EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6587 ISSN 1307-5543 – ejpam.com Published by New York Business Global Random Exact Solutions for the Stochastic Korteweg-de Vries Equation Altaf Alshuhail1, Taher S. Hassan1, Mohamed S. Algolam1, Athar I. Ahmed1, Wael W. Mohammed1,∗ 1 Department of Mathematics, College of Science, University of Ha’il, Ha’il 2440, Saudi Arabia Abstract. This paper considers the stochastic Korteweg-de–de Vries (SKdV) equation perturbed by multiplicative Brownian motion, which is an important model reflecting the nonlinear science. After a systematic change and a rescaling, the SKdV equation is exactly recast into a deterministic KdV equation with random variable coefficients (KdV-RVCs). By using the Jacobi elliptic equation method and the generalized Riccati equation mapping approach, we obtain new exact solutions (rational, hyperbolic, trigonometric, and elliptic) for the KdV-RVCs. After that, we use the obtained solutions to obtain the stochastic solutions for the SKdV equation. Of practical interest, these results are related to specific physical systems: magnetized plasmas in astrophysics and in 1D/2D fusion, soliton propagation in fiber optical communication, and surface-wave dynamics in fluid mechanics. For example, the resulting solutions explain how noise-induced perturbations change soliton propagation in optical fibers and stabilize wave patterns in Turbulence. To give some (visual) impression of how multiplicative noise influences the solution behavior, we use pictures of the probability density distributions and ensemble-averaged trajectories as examples. These findings show that multiplicative Brownian motion has a stabilizing effect on the SKdV solutions by keeping their variations more or less near zero. This connection between stochastic modeling and experimental observations in plasma turbulence, nonlinear optical signal processing, and fluid wave dynamics has implications for the development of predictive theories for noise-driven nonlinear systems. 2020 Mathematics Subject Classifications: 35A20, 60H10, 60H15, 35Q51, 83C15 Key Words and Phrases: Exact stochastic solutions, analytical different methods, Brownian motion, random variable coefficients 1. Introduction The KDV equation, also known as the Korteweg-de Vries equation, is a mathematical model that describes the behavior of waves in shallow water. The KDV equation derives its name from the Dutch mathematicians Diederik Korteweg and Gustav de Vries, who first ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6587 Email address: w.mohammed@uoh.edu.sa (W. W. Mohammed) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. Alshuhail et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6587 2 of 16 introduced it in 1895 [1]. Over the years, the KDV equation has gained plenty of attention for its ability to represent complicated wave dynamics and its relevance in understanding various natural phenomena. It has been widely utilized in different areas, such as nonlinear optics, plasma physics, and fluid dynamics [2]. On the other side, it is crucial to consider the effect of random fluctuations