Exact Solutions for a Class of Wick-Type Stochastic (3+1)-Dimensional Modified Benjamin–Bona–Mahony Equations

In this paper, we investigate the Wick-type stochastic (3+1)-dimensional modified Benjamin–Bona–Mahony (BBM) equations. We present a generalised version of the modified tanh–coth method. Using the generalised, modified tanh–coth method, white noise theory, and Hermite transform, we produce a new set of exact travelling wave solutions for the (3+1)-dimensional modified BBM equations. This set includes solutions of exponential, hyperbolic, and trigonometric types. With the help of inverse Hermite transform, we obtained stochastic travelling wave solutions for the Wick-type stochastic (3+1)-dimensional modified BBM equations. Eventually, by application example, we show how the stochastic solutions can be given as white noise functional solutions.


Introduction
In this paper, with the help of white noise theory, Hermite transform and a generalised, modified tanh-coth method, we deduce stochastic travelling wave solutions for the Wick-type stochastic (3+1)-dimensional modified BBM equations as the forms: and where (x, y, z, t) ∈ R 3 × R + , R i (i = 1, 2, ...9) are non-zero integrable functions from R + to the Kondrative distribution space (S) −1 , which was defined by Holden et al. in [1] as a Banach algebra with the Wick-product Equations (1)- (3) are the perturbations of the (3+1)-dimensional modified BBM equations with variable coefficients: v t + r 4 (t)v x + r 5 (t)v 2 v y + r 6 (t)v xzt = 0, and w t + r 7 (t)w y + r 8 (t)w 2 w z + r 9 (t)w xxt = 0, where r i (i = 1, 2, ...9) are non-zero integrable functions o nR + . The modified BBM equation: which describes the surface long waves in nonlinear dispersive media. It is also used as a character to acoustic-gravity waves in compressible fluids, hydromagnetic waves in cold plasma, and acoustic waves in anharmonic crystals [2]. The study of (3+1)-dimensional nonlinear equations is promising because these equations model the real features in a wide assortment of science, technology, fluid mechanics, wave propagations, electrodynamics, and engineering fields [3][4][5][6]. For this reason, Hereman [4,5] proposed the (3+1)-dimensional nonlinear modified KdV equation. Analogously, and by the same sense, Wazwaz [7] introduced Equations (4)- (6). Moreover, if Equations (4)- (6) are considered in a random environment, we have random (3+1)-dimensional modified BBM equations. In order to obtain the exact solutions of random (3+1)-dimensional modified BBM equations, we only consider them in a white noise environment; that is, we will discuss the Wick-type, stochastic, (3+1)-dimensional modified BBM Equations (1)- (3).
There are many methods to obtain travelling wave solutions to nonlinear PDEs, such as the inverse scattering method [26], the Newton's method [27], the tanh method [28], the Sinc-Galerkin method [29], the residual power series method [30], the semi-inverse variational principle and the first integral method [31], and the Daftardar-Gejji and Jafari method [32]. The tanh method, established by Malfliet [33], pursues a specially straightforward and effective algorithm to obtain exact solutions for a wide class of nonlinear PDEs. Moreover, a variety of research papers have focused on the different applications and extensions of the tanh method. Fan [34] has introduced an extended tanh method and gave new travelling wave solutions that cannot be obtained by the tanh method. Also, Wazwaz extended the tanh method and named it the tanh-coth method [35]. Furthermore, El-Wakil [36] and Soliman [37] modified the tanh-coth method and presented new, exact solutions for some nonlinear PDEs.
Our aim in this work was to obtain new stochastic travelling wave solutions for the Wick-type stochastic (3+1)-dimensional modified BBM equations. Firstly, we give a generalised version of the modified tanh-coth method to make it convenient for the nonlinear (3+1)-dimensional and multi dimensional PDEs. Secondly, we use the generalised, modified tanh-coth method, white noise theory, and Hermite transform to produce a new set of exact travelling wave solutions for the (3+1)-dimensional modified BBM equations, this set includes solutions of exponential, hyperbolic, and trigonometric types. Finally, we use the inverse Hermite transform to obtain stochastic travelling wave solutions for the Wick-type stochastic (3+1)-dimensional modified BBM equations. Moreover, by an application example, we show how the stochastic solutions can be given as white noise functional solutions. In our work, the modified BBM equation describes the surface long waves in nonlinear dispersive media. It is also used as a character of acoustic gravity waves in compressible fluids, hydromagnetic waves in cold plasma, and acoustic waves in harmonic crystals [2]. The study of (3+1)-dimensional nonlinear equations is prospering because these equations model the real features in a wide assortment of science, technology, fluid mechanics, wave propagations, electrodynamics, and engineering fields [3][4][5][6]. The origin and references of Equation (1) are given in Holden [1]. Ghany and Fathallah studied white-noise functional solutions for wick-type stochastic time-fractional Benjamin-Bona-Mahony (BBM) equation in [38]. Recently, Sahoo and Saha Ray studied by other methods the stochastic solutions of wick-type stochastic time-fractional BBM equation for modeling long surface gravity waves of small amplitude, in [39]. The PDE of Benjamin et al. [2] is now often called the BBM equation, although it is also known as the regularised long wave (RLW) equation. Morrison et al proposed the one-dimensional PDE, as an equally valid and accurate model for the same wave phenomena simulated by the KdV and RLW equations [40]. Random waves are an important subject of random PDEs. In essence, to investigate the exact solutions of random BBM equation, we restricted our attention to consider this problem in white noise environment [38].
This paper is organized as follows: In Section 2, we recall some requisites from Gaussian white noise analysis. In Section 3, we give a generalisation to the modified tanh-coth method to make it convenient for the nonlinear (3+1)-dimensional equations. In Section 4, we employ the generalised, modified tanh-coth method, white noise theory, and Hermite transform to obtain a new set of exact travelling wave solutions for the (3+1)-dimensional modified BBM equations. In Section 5, we apply the inverse Hermite transform to explore stochastic travelling wave solutions for the Wick-type stochastic (3+1)-dimensional modified BBM equations. In Section 5, we give some examples to show that the stochastic solutions can be given as Brownian motion functional solutions and white noise functional solutions. In Section 6, we give a summary and discussion.

