EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 7172 ISSN 1307-5543 – ejpam.com Published by New York Business Global New Trigonometric and Hyperbolic Stochastic Fractional Solutions for the Conformable Derivative Coupled Schrödinger-KdV Equations Using Sardar Subequation Method Wael W. Mohammed1,∗, Naveed Iqbal1, Hijyah M. Alshammary1, F. Gassem1, Mohamed S. Algolam1 1 Department of Mathematics, College of Science, University of Ha’il, Ha’il 2440, Saudi Arabia Abstract. In this paper, we look at the stochastic coupled Schrödinger-KdV equations with con- formable derivative operator (SCS-KdVEs-CDO). By using the Sardar subequation method (SS- method), we can obtain results such as periodic soliton, bright soliton, dark-bright soliton and singular soliton. Some previous solutions of coupled Schrödinger-KdV equations are obtained when the order of derivatives is integer and noise is ignored. Due to the important applications of the coupled Schrödinger-KdV equations in dusty plasma, such as dust-acoustic waves, electro- magnetic waves, and Langmuir waves, these obtained solutions might be utilized to the analysis of many fundamental physical phenomena. Furthermore, the effects of the conformable derivative operator and the noise term on the analytical solution of the SCS-KdVEs-CDO were illustrated by simulating some solutions via the MATLAB program. 2020 Mathematics Subject Classifications: 35A20, 60H10, 60H15, 35Q51, 35Q80 Key Words and Phrases: Exact solutions, stochastic process, stochastic soliton solutions, sta- bility by noise 1. Introduction Fractional partial differential equations (FPDEs) represent a powerful mathematical tool for modeling and understanding complex systems. By incorporating fractional deriva- tives, these equations can capture the non-local and memory effects that are often present in real-world phenomena. As research in this area continues to advance, we can anticipate seeing even more applications of fractional partial differential equations in various engi- neering and scientific disciplines. Recently, numerous powerful approaches for discovering ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.7172 Email addresses: w.mohammed@uoh.edu.sa (W. W. Mohammed), n.iqbal@uoh.edu.sa (N. Iqbal) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) W. W. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7172 2 of 22 exact solutions to PDEs have been introduced, such as sine-Gordon expansion technique [1], modified mapping method [2], F-expansion method [3, 4], (G′/G)-expansion [5, 6], sine-cosine method [7], modified extended tanh-function method [8], exp-function method [9], exp(−ϕ(ς))-expansion method [10], Riccati equation method [11], Jacobi elliptic func- tion expansion [12], the Riccati-Bernoulli sub-ODE method [13, 14] and etc. One of these equations is coupled Schrödinger-KdV (CS-KDV) equation . The CS- KDV equations are a powerful mathematical model that combines the KdV equation and the Schrödinger equation to describe the interaction between nonlinear waves and quantum particles. This equation offers a flexible and accurate framework for studying a wide range of physical phenomena, including solitons, quantum chaos, and complex wave dynamics. By incorporating conformable derivative operator, the CS-KDV equations pro- vides a versatile tool for researchers in various scientific disciplines to explore the intricate relationships between quantum mechanics and nonlinear