in the KDV equation in order to accurately describe and predict the behavior of waves. By considering random fluctuations, the KDV equation can better capture the complex interactions and variations in wave dynamics, leading to more accurate predictions and simulations. Waves in nature rarely propagate in isolation, but rather interact with other waves, obstacles, and boundaries. These interactions can lead to wave-breaking, formation of solitons or rogue waves, and the creation of complex interference patterns. Random fluctuations in the KDV equation enable the modeling of such interactions and phenomena, offering insights into the behavior and characteristics of waves in different situations. This understanding is crucial in various fields, including marine engineering, coastal management, and the study of natural disasters such as tsunamis. Here, we consider the following stochastic KDV equation perturbed by multiplicative noise as follows: Pt + aPPx + bPxxx = σPBt, (1) where P(x, t) the profile of waves at position x and time t, aPPx represents the advection of the wave with a velocity of aP, and bPxxx represents the dispersion of the wave; σ is the amplitude of noise, and B(t) is the Brownian motion, Bt = ∂B ∂t . One of the most distinctive properties of the KDV equation is its soliton solutions. Solitons are solitary waves that retain their speed and shape as they propagate without dispersion or attenuation. The KDV equation generates a class of solitons known as KDV solitons. These solitons have a bell-shaped profile and show intriguing features such as stability, particle-like behavior, and non-interacting properties. They arise due to the delicate balance between the wave advection and dispersion terms within the equation. Therefore, many authors, for example [3–10], have extensively studied multiple versions of the KDV equation utilizing diverse methodology and methods from distinct viewpoints, whereas the analytical solutions of SKDV Eq. (1) is acquired by Mohammed et al. [11]. Furthermore, there are other authors, such as [12–18], have obtained the solutions of Eq. (1) in deterministic case. Our aim of this study is to obtain the analytical stochastic solutions of the SKDV Eq. (1). This is accomplished by transforming the SKDV equation into a different KDV equation with random variable coefficients (KDVE-RVCs) by using the appropriate trans- formation. Moreover, the generalized Riccati equation mapping method (GREM-method) and the Jacobi elliptic method (JEF-method) are used to derive exact solutions for KDVE- RVCs. Finally, utilizing the employed transformation, we may obtain the stochastic solu- tions for the SKDV equation. Since all prior studies supposed that the solutions of wave equation of the SKdV equation was deterministic, the novelty of this work is that we assumed it to be stochastic. The achieved solutions are vital in comprehending different complex physical phenomena due to the relevance of the KDV equation (1) in plasma A. Alshuhail et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6587 3 of 16 physics, nonlinear optics, and fluid dynamics. Also, we extend some of