Requisites from Gaussian White Noise Analysis
The Gaussian white noise analysis starts with the rigging S(R d ) ⊂ L 2 (R d ) ⊂ S * (R d ), where S(R d ) is the Schwartz space of rapidly decreasing, infinite differentiable functions on R d , and S * (R d ) is the space of tempered distributions. From the Bochner-Minlos theorem [1], we have a unique white noise measure µ, on S * (R d ), β S * (R d ) . Assume that ξ n (x) = π −1/4 ((n − 1)!) −1/2 e −x 2 /2 h n−1 ( √ 2x), n ∈ N are the Hermite functions, where h n (x) denotes the Hermite polynomials. It is well known that the collection (ξ n ) n∈N forms an orthonormal basis for L 2 (R). Let α = (α 1 , ..., α d ) be a d-dimensional multi-indices with α 1 , ..., α d ∈ N; then, the family of tensor products ξ α := ξ (α 1 ,..., Using this ordering, we define Let n ∈ N, the Kondrative space of stochastic test functions (S) n 1 is defined by: (10) and the Kondrative space of stochastic distributions (S) n −1 is defined by: The family of seminorms f 1,k , k ∈ N produces a topology on (S) n 1 and (S) n −1 can be represented as the dual of (S) n 1 under the action where , by the definition of Hermite transform, we get: for all w such that F(w) and G(w) exist. The multiplication on the right hand side of the above equality is the complex bilinear multiplication in C n which is defined by (w 1 1 , ...w 1 n )(w 2 1 , ..., Hence, The Hermite transform converts the Wick product into the usual product and convergence in (S) n −1 into pointwise and bounded convergence in a specific neighbourhood of zero in C n . For more details about stochastic Kondrative spaces, Wick product, and Hermite transform we refer the reader to [1].

Generalization of the Modified Tanh-Coth Method
Consider a multi dimensional, nonlinear PDE of wave propagation: where u is the dependent variable and t = x 0 , x 1 , x 2 , ..., x m are the independent variables. Introduce the wave transformation: where a i (i = 0, 1, 2, ..., m) are unknown constants. Therefore, Equation (15) can be transformed into a nonlinear ordinary differential equation (NODE): For simplicity, we integrate the NODE (17), provided that all terms include derivatives, and set the integration constants to be zero. Subsequently, the transformed Equation (17) can be solved by expanding its general solution in finite series as follows: where Φ solves the first order Riccati equation [41]: where α 0 ,α 1 , and α 2 are constants to be determined. The positive constant N can be specified by balancing the linear and nonlinear terms of highest order in Equation (17). Inserting Equations (18) and (19) into Equation (17), yields an algebraic equation in Φ and its powers. Equating the coefficients of Φ k to zero, gives an algebraic system of equations in A k and B k . With the help of the computer symbolic system Mathematica, we can obtain A k and B k . The Riccati Equation (19) has the following particular solutions [42]:

Case I
We reduce the system (25) by using α 0 = α 1 = 1 and α 2 = 0. By using Mathematica, we can find a set of solutions for the reduced system as follows: .

