the Itô sense by multiplicative noise. We use an appropriate transformation to convert the SMKdV equation to another MKdV equation with random variable coefficients (MKdV-RVCs). We use the generalizing Riccati equation mapping and Jacobi elliptic functions methods in order to acquire new trigonometric, hyperbolic, and rational solutions for MKdV-RVCs. After that, we can get the solutions to the SMKdV equation. To our knowledge, this is the first time we have assumed that the solution of the wave equation for the SMKdV equation is stochastic, since all earlier research assumed that it was deterministic. Furthermore, we provide different graphic representations to show the influence of multiplicative noise on the exact solutions of the SMKdV equation.

Keywords:

Citation: Wael W. Mohammed, Farah M. Al-Askar. New stochastic solitary solutions for the modified Korteweg-de Vries equation with stochastic term/random variable coefficients[J]. AIMS Mathematics, 2024, 9(8): 20467-20481. doi: 10.3934/math.2024995

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Abstract

1. Introduction

The Modified Korteweg-de Vries (MKdV) equation ^[1] is a nonlinear partial differential equation used to explain dispersive waves in shallow water. It is an extension of the original KdV equation that has qualities such as soliton solutions, integrability, and wave breaking, making it a useful model in many domains of physics and mathematics. The MKdV equation has several applications, involving the investigation of wave propagation in plasma ^[2], the dynamics of traffic flow ^[3], and fluid mechanics ^[4]. Additionally, it is applied in the field of nonlinear optics to describe pulses composed of multiple optical cycles ^[5]. Therefore, numerous authors have tackled exact solutions for (1.1) using various approaches, including bifurcation ^[6], the Riccati equation method ^[7], ( $G^{\prime }/G$ )-expansion ^[8], the Exp-function method ^[9], the first integral method ^[10], and the tanh method ^[11]. While, Mohammed et al. ^[12] obtained the exact solutions of SMKdV Eq (1.1) by using the mapping method.

On the other hand, random fluctuations in the MKdV equation play an important role in the dynamics of nonlinear dispersive waves. They arise from uncertainties, or noise in the initial conditions or from external forces and can lead to deviations from the deterministic predictions of the equation. The study of these fluctuations has both theoretical and practical effects, helping us to get a better understanding of wave statistical behavior in complex systems as well as improve wave dynamics prediction and control in real-world applications. Various approaches, such as stochastic analysis and numerical simulations, have been developed to study and model these fluctuations, providing valuable insights into their effects on wave propagation.

In this study, we look at the stochastic MKdV (SMKdV) equation induced in the Itô sense by multiplicative noise as follows:

$\begin{equation} \mathcal{Y}_{t}+a\mathcal{Y}^{2}\mathcal{Y}_{x}+b\mathcal{Y}_{xxx} = \sigma \mathcal{YB}_{t}, \end{equation}$

(1.1)

where $\mathcal{Y}$ represents the wave amplitude, $t$ is time, and $x$ is position. The nonlinear term, $a\mathcal{Y}^{2}\mathcal{Y}_{x}$ , describes the self-interaction of the wave, while the dispersive term, $b\mathcal{Y} _{xxx}$ , accounts for the effect of higher-order dispersion. $a$ and $b$ are constants, $\mathcal{B}(t)$ is the Brownian motion, $\mathcal{B}_{t} = \frac{ \partial \mathcal{B}}{\partial t}$ and $\sigma$ is the noise strength.

In all previous studies, all authors assumed the solutions to wave equations for some stochastic differential equations were deterministic, such as the Sasa-Satsuma equation ^[13], the nonlinear Schrödinger equation ^[14], the Davey-Stewartson equations ^[15], the potential Yu-Toda-Sasa-Fukuyama equation ^[16], the Burgers' equation ^[17], the Jimbo-Miwa equation ^[18], the coupled stochastic Korteweg-de Vries equations ^[19], the modified Benjamin-Bona-Mahony equation ^[20], and the Fokas system ^[21]. While we consider in this study that the wave equation is stochastic and its solutions are also stochastic.

Our goal in this paper is to find the exact stochastic solutions to the SMKdV Eq (1.1). To achieve this goal, we convert the SMKdV equation to another MKdV equation with random variable coefficients (MKdVE-RVCs) by using a suitable transformation. After that, we get the exact solutions for MKdVE-RVCs by using the generalizing Riccati equation mapping method (GREM-method) and the Jacobi elliptic method (JEF-method). To our knowledge, this is the first time we have assumed that the solution of the wave equation for the SMKdV equation is stochastic, since all earlier research assumed that it was deterministic. Finally, by using the used transformation, we can acquire the stochastic solutions of SMKdV. These acquired solutions are crucial in understanding several difficult physical processes due to the importance of SMKdV Eq (1.1) in fluid dynamics, nonlinear optics, and plasma physics. In order to see the influence of stochastic terms, we provide some figures by utilizing MATLAB tools.

Here is how the rest of the paper is organized: In Section $2$ , we derive MKdVE-RVCs from SMKdV Eq (1.1) and by utilizing the GREM and JEF-methods to find the exact solutions of MKdVE-RVCs. In Section $3$ , we acquire the solutions of SMKdV Eq (1.1). In Section $4,$ we discuss the results that we obtained. Finally, we present the conclusions of this paper.

2. MKdV equation with RVCs and its solutions

In this section, we obtain the MKdV equation with random variable coefficients (MKdVE-RVCs). By using the transformation

$\begin{equation} \mathcal{Y}(t,x) = \mathcal{Z}(t,x)e^{\sigma \mathcal{B}(t)}, \end{equation}$

(2.1)

and the Itô derivatives rule, one can get the MKdVE-RVCs as follows

$\begin{equation} \mathcal{Z}_{t}+b\mathcal{Z}_{xxx}+A(t)\mathcal{Z}^{2}\mathcal{Z}_{x}+\frac{1 }{2}\sigma ^{2}\mathcal{Z} = 0, \end{equation}$

(2.2)

where $\mathcal{Z}$ is a stochastic real function and $A(t) = ae^{2\sigma \mathcal{B}(t)}$ .

