Hypercomplex Systems and Non-Gaussian Stochastic Solutions with Some Numerical Simulation of χ -Wick-Type (2 + 1)-D C-KdV Equations

: In this article, we discuss the (2 + 1)-D coupled Korteweg–De Vries (KdV) equations whose coefﬁcients are variables, and stochastic (2 + 1)-D C-KdV (C-KdV) equations with the χ -Wick-type product. White noise functional solutions (WNFS) are presented with the homogeneous equilibrium principle, Hermite transform (HT), and technicality via the F-expansion procedure. By means of the direct connection between the theory of hypercomplex systems (HCS) and white noise analysis (WNA), we establish non-Gaussian white noise (NGWN) by studying stochastic partial differential equations (PDEs) with NG-parameters. So, by using the F-expansion method we present multiples of exact and stochastic families from variable coefﬁcients of travelling wave and stochastic NG-functional solutions of (2 + 1)-D C-KdV equations. These solutions are Jacobi elliptic functions (JEF), trigonometric, and hyperbolic forms, respectively.


Introduction
This article focuses on stochastic (2 + 1)-D C-KdV equation of the Wick-type with an NG parameter.
The (2 + 1)-D C-KdV system is an important type of nonlinear equations in mathematics and physics. So, it is important to find solutions to this equation. The (2 + 1)-D C-KdV system has various applications in numerous fields of nonlinear science. One of the applications of the (2 + 1)-D C-KdV system is that it can be used to describe generic string dynamic properties in constant curvature spaces for strings and multistrings. Another application can be found in [3][4][5][6][7][8][9]. The (2 + 1)-D C-KdV Equation (2) describes nonlinear wave propagation in polarity-coordination systems. Moreover, if the problem is taken into account in an NG-stochastic framework, we can gain the NG-stochastic (2 + 1)-D C-KdV equation. In order to obtain exact stochastic solutions of the NGS (2 + 1)-D C-KdV equation, we consider this issue in an NGWN framework. Consequently, we dispute variable coefficients of stochastic (2 + 1)-D C-KdV Equation (1). The existence of a solution of non-self-governing fractional differential equations with integral impulse conditions is investigated and studied via the measurement of noncompactness. A novel analytical technique was studied to obtain solitary solutions for the fractional-order equation of nonlinear evolution; hence, the periodic solutions of a certain one-D differential equation were investigated. For more details, see [10][11][12][13][14][15][16]. Furthermore, Dumitru, et al. investigated and studied the fractional differential equation in mathematical physics, which is closely related with our work. For more details, see [17][18][19][20][21]. The functional analysis of stochastic white noise (SWN) in [6,[22][23][24][25][26][27][28][29]: Stochastic WNFS was extensively studied for some nonlinear PDEs [2,[30][31][32].Okb El Bab, Zakarya, et al. [1,2,33], using the theory of HCS and NG-Wick calculus based on HCS with some SPDE and applications, studied some significant subjects related to the construction of NGWN analysis [34,35]. Recently, Zakarya, using HCS and NG-Wick calculus, sought to obtain stochastic Wick-type (3 + 1)-D-modified equations of the Benjamin-Bona-Mahony (BBM) type [36]. Very recently, N-soliton solutions to integrable equations were systematically studied with the Hirota direct method for both (1+1)-dimensional and (2 + 1)-dimensional integrable equations [37], and some important classes of novel equations in (2 + 1)-dimensions [38], and for nonlocal integrable equations [39].
The purpose of this article is giving the exact and stochastic (2 + 1)-D traveling wave solutions for (TWS) C-KdV Equation (2) and NG-stochastic (2 + 1)-D C-KdV Equation (1) in order to obtain NG-stochastic solutions of the (2 + 1)-D C-KdV equation. This problem was only considered in a NGWN framework, which means that the variable coefficients were considered to be stochastic (2 + 1)-D C-KdV Equation (1). We developed an NG-Wick calculus based on the theory of HCS for this purpose (L 1 (Q, dm(x))). We used the specific relation between WNA and theory of HCS [2]. We used this NG-parameter construction and F-expansion method to provide a number of families of (2 + 1)-D C-KdV equation solutions for solitary TWS, namely, (2) and NG-stochastic solutions of χ-Wick-type (2 + 1)-D C-KdV Equation (1).
This article consists of five sections. In Section 1, we present the introduction. In Section 2, we used the main results obtained in [1,2] and the method of F-expansion to obtain the exact TWS for the stochastic style (2 + 1)-D C-KdV Equation (2). In Section 3, for the Wick-type stochastic (2 + 1)-D C-KdV equations, we obtain exact NGWN functional solutions (1). In Section 4, we support our findings with detailed examples. A summary and discussion are in Section 5.

