Impact of Fractional Derivative and Brownian Motion on the Solutions of the Radhakrishnan-Kundu-Lakshmanan Equation

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Researchers and scientists have focused their attention over the last two decades on fractional differential equations (FDEs) that have been found to be more precise than classical differential equations in explaining complex physical phenomena in the real life. The idea of fractional derivative has been used to define various phenomena including fluid dynamics porous medium, signal processing, viscoelastic materials, ocean wave, electromagnetism, photonic, chaotic systems, wave propagation, optical fiber communication, plasma physics, and nuclear physics. Recently, Atangana and Goufo [24] have suggested the new conformable fractional derivative called beta-derivative. From this point, let us define the Atangana conformable derivative (ACD) for the function ψ : ð0,∞Þ ⟶ ℝ of order β ∈ ð0, 1 as follows: The ACD satisfies the following properties for any constant a and b: (1) D β x ½aφðxÞ + bψðxÞ = aD β x φðxÞ + bD β x ψðxÞ, If θ = a/βðx + ð1/ΓðβÞÞÞ β , then D β x ψðθÞ = adψ/dθ. Stochastic partial differential equations (SPDEs), on the other hand, have been widely addressed as theoretical equations for spatial-temporal physical, chemical, and biological systems related to random perturbations. The significance of involving stochastic impacts in complex system modeling has been emphasized. For example, there is gaining awareness in using SPDEs to mathematically model complex phenomena in information systems, condensed matter physics, biology, climate systems, electrical and mechanical engineering, materials sciences, and finance.
It is worth noting that two forms widely utilized for stochastic integral are Itô and Stratonovich [25]. Modeling problems primarily determine which form is acceptable; however, once that form is selected, an equivalent equation of the other form can be produced using the same solutions. As a result, the following relationship can be utilized to switch between Itô (written as Ð t 0 ϕdW) and Stratonovich (written as Ð t 0 ϕ ∘ dW): where WðtÞ is a Brownian motion (BM).
To satisfy a higher degree of quality agreement, the following stochastic Radhakrishnan-Kundu-Lakshmanan equation (FSRKLE) [26][27][28] is considered: where φ ∈ ℂ, γ k for k = 1, 2, 3, 4, 5, 6 are constants and σ is the noise strength and φ ∘ dW is multiplicative Brownian motion in the Stratonovich sense. Recently, many investigators have created exact solutions of FSRKLE (3), with β = 0 and σ = 0, using different methods such as extended simple equation method [29], first integral method [30], sinecosine method [31], Lie group analysis [32], and trial equation method [33]. The motivation of this article is to attain the exact solutions for FSRKLE (3). We use two separate approaches, the sine-cosine and the Jacobi elliptic function methods, to provide a wide range of solutions, including hyperbolic, trigonometric, rational, and elliptic functions. The acquired solutions are helpful for understanding several fascinating scientific events because of the significance of the RKL in describing the propagation of solitons through an optical fiber. Also, by creating 3D representations of the obtained FSRKLE (3) solutions, we examine the effect of BM on these solutions.
The article is in the following format: in Section 2, we determine the wave equation of the FSRKLE (3) by applying a suitable wave transformation. To develop the analytical solutions for the FSRKLE in Section 3, we use two different approaches (3). In Section 4, the impact of the BM on the solutions obtained is examined. The final section of the document is the conclusion.

The Exact Solutions of the FSRKLE
We employ two various methods such as the Jacobi elliptic function [18] and sine-cosine [4], to determine the exact solutions to Equation (16). As a consequence, we can obtain the solutions of the FSRKLE (3).

Sine-Cosine Method. Suppose the solution ψ of Equation (16) takes the form
where Setting Equation (31) into Equation (16), we get rewriting the above equation Balancing the term of Y in Equation (34), we obtain Plugging Equation (35) into Equation (34), We get by setting each coefficient of Y −3 and Y −1 equal to zero By solving Equations (37) and (38), we get Hence, the solution of Equation (16) is Depending on the sign of ℏ 1 and ℏ 2 , there are numerous cases: or Case 2. If ℏ 2 < 0 and ℏ 1 < 0, then the analytical solutions of FSRKLE (3) have the form or Case 3. If ℏ 2 < 0 and ℏ 1 > 0, then the solutions of FSRKLE (3) are
In Figure 2, if the noise appeared, then after small transit behaviors, the surface gets more planer when the intensity of noise increases as follows: Secondly, the impact of fractional derivative: in Figures 3  and 4, if σ = 0, we can observe that as β increases, the surface extends:

Conclusions
In this paper, we obtained the exact solutions of the fractionalstochastic Radhakrishnan-Kundu-Lakshmanan Equation (3).
To obtain rational, elliptic, trigonometric, and hyperbolic stochastic solutions, we used two different methods: the Jacobi elliptic function and the sine-cosine. Because of the priority of the FSRKLE in fluid dynamics and plasma physics, the results produced are useful for understanding some exciting physical phenomena. Finally, we plotted the obtained solutions using MATLAB tools to provide a number 3D diagram to demonstrate the impact of fractional derivative and multiplicative noise on these solutions. In future work, we can consider the FSRKLE (3) with additive noise.  Journal of Function Spaces

Data Availability
All data are available in this paper.

Conflicts of Interest
The authors declare that they have no competing interests.