2.1. GREM-method

Here, we utilize the GREM-method stated in ^[22]. To find the solutions of the MKdVE-RVCs (2.2), assuming the solutions of Eq (2.2) have the form

$\begin{equation} \mathcal{Z}(t,x) = \sum\limits_{k = 0}^{M}\alpha _{k}(t)\mathcal{X}^{k}(\xi ),{ \ } \xi = kx+\int_{0}^{t}\lambda (s)ds, \end{equation}$

(2.3)

where

$\begin{equation} \mathcal{X}^{\prime } = s\mathcal{X}^{2}+r\mathcal{X}+p. \end{equation}$

(2.4)

By balancing $\mathcal{Z}^{\prime \prime \prime }$ with $\mathcal{Z}^{2} \mathcal{Z}^{\prime }$ , $\$ we can calculate the value of $M$ as follows:

$\begin{equation*} M = 1. \end{equation*}$

Rewriting Eq (2.3) as

$\begin{equation} \mathcal{Z}(t,x) = \alpha _{0}(t)+\alpha _{1}(t)\mathcal{X}(\xi ). \end{equation}$

(2.5)

Differentiating Eq (2.5) with regards to $t$ and $x$ , we get

$\begin{eqnarray} \mathcal{Z}_{t}(t,x) & = &(\overset{\cdot }{\alpha }_{0}+p\alpha _{1}\lambda )+(\overset{\cdot }{\alpha }_{1}+\alpha _{1}r\lambda )\mathcal{X}+s\lambda \alpha _{1}\mathcal{X}^{2}, \\ \mathcal{Z}_{x}(t,x) & = &k[s\alpha _{1}\mathcal{X}^{2}+r\alpha _{1}\mathcal{X} +p\alpha _{1}], \\ \mathcal{Z}_{xxx}(t,x) & = &k^{3}[6s^{3}\alpha _{1}\mathcal{X} ^{4}+12rs^{2}\alpha _{1}\mathcal{X}^{3}+(7r^{2}s\alpha _{1}+8ps^{2}\alpha _{1})\mathcal{X}^{2} \\ &&+(r^{3}\alpha _{1}+8rps\alpha _{1})\mathcal{X}+(pr^{2}\alpha _{1}+2p^{2}s\alpha _{1})], \\ \mathcal{Z}^{2}\mathcal{Z}_{x} & = &k[(s\alpha _{1}^{3})\mathcal{X} ^{4}+(2s\alpha _{0}\alpha _{1}^{2}+r\alpha _{1}^{3})\mathcal{X}^{3}+(s\alpha _{0}^{2}\alpha _{1}+2r\alpha _{0}\alpha _{1}^{2}+p\alpha _{1}^{3})\mathcal{X} ^{2} \\ &&+(r\alpha _{0}^{2}\alpha _{1}+2p\alpha _{0}\alpha _{1}^{2})\mathcal{X} +p\alpha _{0}^{2}\alpha _{1}]. \end{eqnarray}$

(2.6)

Substituting Eqs (2.5) and (2.6) into Eq (2.2), we obtain a polynomial of degree $4$ in $\mathcal{X}$ as follows

$\begin{equation*} \lbrack 6bs^{3}k^{3}\alpha _{1}+Aks\alpha _{1}^{3}]\mathcal{X} ^{4}+[12rbk^{3}s^{2}\alpha _{1}+2kAs\alpha _{0}\alpha _{1}^{2}+rkA\alpha _{1}^{3}]\mathcal{X}^{3} \end{equation*}$

$\begin{equation*} +[s\lambda \alpha _{1}+7r^{2}bk^{3}s\alpha _{1}+8pbk^{3}s^{2}\alpha _{1}+skA\alpha _{0}^{2}\alpha _{1}+2rkA\alpha _{0}\alpha _{1}^{2}+pkA\alpha _{1}^{3}]\mathcal{X}^{2} \end{equation*}$

$\begin{equation*} \lbrack \overset{\cdot }{\alpha }_{1}+\alpha _{1}r\lambda +bk^{3}r^{3}\alpha _{1}+8rpsbk^{3}\alpha _{1}+rkA\alpha _{0}^{2}\alpha _{1}+2pkA\alpha _{0}\alpha _{1}^{2}+\frac{1}{2}\sigma ^{2}\alpha _{1}]\mathcal{X} \end{equation*}$

$\begin{equation*} +[\overset{\cdot }{\alpha }_{0}+p\alpha _{1}\lambda +pbk^{3}r^{2}\alpha _{1}+2bk^{3}p^{2}s\alpha _{1}+pkA\alpha _{0}^{2}\alpha _{1}+\frac{1}{2} \sigma ^{2}\alpha _{0}] = 0. \end{equation*}$

Equating each coefficient of $\mathcal{X}^{k}$ to zero, we have

$\begin{equation*} 6bs^{3}k^{3}\alpha _{1}+Aks\alpha _{1}^{3} = 0, \end{equation*}$

$\begin{equation*} 12rbk^{3}s^{2}\alpha _{1}+2kAs\alpha _{0}\alpha _{1}^{2}+rkA\alpha _{1}^{3} = 0, \end{equation*}$

$\begin{equation*} s\lambda \alpha _{1}+7r^{2}bk^{3}s\alpha _{1}+8pbk^{3}s^{2}\alpha _{1}+skA\alpha _{0}^{2}\alpha _{1}+2rkA\alpha _{0}\alpha _{1}^{2}+pkA\alpha _{1}^{3} = 0, \end{equation*}$

$\begin{equation*} \overset{\cdot }{\alpha }_{1}+\alpha _{1}r\lambda +bk^{3}r^{3}\alpha _{1}+8rpsbk^{3}\alpha _{1}+rkA\alpha _{0}^{2}\alpha _{1}+2pkA\alpha _{0}\alpha _{1}^{2}+\frac{1}{2}\sigma ^{2}\alpha _{1} = 0, \end{equation*}$

and

$\begin{equation*} \overset{\cdot }{\alpha }_{0}+p\alpha _{1}\lambda +pbk^{3}r^{2}\alpha _{1}+2bk^{3}p^{2}s\alpha _{1}+pkA\alpha _{0}^{2}\alpha _{1}+\frac{1}{2} \sigma ^{2}\alpha _{0} = 0. \end{equation*}$