Example
In this section, we provide a specific application example to demonstrate the effectiveness of our results and to justify the real contribution of these results. We focus our attention on Equation (1). Concerning the other two equations, Equations (2) and (3), the procedure is similar. We observe that the solutions of Equation (1) are strongly depend on the shape of the given functions R 1 (t) and R 2 (t). So, for dissimilar forms of R 1 (t) and R 2 (t), we can find dissimilar solutions of Equation (1) which come from Equations (70)-(78). We illustrate this by giving the following example.
Assume that R 2 (t) = δ 1 R 1 (t), R 3 (t) = δ 2 R 1 (t) and R 1 (t) = f (t) + δ 3 W t , where δ 1 ,δ 2 , and δ 3 are arbitrary constants, f (t) is a bounded measurable function on R + , and W t is the Gaussian white noise, which is the time derivative (in the strong sense in (S) −1 ) of the Brownian motion B t . The Hermite transform of W t is given by W t (w) = ∑ ∞ i=1 w i t 0 η i (τ)dτ [1]. Using the definition of W t (w), Equations (70)-(78) yield the white noise functional solution of Equation (1) as follows: with Ω 4 (x, y, z, t) = a 1 x + a 2 y + a 3 z + 2a 3

Conclusions
Due to the fact that the stochastic models are more realistic than the deterministic models, we concentrated our study in this paper on the Wick-type, stochastic, (3+1)-dimensional modified BBM equations. Besides that, we investigated and solve the deterministic, (3+1)-dimensional modified BBM equations. In this paper, we set up a new and general version of the modified tanh-coth method to deal with the nonlinear multi dimensional PDEs. By using this generalisation of the modified tanh-coth method, Hermite transform, and white noise theory, we produced a new set of exact travelling wave solutions for the variable coefficients and (3+1)-dimensional modified BBM equations. This set includes solutions of exponential, hyperbolic, and trigonometric types. In [7], Wazwaz has solved the deterministic, (3+1)-dimensional modified BBM equations with constant coefficients, So, our results for this model are more general than the results obtained by him. With the aid of inverse Hermite transform, we obtained stochastic travelling wave solutions for the Wick-type, stochastic, (3+1)-dimensional modified BBM equations. Furthermore, we showed by an example how the stochastic solutions can be given as white noise functional solutions. Note that, the schema proposed in this paper can be used for solving several nonlinear evolution equations in mathematical physics, both Wick-type stochastic and deterministic. Moreover, the Riccati equation (19) has different solutions if we chose different values of α 0 , α 1 , and α 2 . Therefore, we can find many other solutions of the Wick-type stochastic and deterministic (3+1)-dimensional modified BBM equations.
The PDE of Benjamin et al. [2] is now often called the BBM equation, although it is also known as the regularised long wave (RLW) equation. Morrison et al. proposed the one-dimensional PDE, as an equally valid and accurate model for the same wave phenomena simulated by the KdV and RLW equations [40]. Random waves are an important subject of random PDEs. In essence, to investigate the exact solutions of random Benjamin-Bona-Mahony equation, we restricted our attention to consider this problem in a white noise environment [38]. The propagation of nonlinear wave in systems with polarity symmetry can be described by the (3+1)-dimensional modified Benjamin-Bona-Mahony Equation (7). If the problem is considered in a non-Gaussian stochastic environment, we can get non-Gaussian, stochastic, (2+1)-dimensional coupled KdV equation. Obviously, the planner which we have proposed in this paper can be also applied to other non-linear PDEs in mathematical physics such as KdV-Burgers, modified KdV-Burgers, Zhiber-Shabat and Benjamin-Bona-Mahony equations. We observe that the F-expansion method we used has many other particular solutions; this in turn gives many other exact solutions for the considered stochastic, (3+1)-dimensional modified Benjamin-Bona-Mahony equations. Additionally, in this work, we discussed the solutions of SPDEs driven by non-Gaussian white noise; this discussion is less detailed than the Gaussian discussion but more general, because it deals with the dual pairing generated by integration with respect to a non-Gaussian measure. Furthermore, in future work, we will discuss the solutions of SPDEs driven by non-Gaussian white noise to get exact stochastic solutions of the non-Gaussian, stochastic, (3+1)-dimensional modified Benjamin-Bona-Mahony equations; we only considered this problem in a non-Gaussian white noise environment; that is, we investigated the variable coefficients of stochastic, (3+1)-dimensional modified Benjamin-Bona-Mahony equations. For this aim, we developed a non-Gaussian Wick calculus based on the theory of hyper-complex systems to get exact travelling wave solutions of (3+1)-dimensional modified Benjamin-Bona-Mahony equations and non-Gaussian white noise functional solutions of Wick-type stochastic (3+1)-dimensional modified Benjamin-Bona-Mahony equations.