the previous find- ings, such the solutions given in [13]. Finally, we use MATLAB software to provide various graphs that illustrate the influence of the stochastic term on the acquired solutions. The outline of this paper as follows: In Section 2, we conclude KDVE-RVCs from SKDV Eq. (1), then we use the JEF-method and GREM-method to discover the solutions of KDVE-RVCs. Section 3 provides the solutions for SKDV Eq. (1). Section 4, we see the impact of stochastic term on the achieved solutions. Lastly, we present this article’s conclusions. 2. The Derivation and Solutions of KDVE-RVCs To derive the KDVE-RVCs, we take the following transformation P(x, t) = V(x, t)eσB(t). (2) Differentiating Eq. (2) with regards to x and t and utilizing the Itô derivatives rule, we get Px = Vxe σB(t), Pxxx = Vxxxe σB(t). (3) and Pt = Vte σB(t) + (σVBt + 1 2 σ2V)eσB(t), (4) where the term 1 2σ 2V is called the Itô correction term. Now, substituting from Eqs (4) and (3) into Eq. (1), we have the following KDVE-RVCs: Vt + bVxxx +A(t)VVx + 1 2 σ2V = 0, (5) where A(t) = aeσB(t) and V is a real stochastic function. 2.1. JEF-method We apply the JEF-method, as indicated in [19]. Assuming the solutions of KdVE-RVCs (5) have the form V(x, t) = n∑ k=0 ak(t)J k(η), η = kx+ ∫ t 0 λ(s)ds, (6) where J (η) represents one of these elliptic functions: sn(ϖη,m), cn(ϖη,m) or dn(ϖη,m). First let us balance V ′′′ with VV ′ to find n in Eq. (6) as: n+ 3 = n+ n+ 1 =⇒ n = 2. Rewriting Eq. (6) as V(x, t) = a0(t) + a1(t)J (η) + a2(t)J 2(η). (7) A. Alshuhail et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6587 4 of 16 By differentiating Eq. (7) with regards to t and x, we obtain Vt = · a0 + · a1J +ϖλa1J ′ + · a2J 2 + 2ϖλa2J ′J , Vx = (ϖka1 + 2ϖka2J )J ′, Vxx = k2(a1 + 2a2)(B1J +B2J 3) + 2ϖ2k2a2J ′2, Vxxx = ϖk3a1(B1 + 3B2J 2)J ′ + 4ϖk3a2(2B1J + 3B2J 3)J ′, VxV = ϖk(a0a1 + 2a0a2J + a21J + 3a1a2J 2 + 2a22J 3)J ′, (8) where B1 and B2 are constants that based on ϖ, and m, and will be explained later. Putting Eqs. (7) and (8) into KdVE-RVCs (5) and equating all the coefficients of J ′J n to zero, we attain J 0 : · a0 + 1 2 σ2a0 = 0, J : · a1 + 1 2 σ2a1 = 0, J 2 : · a2 + 1 2 σ2a2 = 0, J 0J ′ : ϖa1[λ+ bk3B1 + ka20A(t)] = 0, JJ ′ : 2ϖλa2 + 8bϖk3a2B1 +Aϖka21 + 2ϖkA(t)a0a2 = 0, J 2J ′ : 3bϖk3a1B2 + 3ϖka1a2A(t) = 0, and J 3J ′ : 12bϖk3B2a2 + 2ϖkA(t)a22 = 0. Solving these equations yields a0(t) = ℓ0e − 1 2 σ2t, a1 = 0, a2(t) = ℓ2e − 1 2 σ2t, b = −ℓ2A(t) 6k2B2 e− 1 2 σ2t, and λ(t) = −k( 2ℓ2B1 3B2 + ℓ0)A(t)e− 1 2 σ2t, where ℓ0 and ℓ2 are constants. Hence, the solution of the KdVE-RVCs (5) is V(x, t) = [ℓ0 + ℓ2J 2(η)]e− 1 2 σ2t, η = kx− k( 2ℓ2B1 3B2 + ℓ0) ∫ t 0 eσB(τ)− 1 2 σ2τdτ. (9) In the following, J (η) is defined as: Set 1: When J (η) = sn(ϖη,m), then Eq. (9) takes the form V(x, t) = ( ℓ0 + ℓ2 ( sn(kϖx− kϖ( 2ℓ2B1 3B2 + ℓ0) ∫ t 0 eσB(τ)− 1 2 σ2τdτ,m) )2) e− 1 2 σ2t, (10) where the integral ∫ t 0 e σB(τ)− 1 2 σ2τdτ converges almost surely, B1 = −ϖ2(1 +m2) and B2 = 2ϖ2m2. A. Alshuhail et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6587 5 of 16 Set 2: When J (η) = cn(ϖη,m), then Eq. (9) becomes V(x, t) = ( ℓ0 + ℓ2 ( cn(kϖx− kϖ( 2ℓ2B1 3B2 + ℓ0) ∫ t 0 eσB(τ)− 1 2 σ2τdτ,m) )2) e− 1 2 σ2t, (11) where B1 = ϖ2(1− 2m2) and B2 = −2ϖ2m2. Set 3: When J (η) = dn(ϖη,m), then Eq. (9) becomes V(x, t) = ( ℓ0 + ℓ2 ( dn(kϖx− kϖ( 2ℓ2B1 3B2 + ℓ0) ∫ t 0 eσB(τ)− 1 2 σ2τdτ,m) )2) e− 1 2 σ2t, (12) where B1 = ϖ2(2−m2) and B2 = 2ϖ2. 