Impacts of Brownian motion and fractional derivative on the solutions of the stochastic fractional Davey-Stewartson equations

Abstract: In this article, the stochastic fractional Davey-Stewartson equations (SFDSEs) that result from multiplicative Brownian motion in the Stratonovich sense are discussed. We use two different approaches, namely the Riccati-Bernoulli sub-ordinary differential equations and sine-cosine methods, to obtain novel elliptic, hyperbolic, trigonometric, and rational stochastic solutions. Due to the significance of the DaveyStewartson equations in the theory of turbulence for plasma waves, the discovered solutions are useful in explaining a number of fascinating physical phenomena. Moreover, we illustrate how the fractional derivative and Brownian motion affect the exact solutions of the SFDSEs using MATLAB tools to plot our solutions and display a number of three-dimensional graphs. We demonstrate how the multiplicative Brownian motion stabilizes the SFDSE solutions at around zero.


Introduction
In 1974, Davey and Stewartson developed the Davey-Stewartson equation (DSE) [1]. DSE is used to illustrate how a three-dimensional wave-packet changes over time on water with a limited depth. It is a set of coupled partial differential equations for the complex field (wave amplitude) ( ) u x y t , , and the real field (mean flow) ( ) v x y t , , : (1) where = ± κ 1 and = ± δ 1. 2 The constant κ measures the cubic nonlinearity. The case = δ 1 is known as the DS-I equation, while = δ i is known as the DS-II equation. The DS-I and DS-II are two integrable equations in two space dimensions that originate from higher dimensional versions of the nonlinear Schrödinger  equation. They occur in a variety of applications, including the description of gravity-capillarity surface wave packets in shallow water.
The solutions to the DS equations (1)- (2) have been utilized in plasma physics, nonlinear optics, hydrodynamics, and other disciplines. For instance, the solutions of the DS equation might describe the interaction of a properly matched microwaves and spatiotemporal optical plus. Therefore, many authors have provided exact solutions for this equation utilizing different methods such as double exp-function [2], sine-cosine [3], ( ) ′ / G G -expansion [4], generalized ( ) ′ / G G -expansion [5], the uniform algebraic method [6], the extended Jacobi's elliptic function [7], trial equation method [8], and the first integral method [9].
Many important phenomena, such as image processing, electro magnetic, acoustic, anomalous diffusion, and electrochemical, are now represented by fractional derivatives [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]. One advantage of fractional models is that they may be expressed more explicitly than integer models, which motivates us to create a number of important and useful fractional models. Recently, Khalil et al. [25] proposed a novel concept of fractional derivative that expands the well-known limit definition of the derivatives of a function. This definition is known as the conformable fractional derivative. In contrast to earlier definitions, the new definition is clearly compatible with the classical derivative and appears to meet all of the standards of the standard derivative. It is crucial for fulfilling the product formula, the quotient formula, and it has a less complicated formula for the chain rule. The conformable fractional derivative of the fractional partial differential equations has been studied by a number of researchers, including [26][27][28][29][30] and references theirin.
Conversely, several disciplines, such as engineering, climate dynamics,chemistry, physics, geophysics, the atmosphere, biology, fluid mechanics, and others, have pointed out the benefits of adding random impacts in the statistical features, analysis, simulation, prediction, and modeling of complex processes [31][32][33]. Noise cannot be neglected because it has the potential to initiate interesting phenomena. Finding exact solutions to fractional PDEs with a stochastic term is typically more challenging than finding solutions to classical PDEs.
To reach a better degree of qualitative agreement, we consider DSEs (1) and (2) with fractional space and forced by multiplicative noise in the Stratonovich sense as follows: where α is the conformable fractional derivative [25], σ is the strength of noise, and ( ) B t is the Brownian motion.
We know that there are several possible interpretations for the stochastic integral ∫ X B d t 0 . A stochastic integral is often interpreted using the Stratonovich and Itô calculus [34]. A Stratonovich stochastic integral ) is one that is computed at the middle, whereas the stochastic integral is Itô (written as ) when it is investigated at the left end. Next equation is how the Stratonovich integral and Itô integral are connected: where X is supposed to be sufficiently regular and { } ≥ Z t , 0 t is a stochastic process. The objective of this work is to determine the exact solutions to the stochastic fractional Davey-Stewarts equations (SFDSEs). This study is the first to successfully get the exact solutions of SFDSEs (3) and (4). We used two methods, the Riccati-Bernoulli sub-ordinary differential equation (sub-ODE) approach and the sine-cosine method, to acquire a broad range of solutions, such as those for trigonometric, hyperbolic, and rational functions. Additionally, we use MATLAB package to produce 3D figures for some of the created solutions in this work to examine the impact of the Brownian motion on the solutions of SFDSEs This article is set up as follows: In Section 2, we employ an appropriate wave transformation to provide the wave equation of SFDSEs. In Section 3, to create the analytical solutions of SFDSEs (3) and (4), we use two methods. In Section 4, we examine how the Brownian motion impacts on the produced solutions. Finally, we present conclusions of this article.