We solve these equations to get

$\begin{equation} \alpha _{0}(t) = r = 0,{ \ }\alpha _{1} = \ell e^{-\frac{1}{2}\sigma ^{2}t},\ \ b = \frac{-A\ell ^{2}}{6k^{2}s^{2}}e^{-\sigma ^{2}t},\text{ and }\lambda (t) = \frac{akp\ell ^{2}}{3s}e^{2\sigma \mathcal{B}(t)-\sigma ^{2}t}, \end{equation}$

(2.7)

where $\ell$ is a constant. Hence, by utilizing Eqs (2.5) and (2.7), the solutions of MKdVE-RVCs (2.2) are

$\begin{equation} \mathcal{Z}(t,x) = \ell \mathcal{X}(\xi )e^{-\frac{1}{2}\sigma ^{2}t},\ \xi = kx+\frac{akp\ell ^{2}}{3s}\int_{0}^{t}e^{2\sigma \mathcal{B}(\tau )-\sigma ^{2}\tau }d\tau . \end{equation}$

(2.8)

To find $\mathcal{X}$ , there are three various cases for the solutions of Eq (2.4) based on $p$ and $s$ as follows:

Case 1. If $ps > 0,$ then Eq (2.4) has the solutions:

$\begin{equation*} \mathcal{X}_{1}(\xi ) = \sqrt{\frac{p}{s}}\tan \Big(\sqrt{ps}\xi \Big), \end{equation*}$

$\begin{equation*} \mathcal{X}_{2}(\xi ) = -\sqrt{\frac{p}{s}}\cot \Big(\sqrt{ps}\xi \Big), \end{equation*}$

$\begin{equation*} \mathcal{X}_{3}(\xi ) = \sqrt{\frac{p}{s}}\Big(\tan (\sqrt{4ps}\xi )\pm \sec ( \sqrt{4ps}\xi )\Big), \end{equation*}$

$\begin{equation*} \mathcal{X}_{4}(\xi ) = -\sqrt{\frac{p}{s}}\Big(\cot (\sqrt{4ps}\xi )\pm \csc ( \sqrt{4ps}\xi )\Big), \end{equation*}$

$\begin{equation*} \mathcal{X}_{5}(\xi ) = \frac{1}{2}\sqrt{\frac{p}{s}}\Big(\tan (\frac{1}{2} \sqrt{ps}\xi )-\cot (\frac{1}{2}\sqrt{ps}\xi )\Big). \end{equation*}$

Then, MKdVE-RVCs (2.2) has the trigonometric function solutions:

$\begin{equation} \mathcal{Z}_{1}(t,x) = \ell \sqrt{\frac{p}{s}}\Big(\tan (\sqrt{ps}(kx+\frac{ akp\ell ^{2}}{3s}\int_{0}^{t}e^{2\sigma \mathcal{B}(\tau )-\sigma ^{2}\tau }d\tau ))\Big)e^{-\frac{1}{2}\sigma ^{2}t}, \end{equation}$

(2.9)

$\begin{equation} \mathcal{Z}_{2}(t,x) = -\ell \sqrt{\frac{p}{s}}\Big(\cot (\sqrt{ps}(kx+\frac{ akp\ell ^{2}}{3s}\int_{0}^{t}e^{2\sigma \mathcal{B}(\tau )-\sigma ^{2}\tau }d\tau ))\Big)e^{-\frac{1}{2}\sigma ^{2}t},\text{ } \end{equation}$

(2.10)

$\begin{eqnarray} \mathcal{Z}_{3}(t,x) & = &\ell \sqrt{\frac{p}{s}}\Big(\tan (\sqrt{4ps}(kx+ \frac{akp\ell ^{2}}{3s}\int_{0}^{t}e^{2\sigma \mathcal{B}(\tau )-\sigma ^{2}\tau }d\tau )) \\ &&\pm \sec (\sqrt{4ps}(kx+\frac{akp\ell ^{2}}{3s}\int_{0}^{t}e^{2\sigma \mathcal{B}(\tau )-\sigma ^{2}\tau }d\tau ))\Big)e^{-\frac{1}{2}\sigma ^{2}t}, \end{eqnarray}$

(2.11)

$\begin{eqnarray} \mathcal{Z}_{4}(t,x) & = &-\ell \sqrt{\frac{p}{s}}\Big(\cot (\sqrt{4ps}(kx+ \frac{akp\ell ^{2}}{3s}\int_{0}^{t}e^{2\sigma \mathcal{B}(\tau )-\sigma ^{2}\tau }d\tau )) \\ &&\pm \csc (\sqrt{4ps}(kx+\frac{akp\ell ^{2}}{3s}\int_{0}^{t}e^{2\sigma \mathcal{B}(\tau )-\sigma ^{2}\tau }d\tau ))\Big)e^{-\frac{1}{2}\sigma ^{2}t}, \end{eqnarray}$

(2.12)

$\begin{eqnarray} \mathcal{Z}_{5}(t,x) & = &\ell \sqrt{\frac{p}{s}}\Big(\tan (\frac{1}{2}\sqrt{ps }\sqrt{-ps}(kx+\frac{akp\ell ^{2}}{3s}\int_{0}^{t}e^{2\sigma \mathcal{B} (\tau )-\sigma ^{2}\tau }d\tau )) \\ &&-\cot (\frac{1}{2}\sqrt{ps}(kx+\frac{akp\ell ^{2}}{3s}\int_{0}^{t}e^{2 \sigma \mathcal{B}(\tau )-\sigma ^{2}\tau }d\tau ))\Big)e^{-\frac{1}{2} \sigma ^{2}t}. \end{eqnarray}$

(2.13)

Case 2. If $ps < 0,$ then Eq (2.4) has the solutions:

$\begin{equation*} \mathcal{X}_{6}(\xi ) = -\sqrt{\frac{-p}{s}}\tanh \Big(\sqrt{-ps}\xi \Big), \end{equation*}$

$\begin{equation*} \mathcal{X}_{7}(\xi ) = -\sqrt{\frac{-p}{s}}\coth \Big(\sqrt{-ps}\xi \Big), \end{equation*}$

$\begin{equation*} \mathcal{X}_{8}(\xi ) = -\sqrt{\frac{-p}{s}}\Big(\coth (\sqrt{-4ps}\xi )\pm \text{csch}(\sqrt{-4ps}\xi )\Big), \end{equation*}$

$\begin{equation*} \mathcal{X}_{9}(\xi ) = \frac{-1}{2}\sqrt{\frac{-p}{s}}\Big(\tanh (\frac{1}{2} \sqrt{-ps}\xi )+\coth (\frac{1}{2}\sqrt{-ps}\xi )\Big). \end{equation*}$