2.2. GREM-method In this subsection, we employ the GREM-method reported in [20] to obtain the KDVE- RVC solutions. Let the solutions of Eq. (5), with n = 2, take the form V(x, t) = α0(t) + α1(t)X (η) + α2(t)X 2(η), (13) where X ′ = sX 2 + rX + p. (14) Differentiating Eq. (13) with regards to x and t, we attain Vt = ( · α0 + pα1λ) + ( · α1 + α1rλ+ 2pλα2)X +(sλα1 + · α2 + 2λrα2)X 2 + 2sλα2X 3, Vx = k[2sα2X 3 + (sα1 + 2rα2)X 2 + (rα1 + 2pα2)X + pα1], Vxxx = k3[24α2s 3X 5 + (6s3α1 + 54rs2α2)X 4 +(12rs2α1 + 48ps2α2 + 32sr2α2)X 3 +(7r2sα1 + 8ps2α1 + 50rpsα2 + 8r3α2)X 2 +(r3α1 + 8rpsα1 + 14pr2α2)X +(pr2α1 + 2p2sα1 + 6p2rα2)], VVx = k[2sα2 2X 5 + (sα1α2 + 2rα2 2)X 4 + (15) (2sα0α2 + sα2 1 + 3rα1α2 + 2pα2 2)X 3 +(sα0α1 + 2rα0α2 + rα2 1 + 3pα2α1)X 2 +(rα0α1 + 2pα0α2 + pα2 0)X + pα0α1]. Plugging Eqs. (13) and (15) into Eq. (5), we have the following polynomial of degree 5 in X : [24bk3s3α2 + 2Aksα2 2]X 5 A. Alshuhail et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6587 6 of 16 +[skAα1α2 + 2kArα2 2 + 6bk3s3α1 + 54bk3rs2α2]X 4 +[2λsα2 + 12rbk3s2α1 + 48pbk3s2α2 + 32sbk3r2α2 + 2skAα0α2 +skAα2 1 + 3rkAα1α2 + 2kApα2 2]X 3 +[sλα1 + · α2 + 2rα2 + 7bk3r2sα1 + 8pbk3s2α1 + 50rpsbk3α2 + 8bk3r3α2 +skAα0α1 + 2rkAα0α2 + rkAα2 1 + 3kApα2α1 + 1 2 σ2α2]X 2 +[ · α1 + α1rλ+ 2pλα2 + bk3r3α1 + 8rpsbk3α1 + 14pbk3r2α2 +rkAα0α1 + 2pkAα0α2 + pkAα2 0 + 1 2 σ2α1]X +[ · α0 + pα1λ+ pbk3r2α1 + 2bsk3p2α1 + 6rbk3p2α2 + pkAα0α1 + 1 2 σ2α0] = 0. After putting each coefficient of X k to zero, we attain 24bk3s3α2 + 2Aksα2 2 = 0, skAα1α2 + 2kArα2 2 + 6bk3s3α1 + 54bk3rs2α2 = 0, 2sα2 + 12rbk3s2α1 + 48pbk3s2α2 + 32sbk3r2α2 +2skAα0α2 + skAα2 1 + 3rkAα1α2 + 2kApα2 2 = 0, sλα1 + · α2 + 2rα2 + 7bk3r2sα1 + 8pbk3s2α1 + 50rpsbk3α2 +8bk3r3α2 + skAα0α1 + 2rkAα0α2 + rkAα2 1 + 3kApα2α1 + 1 2σ 2α2 = 0, · α1 + α1rλ+ 2pα2 + bk3r3α1 + 8rpsbk3α1 + 14pbk3r2α2 +rkAα0α1 + 2pkAα0α2 + pkAα2 1 + 1 2σ 2α1 = 0, and · α0 + pα1λ+ pbk3r2α1 + 2bsk3p2α1 +6rbk3p2α1 + pkAα0α1 + 1 2σ 2α0 = 0. Solving these equations yields α0(t) = ℓ0e − 1 2 σ2t, α1 = r = 0, α2 = ℓ2e − 1 2 σ2t, b = −Aℓ2 12k2s2 e− 1 2 σ2t, and λ(t) = −akℓ0e σB(t)− 1 2 σ2t, where ℓ0 and ℓ2 are constants. Therefore, by using Eq. (13), the solution of KDVE-RVCs (5) takes the form V(x, t) = (ℓ0 + ℓ2X 2(η))e− 1 2 σ2t, η = kx− akℓ0 ∫ t 0 eσB(τ)− 1 2 σ2τdτ. (16) A. Alshuhail et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6587 7 of 16 To find X , there are many families for the solutions of Eq. (14) relaying on s and p: Family I: If sp > 0, then the solutions of Eq. (14) are X1(η) = √ p s tan (√ psη ) , X2(η) = − √ p s cot (√ psη ) , X3(η) = √ p s ( tan( √ 4psη)± sec( √ 4psη) ) , X4(η) = − √ p s ( cot( √ 4psη)± csc( √ 4psη) ) , X5(η) = 1 2 √ p s ( tan( 1 2 √ psη)− cot( 1 2 √ psη) ) , Then, KDVE-RVCs (5) possess the trigonometric functions solution: V1(x, t) = ( ℓ0 + ℓ2p s tan2( √ psη) ) e− 1 2 σ2t, (17) V2(x, t) = ( ℓ0 − ℓ2p s cot2( √ psη) ) e− 1 