The Analytical Solutions of the Stochastic Fractional RKL Equation via Jacobi Elliptic Function Method

This article considers the stochastic fractional Radhakrishnan-Kundu-Lakshmanan equation (SFRKLE), which is a higher order nonlinear Schrödinger equation with cubic nonlinear terms in Kerr law. To ﬁ nd novel elliptic, trigonometric, rational, and stochastic fractional solutions, the Jacobi elliptic function technique is applied. Due to the Radhakrishnan-Kundu-Lakshmanan equation ’ s importance in modeling the propagation of solitons along an optical ﬁ ber, the derived solutions are vital for characterizing a number of key physical processes. Additionally, to show the impact of multiplicative noise on these solutions, we employ MATLAB tools to present some of the collected solutions in 2D and 3D graphs. Finally, we demonstrate that multiplicative noise stabilizes the analytical solutions of SFRKLE at zero.

On the other hand, fractional partial differential equations (FPDEs) have been used to explain many physical phenomena in biology, physics, finance, engineering applications, electromagnetic theory, mathematical, signal processing, and different scientific studies; see, for example, [25][26][27][28][29][30][31][32][33][34][35]. These new fractional-order models are better equipped than the previously utilized integer-order models because fractional-order integrals and derivatives allow for the representation of memory and hereditary qualities of different substances [36]. Compared to integer-order models, where such effects are ignored, fractional-order models have the most significant advantage.
It appears that studying stochastic equations with fractional derivative is more essential. As a result, the next stochastic fractional Radhakrishnan-Kundu-Lakshmanan equation (SFRKLE) [37][38][39] perturbed by multiplicative noise in the Stratonovich sense is treated: where Ψ ∈ ℂ, D α x is the conformable fractional derivative (CFD) [40], ℓ 1 is the group-velocity dispersion, ℓ 2 is the intermodal dispersion, ℓ 3 is the coefficient of nonlinearity, ℓ 4 is the higher-order dispersion coefficient, ℓ 5 is the coefficient of self-steepening for short pulses, and ℓ 6 is the thirdorder dispersion term. While σ denotes the noise intensity, W ðtÞ is a standard Wiener process (SWP).
Our motivation of this article is to achieve exact stochastic-fractional solutions for SFRKLE (1). This is the first study to attain the exact solutions of SFRKLE (1) in the existence of a stochastic term and fractional derivative. To get a wide variety of solutions such as trigonometric, hyperbolic, elliptic, and rational functions, we apply the Jacobi elliptic function method. Due to the significance of the RKL in modeling the propagation of solitons through an optical fiber, the solutions obtained are useful for describing some important physical phenomena. In addition, we investigate the impact of BM on the acquired solutions of SFRKLE (1) by generating 3D and 2D diagrams for these solutions.
The outline of this article is as follows. In Section 2, we use a proper wave transformation to deduce the SFRKLE's wave equation (1). While in Section 3, we utilize Jacobi elliptic function method to create the analytic solutions of SFRKLE (1). In Section 4, the influence of the SWP on the obtained solutions is investigated. The conclusion of the document is displayed last.

Wave Equation for SFRKLE
The next wave transformation is used to get the wave equation of SFRKLE (1): where Φ is deterministic function that describes the profile of the pulse, θðx, tÞ is the phase component of the soliton, and ν,k, and ω are nonzero constants. Plugging equation (2) into equation (1) and using where ð1/2Þσ 2 Φ is the Itô correction term, and we get for imaginary part and for real part, Taking expectation Eð·Þ on both sides for equations (5) and (6) and using we have where Φ is deterministic functions. Integrating equation (8) and setting the integration constant to zero, we get

Analytical Solutions of SFRKLE
To determine the solutions to equation (14), we employ the Jacobi elliptic function method [48]. As a result, we are able to acquire the exact solutions of SFRKLE (1).
First set: if p > 0 and ℏ 1 > 0, then the solutions φðηÞ of equation (17) corresponding to Table 1 are as follows.
If m ⟶ 1, then the above table degenerates to the following.
If m ⟶ 1, then Table 3 degenerates to the following. In this case, using Table 4 (or Table 5), we obtain the analytical solutions of SFRKLE (1) as stated in equation (26).
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In Figures 2 and 3, we can see that when noise is introduced after small transit patterns, the surface starts to be flat as the noise intensity increases σ = 0:5,1, 2. Figure 4 shows the 2D shape of equation (27) with σ = 0,0:5,1, 2 which highlight the above results.
We can deduce from Figures 1-4 that   Advances in Mathematical Physics (1) the solutions of SFRKLE (1) are stabilized around zero by the SWP (2) as the fractional order α decreases, the surface shrinks

Conclusions
We considered here the stochastic fractional Radhakrishnan-Kundu-Lakshmanan equation (1) which has never been considered before with fractional derivative and stochastic term. To get hyperbolic, rational, and elliptic stochastic fractional solutions, we used the Jacobi elliptic function method. Because of the importance of SFRKLE in representing the propagation of solitons via an optical fiber, the derived solutions may be utilized to represent a wide range of exciting physical phenomena. Finally, we achieved by plotting the derived solutions to show how multiplicative noise and fractional derivative influence these solutions. We deduced that the SWP stabilizes the solutions around zero when the noise strength increases. In future work, we can try to get the exact solutions of SFRKLE (1) with additive noise or multiplicative color noise.

Data Availability
All data are available in this paper.

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