Wave equation for SFDSEs
The next wave transformation is conducted in order to acquire the wave equation for the SFDSEs (3) and (4): where the functions ψ and φ are deterministic, Putting equation (6) into equation (3) and utilizing where we used (5), and we obtain for real part and for imaginary part From equation (9), we obtain Now, integrating equation (8) once and putting the constant of integral equal zero, we have Plugging equation (11) into equation (7), we obtain where Taking expectation ( ) ⋅ , yields Since ( ) B t is the standard normal process, hence (

Analytical solutions of the SFDSEs
We utilize two distinct methods including the Riccati-Bernoulli sub-ODE [35] and sine-cosine [36,37] to obtain the solutions of SFDSEs (equations (3) and (4)). The methods that have been proposed are useful tools that can solve a wide variety of other nonlinear PDEs. In addition to that, this procedure can provide a brand new infinite sequence of solutions.
Substituting equation (19) into equation (11) and integrating we obtain where  Substituting into equation (11) and integrating, we obtain where  Substituting into equation (11) and integrating, we obtain Thus, the solution of the SFDSEs (3) and (4) is where then the solution of the RBE (16) is Substituting into equation (11) and integrating, we obtain Consequently, the solution of the SFDSEs (3) and (4) is where γ 1 and γ 2 defined in equation (13) and

Sine-cosine method
Let the solutions φ of equation (15) have the form where Putting equation (30) into equation (15) we obtain Equalizing the term of H in equation (32), we have Setting equation (33) into equation (32) Balancing each coefficient of − H 3 and − H 1 to zero, we have and We obtain by solving these equations Hence, the solution of equation (15) is sec or 2 csc . Substituting into equation (11) and integrating, we obtain tan or 4 cot . There are various cases relying on the sign of γ 1 and γ .    MATLAB package to plot many figures for different noise intensity σ and for different value of fractional derivative order σ. First, the impact of noise: In Figure 1, for various values of α, we observe that the surface fluctuates when = σ 0. Figure 2 shows that when the noise intensity increases, after these short periods of transit, the surface becomes more flat. Figures 1 and 2 allow us to draw the following conclusion: when the noise is not taken into account (i.e., = σ 0), there are several distinct kinds of solutions, including periodic solution, kink solution, and others. When noise is added, and its intensity is increased by a factor of = σ 1, 2, the surface, which before had only modest transit patterns, becomes substantially flatter. This illustrates that the SFDSE solutions are affected by the white noise and that it stabilizes them around zero.
Second the impact of fractional derivative: In Figures 3 and 4, if we set = σ 0, we see that the surface grows with increasing α.
We inferred from Figures 3 and 4 that the solution curves do not overlap. Additionally, the surface gets smaller as the order of the fractional derivative gets smaller.

Conclusion
The stochastic-fractional ( ) + 2 1 -dimensional DSEs (3) and (4) were taken into consideration in this article. To acquire stochastic fractional hyperbolic, rational, trigonometric solutions, we applied the Riccati- Bernoulli sub-ODE and sine-cosine methods. These solutions are essential for comprehending many fundamentally complex basic phenomena. Additionally, the obtained answers will be very helpful for further research in subjects like hydrodynamics, nonlinear optics, plasma physics, and other areas. Finally, the influence of multiplicative Brownian motion on the exact solutions of the SFDSEs (3) and (4) is illustrated. We came to the conclusion that the noise makes the solutions stable around zero. In the future work, we might think about solving the ( ) + 2 1 -dimensional DSEs equations using either additive noise or color noise.