2 σ2t, (18) V3(x, t) = ( ℓ0 + ℓ2p s ( tan( √ 4psη)± sec( √ 4psη) )2) e− 1 2 σ2t, (19) V4(x, t) = ( ℓ0 − ℓ2p s ( cot( √ 4psη)± csc( √ 4psη) )2) e− 1 2 σ2t, (20) V5(x, t) = ( ℓ0 + ℓ2p 2s ( tan( 1 2 √ psη)− cot( 1 2 √ psη) )2) e− 1 2 σ2t, (21) where η = kx− akℓ0 ∫ t 0 e σB(τ)− 1 2 σ2τdτ. Family II: If sp < 0, then the solutions of Eq. (14) are X6(η) = − √ −p s tanh (√ −psη ) , X7(η) = − √ −p s coth (√ −psη ) , X8(η) = − √ −p s ( coth( √ −4psη)± csch( √ −4psη) ) , X9(η) = −1 2 √ −p s ( tanh( 1 2 √ −psη) + coth( 1 2 √ −psη) ) . A. Alshuhail et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6587 8 of 16 Hence, KDVE-RVCs (5) possess the following hyperbolic functions solution: V6(x, t) = ( ℓ0 + ℓ2p s tanh2( √ −psη) ) e− 1 2 σ2t, (22) V7(x, t) = ( ℓ0 + ℓ2p s coth2( √ −psη) ) e− 1 2 σ2t, (23) V8(x, t) = ( ℓ0 + ℓ2p s ( coth( √ −4psη)± csch( √ −4psη) )2) e− 1 2 σ2t, (24) V9(x, t) = ( ℓ0 + ℓ2p 2s ( tanh( 1 2 √ −psη) + coth( 1 2 √ −psη) )) e− 1 2 σ2t, (25) where η = kx− akℓ0 ∫ t 0 e σB(τ)− 1 2 σ2τdτ. Family III: If s ̸= 0 and p = 0, then Eq. (14) has the solution: X10(η) = −1 sη . Therefore, the KDVE-RVCs (5) has the following rational function solution V10(x, t) = ( ℓ0 + ℓ2 s2η2 ) e− 1 2 σ2t, (26) where η = kx− akℓ0 ∫ t 0 e σB(τ)− 1 2 σ2τdτ. 3. Exact Solutions of SKDV Equation Here, we apply the results from the preceding section to derive solutions to the SKDV Eq. (1). 3.1. JEF-method Substituting Eqs (10)-(12) into Eq. (2), we have the solutions of SKDV Eq. (1): P(x, t) = eσB(t)− 1 2 σ2t ( ℓ0 +ℓ2sn 2 ( kϖx− kϖ( −ℓ2(1 +m2) 3m2 + ℓ0) ∫ t 0 eσB(τ)− 1 2 σ2τdτ,m )) , (27) P(x, t) = eσB(t)− 1 2 σ2t ( ℓ0 +ℓ2cn 2 ( kϖx− kϖ( ℓ2(2m 2 − 1) 3m2 + ℓ0) ∫ t 0 eσB(τ)− 1 2 σ2τdτ,m )) , (28) and P(x, t) = eσB(t)− 1 2 σ2t ( ℓ0 A. Alshuhail et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6587 9 of 16 +ℓ2dn 2 ( kϖx− kϖ( ℓ2(2−m2) 3B2 + ℓ0) ∫ t 0 eσB(τ)− 1 2 σ2τdτ,m )) . (29) If m → 1, then the Eqs (27)-(29) turn into P(x, t) = eσB(t)− 1 2 σ2t ( ℓ0 +ℓ2tanh 2 ( kϖx− kϖ( −2ℓ2 3 + ℓ0) ∫ t 0 eσB(τ)− 1 2 σ2τdτ )) , (30) and P(x, t) = eσB(t)− 1 2 σ2t ( ℓ0 +ℓ2sech 2 ( kϖx− kϖ( ℓ2 3 + ℓ0) ∫ t 0 eσB(τ)− 1 2 σ2τdτ) )) . (31) 3.2. GREM-method Plugging Eq. (16) into Eqs (2), we acquire the solutions of SKDV Eq. (1) as P(x, t) = V(η)e[σB(t)− 1 2 σ2t], η = kx− akℓ0 ∫ t 0 eσB(τ)− 1 2 σ2τdτ. (32) If ps > 0, then the SKDV Eq. (1), utilizing (17)-(21), has the solutions: P1(x, t) = ( ℓ0 + ℓ2p s tan2( √ psη) ) e[σB(t)− 1 2 σ2t], (33) P2(x, t) = ( ℓ0 − ℓ2p s cot2( √ psη) ) e[σB(t)− 1 2 σ2t], (34) P3(x, t) = ( ℓ0 + ℓ2p s ( tan( √ 4psη)± sec( √ 4psη) )2) e[σB(t)− 1 2 σ2t], (35) P4(x, t) = ( ℓ0 − ℓ2p s ( cot( √ 4psη)± csc( √ 4psη) )2) e[σB(t)− 1 2 σ2t], (36) P5(x, t) = ( ℓ0 + ℓ2p 2s ( tan( 1 2 √ psη)− cot( 1 2 √ psη) )2) e[σB(t)− 1 2 σ2t]. (37) While, if ps < 0, then the SKDV Eq. (1), utilizing (22)-(25), has the solutions: P6(x, t) = ( ℓ0 + ℓ2p s tanh2( √ −psη) ) e[σB(t)− 1 2 σ2t], (38) P7(x, t) = ( ℓ0 + ℓ2p s coth2( √ −psη) ) e[σB(t)− 1 2 σ2t], (39) P8(x, t) = ( ℓ0 + ℓ2p s ( coth( √ −4psη)± csch( √ −4psη) )2) e[σB(t)− 1 2 σ2t], (40) A. Alshuhail et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6587 10 of 16 P9(x, t) = ( ℓ0 + ℓ2p 2s ( tanh( 1 2 √ −psη) + coth( 1 2 √ −psη) )2) e[σB(t)− 1 2 σ2t]. (41) If s ̸= 0 and p = 0, then SKDV Eq. (1), utilizing (26), has the solution: P10(x, t) = (−1 sη ) e[σB(t)− 1 2 σ2t], (42) where η = kx− akℓ0 ∫ t 0 e σB(τ)− 1 2 σ2τdτ. Remark 1. Putting p = s = 3 2 , ℓ0 = −c 6 , ℓ2 = −c 2 , k = √ c 3 and σ = 0 (i.e. no noise) in Eqs (33), (34), (38) and (39), we acquired the results that reported in [13] as follows: P(x, t) = −c 6 ( 1 + 3 tan2( √ −c 2 (x− ct)) ) , P(x, t) = −c 6 ( 1 + 3 cot2( √ −c 2 (x− ct)) ) , P(x, t) = −c 6 ( 1− 3 tanh2( √ −c 2 (x− ct)) ) , and P(x, t) = −c 6 ( 1− 3 coth2( √ −c 2 (x− ct)) ) . Remark 2. If we choose ℓ0 = −λ 6 and ℓ2 = −λ 2 in Eqs (33), (34), (38) and (39), then we get P(x, t) = (−λ 6 + λ 2 tan2( √ λ 2 (x− λ ∫ t 0 eσB(τ)− 1 2 σ2τdτ)) ) e[σB(t)− 1 2 σ2t] P(x, t) = (−λ 6 + λ 2 cot2( √ λ 2 (x− λ ∫ t 0 eσB(τ)− 1 2 σ2τdτ)) ) e[σB(t)− 1 2 σ2t], P(x, t) = (−λ 6 + λ 2 tanh2( √ −λ 2 (x+ λ ∫ t 0 eσB(τ)− 1 2 σ2τdτ)) ) e[σB(t)− 1 2 σ2t], and P(x, t) = (−λ 6 + λ 2 coth2(k √ −ps(x− 6ℓ0 ∫ t 0 eσB(τ)− 1 2 σ2τdτ)) ) e[σB(t)− 1 2 σ2t]. These results extend the results that obtained in [11] as follows: P(x, t) = (−λ 6 + λ 2 tan2( √ λ 2 (x− λt)) ) e[σB(t)− 1 2 σ2t] P(x, t) = (−λ 6 + λ 2 cot2( √ λ 2 (x− λt)) ) e[σB(t)− 1 2 σ2t], P(x, t) = (−λ 6 + λ 2 tanh2( √ −λ 2 (x− λt)) ) e[σB(t)− 1 2 σ2t], and P(x, t) = (−λ 6 + λ 2 coth2(k √ −ps(x− λt)) ) e[σB(t)− 1 2 σ2t]. A. Alshuhail et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6587 11 of 16 4. Discussion and effect of noise Discussion: In this study, we acquired the stochastic solutions of the SKDV Eq. (1). We employed two methods, namely the JEF-method and the GREM-method. The JEF- method is suggested to create the exact periodic solutions of nonlinear wave equations. Moreover, the JEF-method is more general than the hyperbolic tangent function expansion method. While the GREM-method is a powerful and effective approach that provides solutions in several forms such as the hyperbolic function, the trigonometric function, and the rational functional form. By using these methods, we acquired a variety of solutions, including the elliptic solutions (27)-(29), kink solution (38), singular solution (39), singular periodic (33) and (34) and etc. One of the key aspects of singular solitons in the KdV equation is their stability properties. Unlike traditional wave solutions that disperse and dissipate over time, singular solitons have the remarkable feature of maintaining their form and amplitude as they propagate through a medium. This stability allows solitons to travel long distances without distortion, making them valuable in modeling the behavior of waves in oceans, rivers, and other fluid systems. Effect of noise: The impact of multiplicative noise on the exact solutions of the SKDV Eq. (1) is examined here. A number of graphs representing different solutions with distinct value of noise intensity are shown. The main distinction between the solutions supplied here and those obtained in [11] is the amplitude functions V(x, t). Here V(x, t) is a stochastic functions, while V(x, t) is supposed deterministic function in [11]. Figures 1, 2 and 3 