Exact Solutions and Conservation Laws of the (3 + 1)-Dimensional B-Type Kadomstev–Petviashvili (BKP)-Boussinesq Equation

: In this paper, Lie symmetry analysis is presented for the (3 + 1)-dimensional BKP-Boussinesq equation, which seriously affects the dispersion relation and the phase shift. To start with, we derive the Lie point symmetry and construct the optimal system of one-dimensional subalgebras. Moreover, according to the optimal system, similarity reductions are investigated and we obtain exact solutions of reduced equations by means of the Tanh method. In the end, we establish conservation laws using Ibragimov’s approach.


Introduction
In the past few years, nonlinear evolution equations have been used to explore physical phenomena, such as marine engineering, plasma physics, fluid dynamics, etc. In order to understand many complex physical phenomena better, we need to research explicit solutions of nonlinear evolution equations. Wazwaz [1] proposed the (3 + 1)-dimensional generalized BKP equation − u xxxy + u ty + 3u xz − 3(u x u y ) x = 0, (1) which explains evolution of quasi-one dimensional shallow water waves when the effects of viscosity and surface tension are taken to be negligible [2]. Recently, a host of exact solutions of Equation (1), including grammian-type determinant solutions [3], periodic wave solutions [4], lump solutions [5] and multiple wave solutions [6], have been discussed. Moreover, adding an extra term u tt , Wazwaz and El-Tantawy got an expansion of Equation (1), that is a new form of the (3 + 1)-dimensional BKP-Boussinesq Equation [7] u ty − u xxxy − 3(u x u y ) x + u tt + 3u xz = 0, where u(x, y, t, z) is an unknown function and subscripts denote the partial derivatives. This equation possesses the properties of both the Boussinesq and the BKP equations, which can be used to describe the propagation of long waves in shallow water [8]. By using analysis of the Painlevé property, the integrability properties of Equation (2) have been proved [2]. One and two soliton solutions were derived by utilizing the simplified Hirota's method in [7]. It was reported that coefficients of spatial variables were left as free parameters. Based on the Bäcklund transformation, the rational solutions and exponential wave solutions of Equation (2) were obtained in [8].
and its fourth-order prolongation is pr (4) , η 1 y = D y (η 1 ) − u x D y (ξ 1 ) − u y D y (ξ 2 ) − u t D y (ξ 3 ) − u z D y (ξ 4 ), η 1 ty = D t D y (η 1 − ξ 1 u x − ξ 2 u y − ξ 3 u t − ξ 4 u z ) + ξ 1 u xty + ξ 2 u yty + ξ 3 u tty + ξ 4 u zty , and D x , D y , D t , D z , respectively, represent the total derivatives concerning x, y, t and z. Then, the determining equations generated by the invariance conditions can be written as where ∆ = u ty − u xxxy − 3(u x u y ) x + u tt + 3u xz . Furthermore, we obtain the following system of overdetermined equations Solving this system, we can get where c 1 , c 2 are arbitrary constants and F 1 (z), F 2 (z), F 3 (z), F 4 (z) and F 5 (z) are arbitrary functions. To obtain physically crucial solutions, we take then substituting the above and obtaining where c 1 , c 2 , c 3 , c 4 , c 5 , c 6 and c 7 are arbitrary constants. Thus seven-dimensional Lie algebra made up of infinitesimal symmetries is spanned by the following generators After getting the infinitesimal generators, the following group transformations, which are formed by the X i for i = 1, 2, 3, 4, 5, 6, 7 can be given as G 5 : (x, y, t, z, u) → (x + , y, t, z, u), G 6 : (x, y, t, z, u) → (x, y, t, z, t + u), where is any real number. We discover that G 1 is a scalar transformation, G 2 is a z -translation, G 3 is a t -translation, G 4 is a y -translation, G 5 is an x -translation, G 6 and G 7 are Galilean transformations. Therefore, if u(x, y, t, z) is a solution of Equation (2), the following solutions are equivalent to the solutions of Equation (2) G 1 ( ) · u(x, y, t, z) = e 1 3 u(e − 1 3 x, e − y, e − t, e − 5 3 z), where is any real number.