Solitary TWS in Equation (2)
In this part of the report, we minimize Equation (1) into deterministic partial differential equations (PDEs) by using χ-HT. In addition, the obtained PDEs could be translated into ordinary nonlinear differential equations (ODEs) by applying proper transformation. So, by using the proposed F-expansion method, we obtained many accurate solutions of Equation (2). Influenced by the χ-HT of Equation (1), we obtained the following deterministic equations.
where z = (z 1 , z 2 , ...) ∈ (C N ) c is a vector parameter. To seek solitary TWS of Equation (3), we took the transformations as follows: where k, l, µ and c are free constants, and klµ = 0 , ω(t, z) is a nonzero variable function that is subsequently determined. As a result, Equation (3) can be converted into ODE as follows: where indicates the derivative for . According to the F-expansion method [31], we could express the solutions of Equation (3) as follows.
where a i and b i are constants that are established later. Using the principle of homogeneous balance in Equation (4), we could obtain N = M = 2. So, Equation (5) becomes where b 0 , b 1 , b 2 , a 0 , a 1 and a 2 , are constants that are found later. Using the F-expansion method and substituting (6) into (4) and the collection of all expressions that had the same power of F i ( )[F ( )] j , (i = 0 ± 1, ±2, ..., j = 0, 1) as follows: By vanishing the coefficients of F i ( )[F ( )] j , we obtain an algebraic system as follows: By solving system 8, we obtain the following coefficients: By substituting from (9) into (6), we obtain the following general forms of the solutions of Equation (2).
From Appendix A, we obtain special cases of solutions as follows: Case I.

Remark 1.
There are many results for Equation (2) that are based on checking various coefficients values P 1 , P 2 and P 3 (see Appendices A-C). The above cases clearly illustrate the degree to which our approach is valid.

NG-Stochastic Solutions of Equation (2)
In this section, by applying HT to obtain NG-stochastic solutions, we use the results of the above section to Wick-type (2 + 1)-D C-KdV Equation (2). The characterization of trigonometric and exponential functions ensures that there is a bounded set open G ⊂ R + × R 2 , q < ∞, M > 0., such that the solution u(t, x 1 , y 1 , z) of Equation (3) and all its partial derivatives in Equation (3) are uniformly bounded for (t, x 1 , y 1 , z) ∈ G × O q (M), continuous with respect to (t, x 1 , y 1 ) ∈ G for all z ∈ O q (M) and analytic with respect to z ∈ O q (M), for all (t, x 1 , y 1 ) ∈ G. From Theorem 2.1 in [34], there exists U(t, x 1 , y 1 ) ∈ H χ −q such that u(t, x 1 , y 1 , z) = U(t, x 1 , y 1 )(z) for all (t, x 1 , y 1 , z) ∈ G × O q (M) and U(t, x 1 , y 1 ) solves Equation (2) in H χ −q . For this reason, by using the inverse χ-HT to the above results of Section 2, we have NG-stochastic functional solutions of Equation (1) as follows.
(I) NG-Stochastic Solutions of JEF Type:

Conclusions
The (2 + 1)-D coupled KdV Equation (2) can describe the diffusion of nonlinear waves in polarity symmetry schemes. When a problem is considered in a NG-stochastic system, we can obtain the NG-stochastic (2 + 1)-D C-KdV equation [1,2,33]. In order to obtain NG-stochastic solutions of the (2 + 1)-D C-KdV equation, we just considered this topic in a NG-white noise climate, that is, we considered variable coefficient stochastic (2 + 1)-D C-KdV Equation (1). For this reason, we developed an NG-Wick calculus-based theory of HCS L 1 (Q, dm(x)). We used a direct relation to the study of white noise and HCS [2]. We employed this construction of NG-parameters and the F-expansion method to provide several families with solitary TWS of (2 + 1)-D C-KdV Equation (2) and NGstochastic solutions of χ-Wick-type (2 + 1)-D C-KdV Equation (1). The obtained solutions were JEF functional solutions, and trigonometric and hyperbolic forms. Through suitable parametric choices, we also plotted three-dimensional graphics of some NG-white-noiseinduced solutions for SPDEs. In future work, we will apply this solution approach by using fractional differential equations.

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A. ODE and Jacobi Elliptic Functions
Relation between values of (P 1 , P 2 , P 3 ) and corresponding F( ) in ODE.