illustrate the solutions P(x, t) given in Eqs (28), (31) and (38) for distinct value of the noise strength σ as follows: (i) σ = 0 (ii) σ = 0.1 A. Alshuhail et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6587 12 of 16 (iii) σ = 0.3 (iv) σ = 1 (v) σ = 2 (vi) σ = 0, 0.1, 0.3, 1, 2 Figure 1. (i-v) depict 3D periodic solution of P(x, t) stated in Eq (28) with ϖ = k = ℓ0 = ℓ2 = a = 1, t ∈ [0, 4] and x ∈ [−4, 4] (vi) shows 2D-shape of Eq. (28) with distinct value of σ and x = 0.8. (i) σ = 0 (ii) σ = 0.1 A. Alshuhail et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6587 13 of 16 (iii) σ = 0.3 (iv) σ = 1 (v) σ = 2 (vi) σ = 0, 0.1, 0.3, 1, 2 Figure 2. (i-v) depict bell-shape bright solution of P(x, t) stated in Eq. (31) with p = ℓ0 = a = 1, s = ℓ2 = −1, t ∈ [0, 4] and x ∈ [−4, 4] (vi) shows 2D-shape of Eq. (31) with distinct value of σ and x = 1.2. (i) σ = 0 (ii) σ = 1 A. Alshuhail et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6587 14 of 16 (iii) σ = 0.3 (iv) σ = 1 (v) σ = 2 (vi) σ = 0, 0.1, 0.3, 1, 2 Figure 3. (i-v) depict bell-shape dark solution of P(x, t) stated in Eq (38) with m = 0.5, ϖ = k = ℓ0 = ℓ2 = a = 1, t ∈ [0, 4] and x ∈ [−4, 4] (vi) shows 2D-shape of Eq. (38) with distinct value of σ and x = 0.01. As mentioned before, in the absence of noise (i.e., σ = 0), a variety of solutions appear including periodic solutions, bell-shape bright solutions, and bell-shape dark solutions, as illustrated in Figures 1(i)-3(i). When noise is introduced as illustrated in Figures 1(ii)- 1(v), 2(ii)-2(v) and 3(ii)-3(v), the surface flattens after a few transit patterns. This work shows that the SKDV solutions of Eq. (1) may be stabilized around zero when we added the multiplicative noise term into Eq. (1). 5. Conclusions The stochastic KDV Eq. (1) forced by multiplicative noise in the Itô sense was stud- ied in this study. We converted the SKDV equation into a KDV-RVCs (5) by using a suitable transformation. Abundant exact stochastic solutions for KDV-RVCs in the type of elliptic, rational, trigonometric, and hyperbolic functions was discovered by employing A. Alshuhail et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6587 15 of 16 the JEF-method and GREM-method. Next, we obtained the solutions of SKDV Eq. (1). Additionally, we expanded on a few earlier solutions, such the solutions stated in [13]. Due to the relevance of the KDV equation in nonlinear optics, plasma physics, and fluid dynamics, the obtained solutions are essential in comprehending various complex physical phenomena. Finally, several illustrations were provided to show how the multiplicative noise term affected the exact stochastic solutions of the SKDV equation. Acknowledgements This research has been funded by Scientific Research Deanship at the University of Ha’il-Saudi Arabia through project number RG-25005. References [1] D.J. Korteweg and G. de Vries. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 39:422–443, 1895. [2] A. Saha U. K. Samanta and P. Chatterjee. ifurcations of dust ion acoustic trav- elling waves in a magnetized dusty plasma with a q-nonextensive electron velocity distribution. Physics of Plasmas, 20:022111, 2013. [3] M. J. Ablowitz and H. Segur. ifurcations of dust ion acoustic travelling waves in a magnetized dusty plasma with a q-nonextensive electron velocity distribution. Physics of Plasmas, 20:022111, 2013. [4] R. Hirota. The Direct method in Soliton theory. Cambridge University Press, Japan, Osaka City University, 2004. [5] P. J. Olver. Application of Lie Group to differential equation. Springer New York, NY, USA, 1986. [6] A. M. Wazwaz. A kdv6 hierarchy: integrable members with distinct dispersion rela- tions. Applied Mathematics and Computation, 45:86–92, 2015. [7] X. Geng and B. Xue. N-soliton and quasi-periodic solutions of the kdv6 equations. Applied Mathematics and Computation, 219:3504–3510, 2012. [8] A. M.Wazwaz and G.Q. Xu. An extended modified kdv equation and its painlevé integrability. Nonlinear Dynamics, 86:1455–1460, 2016. [9] X.L. Dang Y. Zhang and H. X. Xu. Backlund transformations and soliton solutions for the kdv6 equation. Applied Mathematics and Computation, 217:6230–6236, 2011. [10] Y.T. Gao X.Y. Wen and L. Wang. Darboux transformation and explicit solutions for the integrable sixth-order kdv equation for nonlinear waves. Applied Mathematics and Computation, 218:55–60, 2011. [11] F. M. Al-Askar W. W. Mohammed, C. Cesarano and M. El-Morshedy. Solitary wave solutions for the stochastic fractional-space kdv in the sense of the m-truncated derivative. Mathematics, 10:4792, 2022. A. Alshuhail et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6587 16 of 16 [12] S.F.C. Maureen S. Masitah and H.M.N. Hajah. Applying explicit schemes to the korteweg-devries equation. Modern Applied Science, 9:200, 2015. [13] A. M.Wazwaz. The extended tanh method for abundant solitary wave solutions of nonlinear wave equations. Applied Mathematics and Computation, 187:1131–1142, 2007. [14] E. Ayankop-Andi H. Orapine and G. Ibeh. Analytical and numerical computations of multi-solitons in the korteweg-de vries (kdv)equation. Applied Mathematics, 11:511– 531, 2020. [15] W. W. Mohammed N. Iqbal, T. Botmart and A. Ali. Numerical investigation of fractional-order kersten-krasil’shchik coupled kdv–mkdv system with atangana– baleanu derivative. Advances in Continuous and Discrete Models, 2020:37, 2022. [16] W. W. Mohammed M. Alshammari, N. Iqbal and T. Botmart. The solution of fractional-order system of kdv equations with exponential-decay kernel. Results in Physics, 38:105615, 2022. [17] F. M. Al-Askar W. W. Mohammed and C. Cesarano. On the dynamical behavior of solitary waves for coupled stochastic korteweg–de vries equations. Mathematics, 11:3506, 2023. [18] A. Saha R. Ali and P. Chatterjee. Analytical electron acoustic solitary wave solution for the forced kdv equation in superthermal plasmas. Physics of Plasmas, 24:122106, 2017. [19] E. Fan and J. Zhang. Applications of the jacobi elliptic function method to special- type nonlinear equations. Physics Letters A, 305:383–392, 2002. [20] S. D. Zhu. The generalizing riccati equation mapping method in non-linear evolution equation: application to (2 +1)-dimensional boiti–leon–pempinelle equation. Chaos, Solitons Fractals, 37:1335–1342, 2008.