The Optimal System of One-Dimensional Subalgebras
It is impractical for us to list all possible group-invariant solutions. Consequently, we need an effective and systematic way to classify these solutions; after doing this we form an optimal system of group-invariant solutions. Ibragimov et al. introduced a succinct method that relies only on the commutator table [25] to obtain the optimal system of one-dimensional subalgebras. The commutation relations about Lie algebra determined by X 1 , X 2 , X 3 , X 4 , X 5 , X 6 , X 7 are given in Table 1. Evidently, {X 1 , X 2 , X 3 , X 4 , X 5 , X 6 , X 7 } is closed under the Lie bracket.
Taking all the possible combinations, we derive the following representatives
Taking all the possible combinations, we derive the following representatives Ultimately, by collecting Operators (7), (9) and (10), we reach the following theorem: Theorem 1. An optimal system of subalgebras of seven-dimensional Lie algebras of Equation (2) is offered in the following operators:

Similarity Reductions of the BKP-Boussinesq Equation
In the preceding sections, we studied Lie symmetry analysis and constructed the optimal system of Equation (2). Next, we cope with the similarity reductions and obtain the reduced equations.

Case 4.1
For the generator X 2 , we have similarity variables and the group invariant solution is written as Substituting Equation (11) into Equation (2), we obtain the following reduced equation

Case 4.2
For generator X 4 + X 5 + X 7 , we have similarity variables u = f ( y, t, z) + x where y = −x + y, z = z, t = t. Substituting them into Equation (2) enables f to satisfy the following reduced equation

Case 4.3
For generator The reduced equation is given as follows

Case 4.4
For generator X 3 + X 4 + X 7 , we have x = x, t = −y + t, z = z, u = f ( x, z, t) + y. Substituting them into Equation (2) enables f to satisfy the following reduced equation

Case 4.5
For generator Substituting them into Equation (2) causes f to satisfy the following reduced equation

Case 4.6
For generator X 2 + X 7 , we have x = x, y = y, t = t, u = f ( x, y, t) + z. By substituting them into Equation (2), we have the following reduced equation

Case 4.7
For generator X 2 + X 3 + X 4 , we have u = f ( x, y, z) where x = x, y = −y + t, z = −y + z. By substituting them into Equation (2), the following reduced equation is expressed as follows

Case 4.8
For generator X 3 + X 5 , we have x = −x + t, y = y, z = z, u = f ( x, y, z). By substituting them into Equation (2), we obtain the following reduced equation

Case 4.9
For generator X 3 + X 4 , we have x = x, y = −y + t, z = z, u = f ( x, y, z). The form of the reduced equation is

Case 4.10
For generator X 2 + X 6 , we obtain x = x, y = y, t = t, u = f ( x, y, t) + tz. The corresponding reduced equation is

The Explicit Solutions of Reduced Equations
In the previous section, we have dealt with the similarity reductions and derived the corresponding reduced equations. In this section, we perform the Tanh method on reduced equations and obtain exact solutions of Equation (2). With the help of exact solutions, we can clearly understand the properties and applications of the (3 + 1)-dimensional BKP-Boussinesq equation. Here, we consider Equations (12)- (16); the others can be obtained in the same way.

Description of the Tanh Method
The main steps of the Tanh method [20,21] are expressed as follows: 1. Consider a general form of nonlinear partial differential equation F(u, u x , u xx , ..., u y , ..., u z , ..., u t ) = 0, where F is a polynomial of the u and its derivatives.

By using wave transformation
where l, m, n and c are unknown constants. Substituting Equation (23) into Equation (22), we obtain the following nonlinear ordinary differential equation 3. Next, we introduce an independent variable which has the following changes 4. We assume that the solution of Equation (24) is written in the following form where k is an integer, which is determined by balancing the highest order derivative terms with the nonlinear terms in the resulting equation. After determining k, putting Equations (26) and (27) into Equation (24), we get a polynomial concerning Y i (i = 0, 1, 2, · · · ). Then we collect all terms of Y i (i = 0, 1, 2, · · · ) and make each of them equal to zero, which obtaina the algebraic equations containing unknown numbers a i (i = 0, 1, · · · ), l, m, n, and c. Solving these equations, we get the values of unknowns. Finally, plugging these values into equations, we derive exact solutions of equations.

Exact Solution of Equation (12)
For Equation (12), substituting Equation (23) into Equation (12), we obtain the following ordinary differential equation Concerning Equation (28), balancing Φ (4) with Φ Φ , we have Hence, according to the Equation (27), the solution of Equation (12) is assumed as Then, substituting Equations (26) and (29) into Equation (28), we collect all terms of Y i and obtain the algebraic equations including unknown numbers a i (i = 0, 1), l, m and c. By solving these equations, we have the following solutions Putting Equation (30) into Equation (12), we obtain the exact solution of Equation (12) as follows where c = 4l 3 , a 0 and l are arbitrary constants. By using similarity variables x = x, y = y, t = t, and the group invariant solution u = f ( x, y, t), we obtain the exact solution of Equation (2) as follows where c = 4l 3 , a 0 and l are arbitrary constants. Figure 1 depicts the kink solution of Equation (2), which is obtained by taking a 0 = 0, l = 1, c = 1 at y = 1.
So, the exact solution of Equation (2) is where m = 0, a 0 and c are arbitrary constants. When we take a 0 = 0, m = 1, c = 1 at y = 1, x = 1, the value of u is as illustrated in Figure 2 below.