Then, MKdVE-RVCs (2.2) has the hyperbolic function solution:

$\begin{equation} \mathcal{Z}_{6}(t,x) = -\ell \sqrt{\frac{-p}{s}}\Big(\tanh (\sqrt{-ps}(kx+ \frac{akp\ell ^{2}}{3s}\int_{0}^{t}e^{2\sigma \mathcal{B}(\tau )-\sigma ^{2}\tau }d\tau ))\Big)e^{-\frac{1}{2}\sigma ^{2}t}, \end{equation}$

(2.14)

$\begin{equation} \mathcal{Z}_{7}(t,x) = -\ell \sqrt{\frac{-p}{s}}\Big(\coth (\sqrt{-ps}\xi (kx+ \frac{akp\ell ^{2}}{3s}\int_{0}^{t}e^{2\sigma \mathcal{B}(\tau )-\sigma ^{2}\tau }d\tau ))\Big)e^{-\frac{1}{2}\sigma ^{2}t}, \end{equation}$

(2.15)

$\begin{eqnarray} \mathcal{Z}_{8}(t,x) & = &-\ell \sqrt{\frac{-p}{s}}\Big(\coth (\sqrt{-4ps}(kx+ \frac{akp\ell ^{2}}{3s}\int_{0}^{t}e^{2\sigma \mathcal{B}(\tau )-\sigma ^{2}\tau }d\tau )) \\ &&\pm \text{csch}(\sqrt{-4ps}(kx+\frac{akp\ell ^{2}}{3s}\int_{0}^{t}e^{2 \sigma \mathcal{B}(\tau )-\sigma ^{2}\tau }d\tau ))\Big)e^{-\frac{1}{2} \sigma ^{2}t}, \end{eqnarray}$

(2.16)

$\begin{eqnarray} \mathcal{Z}_{9}(t,x) & = &-\frac{\ell }{2}\sqrt{\frac{-p}{s}}\Big(\tanh (\frac{ 1}{2}\sqrt{-ps}(kx+\frac{akp\ell ^{2}}{3s}\int_{0}^{t}e^{2\sigma \mathcal{B} (\tau )-\sigma ^{2}\tau }d\tau )) \\ &&+\coth (\frac{1}{2}\sqrt{-ps}(kx+\frac{akp\ell ^{2}}{3s} \int_{0}^{t}e^{2\sigma \mathcal{B}(\tau )-\sigma ^{2}\tau }d\tau ))\Big)e^{- \frac{1}{2}\sigma ^{2}t}. \end{eqnarray}$

(2.17)

Case 3. If $p = 0, \$ and $s\neq 0,$ then the solution of Eq (2.4) is

$\begin{equation*} \mathcal{X}_{10}(\xi ) = \frac{-1}{s\xi }. \end{equation*}$

Hence, the MKdVE-RVCs (2.2) have the rational function solution:

$\begin{equation} \mathcal{Z}_{10}(t,x) = \Big(\frac{-\ell }{(skx+\frac{akp\ell ^{2}}{3} \int_{0}^{t}e^{2\sigma \mathcal{B}(\tau )-\sigma ^{2}\tau }d\tau )}\Big)e^{- \frac{1}{2}\sigma ^{2}t}. \end{equation}$

(2.18)

2.2. JEF-method

We use here the JEF-method stated in ^[23]. Supposing the solutions of MKdVE-RVCs (2.2), with $N = 1$ , have the form

$\begin{equation} \mathcal{Z}(t,x) = a_{0}(t)+a_{1}(t)J(\eta ), \end{equation}$

(2.19)

where $J(\eta)$ is one of the following elliptic functions: $sn(\omega \eta, \check{n}), \ cn(\omega \eta, \check{n})$ , or $dn(\omega \eta, \check{n})$ . Differentiating Eq (2.19) with respect to $t, \; x$ , and $y$ , we get

$\begin{eqnarray} \mathcal{Z}_{t} & = &\overset{\cdot }{a}_{0}+\overset{\cdot }{a}_{1}J+\omega \lambda a_{1}J^{\prime },\ \ \mathcal{Z}_{x} = \omega ka_{1}J^{\prime },\ \ \\ \mathcal{Z}_{xx} & = &k^{2}a_{1}(B_{1}J+B_{2}J^{3}),\ \mathcal{Z} _{xxx} = k^{3}a_{1}(B_{1}+2B_{2}J^{2})J^{\prime }, \\ \mathcal{Z}_{x}\mathcal{Z}^{2} & = &\omega ka_{1}(a_{1}^{2}J^{2}+2a_{0}a_{1}J+a_{0}^{2})J^{\prime }, \end{eqnarray}$

(2.20)

where $B_{1}$ and $B_{2}$ are constants depending on $\omega, \ \check{n},$ and they will be defined later. Plugging Eqs (2.19) and (2.20) into MKdVE-RVCs (2.2). After that, by putting each coefficient of $J^{\prime }J^{k}$ equal to zero, we acquire

$\begin{eqnarray*} J^{0} &:&\ \ \overset{\cdot }{a}_{0}+\frac{1}{2}\sigma ^{2}a_{0} = 0, \\ J^{1} &:&\ \ \overset{\cdot }{a}_{1}+\frac{1}{2}\sigma ^{2}a_{1} = 0, \\ J^{0}J^{\prime } &:&\omega a_{1}[\lambda +bk^{3}B_{1}+ka_{0}^{2}A(t)] = 0, \\ JJ^{\prime } &:&\ 2\omega ka_{0}a_{1}^{2}A(t) = 0, \end{eqnarray*}$

and

$\begin{equation*} J^{2}J^{\prime }:\ \ \ \ 2b\omega k^{3}a_{1}B_{2}+\omega ka_{1}^{3}A(t) = 0. \end{equation*}$

