The new exact solutions for the deterministic and stochastic (2+1)-dimensional equations in natural sciences

ABSTRACT This paper poses the Riccati–Bernoulli sub-ODE method in order to find the exact (random) travelling wave solutions for the (2+1)-dimensional cubic nonlinear Klein–Gordon (cKG) equation and the (2+1)-dimensional nonlinear Zakharov–Kuznetsov modified equal width (ZK-MEW) equation. The obtained travelling wave solutions are expressed by the hyperbolic, trigonometric and rational functions. Indeed, these solutions reflect some interesting physical interpretation for nonlinear phenomena. We discuss our method in deterministic case and in a random case. Additionally, we can show and discuss this method under some random distributions. Finally, some three-dimensional graphics of some solutions have been illustrated.

In this paper, we use the Riccati-Bernoulli sub-ODE method [11,12,[45][46][47], to construct exact solutions, solitary wave solutions for the (2+1)-dimensional cubic nonlinear Klein-Gordon (cKG) equation and the (2+1)dimensional nonlinear Zakharov-Kuznetsov modified equal width (ZK-MEW) equation. If we get a solution of NPDEs, we obtain new infinite sequence of solutions of these equations by using a Bäcklund transformation. Actually, we give new solutions and show that this method is efficient, powerful and vital for solving other type of NPDEs. Moreover the Riccati-Bernoulli sub-ODE method has an interesting feature, namely it is give infinite solutions. Indeed all presented solutions have so important contribution for the explanation of some practical physical phenomena and further nonlinear problems. To the best of our knowledge, no previous research work has been done using Riccati-Bernoulli sub-ODE method for solving the cKG equation and the ZK-MEW equation.
In many recent models, studying has ensued about the role of random or noise variables that end up as basic variables in predictive models. Therefore, one might think that the randomness part or the randomness effectiveness might be a normal outcome in many models. There are several difficulties in the study of stochastic models which may be as stochastic differential equations (SDEs) [48,49], or stochastic partial differential equations (SPDEs) [50,51]. As result, we implemented the Riccati-Bernoulli sub-ODE method for finding the exact stochastic solutions of the (2+1)dimensional stochastic cKG equation and the (2+1)dimensional nonlinear stochastic ZK-MEW equation, when the parameters are assigned random variables.
In order to find the conditions for our method in random case we will state the stability and convergence theorem.
The rest of the paper is given as follows. Section 2 describes the Riccati-Bernoulli sub-ODE method and a Bäcklund transformation of the Riccati-Bernoulli equation. In Sections 3, we apply the Riccati-Bernoulli sub-ODE method to solve the (2+1)-dimensional cGK equation and the (2+1)-dimensional nonlinear ZK-MEW equation. Additionally, in Section 4 we can discuss the Random Riccati-Bernoulli sub-RODE method. Finally, in Section 6 we give the conclusions.

The Riccati-Bernoulli sub-ODE method
Consider the following nonlinear evolution equation where P is a polynomial in φ(x, y, t) and its partial derivatives with even highest order derivatives and nonlinear terms.
Step 1. Let then Equation (1) reduces to a nonlinear ordinary differential equation ODE: Step 2. Assume the solution of Equation (3) takes the form where a,b,c and n are constants to be determined. From Equation (4), we get
Step 3. Substituting the derivatives of u into Equation (3) gives an algebraic equation of u. We can determine n, by using the symmetry of the right-hand item of Equation (4) and setting the highest power exponents of u to equivalence in Equation (3). We compare the coefficients of u i gives a set of algebraic equations for a, b, c, and v. Solving these equations and superseding n,a,b,c,v, and ξ = kx + βy − ςt into Equations (7)- (14)), then the solutions of Equation (1) is obtained.
(16) where A 1 and A 2 are arbitrary constants.
The Equation (16) is used to get infinite sequence of solutions for Equation (4) and consequently Equation (1).
Case III. When b = 0 and c = 0, the solution of Equation (17) is where λ and μ are arbitrary constants.

Physical interpretation
Here, physical interpretations of the solutions are illustrated to show the effectiveness of the Riccati-Bernoulli sub-ODE method for seeking exact solitary wave solution and periodic travelling wave solution of Equations (17). The wave speed λ = ± √ 2 or λ = ±0 and the nonzero constants α and β play an useful role in the physical structure of the solutions obtained in Equations (29)-(32) as we clarify: (1) If − √ 2 < λ < √ 2 and α > 0 and β > 0, then from Equations (29) and (30), we get complex soliton solutions.

