Abstract
In this paper, the ()-dimensional nonlinear conformable fractional stochastic Schrödinger system (NCFSSS) generated by the multiplicative Brownian motion is treated. To get new rational, trigonometric, hyperbolic, and elliptic stochastic solutions, we use two different methods: the sine-cosine and the Jacobi elliptic function methods. Moreover, we use the MATLAB tools to plot our figures to introduce a variety of 2D and 3D graphs to highlight the effect of the multiplicative noise on the exact solutions of the NCFSSS. Finally, we illustrate that the multiplicative Brownian motion stabilizes the solutions of NCFSSS a round zero.
1. 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 [1–3].
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 [4–9].
The exact solutions of PDEs are important in nonlinear science. As a result, various analytical techniques, such as tanh-sech [10, 11], Darboux transformation [12], sine-cosine [13, 14], extended simple equation [15], extended sinh-Gordon equation expansion [16], -expansion [17], Kudryashov technique [18], generalized Kudryashov [19–21], -expansion [22], -expansion [23–25], Hirota’s function [26], perturbation [5, 27], the Jacobi elliptic function [28, 29], and Riccati-Bernoulli sub-ODE [30], have been created to deal with these types of equations.
To reach a better level of qualitative agreement, the following ()-dimensional nonlinear conformable fractional stochastic Schrödinger system (NCFSSS) is addressed: where while . is the conformable derivative (CD) [31], and are arbitrary constants for . is a Brownian motion (BM), and is multiplicative noise in the Itô sense.
The NCFSSS ((1) and (2)) is crucial in atomic physics, and the functions and have diverse physical meanings in various disciplines of physics such as plasma physics [32] and fluid dynamics [33]. In the hydrodynamic context, is the induced mean flow, and is the envelope of the wave packet [33], while, in the context of water waves, is the velocity potential of the mean flow interacting with the surface waves and is the amplitude of a surface wave packet [34]. The multiplicative noise 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.
Recently, many authors have established exact solutions of NCFSSS ((1) and (2)) with and by employing various techniques, such as applied Kudryashov approach [37], direct approach [38], and the extended modified auxiliary equation [39]. Moreover, Bilal and Ahmad [40] applied three methods such as generalized Kudryashov, modified direct algebraic, and -expansion function to attain diverse forms of optical solutions of the NCFSSS ((1) and (2)) with , while the exact solutions of the NCFSSS ((1) and (2)) has not yet been studied.
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.
2. Preliminaries
Here, we define and state some features of the CD and BM.
Definition 1 (see [31]). Let , then the CD of of order is defined as
Theorem 2. Let be differentiable and also be differentiable functions, then Let us state some properties of the CD. If and are constant, then (1)(2)(3)(4)
In next definition, we define Brownian motion .
Definition 3. Stochastic process is said a Brownian motion if it satisfies: (1)(2) is continuous function of (3) is independent for (4) has a normal distribution
3. Wave Equation for NCFSSS
The next wave transformation is used to get the wave equation of the NCFSSS ((1) and (2)): with where and are deterministic functions and and for are nonzero constants. Plugging Equation (5) into Equations (1) and (2) and using we get for imaginary part and for real part
Taking expectation on both sides for Equations (9) and (10), we have
Since is standard Gaussian process, hence . 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
4. The Exact Solutions of the NCFSSS
To find the exact solutions of Equation (16), we use two different methods such as sine-cosine [13, 14] and the Jacobi elliptic function [29] methods. As a result, we are able to obtain the exact solutions of the NCFSSS ((1) and (2)).
4.1. Sine-Cosine Method
Assume the solution of Equation (16) has the form where
Setting Equation (18) into Equation (16) we get rewriting the above equation
Equalizing the term of in Equation (21), we attain
Substituting Equation (22) into Equation (21)
Equating each coefficient of and to zero, we have
By solving these equations, we get
Hence, the solution of Equation (16) is
There are many cases depending on the sign of and
Case 1. If and , then the exact solutions of the NCFSSS ((1) and (2)) are or
Case 2. If and , then the exact solutions of the NCFSSS ((1) and (2)) are or
Case 3. If and , then the exact solutions of the NCFSSS ((1) and (2)) are or
Case 4. If and then the exact solution of the NCFSSS ((1) and (2)) are or where and are defined in (17) and and
4.2. The Jacobi Elliptic Function Method
We suppose that the solution to Equation (16) has the type where , and are undefined constants and is the Jacobi elliptic sine function (Latin: sinus amplitudinis) for . Differentiate Equation (35) two times, we have
Plugging Equations (35) and (36) into Equation (16), we attain
Putting each coefficient of to be zero for , we have
When we solve the previous equations, we get
As a result, using (35), the solution of Equation (16) is
Therefore, the exact solution of the NCFSSS ((1) and (2)) is for and If , then the solutions (41) and (42) turn to:
In the same way, we can substitute in (35) with (whereis the elliptic cosine (Latin: cosinus amplitudinis)) and (where is the delta amplitude (Latin: delta amplitudinis)) to derive the solutions of Equation (16) as follows:
Therefore, the solutions of the NCFSSS ((1) and (2)) are as follows: for , and for and , respectively. If , then the solutions (45) and (46) and (47) and (48) turn to: for and
5. The Effect of BM on NCFSSS Solutions
The effect of BM on the exact solutions of the NCFSSS ((1) and (2)) is discussed here. Fix the parameters , and . Hence, , and . Now, for various values of (the fractional derivative order) and (noise intensity), we provide a number of graphs for and . To draw these graphs, we use the MATLAB tools. In the following Figure 1, if we can observe how the surface fluctuates as the value of changes:
While in Figures 2 and 3, we can observe that after small transit patterns, the surface smooths significantly when noise is incorporated, and its intensity increases for different value of .
Figure 4 shows 2D graphs of Equation (41) with and with , which highlight the above results.
We may infer from Figures 1–4 the following: (1)As fractional-order decreases, the surface shrinks(2)The solutions of NCFSSS are stabilized by BM around zero
6. Conclusions
In this paper, we considered the ()-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, 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. In future study, we can address the NCFSSS ((1) and (2)) with multidimensional multiplicative noise.
Data Availability
All data are available in this paper.
Conflicts of Interest
The authors declare that they have no competing interests.
Authors’ Contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Acknowledgments
This research has been funded by the Scientific Research Deanship at University of Hail, Saudi Arabia, through project number RG-21046.