wave propagation. On other side, including a random term into coupled Schrödinger-KdV equations is important because it introduces an element of realism and flexibility. In real-world sce- narios, systems often experience random disturbances and fluctuations, such as noise or variability in the environment. By incorporating a random term, mathematicians and physicists can simulate and analyze how these unpredictable influences affect the behavior and evolution of waves within these systems. This enhances the accuracy and reliability of the models used to predict wave phenomena.. Here, we take into account the stochastic coupled Schrödinger-KdV equations with conformable derivative operator (SCS-KdVEs-CDO) as follows: Zt + 6ZT α xZ + T α xxxZ = T α x ( |Y|2 ) + ρZW t, iYt = T α xxY − YZ + iρYWt, (1) where Z(x, t) is the real-valued function and presents a long wave profile while Y(x, t) is the complex function and presents the short wave profile; T α is the conformable derivative operator [15] for 0 < α ≤ 1; ρ is the amplitude of noise; W is a standard Wiener process and Wt = ∂W ∂t . Due to the importance of the CS-KdV equation in soliton theory, mathematical physics and hydrodynamics, many authors have obtained its solutions by applying different meth- ods, including mapping method [16], finite element method [17] Kudryashov method [18], adomian’s decomposition method [19], Petrov-Galerkin Method [20], F-expansion method [21], complex hyperbolic-function method [22], (G′/G)-expansion method [23], exp(−φ(ξ))-expansion method [24], and the mapping method [25]. While the stochastic coupled Schrödinger-KdV equations with conformable derivative operator have not been investigated before. Our novelty of this work is to obtain the exact solutions for the SCS-KdVEs-CDO (1). The Sardar subequation method (SS-Method) is used to find various types of solutions such as periodic soliton, bright soliton, dark-bright soliton and singular soliton. Moreover, we provide a number of three-and two-dimensional graphs to examine how the Wiener W. W. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7172 3 of 22 process term affects on the obtained solutions. Also, we generalize some early results such as the results stated in [16, 21]. The following is an outline of the paper: The SCS-KdVEs-CDO (1) wave equation is given in Sec. 2 , while Sec. 3 states the Sardar subequation method. In Sec. 4, we get the solutions of (1). In Sec. 5, we address the effect of the Wiener process term on the solutions of the SCS-KdVEs-CDO (1). The results acquired for this research are finally given. 2. Traveling Wave Equation for the SCS-KdVEs-CDO To have the wave equation for the SCS-KdVEs-CDO (1), we apply Y(x, t) = u(ζ)eiΘ+ρW− 1 2 ρ2t and Z(x, t) = v(ζ)eρW− 1 2 ρ2t, (2) Θ = k α xα + δt, ζ = 1 α xα + 2kt, where u and v are deterministic and real functions; k and δ are non-zero real constants. It is observed that Yt = [2ku′ + iδu+ ρuWt − 1 2 ρ2u+ 1 2 ρ2u]eiΘ+ρW− 1 2 ρ2t = [2ku′ + iδu+ ρuWt]e iΘ+ρW− 1 2 ρ2t, (3) Yx = (u′ + iku)eiΘ+ρW− 1 2 ρ2t, Yxx = (u′′ + 2iku′ − k2u)eiΘ+ρW− 1 2 ρ2t, where 1 2ρ 2u is the Itô correction term. Similarly, Zt = [2kv′ + ρvWt]e ρW− 1 2 ρ2t, Zx = v′eρW− 1 2 ρ2t, Zxxx = v′′′eρW− 1 2 ρ2t. (4) Substituting Eqs (2), (3) and (4) into Eq. (1), we have the next system: 2kv‵ + v‵‵‵ + [6vv‵ − ( u2 )‵ ]eρW− 1 2 ρ2t = 0, u‵‵ + ( δ − k2 ) u+ uveρW− 1 2 ρ2t = 0. (5) Considering the expectation on both sides of Eq. (5), yields 2kv‵ + v‵‵‵ + [6vv‵ − ( u2 )‵ ]e− 1 2 ρ2tE(eρW) = 0, u‵‵ + ( δ − k2 ) u+ uve− 1 2 ρ2tE(eρW) = 0. (6) Since W(t) is a Wiener process, hence E(eρW(t)) = e 1 2 ρ2t for any real number ρ. Therefore, Eq. (6) tends into 2kv‵ + v‵‵‵ + 6vv‵ − ( u2 )‵ = 0, u‵‵ + ( δ − k2 ) u+ uv = 0. (7) W. W. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7172 4 of 22 Integrating first equation in Eq. (7) once, we get v′′ + 2kv + 3v2 − u2 = 0, u‵‵ + ( δ − k2 ) u+ uv = 0. (8) 3. The clarification of SS-method Let us describe the SS-method that we use here ( for more detail see [16, 26]). Let the solution of Eq. (8) have the form u(ζ) = ∑m1 i=0 ℓiF i(ζ), v(ζ) = ∑m2 i=0ϖiF i(ζ), (9) where ℓi and ϖi are undetermined constants to be established, and F solves the auxiliary equation: (F ′)2 = F4 + ℏ1F2 + ℏ2, (10) where ℏ1, and ℏ2 are real constants. Many cases rely on ℏ1, and ℏ2 for the solutions of Eq. (10) as follows: Case 1: If ℏ1 > 0, and ℏ2 = 0, then F1(ζ) = ± √ pqℏ1sechpq( √ ℏ1ζ), (11) and F2(ζ) = ± √ pqℏ1cschpq( √ ℏ1ζ), (12) where sechpq(θ) = 2 peθ + qe−θ and cschpq(θ) = 2 peθ − qe−θ . Case 2: If ℏ1 < 0, and ℏ2 = 0, then F3(ζ) = ± √ −pqℏ1 secpq( √ −ℏ1ζ), (13) and F4(ζ) = ± √ −pqℏ1 cscpq( √ −ℏ1ζ), (14) where secpq(θ) = 2 peθi + qe−θi and cscpq(θ) = 2 peθi − qe−θi . Case 3: If ℏ1 < 0, and ℏ2 = ℏ21 4 , then F5(ζ) = ± √ −ℏ1 2 tanhpq( √ −ℏ1 2 ζ), (15) W. W. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7172 5 of 22 F6(ζ) = ± √ −ℏ1 2 cothpq( √ −ℏ1 2 ζ), (16) F7(ζ) = ± √ −ℏ1 2 [cothpq( √ −2ℏ1ζ)± √ pqcschpq( √ −2ℏ1ζ)], (17) and F8(ζ) = ± √ −ℏ1 8 [tanhpq( √ −ℏ1 8 ζ) + cothpq( √ −ℏ1 8 ζ)], (18) where tanhpq(θ) = peθ − qe−θ peθ + qe−θ and cothpq(θ) = peθ + qe−θ peθ − qe−θ . Case 4: If ℏ1 > 0, and ℏ2 = ℏ21 4 , then F9(ζ) = ± √ ℏ1 2 tanpq( √ ℏ1 2 ζ), (19) F10(ζ) = ± √ ℏ1 2 cotpq( √ ℏ1 2 ζ), (20) F11(ζ) = ± √ ℏ1 2 [tanpq( √ 2ℏ1ζ)± √ pq secpq( √ 2ℏ1ζ)], (21) F12(ζ) = ± √ ℏ1 2 [cotpq( √ 2ℏ1ζ)± √ pq cscpq( √ 2ℏ1ζ)], (22) and F13(ζ) = ± √ ℏ1 8 [tanpq( √ ℏ1 8 ζ) + cotpq( √ ℏ1 8 ζ)], (23) where tanpq(θ) = −i peθi − qe−θi peθi + qe−θi and cotpq(θ) = i peθi + qe−θi peθi − qe−θi . 4. Exact solutions of the the SCS-KdVEs-CDO First, let us equate u′′ with uv and v′′ with v2 in Eq. (8) to determine the parameters m1 and m2 as m1 = 2 and m2 = 2. With m1 = 2 and m2 = 2, Eq. (9) has the form u(ζ) = ℓ0 + ℓ1F(ζ) + ℓ2F2(ζ), v(ζ) = ϖ0 +ϖ1F(ζ) +ϖ2F2(ζ). (24) Substituting Eq. (24) into Eq. (8), we obtain [ℓ2ϖ2 + 6ℓ2]F4 + [2ℓ1 + ℓ1ϖ2 + ℓ2ϖ1]F3 W. W. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7172 6 of 22 +[4ℏ1ℓ2 + ( δ − k2 ) ℓ2 + ℓ0ϖ2 + ℓ2ϖ0 + ℓ1ϖ1]F2 +[ℏ1ℓ1 + ( δ − k2 ) ℓ1 + ℓ1ϖ0 + ℓ0ϖ1]F +[2ℏ2ℓ2 + ( δ − k2 ) ℓ0 + ℓ0ϖ0] = 0, and (6ϖ2 − ℓ22 + 3ϖ2 2)F4 + (2ϖ1 + 6ϖ1ϖ2 − 2ℓ1ℓ2)F3 +(4ϖ2ℏ1 + 2kϖ2 + 6ϖ0ϖ2 − 2ℓ0ℓ2 + 3ϖ2 1 − ℓ21)F2 +(ℏ1ϖ1 + 2kϖ1 + 6ϖ0ϖ1 − 2ℓ0ℓ1)F +(2ℏ2ϖ2 + 2kϖ0 + 3ϖ2 0 − ℓ20) = 0. For j = 4, 3, 2, 1, 0, we balance each coefficient of F j with zero to get ℓ2ϖ2 + 6ℓ2 = 0, 2ℓ1 + ℓ1ϖ2 + ℓ2ϖ1 = 0, 4ℏ1ℓ2 + ( δ − k2 ) ℓ2 + ℓ0ϖ2 + ℓ2ϖ0 + ℓ1ϖ1 = 0, ℏ1ℓ1 + ( δ − k2 ) ℓ1 + ℓ1ϖ0 + ℓ0ϖ1 = 0, 2ℏ2ℓ2 + ( δ − k2 ) ℓ0 + ℓ0ϖ0 = 0, and 6ϖ2 − ℓ22 + 3ϖ2 2 = 0, 2ϖ1 + 6ϖ1ϖ2 − 2ℓ1ℓ2 = 0, 4ϖ2ℏ1 + 2kϖ2 + 6ϖ0ϖ2 − 2ℓ0ℓ2 + 3ϖ2 1 − ℓ21 = 0, ℏ1ϖ1 + 2kϖ1 + 6ϖ0ϖ1 − 2ℓ0ℓ1 = 0, 2ℏ2ϖ2 + 2kϖ0 + 3ϖ2 0 − ℓ20 = 0. By solving the above equations, we have two sets of solutions as follows: Set I: ℓ0 = ℓ2 = 0, ℓ1 = ±2 √ (ℏ1 − k − 3k2 + 3δ), ϖ0 = k2 − δ − ℏ1, ϖ1 = 0, ϖ2 = −2. (25) Set II: ℓ0 = ±2 √ 2 ( 10ℏ1 − √ ℏ1 − 3ℏ2 ) , ℓ1 = 0, ℓ2 = ±6 √ 2 ϖ0 = −(2ℏ1 + 1 3k + 4 3 √ ℏ1 − 3ℏ2), ϖ1 = 0, ϖ2 = −6, δ = −k2 − 1 3k ± 10 3 √ ℏ1 − 3ℏ2. (26) Set I: Plugging (25) into Eq. (24), we obtain the next solution for the traveling wave Eq. (8): u = ±2 √ (ℏ1 − k − 3k2 + 3δ)F (ζ) , (27) W. W. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7172 7 of 22 v = k2 − δ − ℏ1 − 2F2 (ζ) , For ( ℏ1 − k − 3k2 + 3δ ) > 0. Similarly, putting Eq. (27) into equation Eq. (2), we attain the solutions of the SCS-KdVEs-CDO (1) as follows: Y(x, t) = ±2 √ (ℏ1 − k − 3k2 + 3δ)F (ζ) eiΘ+ρW− 1 2 ρ2t, (28) Z(x, t) = {k2 − δ − ℏ1 − 2F2 (ζ)}eρW− 1 2 ρ2t, where ζ = 1 αx α + 2kt and Θ = k αx α + δt. Now, substituting from Eqs (11) to (23) into Eq. (28), we have the next solutions of the the SCS-KdVEs-CDO (1): Case I-1: If ℏ1 > 0, and ℏ2 = 0, then Y1,1 = ±2 √ pqℏ1 (ℏ1 − k − 3k2 + 3δ)sechpq( √ ℏ1ζ)eiΘ+ρW− 1 2 ρ2t, (29) Z1,1 = {k2 − δ − ℏ1 − 2pqℏ1sech2 pq( √ ℏ1ζ)}eρW− 1 2 ρ2t, (30) and Y1,2 = ±2 √ pqℏ1 (ℏ1 − k − 3k2 + 3δ)cschpq( √ ℏ1ζ)eiΘ+ρW− 1 2 ρ2t, (31) Z1,2 = {k2 − δ − ℏ1 − 2pqℏ1csch2 pq( √ ℏ1ζ)}eρW− 1 2 ρ2t, (32) for ( ℏ1 − k − 3k2 + 3δ ) > 0. Case I-2: If ℏ1 < 0, and ℏ2 = 0, then Y1,3 = ±2 √ −pqℏ1 (ℏ1 − k − 3k2 + 3δ) secpq(− √ ℏ1ζ)eiΘ+ρW− 1 2 ρ2t, (33) Z1,3 = {k2 − δ − ℏ1 + 2pqℏ1 sec2pq( √ ℏ1ζ)}eρW− 1 2 ρ2t, (34) and Y1,4 = ±2 √ −pqℏ1 (ℏ1 − k − 3k2 + 3δ) cscpq(− √ ℏ1ζ)eiΘ+ρW− 1 2 ρ2t, (35) Z1,4 = {k2 − δ − ℏ1 + 2pqℏ1 csc2pq( √ ℏ1ζ)}eρW− 1 2 ρ2t, (36) for ( ℏ1 − k − 3k2 + 3δ ) > 0. Case I-3: If ℏ1 < 0, and ℏ2 = ℏ21 4 , then Y1,5 = ± √ −2ℏ1 (ℏ1 − k − 3k2 + 3δ) tanhpq( √ −ℏ1 2 ζ)eiΘ+ρW− 1 2 ρ2t, (37) Z1,5 = {k2 − δ − ℏ1 + ℏ1 tanh2pq( √ −ℏ1 2 ζ)}eρW− 1 2 ρ2t, (38) Y1,6 = ± √ −2ℏ1 (ℏ1 − k − 3k2 + 3δ) cothpq( √ −ℏ1 2 ζ)eiΘ+ρW− 1 2 ρ2t, (39) Z1,6 = {k2 − δ − ℏ1 + ℏ1 coth2pq( √ −ℏ1 2 ζ)}eρW− 1 2 ρ2t, (40) W. W. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7172 8 of 22 Y1,7 = ± √ −2ℏ1 (ℏ1 − k − 3k2 + 3δ)[cothpq( √ −2ℏ1ζ)± √ pqcschpq( √ −2ℏ1ζ)]eiΘ+ρW− 1 2 ρ2t, (41) Z1,7 = {k2 − δ − ℏ1 + ℏ1[cothpq( √ −2ℏ1ζ)± √ pqcschpq( √ −2ℏ1ζ)]2}eρW− 1 2 ρ2t, (42) and Y1,8 = ±1 2 √ −ℏ1 (ℏ1 − k − 3k2 + 3δ)[tanhpq( √ −ℏ1 8 ζ) + cothpq( √ −ℏ1 8 ζ)]eiΘ+ρW− 1 2 ρ2t, (43) Z1,8 = {k2 − δ − ℏ1 + 1 4 ℏ1[tanhpq( √ −ℏ1 8 ζ) + cothpq( √ −ℏ1 8 ζ)]2}eρW− 1 2 ρ2t, (44) for ( ℏ1 − k − 3k2 + 3δ ) > 0. Case I-4: If ℏ1 > 0, and ℏ2 = ℏ21 4 , then Y1,9 = ± √ 2ℏ1 (ℏ1 − k − 3k2 + 3δ) tanpq( √ ℏ1 2 ζ)eiΘ+ρW− 1 2 ρ2t, (45) Z1,9 = {k2 − δ − ℏ1 − ℏ1 tan2pq( √ ℏ1 2 ζ)}eρW− 1 2 ρ2t, (46) Y1,10 = ± √ 2ℏ1 (ℏ1 − k − 3k2 + 3δ) cotpq( √ ℏ1 2 ζ)eiΘ+ρW− 1 2 ρ2t, (47) Z1,10 = {k2 − δ − ℏ1 − ℏ1 cot2pq( √ ℏ1 2 ζ)}eρW− 1 2 ρ2t, (48) Y1,11 = ± √ 2ℏ1 (ℏ1 − k − 3k2 + 3δ)[tanpq( √ 2ℏ1ζ)± √ pq secpq( √ 2ℏ1ζ)]eiΘ+ρW− 1 2 ρ2t, (49) Z1,11 = {k2 − δ − ℏ1 − ℏ1[tanpq( √ 2ℏ1ζ)± √ pq secpq( √ 2ℏ1ζ)]2}eρW− 1 2 ρ2t, (50) Y1,12 = ± √ 2ℏ1 (ℏ1 − k − 3k2 + 3δ)[cotpq( √ 2ℏ1ζ)± √ pq cscpq( √ 2ℏ1ζ)]eiΘ+ρW− 1 2 ρ2t, (51) Z1,12 = {k2 − δ − ℏ1 − ℏ1[cotpq( √ 2ℏ1ζ)± √ pq cscpq( √ 2ℏ1ζ)]2}eρW− 1 2 ρ2t, (52) and Y1,13 = ±1 2 √ ℏ1 (ℏ1 − k − 3k2 + 3δ)[tanpq( √ ℏ1 8 ζ) + cotpq( √ ℏ1 8 ζ)]eiΘ+ρW− 1 2 ρ2t, (53) Z1,13 = {k2 − δ − ℏ1 − ℏ1 4 [tanpq( √ ℏ1 8 ζ) + cotpq( √ ℏ1 8 ζ)]2}eρW− 1 2 ρ2t, (54) for ( ℏ1 − k − 3k2 + 3δ ) > 0. Remark 1. Putting p = q = 1, ρ = 0 and α = 1 from (29) to (54) yields the identical results as those expressed in [16]. W. W. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7172 9 of 22 Set II: Putting (26) into Eq. (24), we get the following solution for the traveling wave Eq. (8): u (η) = 2 √ 2 ( 10ℏ1 − √ ℏ1 − 3ℏ2 ) + 6 √ 2M2 (η) , (55) v (η) = −(2ℏ1 + 1 3 k + 4 3 √ ℏ1 − 3ℏ2)− 6M2 (η) . Consequently, setting Eq. (55) into Eq. (2), we obtain the solutions of the SCS-KdVEs- CDO (1) for ℏ1 ≥ 3ℏ2 as follows: Y = [2 √ 2 ( 10ℏ1 − √ ℏ1 − 3ℏ2 ) + 6 √ 2M2 (η)]eiΘ+ρB− 1 2 ρ2t, (56) Z = −[2ℏ1 + 1 3 k + 4 3 √ ℏ1 − 3ℏ2 + 6M2 (η)]eρB− 1 2 ρ2t, where η = 1 αx α + 2kt and Θ = k αx α + δt. Now, substituting from Eqs (11) to (23) into Eq. (56), we have the next solutions of the SCS-KdVEs-CDO (1): Case II-1: If ℏ1 > 0, and ℏ2 = 0, then Y2,1 = [18 √ 2ℏ1 + 6 √ 2sech2 pq( √ ℏ1ζ)]eiΘ+ρW− 1 2 ρ2t, (57) Z2,1 = −[ 10 3 ℏ1 + 1 3 k + 6sech2 pq( √ ℏ1ζ)]eρW− 1 2 ρ2t, (58) and Y2,2 = [18 √ 2ℏ1 + 6 √ 2csch2 pq( √ ℏ1ζ)]eiΘ+ρW− 1 2 ρ2t, (59) Z2,2 = −[ 10 3 ℏ1 + 1 3 k + 6csch2 pq( √ ℏ1ζ)]eρW− 1 2 ρ2t. (60) Case II-2: If ℏ1 < 0, and ℏ2 = 0, then Y2,3 = [22 √ 2ℏ1 + 6 √ 2 sec2pq( √ −ℏ1ζ)]eiΘ+ρW− 1 2 ρ2t, (61) Z2,3 = −[ 2 3 ℏ1 + 1 3 k + 6 sec2pq( √ −ℏ1ζ)]eρW− 1 2 ρ2t, (62) and Y2,4 = [22 √ 2ℏ1 + 6 √ 2 csc2pq( √ −ℏ1ζ)]eiΘ+ρW− 1 2 ρ2t, (63) Z2,4 = −[ 2 3 ℏ1 + 1 3 k + 6 csc2pq( √ −ℏ1ζ)]eρW− 1 2 ρ2t. (64) Case II-3: If ℏ1 < 0, and ℏ2 = ℏ21 4 , then Y2,5 = [19 √ 2ℏ1 + 6 √ 2 tanh2pq( √ −ℏ1 2 ζ)]eiΘ+ρW− 1 2 ρ2t, (65) Z2,5 = −[ 1 3 k + 4 3 ℏ1 + 6 tanh2pq( √ −ℏ1 2 ζ)]eρW− 1 2 ρ2t, (66) W. W. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7172 10 of 22 Y2,6 = [19 √ 2ℏ1 + 6 √ 2 coth2pq( √ −ℏ1 2 ζ)]eiΘ+ρW− 1 2 ρ2t, (67) Z1,6 = [ 1 3 k + 4 3 ℏ1 + 6 coth2pq( √ −ℏ1 2 ζ)]eρW− 1 2 ρ2t, (68) Y2,7 = [19 √ 2ℏ1 + 6 √ 2(cothpq( √ −2ℏ1ζ)± √ pqcschpq( √ −2ℏ1ζ))2]eiΘ+ρW− 1 2 ρ2t, (69) Z2,7 = −[ 1 3 k + 4 3 ℏ1 + 6(cothpq( √ −2ℏ1ζ)± √ pqcschpq( √ −2ℏ1ζ))2]eρW− 1 2 ρ2t, (70) and Y2,8 = [19 √ 2ℏ1 + 6 √ 2(tanhpq( √ −ℏ1 8 ζ) + cothpq( √ −ℏ1 8 ζ))2]eiΘ+ρW− 1 2 ρ2t, (71) Z2,8 = −[ 1 3 k + 4 3 ℏ1 + 6(tanhpq( √ −ℏ1 8 ζ) + cothpq( √ −ℏ1 8 ζ))2]eρW− 1 2 ρ2t. (72) Case II-4: If ℏ1 > 0, and ℏ2 = ℏ21 4 , then Y2,9 = [19 √ 2ℏ1 + 6 √ 2 tan2pq( √ ℏ1 2 ζ)]eiΘ+ρW− 1 2 ρ2t, (73) Z2,9 = −[ 8 3 ℏ1 + 1 3 k + 6 tan2pq( √ ℏ1 2 ζ)]eρW− 1 2 ρ2t, (74) Y2,10 = [19 √ 2ℏ1 + 6 √ 2 cot2pq( √ ℏ1 2 ζ)]eiΘ+ρW− 1 2 ρ2t, (75) Z2,10 = −[ 8 3 ℏ1 + 1 3 k + 6 cot2pq( √ ℏ1 2 ζ)]eρW− 1 2 ρ2t, (76) Y2,11 = [19 √ 2ℏ1 + 6 √ 2(tanpq( √ 2ℏ1ζ)± √ pq secpq( √ 2ℏ1ζ))2]eiΘ+ρW− 1 2 ρ2t, (77) Z2,11 = −[ 8 3 ℏ1 + 1 3 k + 6(tanpq( √ 2ℏ1ζ)± √ pq secpq( √ 2ℏ1ζ))2]eρW− 1 2 ρ2t, (78) Y2,12 = [19 √ 2ℏ1 + 6 √ 2(cotpq( √ 2ℏ1ζ)± √ pq cscpq( √ 2ℏ1ζ))2]eiΘ+ρW− 1 2 ρ2t, (79) Z2,12 = −[ 8 3 ℏ1 + 1 3 k + 6(cotpq( √ 2ℏ1ζ)± √ pq cscpq( √ 2ℏ1ζ))2]eρW− 1 2 ρ2t, (80) and Y2,13 = [19 √ 2ℏ1 + 6 √ 2(tanpq( √ ℏ1 8 ζ) + cotpq( √ ℏ1 8 ζ))2]eiΘ+ρW− 1 2 ρ2t, (81) Z2,13 = −[ 8 3 ℏ1 + 1 3 k + 6(tanpq( √ ℏ1 8 ζ) + cotpq( √ ℏ1 8 ζ))2]eρW− 1 2 ρ2t. (82) Remark 2. Putting p = q = 1, ρ = 0 and α = 1 from (57) to (82) yields the identical results as those expressed in [21]. W. W. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7172 11 of 22 5. Effect of Noise and Fractional Derivatives The recent advances in experimental techniques and numerical simulations of nonlin- ear wave dynamics offer the possibility of confirming the analytical results presented in this study. In particular, the bright and dark soliton solutions obtained herein under the influence of stochasticity can serve as test beds for the validation of numerical ap- proaches to noisy fractional systems. For instance, the application of fractional spectral methods and stochastic finite-difference techniques can model the same conditions and be contrasted with our exact solutions in order to understand the robustness and validity of computational approaches. Additionally, in