Stochastic models
In fact, the stochastic models has many applications in our life and it can be as stochastic dynamical system so, it is important for us to discuss what is the effect when dealing with these stochastic models and how to find the control of stability or the constraints on the randomness part in order to find the solution for these models. In this section, we investigate stochastic process solution of the (2+1)-dimensional Stochastic cubic nonlinear Klein-Gordon model with two random variables input and (2+1)-dimensional nonlinear stochastic ZK-MEW model with three random variables by using the Riccati-Bernoulli sub-RODE method.

The (2+1)-dimensional cubic stochastic nonlinear Klein-Gordon model
The nonlinear stochastic Klein-Gordon equation is used to model many nonlinear phenomena. In this part of the paper, a new Riccati-Bernoulli sub-RODE method for solving the (2+1)-dimensional cubic stochastic nonlinear Klein-Gordon model is proposed as follow: where α and β are non zero random variables.

Theory of stability
The stochastic process solution for our problem, the (2+1)-dimensional stochastic cubic nonlinear Klein-Gordon equation is will be stable under the conditions: (1) α is bounded and second order random variable (E[α 2 ] < ∞).
Also, for the Riccati-Bernoulli sub-ODE method that we used must be with second order the travelling wave transformation since (E[λ 2 ] < ∞).

The (2+1)-dimensional stochastic nonlinear ZK-MEW equation
Stochastic nonlinear ZK-MEW equation in many areas play an important role. Therefore, solving stochastic or deterministic nonlinear ZK-MEW equation or generally, the nonlinear evolution equations has become a valuable task. For this purpose, we will try to deal with the stochastic case as follow. This equation given as follows: where α, β and γ are non zero random variables. AS the same as in deterministic case we can find the stochastic solution relation as follow, When α, β and γ are positive bounded random variables i.e. 0 < α(ω) ≤ α 1 , 0 < β(ω) ≤ β 1 , λ/((γ 2 − βλ)) > 0 and 0 < γ (ω) ≤ γ 1 , we get stochastic exact travelling wave solutions as follow, and φ 3,4 (x, y, t) where α, β, γ are random variables, λ, ε, and μ are arbitrary constants. The expected value operator of the stochastic solutionφ 1 ,φ 3 is depicted in Figure 7. The variance of the stochastic solutionφ 1 ,φ 3 is depicted in Figure 8.

Theory of stability
The stochastic process solution for our problem, the (2+1)-dimensional nonlinear stochastic Zakharov-Kuznetsov modified equal width (SZK-MEW) equation is will be stable under the conditions: (1) α is bounded and second order random variable (E[α 2 ] < ∞).
Also, for the Riccati-Bernoulli sub-ODE method that we used must be with second order the travelling wave transformation since (E[λ 2 ] < ∞).   concluded that this method is a very robust and efficient technique to find the exact solutions for a large class of NPDEs. Moreover, from Remark 3.1 we find that the Riccati-Bernoulli sub-ODE method gives an infinite sequence of solutions. Moreover, we implemented the Riccati-Bernoulli sub-ODE method for finding the exact stochastic solutions of the proposed models, when the parameters are assigned random variables. We also state the stability and convergence theorem to find the conditions for the proposed method in random case.

Comparisons
We compare the results presented in this paper with other results in order to show that the Riccati-Bernoulli sub-ODE is powerful,efficient and adequate.
(1) Wang et al. [52] have presented only five solutions for the cKG equation, using the multi-function expansion method. Whereas Khan et al. [55] given eight solutions, using the modified simple equation (MSE) method. Comparing these results with presented result in this paper, we deduce that the Riccati-Bernoulli sub-ODE method gives many new exact travelling wave solutions along with additional free parameters. Thus, the Riccati-Bernoulli sub-ODE method is more effective in providing many new solutions than these two methods.
(2) Wazwaz [54] has presented only four solutions for the ZK-MEW equation, using the sine-cosine method and has introduced four solutions, using the tanh-method. On the other hand, we give many new exact travelling wave solutions with numerous free parameters.
Thus, the Riccati-Bernoulli sub-ODE method superior to other methods.

Conclusions
We have introduced the (2+1)-dimensional cubic nonlinear Klein-Gordon equation and the (2+1)-dimensional nonlinear Zakharov-Kuznetsov modified equal width (ZK-MEW) models in deterministic case and also, if we have some disturbance in their coefficients. We have proposed Riccati-Bernoulli sub-ODE method in order to find the exact travelling wave solutions for these models, additionally, to find the stochastic process solution. The physical cases for the solution of the (2+1)dimensional cKG equation is discussed. The stochastic process solutions are studied for our models by stability control on the randomness part. The statistical moments are computed.