We solve these equations to get

$\begin{equation*} a_{0}(t) = 0,{ \ }a_{1} = \ell e^{-\frac{1}{2}\sigma ^{2}t},\ b = \frac{-\ell ^{2}A(t)}{2k^{2}B_{2}}e^{-\sigma ^{2}t}\text{, }\ \ \lambda (t) = \frac{\ell ^{2}kB_{1}}{2B_{2}}A(t)e^{-\sigma ^{2}t}, \end{equation*}$

where $\ell$ is a constant. Hence, the solution of the MKdVE-RVCs (2.2) is

$\begin{equation} \mathcal{Z}(t,x) = \ell J(\eta ),\ \eta = kx+\frac{a\ell ^{2}kB_{1}}{2B_{2}} \int_{0}^{t}e^{2\sigma \mathcal{B}(\tau )-\sigma ^{2}\tau }d\tau . \end{equation}$

(2.21)

Let us now define $J(\eta)$ as follows:

Set 1. If $J(\eta) = sn(\omega \eta, \check{n})$ , then Eq (2.21) takes the form

$\begin{equation} \mathcal{Z}(t,x) = \ell \Big(sn(k\omega x+\frac{a\omega \ell ^{2}kB_{1}}{2B_{2} }\int_{0}^{t}e^{2\sigma \mathcal{B}(\tau )-\sigma ^{2}\tau }d\tau ,\check{n}) \Big)e^{-\frac{1}{2}\sigma ^{2}t}, \end{equation}$

(2.22)

where

$\begin{equation*} B_{1} = -\omega ^{2}(1+\check{n}^{2})\text{ and }B_{2} = 2\omega ^{2}\check{n} ^{2}. \end{equation*}$

Set 2. If $J(\eta) = cn(\omega \eta, \check{n})$ , then Eq (2.21) takes the form

$\begin{equation} \mathcal{Z}(t,x) = \ell \Big(cn(k\omega x+\frac{a\omega \ell ^{2}kB_{1}}{2B_{2} }\int_{0}^{t}e^{2\sigma \mathcal{B}(\tau )-\sigma ^{2}\tau }d\tau ,\check{n}) \Big)e^{-\frac{1}{2}\sigma ^{2}t}, \end{equation}$

(2.23)

where

$\begin{equation*} B_{1} = \omega ^{2}(1-2\check{n}^{2})\text{ and }B_{2} = -2\omega ^{2}\check{n} ^{2}. \end{equation*}$

Set 3. If $J(\eta) = dn(\omega \eta, \check{n})$ , then Eq (2.21) takes the form

$\begin{equation} \mathcal{Z}(t,x) = \ell \Big(dn(k\omega x+\frac{a\omega \ell ^{2}kB_{1}}{2B_{2} }\int_{0}^{t}e^{2\sigma \mathcal{B}(\tau )-\sigma ^{2}\tau }d\tau ,\check{n}) \Big)e^{-\frac{1}{2}\sigma ^{2}t}, \end{equation}$

(2.24)

where

$\begin{equation*} B_{1} = \omega ^{2}(2-\check{n}^{2})\text{ and }B_{2} = -2\omega ^{2}. \end{equation*}$

3. Exact solutions of stochastic MKdV equation

Now, we can use the solutions of MKdVE-RVCs (2.2) that we obtained in the previous section to get the solutions of SMKdV Eq (1.1) as follows:

3.1. GREM-method

Substituting Eqs (2.9)–(2.18) into Eq (2.1), we get the solutions of SMKdV Eq (1.1) as:

$\begin{equation} \mathcal{Y}_{1}(t,x) = \ell \sqrt{\frac{p}{s}}\Big(\tan (\sqrt{ps}(kx+\frac{ akp\ell ^{2}}{3s}\int_{0}^{t}e^{2\sigma \mathcal{B}(\tau )-\sigma ^{2}\tau }d\tau ))\Big)e^{\sigma \mathcal{B}(t)-\frac{1}{2}\sigma ^{2}t}, \end{equation}$

(3.1)

$\begin{equation} \mathcal{Y}_{2}(t,x) = -\ell \sqrt{\frac{p}{s}}\Big(\cot (\sqrt{ps}(kx+\frac{ akp\ell ^{2}}{3s}\int_{0}^{t}e^{2\sigma \mathcal{B}(\tau )-\sigma ^{2}\tau }d\tau ))\Big)e^{\sigma \mathcal{B}(t)-\frac{1}{2}\sigma ^{2}t},\text{ } \end{equation}$

(3.2)

$\begin{eqnarray} \mathcal{Y}_{3}(t,x) & = &\ell \sqrt{\frac{p}{s}}\Big(\tan (\sqrt{4ps}(kx+ \frac{akp\ell ^{2}}{3s}\int_{0}^{t}e^{2\sigma \mathcal{B}(\tau )-\sigma ^{2}\tau }d\tau )) \\ &&\pm \sec (\sqrt{4ps}(kx+\frac{akp\ell ^{2}}{3s}\int_{0}^{t}e^{2\sigma \mathcal{B}(\tau )-\sigma ^{2}\tau }d\tau ))\Big)e^{\sigma \mathcal{B}(t)- \frac{1}{2}\sigma ^{2}t}, \end{eqnarray}$

(3.3)

$\begin{eqnarray} \mathcal{Y}_{4}(t,x) & = &-\ell \sqrt{\frac{p}{s}}\Big(\cot (\sqrt{4ps}(kx+ \frac{akp\ell ^{2}}{3s}\int_{0}^{t}e^{2\sigma \mathcal{B}(\tau )-\sigma ^{2}\tau }d\tau )) \\ &&\pm \csc (\sqrt{4ps}(kx+\frac{akp\ell ^{2}}{3s}\int_{0}^{t}e^{2\sigma \mathcal{B}(\tau )-\sigma ^{2}\tau }d\tau ))\Big)e^{\sigma \mathcal{B}(t)- \frac{1}{2}\sigma ^{2}t}, \end{eqnarray}$

(3.4)

$\begin{eqnarray} \mathcal{Y}_{5}(t,x) & = &\ell \sqrt{\frac{p}{s}}\Big(\tan (\frac{1}{2}\sqrt{ps }\sqrt{-ps}(kx+\frac{akp\ell ^{2}}{3s}\int_{0}^{t}e^{2\sigma \mathcal{B} (\tau )-\sigma ^{2}\tau }d\tau )) \\ &&-\cot (\frac{1}{2}\sqrt{ps}(kx+\frac{akp\ell ^{2}}{3s}\int_{0}^{t}e^{2 \sigma \mathcal{B}(\tau )-\sigma ^{2}\tau }d\tau ))\Big)e^{\sigma \mathcal{B} (t)-\frac{1}{2}\sigma ^{2}t}, \end{eqnarray}$

