)-Dimensional Nonlinear Conformable Fractional Schrödinger System Forced by Multiplicative Brownian Motion

Department of Mathematics, Faculty of Science, University of Ha’il, Ha’il 2440, Saudi Arabia Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt Mathematics Department, Faculty of Science, Jazan University, Jazan 45142, Saudi Arabia Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia Department of Statistics and Computer Science, Faculty of Science, Mansoura University, Mansoura 35516, Egypt


Introduction
Stochastic partial differential equations (SPDEs) can be used to represent a wide range of complicated nonlinear physical processes. These kinds of equations appear in a variety of areas including physics, finance, climate dynamics, chemistry, biology, geophysical, engineering, and other fields [1][2][3].
On the other side, fractional partial differential equations (FPDEs) have gotten a lot of interest because they may illustrate the fundamental components underlying real-world issues. They have been seen in a number of physical phenomena, such as viscoelastic materials with relaxation and creeping functions, the motion of a heavy meager surface in a Newtonian fluid, and relapse subordinate dissipative occupancy of components. As a result, FPDEs are employed in a range of fields, including predicting, describing, and modeling the mechanisms engaged in finance, polymeric materials, a kinematic model of neutron points, engineering, electrical circuits, solid-state physics, optical fibers, chemical kinematics, biogenetics, plasma physics, physics of condensed matter, meteorology, electromagnetic, elasticity, and oceanic spectacles [4][5][6][7][8][9].
To reach a better level of qualitative agreement, the following (2 + 1)-dimensional nonlinear conformable fractional stochastic Schrödinger system (NCFSSS) is addressed: where v ∈ ℝ while u ∈ ℂ. D α is the conformable derivative (CD) [31], and γ i are arbitrary constants for i = 1, ::, 4. BðtÞ is a Brownian motion (BM), and udB is multiplicative noise in the Itô sense. The NCFSSS ( (1) and (2)) is crucial in atomic physics, and the functions v and u have diverse physical meanings in various disciplines of physics such as plasma physics [32] and fluid dynamics [33]. In the hydrodynamic context, v is the induced mean flow, and u is the envelope of the wave packet [33], while, in the context of water waves, v is the velocity potential of the mean flow interacting with the surface waves and u is the amplitude of a surface wave packet [34]. The multiplicative noise iσudB plays an important role in the theory of measurements continuous in time in open quantum systems. For more physical interpretations, we refer to [35,36] and the references therein.
The motivations of this work are to obtain the exact fractional stochastic solutions of NCFSSS ((1) and (2)). This is the first investigation to acquire the exact solutions of NCFSSS ((1) and (2)) in the presence of stochastic term and fractional-space derivatives. To accomplish a wide variety of solutions, such as trigonometric, hyperbolic, elliptic, and rational functions, we apply two different methods such as the Jacobi elliptic function and the sine-cosine methods. Also, we study the effect of BM on the obtained solutions of NCFSSS ((1) and (2)) by using MATLAB to create 3D and 2D diagrams for some of the obtained solutions here.
The document is laid out as follows: we define and state some features of the CD and BM in Section 2. We employ an appropriate wave transformation in Section 3 to derive the wave equation of NCFSSS ((1) and (2)). While in Section 4, we utilize two methods to create the analytic solutions of the NCFSSS ((1) and (2)). In Section 5, the influence of the BM on the obtained solutions is investigated. The conclusion of the document is displayed last.

Preliminaries
Here, we define and state some features of the CD and BM.
Definition 1 (see [31]). Let ϕ : ð0,∞Þ ⟶ ℝ, then the CD of ϕ of order α ∈ ð0, 1 is defined as Theorem 2. Let ϕ, H : ð0,∞Þ ⟶ ℝ be differentiable and also α be differentiable functions, then Let us state some properties of the CD. If a and b are constant, then In next definition, we define Brownian motion BðtÞ.

Wave Equation for NCFSSS
The next wave transformation is used to get the wave equation of the NCFSSS ( (1) and (2)): where φ and ψ are deterministic functions and ζ k and ℏ k for k = 1, 2, 3, are nonzero constants. Plugging Equation (5) into 2

Journal of Function Spaces
Equations (1) and (2) and using we get for imaginary part and for real part Taking expectation Eð·Þ on both sides for Equations (9) and (10), we have Since BðtÞ is standard Gaussian process, hence Eð e −σBðtÞ Þ = e ðσ 2 /2Þt . Now Equations (11) and (12) have the form Integrating Equation (14) once and setting the integral constant equal zero yields Plugging Equation (15) into Equation (13), we get the following wave equation where
We may infer from Figures 1-4 the following: (1) As fractional-order α decreases, the surface shrinks (2) The solutions of NCFSSS are stabilized by BM around zero

Conclusions
In this paper, we considered the (2 + 1)-dimensional nonlinear conformable fractional stochastic Schrödinger system ( (1) and (2)) which has never been examined before with stochastic term and fractional space at the same time. We employed two different methods such as the sine-cosine and the Jacobi elliptic function methods to get elliptic,  Journal of Function Spaces trigonometric, rational, and hyperbolic fractional stochastic solutions. These obtained solutions are useful in describing some of interesting physical phenomena due to the importance of the NCFSSS in plasma physics and fluid dynamics. Finally, the effect of BM on the exact solution of the NCFSSS ( (1) and (2)) is demonstrated by introducing 3D and 2D graphs for some analytical fractional stochastic solutions.

Data Availability
All data are available in this paper.

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