experimental settings such as nonlinear op- tics and Bose-Einstein condensates, setups with tunable noise and dispersion parameters can provide feasible arenas for tracking the qualitative dynamics predicted by our work. Hence, our results not only enrich the theory of nonlinear FPDEs but also pave the way for future experimental verification and numerical exploration. Effect of Noise: The existence of multiplicative noise in the Schrödinger-KdV equation can have significant implications for the dynamics of the system. In the presence of noise, the solutions of the equation may exhibit stochastic behavior, leading to fluctuations in the amplitude and phase of the waves. These fluctuations can affect the stability of the solutions and the overall dynamics of the system, making it challenging to accurately predict the behavior of the waves over time. Let us now display the noise impacts on the exact solution of the SCS-KdVEs-CDO (1). Many figures are included to illustrate the behavior of some of the solutions that were produced, such as Eqs (29), (30), (37), and (38) as follows: W. W. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7172 12 of 22 (a) ρ = 0 (b) ρ = 0.1 (c) ρ = 0.3 (d) ρ = 1 (e) ρ = 2 (f) ρ = 0, 0.1, 0.3, 1, 2 Figure 1. (a-e) 3D profiles of the bright soliton solution Y(x, t) of Eq. (29) under different noise intensities ρ. As ρ increases from 0 to 2, the soliton peak deforms significantly: small noise (ρ = 0.1, 0.3) induces slight oscillations, while stronger noise (ρ = 1, 2) destabilizes and suppresses the soliton, reflecting the dissipative role of stochasticity. (f) 2D cross-sectional profiles showing how increasing ρ reduces amplitude and broadens the wave, indicating that noise progressively stabilizes the solution toward zero. Simulations are performed with k = −1, q = 0.8, p = 0.9, δ = 1, x ∈ [0, 4], W. W. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7172 13 of 22 t ∈ [0, 2.5], and derivative order α = 1. (a) ρ = 0 (b) ρ = 0.1 (c) ρ = 0.3 (d) ρ = 1 (e) ρ = 2 (f) ρ = 0, 0.1, 0.3, 1, 2 Figure 2. (a-e) 3D views of the bright soliton solution Z(x, t) of Eq. (30) under varying noise intensities ρ = 0, 0.1, 0.3, 1, 2. Thes plots are drawn when k = −1, q = 0.8, p = 0.9, δ = 1, x ∈ [0, 4], t ∈ [0, 2.5]. W. W. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7172 14 of 22 (a) ρ = 0 (b) ρ = 0.1 (c) ρ = 0.3 (d) ρ = 1 (e) ρ = 2 (f) ρ = 0, 0.1, 0.3, 1, 2 Figure 3. (a-e) 3D profiles of the dark soliton solution Y(x, t) of Eq. (37) for different noise strengths ρ = 0, 0.1, 0.3, 1, 2. Thes plots are drawn when k = −1, q = 0.8, p = 0.9, δ = 1, x ∈ [0, 4], t ∈ [0, 2.5]. W. W. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7172 15 of 22 (a) ρ = 0 (b) ρ = 0.1 (c) ρ = 0.3 (d) ρ = 1 (e) ρ = 2 (f) ρ = 0, 0.1, 0.3, 1, 2 Figure 4.(a-e) The 3D plots present the dark soliton solution Z(x, t) of Eq. (38) under varying noise intensities ρ = 0, 0.1, 0.3, 1, 2. Thes plots are drawn when k = −1, q = 0.8, p = 0.9, δ = 1, x ∈ [0, 4], t ∈ [0, 2.5]. Now, Figures 1-4 show some different kinds of solutions, such as bright soliton, dark soliton solutions, when the multiplicative noise is disregarded (i.e. when ρ = 0). When noise is introduced and its amplitude is increased by ρ = 0.1, 0.3, 1, 2, the surface flattens out after brief transit patterns. This indicates that the solutions of the SCS-KdVEs-CDO W. W. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7172 16 of 22 are impacted by multiplicative noise and are kept stable around zero. Effect of Fractional derivatives: The presence of conformable derivative operator in the Schrödinger-KdV equation allows for a more accurate and comprehensive description of wave propagation in one-dimensional media. By incorporating fractional derivatives, re- searchers can better understand the complex dynamics and emergent behavior that arise in nonlinear systems. This highlights the importance of considering fractional calculus in the study of physical phenomena, and suggests that it may play a key role in the development of future models and theories in physics and applied mathematics. The leftward displacement of wave profiles with decreasing fractional order α signifies a fundamental alteration of propagation dynamics, in direct result of the memory and hereditary effects introduced by the fractional derivative. Physically, the displacement reflects the reduction of the effective wave speed and the enhancement of attenuation, as the non-local operator ∂α ∂tα imposes more dissipation and frequency-dependent disper- sion, characteristics shared with the propagation of waves in complex viscoelastic media. Mathematically, the enhanced sensitivity of the system to its previous history hinders the advancement of the wave, which appears as an apparent forward shift of the waveform that demonstrates the model’s ability to capture anomalous transport behavior. Now, let us show the effect of the conformable derivative operator on the exact solution of the SCS-KdVEs-CDO (1). To address the behavior of some of the obtained solutions, many figures are provided, such as Eqs (29), (30), (37), and (38) as follows: (a) α = 1, ρ = 0, (b) α = 0.8, ρ = 0, W. W. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7172 17 of 22 (c) α = 0.6, ρ = 0, (d) α = 1, 0.8, 0.6 and ρ = 0, Figure 5. (a-c) exhibit 3D-profile of solution Y(x, t) in Eq (29) with k = −1, q = 0.8, p = 0.9, δ = 1, x ∈ [0, 4], t ∈ [0, 2.5] and ρ = 0, (d) shows 2D-profile of Eq. (29) with different α demonstrating how reducing the fractional derivative order α attenuates the wave amplitude and smooths the wave profile. (a) α = 1, ρ = 0, (b) α = 0.8, ρ = 0, (c) α = 0.6, ρ = 0, (d) α = 1, 0.8, 0.6 and ρ = 0, Figure 6. (a-c) exhibit 3D-profile of solution Z(x, t) in Eq. (30) with k = −1, q = 0.8, p = 0.9, δ = 1, x ∈ [0, 4], t ∈ [0, 2.5] and ρ = 0 (d) 2D profile which demonstrating that a reduction in the fractional derivative order α introduces significant W. W. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7172 18 of 22 diffusivity, smoothing sharp wave features and reducing peak intensity, which reflects enhanced dissipative behavior typical of fractional-order systems in physical contexts such as anomalous diffusion or viscoelastic damping for Eq. (30. (a) α = 1, ρ = 0, (b) α = 0.8, ρ = 0, (c) α = 0.6, ρ = 0, (d) α = 1, 0.8, 0.6 and ρ = 0 Figure 7. (a-c) exhibit 3D-profile of solution Y(x, t) in Eq. (37) with k = −1, q = 0.8, p = 0.9, δ = 1, x ∈ [0, 4], t ∈ [0, 2.5] and ρ = 0 (d) 2D profile showing that decreasing α dampens wave oscillations and flattens the amplitude of Eq. (37). (a)α = 1, ρ = 0, (b) α = 0.8, ρ = 0, W. W. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7172 19 of 22 (c) α = 0.6, ρ = 0, (d) α = 1, 0.8, 0.6 and ρ = 0, Figure 8. (a-c) exhibit the 3D profile of solution Z(x, t) in Eq. (38) with k = −1, q = 0.8, p = 0.9, δ = 1, x ∈ [0, 4], t ∈ [0, 2.5] and ρ = 0 (d) 2D profile showing that decreasing α dampens wave oscillations and flattens the amplitude of Eq. (38). From the previous Figures 5− 8, we conclude that the surface shifts to the left as the derivative order α of conformable derivative operator decreases. Physical meaning: The SCS-KdVEs-CDO (1) represent a compelling intersection be- tween quantum mechanics, nonlinear dynamics, and fractional calculus. It describes a generalization of the classical Schrödinger equation combined with the KdV equation, incorporating random perturbations and fractional derivatives. Therefore, the obtained solutions for these equations are very important. We obtained various soliton solutions, including a periodic soliton, a bright soliton, a dark-bright soliton and a singular soli- ton solution, by using the Sardar subequation method. The soliton solutions of the Schrödinger-KdV equation have implications for quantum field theory and soliton the- ory itself. In quantum mechanics, certain soliton states can represent bound states of particles, providing a novel perspective on particle interactions in quantum fields. The integrability properties of the Schrödinger-KdV equation allow physicists to derive exact solutions, facilitating the exploration of new physical phenomena such as rogue waves and soliton gas formations. These solutions can aid in the mathematical formulation of highly non-linear dynamics and enrich our theoretical understanding of nature, where complex interactions often lead to emergent phenomena. 6. Conclusions In this paper, the stochastic coupled Schrödinger-KdV equations with conformable derivative operator (1) were considered. By using the Sardar subequation method, we were able to obtain a periodic soliton, a bright soliton, a dark-bright soliton, and a sin- gular soliton solution. When ρ = 0 and α = 1, we get some previously known solutions of the SCS-KdVEs-CDO (1) as stated in previous remarks. These obtained solutions may be beneficial, practical, and can explain a wide variety of crucial physical phenomena because the SCS-KdVEs-CDO equations have significant uses in dusty plasma, including W. W. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7172 20 of 22 Langmuir, electromagnetic waves, and dust-acoustic waves. Compared to former deter- ministic approaches such as the Petrov-Galerkin approach for coupled Schrödinger-KdV equations [20] and Sardar sub-equation approach for the Boussinesq equation [26], the present paper provides a broader solution space. Exactly, new solution patterns, perverse as well as systematic extensions of the conformable deterministic counterpart (ρ = 0) are all encompassed in our results, which thereby demonstrates the novelty of the proposed method. Additionally, by drawing several graphs in both two and three dimensions using the MATLAB program (version R2022b), the effects of the conformable derivative oper- ator and noise term on the analytical solution of the SCS-KdVEs-CDO (1) were studied. We showed that the multiplicative Wiener process stabilized the solutions around zero, while the conformable derivative operator shifts the surface to the left when the derivative order α decreases. We can investigate the exact solutions of coupled Schrödinger-KdV equations with additive noise in further research. Acknowledgements This research has been funded by Scientific Research Deanship at the University of Ha’il-Saudi Arabia through project number RG-25050. 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