(3.5)

for $ps > 0,$

$\begin{equation} \mathcal{Y}_{6}(t,x) = -\ell \sqrt{\frac{-p}{s}}\Big(\tanh (\sqrt{-ps}(kx+ \frac{akp\ell ^{2}}{3s}\int_{0}^{t}e^{2\sigma \mathcal{B}(\tau )-\sigma ^{2}\tau }d\tau ))\Big)e^{\sigma \mathcal{B}(t)-\frac{1}{2}\sigma ^{2}t}, \end{equation}$

(3.6)

$\begin{equation} \mathcal{Y}_{7}(t,x) = -\ell \sqrt{\frac{-p}{s}}\Big(\coth (\sqrt{-ps}\xi (kx+ \frac{akp\ell ^{2}}{3s}\int_{0}^{t}e^{2\sigma \mathcal{B}(\tau )-\sigma ^{2}\tau }d\tau ))\Big)e^{\sigma \mathcal{B}(t)-\frac{1}{2}\sigma ^{2}t}, \end{equation}$

(3.7)

$\begin{eqnarray} \mathcal{Y}_{8}(t,x) & = &-\ell \sqrt{\frac{-p}{s}}\Big(\coth (\sqrt{-4ps}(kx+ \frac{akp\ell ^{2}}{3s}\int_{0}^{t}e^{2\sigma \mathcal{B}(\tau )-\sigma ^{2}\tau }d\tau )) \\ &&\pm \text{csch}(\sqrt{-4ps}(kx+\frac{akp\ell ^{2}}{3s}\int_{0}^{t}e^{2 \sigma \mathcal{B}(\tau )-\sigma ^{2}\tau }d\tau ))\Big)e^{\sigma \mathcal{B} (t)-\frac{1}{2}\sigma ^{2}t}, \end{eqnarray}$

(3.8)

$\begin{eqnarray} \mathcal{Y}_{9}(t,x) & = &-\frac{\ell }{2}\sqrt{\frac{-p}{s}}\Big(\tanh (\frac{ 1}{2}\sqrt{-ps}(kx+\frac{akp\ell ^{2}}{3s}\int_{0}^{t}e^{2\sigma \mathcal{B} (\tau )-\sigma ^{2}\tau }d\tau )) \\ &&+\coth (\frac{1}{2}\sqrt{-ps}(kx+\frac{akp\ell ^{2}}{3s} \int_{0}^{t}e^{2\sigma \mathcal{B}(\tau )-\sigma ^{2}\tau }d\tau ))\Big) e^{\sigma \mathcal{B}(t)-\frac{1}{2}\sigma ^{2}t}, \end{eqnarray}$

(3.9)

for $ps < 0,$ and

$\begin{equation} \mathcal{Y}_{10}(t,x) = \Big(\frac{-\ell }{(skx+\frac{akp\ell ^{2}}{3} \int_{0}^{t}e^{2\sigma \mathcal{B}(\tau )-\sigma ^{2}\tau }d\tau )}\Big) e^{\sigma \mathcal{B}(t)-\frac{1}{2}\sigma ^{2}t}, \end{equation}$

(3.10)

for $p = 0$ and $s\neq 0.$

Remark 1. Putting $p = -1, \ s = a, \ \ell _{1} = -\sqrt{c}, \; k = \sqrt{\frac{-c}{2b}}$ , and $\sigma = 0$ (i.e., no noise) in Eqs (3.1) and (3.2) we have the results stated in ^[11] as follows:

$\begin{equation*} \mathcal{Y}(t,x) = \sqrt{\frac{c}{a}}\tanh (\sqrt{\frac{-c}{2b}}(x-ct))\Big), \end{equation*}$

and

$\begin{equation*} \mathcal{Y}(t,x) = \sqrt{\frac{c}{a}}\coth (\sqrt{\frac{-c}{2b}}(x-ct))\Big). \end{equation*}$

3.2. JEF-method

Substituting Eqs (2.22)–(2.24) into Eq (2.1), we have the SMKdV Eq (1.1):

$\begin{equation} \mathcal{Y}(t,x) = \ell \Big(sn(k\omega x-\frac{a\omega \ell ^{2}k(1+\check{n} ^{2})}{4\check{n}^{2}}\int_{0}^{t}e^{2\sigma \mathcal{B}(\tau )-\sigma ^{2}\tau }d\tau ,\check{n})\Big)e^{\sigma \mathcal{B}(t)-\frac{1}{2}\sigma ^{2}t}, \end{equation}$

(3.11)

$\begin{equation} \mathcal{Y}(t,x) = \ell \Big(cn(k\omega x+\frac{a\omega \ell ^{2}k(1-2\check{n} ^{2})}{4\check{n}^{2}}\int_{0}^{t}e^{2\sigma \mathcal{B}(\tau )-\sigma ^{2}\tau }d\tau ,\check{n})\Big)e^{\sigma \mathcal{B}(t)-\frac{1}{2}\sigma ^{2}t}, \end{equation}$

(3.12)

and

$\begin{equation} \mathcal{Y}(t,x) = \ell \Big(dn(k\omega x-\frac{a\omega \ell ^{2}k(2-\check{n} ^{2})}{4}\int_{0}^{t}e^{2\sigma \mathcal{B}(\tau )-\sigma ^{2}\tau }d\tau , \check{n})\Big)e^{\sigma \mathcal{B}(t)-\frac{1}{2}\sigma ^{2}t}. \end{equation}$

(3.13)

If $\check{n}\rightarrow 1, \$ then the Eqs (3.11)–(3.13) become

$\begin{equation} \mathcal{Y}(t,x) = \ell \Big(\tanh (k\omega x-\frac{a\omega \ell ^{2}k}{2} \int_{0}^{t}e^{2\sigma \mathcal{B}(\tau )-\sigma ^{2}\tau }d\tau ,\check{n}) \Big)e^{\sigma \mathcal{B}(t)-\frac{1}{2}\sigma ^{2}t}, \end{equation}$

(3.14)

and

$\begin{equation} \mathcal{Y}(t,x) = \ell \Big(\text{sech}(k\omega x-\frac{a\omega \ell ^{2}k}{2} \int_{0}^{t}e^{2\sigma \mathcal{B}(\tau )-\sigma ^{2}\tau }d\tau ,\check{n}) \Big)e^{\sigma \mathcal{B}(t)-\frac{1}{2}\sigma ^{2}t}. \end{equation}$