Exact Solution of Equation (14)
Equally, substituting Equation (23) into Equation (14), we get the following ordinary differential equation Furthermore, balancing Φ (4) with Φ Φ for (34), we have k = 1. Therefore, based on Equation (27), the solution of Equation (14) can be assumed as Next, substituting Equation (26) and Equation (35) into Equation (34), we make all coefficients of Y i vanish and obtain the algebraic equations including unknown numbers a i (i = 0, 1), l, m, and n. Solving these equations, we have the following solutions So, the exact solution of Equation (2) is where a 0 , a 1 and m are arbitrary constants. Figure 3 portrays the solution of Equation (2), which is obtained by taking a 0 = 0, a 1 = 2, m = 1 at y = 1, z = 1.

Exact Solution of Equation (15)
In the same way, substituting Equation (23) into Equation (15), we have the following ordinary differential equation Then, balancing Φ (4) with Φ Φ for (36), we have k = 1. Therefore, based on the Equation (27), the solution of Equation (15) can be assumed to be Next, substituting Equations (26) and (37) into Equation (36), we make all coefficients of Y i vanish and obtain the algebraic equations including unknown numbers a i (i = 0, 1), l, n, and c. Solving these equations, we have the following solutions c = 3(−n + 1) 4l 2 , l = l, n = n, a 0 = a 0 , a 1 = 2l.
So, the exact solution of Equation (2) is where l = 0, a 0 and n are arbitrary constants. When we take a 0 = 0, l = −1, n = 2 at y = 1, z = 2, the value of u is illustrated in Figure 4 below.

Exact Solution of Equation (16)
Likewise, substituting Equation (23) into Equation (16), we get the following ordinary differential equation Then, balancing Φ (4) with Φ Φ for Equation (38), we have k = 1. Therefore, based on Equation (27), the solution of Equation (16) can be assumed to be Next, substituting Equations (26) and (39) into Equation (38), we make all coefficients of Y i vanish and obtain the algebraic equations including unknown numbers a i (i = 0, 1), l, m, and n. Solving these equations, we have the following solutions l = l, m = n(n + 3l) n + 4l 3 , n = n, a 0 = a 0 , a 1 = 2l.

Construction of Conservation Laws
In this section, we chiefly construct conservation laws of Equation (2) using Ibragimov's method [24,26]. First, we prove that Equation (2) is nonlinear self-adjoint.
The adjoint equations of Equation (42) are written as Besides, where L is a formal Lagrangian of the following form L = v β R (x, u, · · · , u (k) ), β = 1, 2, ..., m, and the Euler-Lagrange operator is expressed as
If the formal Lagrangian of Equation (2) is given as based on Theorem 2, we can get Therefore, Equation (2) is nonlinearly self-adjoint with Equation (43).

Construction of Conservation Laws
Theorem 3 ([28]). The system of differential Equation (42) is nonlinearly self-adjoint, so every Lie point, Lie-Bäcklund, nonlocal symmetry admitted by the system of Equation (42) gives rise to a conservation law, where the components C i of the conserved vector C = (C 1 , · · · , C n ) are determined by and W α = η α − ξ j u α j . The formal Lagrangian L should be written in the symmetric form concerning all mixed derivatives u α ij , u α ijk , · · · .

Conclusions
In this paper, the Lie symmetry analysis method is applied to the (3 + 1)-dimensional BKP-Boussinesq equation. Based on this method, we construct the optimal system of one-dimensional subalgebras. Furthermore, some similarity reductions are handled and exact solutions of the reduced equations are obtained by means of the Tanh method. Finally, it is shown that Equation (2) is nonlinearly self-adjoint. Meanwhile, using Ibragimov's method, we derive the conservation laws widely used in the field of mathematical physics. After obtaining the exact solutions of Equation (2), we can depict the propagation of long waves in shallow water better and know more applications in the physical field, such as the percolation of water in porous subsurfaces of a horizontal layer of material.