The Influence of Multiplicative Noise and Fractional Derivative on the Solutions of the Stochastic Fractional Hirota–Maccari System

We address here the space-fractional stochastic Hirota–Maccari system (SFSHMs) derived by the multiplicative Brownian motion in the Stratonovich sense. To acquire innovative elliptic, trigonometric and rational stochastic fractional solutions, we employ the Jacobi elliptic functions method. The attained solutions are useful in describing certain fascinating physical phenomena due to the significance of the Hirota–Maccari system in optical fibers. We use MATLAB programm to draw our figures and exhibit several 3D graphs in order to demonstrate how the multiplicative Brownian motion and fractional derivative affect the exact solutions of the SFSHMs. We prove that the solutions of SFSHMs are stabilized by the multiplicative Brownian motion around zero.


Introduction
Recently, numerous significant phenomena have been represented by fractional derivatives, including electro-magnetic, image processing, acoustics, electrochemistry and anomalous diffusion phenomena [1][2][3][4][5][6]. One benefit of fractional models is that they may be stated more specifically than integer models, which encourages us to construct a number of significant and practical fractional models. On the other hand, the advantages of taking random influences into account in the analysis, simulation, prediction and modeling of complex processes have been highlighted in several fields including chemistry, geophysics, fluid mechanics, biology, atmosphere, physics, climate dynamics, engineering and other fields [7][8][9][10]. Since noise may produce statistical features and significant phenomena, it cannot be ignored. In general, it is more difficult to obtain exact solutions to fractional PDEs forced by a stochastic term than to classical ones.
As a result, we study here the following stochastic fractional-space Hirota-Maccari system (SFSHMs) with multiplicative noise in the Stratonovich sense: where Ψ(x, y, t) denotes the real field of scalars and Φ(x, y, t) is the complex scalar field, x, y are independent spatial variables and t is the temporal variable. D α x is the conformable derivative (CD) for α ∈ (0, 1] [25]. W t = dW dt is the time derivative of Brownian motion W(t) and σ is a noise strength.
The stochastic integral t 0 Φ(s)dW(s) is called the Stratonovich stochastic integral (denoted by t 0 Φ(s) • dW(s)), if we calculate the stochastic integral at the middle, while the stochastic integral t 0 Φ(s)dW(s) is called Itô (denoted by t 0 Φ(s)dW(s)) when we calculate it at the left end [26]. The relation between the Stratonovich integral and Itô integral is: The conformable derivative for the function φ : The important property of CD is the following chain rule: The Hirota-Maccari system (1-2), with σ = 0 and α = 1, was derived by Maccari [27]. There are several physical applications of the integrable Hirota-Maccari system including the transmission of optical pulses across nematic liquid crystal waveguides and for a certain parameter regime, the transmission of femtosecond pulses through optical fibers. Due to the importance of the Hirota-Maccari system, many researchers have examined a lot of techniques in order to find the exact solutions for this system, such as the extended trial equation and the generalized Kudryashov [28], tanh-coth, sec-tan, rational sinh-cosh and sech-csch methods [29], (G /G)-expansion [30], Hirota bilinear method [31], Weierstrass elliptic function expansion [32], Painleve approach [33], Painleve test [34], general projective Riccati equation and improved tan( 2 )-expansion method [35] and complex hyperbolicfunction [36]. While the exact solutions of stochastic Hirota-Maccari system have been studied in [37] in the Itô sense by using three different methods: Riccati-Bernoulli sub-ODE, sine-cosine and He's semi-inverse.
The originality of this paper is to acquire the analytical solutions of the SFSHMs (1-2). This work is the first to attain the exact solutions of the SFSHMs (1-2). We employ the Jacobi elliptic functions approach to obtain a broad range of solutions, including hyperbolic, trigonometric and rational functions. Moreover, to study the effects of Brownian motion on the solutions of the SFSHMs (1-2), we build 3D graphs for some of the developed solutions by using MATLAB tools. This is how the paper is organized: We use a suitable wave transformation in Section 2 to provide the wave equation of SFSHMs. We employ the Jacobi elliptic functions approach in Section 3 to obtain the analytical solutions of the SFSHMs (1-2). In Section 4, we look at how the Brownian motion affects the generated solutions. Finally, we state the conclusions of this paper.

Wave Equation for SFSHMs
To get the wave equation of the SFSHMs (1-2), let us utilize the following transformation: with where θ k , ζ k for k = 1, 2, 3 are nonzero constants. We substitute Equation (5) into Equations (1-2), and use to obtain for the real part Integrating Equation (7), we have Setting Equation (8) into Equation (6) we obtain where Taking expectation E(·) on both sides for Equation (9), we attain Since W(t) is a normal process, then E(e −2σW(t) ) = e 2σ 2 t . Therefore Equation (11) becomes

The Analytical Solutions of the SFSHMs
In this section, we use the Jacobi elliptic functions method [38] to acquire the solutions to Equation (12). Consequently, we obtain the analytical solutions of the SFSHMs (1-2).

Method Description
Let the solutions of Equation (12) have the form where Z solves where 1 , 2 and 3 are real parameters and N is a positive integer number. We notice that Equation (14) has a variety of solutions depending on 1 , 2 and 3 as in the following Table 1 : Table 1. All possible solutions for Equation (14) for different values of 1 , 2 and 3 .

Solutions of SFSHMs
Let us balance Q with Q 3 in Equation (12) to define N as follows: Equation (14) is rewritten with N = 1 as Differentiating Equation (16) twice, we have, by using (14), Plugging Equation (16) and Equation (17) into Equation (12) we have Setting each coefficient of Z k for k = 0, 1, 2, 3 equal to zero, we attain We obtain by solving these equations Thus, Equation (12) has the following solution The following are two sets that depend on 1 and A 1 : First set: If 1 > 0 (from Table 1)and A 1 > 0, then the wave Equation (12) has the solution Q(ζ) as in the following If m → 1, then the previous Table 2 becomes  Table 3. All possible solutions for wave Equation (12) when 1 > 0 and m → 1.
If m → 1, then the previous Table 4 becomes   Table 5. All possible solutions for wave Equation (12) when 1 < 0 and m → 1.
In this situation, we may obtain the analytical solutions of the SFSHMs (1-2) as reported in Equations (19) and (20) by utilizing the previous Table 4 (or Table 5 when m → 1).
Furthermore, in Figure 2, if the noise intensity is raised, the surface becomes more planar after small transit behaviors as follows: Equation (21)  Secondly the effect of fractional order: In Figures 3 and 4, if σ = 0, we can observe that as α increases, the surface extends:

Conclusions
The stochastic fractional-space Hirota-Maccari system (1-2) were taken into consideration in this work. To obtain stochastic trigonometric, elliptic, rational solutions, we used the Jacobi elliptic functions approach. The obtained solutions will be very helpful for further research in disciplines such as optical fibers and others. Finally, an illustration is provided of how multiplicative Brownian motion affects the exact solutions of the SFSHMs (1-2). In future studies, we can consider SDSEs with additive noise.