(3.15)

4. Discussion and impacts of noise

4.1. Discussion

Here, we obtained the solutions of the SMKdV Eq (1.1). We used the METF and JEF methods, which produced a wide range of solutions, including optical trigonometric solutions (3.1)–(3.5), optical hyperbolic solutions (3.6)–(3.9), optical rational solution (3.10), and optical elliptic solutions (3.11)–(3.13). Optical solutions are an effective tool for studying the behavior of solutions to the modified KdV equations, as they provide a unique viewpoint on wave dynamics and interactions in complex systems. Furthermore, optical solutions enable researchers to look at the stability and nonlinear dynamics of solutions to MKdV equations. Using optical solutions, researchers may investigate the system's nonlinear effects, such as wave breaking and soliton formation, which are important for understanding the solutions' long-term behavior. By investigating the stability of optical systems, researchers may gain a better understanding of the system's behavior and make more accurate predictions about its future evolution.

4.2. Impacts of noise

The impact of multiplicative noise on the exact solution of SMKdV Eq (1.1) is examined in this section. For our knowledge, the key difference between the solutions provided here and the ones acquired in ^[12] is the amplitude function $\mathcal{Z} (t, x).$ Here, $\mathcal{Z}(t, x)$ is a stochastic function, while $\mathcal{Z }(t, x)$ is an assumed deterministic function in ^[12]. Numerous numerical simulations of various solutions with different intensities of noise are shown. – display the solutions $\mathcal{Z}(t, x)$ described in Eqs (3.11), (3.14), and (3.15), respectively, for various amplitudes of noise $\sigma$ as follows:

Figure 1. (a–e) show 3D-profile of

$\mathcal{Z}(t, x)$ described in Eq (3.11) with

$\ k = 1$ ,

$\ \ell = \check{n} = 0.5, \ a = \omega = 1, \ x\in \lbrack -4, 4]$ ,

$t\in \lbrack 0, 2]$ , (f) exhibits 2D-profile of Eq (3.11) with distinct

$\sigma$ .

DownLoad: Full-Size Img PowerPoint

Figure 2. (a–e) display 3D-shape of

$\mathcal{Z}(t, x)$ described in Eq (3.14) with

$\ell = \omega = k = 1, \ a = 1, \ x\in \lbrack -4, 4]$ , and

$t\in \lbrack 0, 2]$ , (f) exhibits 2D-shape of Eq (3.14) with different

$\sigma$ .

DownLoad: Full-Size Img PowerPoint

Figure 3. (a–e) show 3D-shape of

$\mathcal{Z}(t, x)$ described in Eq (3.15) with

$\ell = \omega = k = 1, \ a = 1, \ x\in \lbrack -4, 4]$ , and

$t\in \lbrack 0, 2]$ , (f) exhibits 2D-shape of Eq (3.15) with various

$\sigma$ .

DownLoad: Full-Size Img PowerPoint

– show that when noise is ignored (i.e., $\sigma = 0$ ), numerous types of solutions emerge, including optical periodic solutions, optical singular solutions, optical kink solutions, and so on. When noise is introduced at $\sigma = 0.1, \ 0.4, \ 1, \ 2$ , the surface flattens after some transit patterns. This result shows how multiplicative Brownian motion affects the SMKdV Eq (1.1) solutions, stabilizing them around zero.

5. Conclusions

In this paper, we looked at the SMKdV Eq (1.1) driven by multiplicative noise in the Itô sense. By using appropriate transformations, we converted the SMKdV equation to another MKdV equation with random variable coefficients (MKdV-RVCs) (2.2). Using the GREM-method and the JEF-method, we obtained a new stochastic exact solutions for MKdV-RVCs in the form of trigonometric, hyperbolic, and rational functions. After that, we acquired the obtained solutions of SMKdV (1.1). Moreover, we generated some previous solutions, such as the results reported in ^[11]. Because of the importance of MKdV equation used in fluid dynamics, nonlinear optics, and plasma physics, the acquired solutions are crucial in understanding several difficult physical processes. Finally, some graphics were included to demonstrate the effect of the stochastic term on the stochastic exact solutions of the SMKdV equation.

Author contributions

Wael W. Mohammed: Conceptualization, Methodology, Software, Formal analysis, Writing-original draft; Farah M. Al-Askar: Conceptualization, Formal analysis, Writing-original draft. All authors have read and approved the final version of the manuscript for publication.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

Princess Nourah bint Abdulrahman University Researcher Supporting Project number (PNURSP2024R 273), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Conflict of interest

The authors declare that they have no competing interests.

References

[1]	S. Tanaka, Modified Korteweg-de Vries equation and scattering theory, P. Jpn. Acad., 48 (1972), 466–469. http://dx.doi.org/10.3792/pja/1195519590 doi: 10.3792/pja/1195519590
[2]	A. H. Khater, O. H. El-Kalaawy, D. K. Callebaut, Bäcklund transformations and exact solutions for Alfvén solitons in a relativistic electron-positron plasma, Phys. Scripta, 58 (1998), 545. http://dx.doi.org/10.1088/0031-8949/58/6/001 doi: 10.1088/0031-8949/58/6/001
[3]	Z. P. Li, Y. C. Liu, Analysis of stability and density waves of traffic flow model in an ITS environment, Eur. Phys. J. B, 53 (2006), 367–374. https://doi.org/10.1140/epjb/e2006-00382-7 doi: 10.1140/epjb/e2006-00382-7
[4]	M. A. Helal, Soliton solution of some nonlinear partial differential equations and its applications in fluid mechanics, Chaos Solition. Fract., 13 (2002), 1917–1929. https://doi.org/10.1016/S0960-0779(01)00189-8 doi: 10.1016/S0960-0779(01)00189-8
[5]	H. Leblond, D. Mihalache, Few-optical-cycle solitons: Modified Kortewegde Vries sine-Gordon equation versus other non-slowly-varying-envelopeapproximation models, Phys. Rev. A, 79 (2009).
[6]	A. A. Elmandouha, A. G. Ibrahim, Bifurcation and travelling wave solutions for a (2+1)-dimensional KdV equation, J. Taibah Univ. Sci., 14 (2020), 139–147. https://doi.org/10.1080/16583655.2019.1709271 doi: 10.1080/16583655.2019.1709271
[7]	F. M. Al-Askar, C. Cesarano, W. W. Mohammed, Effects of the wiener process and beta derivative on the exact solutions of the kadomtsev-petviashvili equation, Axioms, 12 (2023), 748. https://doi.org/10.3390/axioms12080748 doi: 10.3390/axioms12080748
[8]	N. Taghizadeh, Comparison of solutions of mKdV equation by using the first integral method and ( $G^{\prime }/G$ )-expansion method, Math. Aeterna, 2 (2012), 309–320.
[9]	K. R. Raslan, The application of He's Exp-function method for mKdV and Burgers' equations with variable coefficients, Int. J. Nonlin. Sci., 7 (2009), 174–181. https://doi.org/10.1016/j.camwa.2009.03.019 doi: 10.1016/j.camwa.2009.03.019
[10]	Y. Yang, Exact solutions of the mKdV equation, IOP Conference Series: Earth and Environmental Science, IOP Publishing, 769 (2021), 042040. https://doi.org/10.1088/1755-1315/769/4/042040
[11]	A. M. Wazwaz, The tanh method for generalized forms of nonlinear heat conduction and Burgers-Fisher equations, Appl. Math. Comput., 169 (2005), 321–338. https://doi.org/10.1016/j.amc.2004.09.054 doi: 10.1016/j.amc.2004.09.054
[12]	W. W. Mohammed, F. M. Al-Askar, C. Cesarano, The analytical solutions of the stochastic mKdV equation via the mapping method, Mathematics, 10 (2022), 4212. https://doi.org/10.3390/math10224212 doi: 10.3390/math10224212
[13]	C. Liu, Z. Li, The dynamical behavior analysis and the traveling wave solutions of the stochastic Sasa-Satsuma equation, Qual. Theory Dyn. Syst., 23 (2024), 157. https://doi.org/10.1007/s12346-024-01022-y doi: 10.1007/s12346-024-01022-y
[14]	Z. Li, C. Liu, Chaotic pattern and traveling wave solution of the perturbed stochastic nonlinear Schrödinger equation with generalized anti-cubic law nonlinearity and spatio-temporal dispersion, Results Phys., 56 (2024), 107305. https://doi.org/10.1016/j.rinp.2023.107305 doi: 10.1016/j.rinp.2023.107305
[15]	C. Liu, Z. Li, Multiplicative brownian motion stabilizes traveling wave solutions and dynamical behavior analysis of the stochastic Davey-Stewartson equations, Results Phys., 53 (2023), 106941. https://doi.org/10.1016/j.rinp.2023.106941 doi: 10.1016/j.rinp.2023.106941
[16]	S. Albosaily, E. M. Elsayed, M. D. Albalwi, M. Alesemi, W. W. Mohammed, The analytical stochastic solutions for the stochastic Potential Yu-Toda-Sasa-Fukuyama equation with conformable derivative using different methods, Fractal Fract., 7 (2023), 787. https://doi.org/10.3390/fractalfract7110787 doi: 10.3390/fractalfract7110787
[17]	M. Z. Baber, N. Ahmed, M. S. Iqbal, Exact solitary wave propagations for the stochastic Burgers' equation under the influence of white noise and its comparison with computational scheme, Sci. Rep., 14 (2024), 10629. https://doi.org/10.1038/s41598-024-58553-2 doi: 10.1038/s41598-024-58553-2
[18]	F. M. Al-Askar, C. Cesarano, W. W. Mohammed, The solitary solutions for the stochastic Jimbo-Miwa equation perturbed by White noise, Symmetry, 15 (2023), 1153. https://doi.org/10.3390/sym15061153 doi: 10.3390/sym15061153
[19]	W. W. Mohammed, F. M. Al-Askar, C. Cesarano, On the dynamical behavior of solitary waves for coupled stochastic Korteweg-De Vries equations, Mathematics, 11 (2023), 3506. https://doi.org/10.3390/math11163506 doi: 10.3390/math11163506
[20]	A. Elmandouh, E. Fadhal, Bifurcation of exact solutions for the space-fractional stochastic modified Benjamin-Bona-Mahony equation, Fractal Fract., 6 (2022), 718. https://doi.org/10.3390/fractalfract6120718 doi: 10.3390/fractalfract6120718
[21]	W. W. Mohammed, C. Cesarano, A. A. Elmandouh, I. Alqsair, R. Sidaoui, H. W. Alshammari, Abundant optical soliton solutions for the stochastic fractional fokas system using bifurcation analysis, Phys. Scripta, 99 (2024), 045233. https://doi.org/10.1088/1402-4896/ad30fd doi: 10.1088/1402-4896/ad30fd
[22]	S. D. Zhu, The generalizing Riccati equation mapping method in non-linear evolution equation: Application to (2+1)-dimensional Boiti-Leon-Pempinelle equation, Chaos Solition. Fract., 37 (2008), 1335–1342.
[23]	E. Fan, J. Zhang, Applications of the Jacobi elliptic function method to special-type nonlinear equations, Phys. Lett. A, 305 (2002), 383–392. https://doi.org/10.1016/S0375-9601(02)01516-5 doi: 10.1016/S0375-9601(02)01516-5

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AIMS Mathematics

New stochastic solitary solutions for the modified Korteweg-de Vries equation with stochastic term/random variable coefficients

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Abstract

1. Introduction

2. MKdV equation with RVCs and its solutions

2.1. GREM-method

2.2. JEF-method

3. Exact solutions of stochastic MKdV equation

3.1. GREM-method

3.2. JEF-method

4. Discussion and impacts of noise

4.1. Discussion

4.2. Impacts of noise

5. Conclusions

Author contributions

Use of AI tools declaration

Acknowledgments

Conflict of interest

References

This article has been cited by:

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Abstract

1. Introduction

2. MKdV equation with RVCs and its solutions

2.1. GREM-method

2.2. JEF-method

3. Exact solutions of stochastic MKdV equation

3.1. GREM-method

3.2. JEF-method

4. Discussion and impacts of noise

4.1. Discussion

4.2. Impacts of noise

5. Conclusions

Author contributions

Use of AI tools declaration

Acknowledgments

Conflict of interest

References